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+{"text":"\\section{Introduction}\n\nSince oxygen is the third most abundant element in the protosolar nebula \\citep{An89,Lo09}, this naturally makes water as the most abundant volatile compound in planetary bodies of our solar system, if one excepts the hydrogen and helium presents in the envelopes of the giant planets \\citep{En08,Boc17,Gr17}.\n\nWater-rich worlds (Europa, Titan, Enceladus, Pluto, Triton, etc) are ubiquitous in our solar system, and the building blocks of Uranus and Neptune are also supposed to be water rich \\citep{Mo18}. These properties led astronomers to consider the possible existence of massive water-rich planets around other stars, i.e. the so-called ocean planets \\citep{Le04}. Those planets would have grown from ice-rich building embryos formed beyond the snowline in protoplanetary disks, and would have subsequently migrated inward up to their current orbital location nearby their host star \\citep{Ra18a,Ra18b}. This motivated the implementation of an H$_2$O layer to existing internal structure models, in which the liquid water had a simple prescription for the temperature profile (often isothermal), which often led to the coexistence of liquid water with high pressure ices \\citep{So07,Va07,Fo07,Ze13,Ze19}. At that time, it was believed that the temperature structure had a minor impact on the radii as it is the case for telluric planets \\citep{Va06,Fo07}.\n\nHowever, exoplanets considered today as good candidates for being water-rich worlds are also subject to important irradiation from their host star due to their short orbital periods. For such conditions at the surface of the planet, assuming an adiabatic temperature gradient produces very shallow P(T) profiles \\citep{Th16}. As a consequence, water is not in condensed phase, but rather in supercritical state in most of their hydrospheres, making ocean planets way more inflated with an adiabatic prescription compared to an isothermal one \\citep{Tu20,Mo20,Ha20}.\n\nThe inflated hydrospheres of irradiated supercritical ocean planets have been recently shown to be good candidates to account for the large radii of sub-Neptunes planets \\citep{Mo20}. They could also provide a possible explanation for the bimodal distribution of super-Earth and sub-Neptune populations, also known as the Fulton gap \\citep{Fu17}. These physical properties, along with the availability of several sets of thermodynamic data for H$_2$O \\citep{Wa11,Du06,Ma19,Jo20}, has recently motivated the modeling of the equation of state (EoS) of water in conditions relevant to planetary interiors, from 0 to a few TPa, the latter value corresponding to a Jupiter-mass planet fully made of water \\citep{Ma19,Tu20,Ha20}.\n\nFor the sake of precision, mass-radius relationships of supercritical ocean planets must be calculated via the simultaneous use of atmosphere and interior structure models that are both connected at their boundaries. For example, \\cite{Tu20} focused on planets of masses $0.2$--$2$ $M_\\Earth$ and water contents of $0.01$--$5$ wt\\%, to investigate the presence of water in the planets of the TRAPPIST-1 system. They added an irradiated steam atmosphere on top of rocky cores, using tabulated mass-radius relationships of \\cite{Ze16}. These latters were computed at a 1 bar surface pressure, and might become invalid in the case of heavy H$_2$O layers (surface pressures considered up to 10 GPa). In the approach presented in \\cite{Mo20}, the atmosphere model from \\cite{Mar19} only considers the uppermost part of the hydrosphere up to a given pressure. The rest of the interior structure, including extreme phases of H$_2$O, is computed via an interior model \\citep{Br17}, allowing to compute planets with any water content. The aim of our work is to update this model by using state of the art equations of state, and to include a better connection between the atmosphere and the interior models.\n\nTo do so, we combine the three parts of an hypothetical supercritical planet (refractory interior, condensed-fluid H$_2$O layer, and steam atmosphere) in a self-consistent framework to provide analytical descriptions of mass-radius relationships, which depend on the planetary mass, water mass fraction (WMF) and the equilibrium temperature. Such a derivation will allow estimating the WMF of irradiated ocean exoplanets from ground- or space-based mass-radius observations.\n\nWe also discuss the possible existence of these supercritical planets in light of hydrodynamic and Jeans' atmospheric escapes, and provide the mass-radius domains where escape is efficient. We finally use our model to compute the WMF of exoplanets b, c, and d of the system GJ-9827, chosen as a test case, and find that planet d could be a planet in supercritical state made of $20\\pm 10\\%$ of H$_2$O by mass.\n\nSection \\ref{sec:previous} reviews the model from \\cite{Mo20}, presenting its main features, inputs and outputs. Section \\ref{sec:improvements} details the work that has been made to update the model's EoS and make a consistent connection between the interior and the atmosphere model. Results are shown in Section \\ref{sec:results} in the form of mass-radius relationships, and ternary diagrams, and a conclusion is made in Section \\ref{sec:ccls}.\n\n\\section{Underlying interior and atmospheric models} \\label{sec:previous}\n\nWe follow the approach of \\cite{Mo20} consisting in coupling a Super-Earth interior model derived from \\cite{Br17} and the atmospheric model described in \\cite{Ma17,Mar19}. Here we recall the basic assumptions of these models.\n\n\\subsection{Interior Model}\n\nOur model solves iteratively the equations describing the interior of a planet:\n\n\\begin{eqnarray}\n\\frac{\\mathrm{d} g}{\\mathrm{d} r}&=&4 \\pi G \\rho-\\frac{2 G m}{r^{3}}, \\label{eq:gauss}\\\\\n\\frac{\\mathrm{d} P}{\\mathrm{d} r}&=&-\\rho g, \\label{eq:hydrostatic}\\\\\n\\frac{\\mathrm{d} T}{\\mathrm{d} r}&=&-g\\gamma T\\frac{\\mathrm{d} \\rho}{\\mathrm{d} P}, \\label{eq:temp_grad}\\\\\nP &=& f(\\rho,T), \\label{eq:solve_eos}\n\\end{eqnarray}\n\n\\noindent which correspond to the Gauss's theorem, hydrostatic equilibrium, adiabatic profile with use of Adams-Williamson equation, and the EoS of the considered medium, respectively. $g$, $P$, $T$ and $\\rho$ are gravity, pressure, temperature and density profiles, respectively. $m$ is the mass encapsulated within the radius $r$, $G$ is the gravitational constant, and $\\gamma$ is the Gr\\\"uneisen parameter. The Grune\u00efsen parameter is key to compute the thermal profile of the planet, and the literature sometimes refers to the adiabatic gradient instead, expressed as follows \\citep{Ki12,Ma19,Ha20}:\n\n\\begin{eqnarray}\n\t\\nabla_\\mathrm{ad} = \\left(\\frac{\\partial \\ln T}{\\partial \\ln P}\\right)_S = \\gamma \\frac{P}{\\rho} \\frac{1}{c^2},\n\\end{eqnarray}\nwhere $S$ is the entropy, and $c$ the speed of sound.\\\\\n\nThe interior model can display up to five distinct layers, depending on the planet's characteristics:\n\n\\begin{itemize}\n\t\\item a core made of metallic Fe and FeS alloy;\n\t\\item a lower mantle made of bridgmanite and periclase;\n\t\\item an upper mantle made of olivine and enstatite;\n\t\\item an ice VII phase;\n\t\\item a hydrosphere covering the whole fluid region of H$_2$O.\n\\end{itemize}\n\nThe Vinet EoS \\citep{Vi89} with thermal Debye correction is used for all solid phases:\n\\begin{eqnarray}\n\\begin{aligned}\nP\\left(\\rho, T\\right)=& 3 K_{0}\\left[\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\frac{2}{3}}-\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\frac{1}{3}}\\right] \\times \\\\\n& \\exp\\left\\{\\frac{3}{2}\\left(K_{0}^{\\prime}-1\\right)\\left[1-\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{-\\frac{1}{3}}\\right]\\right\\}\\\\\n& + \\Delta P,\n\\end{aligned}\n\\end{eqnarray}\n\n\\noindent with\n\n\\begin{eqnarray}\n\\begin{aligned}\n\\Delta P = &9 \\frac{\\gamma \\rho R}{M_\\mathrm{mol} \\theta^{3}} \\times\\\\\n&\\left[T^{4} \\int_{0}^{\\frac{\\theta}{T}} \\frac{t^{3}}{e^{t}-1} d t-T_{0}^{4} \\int_{0}^{\\frac{\\theta}{T_{0}}} \\frac{t^{3}}{e^{t}-1} d t\\right],\n\\end{aligned}\n\\end{eqnarray}\n\n\\noindent where $\\theta=\\theta_{0}\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\gamma}$, $\\gamma=\\gamma_{0}\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{-q}$, $R$ the ideal gas constant and $M_\\mathrm{mol}$ the molar mass of the considered material. All quantities with an index 0 are reference parameters obtained by fit on experimental data, given in table \\ref{tab:parameters}. The EoS used by \\cite{Mo20} to solve Eq. (\\ref{eq:solve_eos}) is the one formulated by \\cite{Du06}, valid up to 10 GPa and 2573.15 K.\n\nAll thermodynamic and compositional parameters of mineral layers are taken equal to those of Earth \\citep{St05,So07,So10}, and summarized in table \\ref{tab:parameters}. We refer the reader to \\cite{Br17} to get all the computational details. \n\n\\cite{Mo20} computed the Gr\\\"uneisen parameter for water via a bilinear interpolation in a grid generated from the python library of the IAPWS formulation\\footnote{https:\/\/pypi.org\/project\/iapws\/\\#description}, computing the Gr\\\"uneisen parameter in the form $\\gamma = f(\\rho,T)$ with $\\rho$ and $T$ varying in the 316--2500 kg.m$^{-3}$ and 650--10,000 K ranges, respectively. An important issue is that the density range is very limited, since this quantity can easily vary from $\\sim$10 kg.m$^{-3}$ at the planetary surface to $\\sim$5000  kg.m$^{-3}$ at the center of a 100\\% water planet of 1 $M_{\\Earth}$, implying that the computation of $\\gamma$ is erroneous at the top and at the bottom of the hydrosphere. A solution for overcoming this limitation is provided in Section \\ref{sec:gruneisen}.\n\nApart from compositional inputs, the main physical inputs of the model are the core mass fraction (CMF) $x_\\mathrm{core}$ and water mass fraction (WMF) $x_{\\mathrm{H}_2 \\mathrm{O}}$, the mantle mass fraction is then $x_\\mathrm{mantle} = 1 - x_\\mathrm{core} -x_{\\mathrm{H}_2 \\mathrm{O}}$. Pressure and temperature profiles are integrated from outside, and require the inputs of the boundary pressure $P_\\mathrm{b}$ and boundary temperature $T_\\mathrm{b}$. Finally, the model also requires the input of the planet's mass $M_\\mathrm{b}$ (subscript $b$ denotes the mass encapsulated within the boundary of the interior model, excluding the contribution of any potential atmosphere). Once defined, these input parameters allow for the computation of the planet's internal structure and associated boundary radius. In the case of the Earth ($x_\\mathrm{core}=0.325$, $x_{\\mathrm{H}_2 \\mathrm{O}}=0.0005$, $M_\\mathrm{b}=1$ M$_\\Earth$), the model computes a radius $R_\\mathrm{b}$ equal to 0.992 $R_\\Earth$, which is less than 1\\% of error, indicating that errors from the model are negligible compared to errors on measurements. In the following, subscript $b$ refers to quantities at the boundary between the interior model and the atmosphere model, such as bulk mass $M_\\mathrm{b}$, radius $R_\\mathrm{b}$, gravity $g_\\mathrm{b}$, pressure $P_\\mathrm{b}$ and temperature $T_\\mathrm{b}$.\n\n\\begin{table*}[!ht]\n\t\\movetabledown=5cm\n\t\\begin{rotatetable*}\n\t\\centering\n\t\\caption{List of thermodynamic and compositional parameters used in the interior model.} \n\t\\label{tab:parameters}\n\t\\begin{tabular}{llllllllllll}\n\t\t\\hline\n\t\t\\multicolumn{2}{l}{\\textit{Layer}}                     & \\multicolumn{2}{l}{\\textit{Core}}   & \\multicolumn{4}{l}{\\textit{Lower mantle}}         \n             & \\multicolumn{4}{l}{\\textit{Upper Mantle}}                  \n           \\\\ \\hline\n\t\tPhases                              &                  & \\multicolumn{2}{l}{Iron rich phase} & \\multicolumn{2}{l}{Perovskite} & \\multicolumn{2}{l}{Periclase} & \\multicolumn{2}{l}{Olivine}   & \\multicolumn{2}{l}{Enstatite}         \\\\\n\t\tComposition (\\%)                    &                  & \\multicolumn{2}{l}{100}             & \\multicolumn{2}{l}{79.5}       & \\multicolumn{2}{l}{20.5}      & \\multicolumn{2}{l}{41}        & \\multicolumn{2}{l}{59}    \n            \\\\ \\cline{3-12} \n\t\tComponents                          &                  & Fe             \n  & FeS              & FeSiO$_3$      & MgSiO$_3$     & FeO           & MgO           & Fe$_2$SiO$_4$ & Mg$_2$SiO$_4$ & Fe$_2$Si$_2$O$_6$ & Mg$_2$Si$_2$O$_6$ \\\\\n\t\tComposition (\\%)                    &                  & 87             \n  & 13               & 10             & 90            & 10            & 90            & 10            & 90            & 10                & 90     \n           \\\\ \\cline{3-12} \n\t\tMolar mass (g.mol$^{-1}$)           & $M_\\mathrm{mol}$ & 55.8457        \n  & 87.9117          & 131.9294       & 100.3887      & 71.8451       & 40.3044       & 203.7745      & 140.6931      & 263.8588          & 200.7774          \\\\\n\t\tReference density (kg.m$^{-3}$)     & $\\rho_0$         & 8340           \n  & 4900             & 5178           & 4108          & 5864          & 3584          & 4404          & 3222          & 4014              & 3215   \n           \\\\\n\t\tReference temperature (K)           & $T_0$            & 300            \n  &                  & 300            &               & 300           &   \n            & 300           &               & 300               &         \n          \\\\\n\t\tReference bulk modulus (GPa)        & $K_0$            & 135            \n  &                  & 254.7          &               & 157           &   \n            & 128           &               & 105.8             &         \n          \\\\\n\t\tPressure derivative of bulk modulus & $K_0^\\prime$     & 6              \n  &                  & 4.3            &               & 4             &   \n            & 4.3           &               & 8.5               &         \n          \\\\\n\t\tReference Debye temperature (K)     & $\\theta_0$       & 474            \n  &                  & 736            &               & 936           &   \n            & 757           &               & 710               &         \n          \\\\\n\t\tReference Gr\u00fcneisen parameter       & $\\gamma_0$       & 1.36      \n       &                  & 2.23           &               & 1.45         \n &               & 1.11          &               & 1.009             &    \n               \\\\\n\t\tAdiabatic power exponent            & $q$              & 0.91           \n  &                  & 1.83           &               & 3             &   \n            & 0.54          &               & 1                 &         \n          \\\\ \\hline\n\t\\end{tabular}\n\t\\tablecomments{Thermodynamic data are summarized in Table 1 of \\cite{So07}, and compositional data are the results of model calibration for Earth \nby \\cite{St05,So07}.}\n\t\\end{rotatetable*}\n\\end{table*}\n\n\\subsection{Atmospheric model} \\label{sec:atmo}\n\nThe atmospheric model generates the properties of a 1D spherical atmosphere of H$_2$O by integrating the thermodynamic profiles bottom to top. The model takes as inputs the planet's mass and radius, as well as the thermodynamic conditions at its bottom. We choose to connect the atmospheric model with the interior model at a pressure $P$ = $P_\\mathrm{b}=300$ bar (slightly above the critical pressure $P_\\mathrm{crit}=220.67$ bar) and at a temperature $T$ = $T_\\mathrm{b}$. $(P,T,\\rho)$ profiles are then integrated upward via the prescription from \\cite{Ka88} in the case of an adiabat at hydrostatic equilibrium. Once the temperature reaches the top temperature of the atmospheric layer, here set to $T_\\mathrm{top}=200$ K, an isothermal radiative mesosphere at $T=T_\\mathrm{top}$ is assumed. Figure \\ref{fig:profiles} shows several $(P,T)$ profiles representing the whole hydrospheres of planets (under Ma19+ parametrization, see section \\ref{sec:gruneisen}) for masses and irradiation temperatures in the range 1--20 $M_\\Earth$ and  $T_\\mathrm{irr}=$400-1300 K (see Eq. (\\ref{eq:teq})), respectively.\n\nThe atmosphere transition radius is controlled by the altitude of the top of the H$_2$O clouds, corresponding to the top of the moist convective layer, assumed to be at a pressure $P_\\mathrm{top}=0.1$ Pa. We choose this limit as the observable transiting radius, assuming that results are similar for cloudy and cloud-free atmospheres \\citep{Tu19,Tu20}. The EOS is taken from the NBS\/NRC steam tables \\citep{Ha84}, implying the atmosphere is not treated as an ideal gas. The discontinuities in $(P,T)$ profiles occuring for $T_\\mathrm{irr}=1300$ K are due to the limited range of these tables, but the height of this region ($P=$ 100-300 bar) is negligible compared to the thickness of the atmosphere. Increasing the $T_\\mathrm{top}$ temperature will impact the final structure of the atmosphere, decreasing both the thickness of the atmosphere and the interior. Numerical tests with $T_\\mathrm{top}$ varying from 200 K to $T_\\mathrm{skin}=T_\\mathrm{eff}\/ 2^{0.25}$ decrease the final radius of the planet of at most $\\sim 200$ km for the cases considered in this study. It corresponds to a difference of $2\\%$ in radius at most, but this difference is mainly below $1\\%$.\n\nShortwave and thermal fluxes are then computed using 4-stream approximation. Gaseous (line and continuum) absorptions are computed using the $k$-correlated method on 38 spectral bands in the thermal infrared, and 36 in the visible domain. Absorption coefficients are exactly the same as those in \\cite{Le13} and \\cite{Tu19} which includes several databases, specificaly designed for H$_2$O-dominated atmospheres. Rayleigh opacity is also included. This method computes the total outgoing longwave radiation (OLR, in W.m$^{-2}$) of the planet that gives the temperature that the planet would have if it was a blackbody:\n\n\\begin{eqnarray}\n\tT_\\mathrm{p} = \\left(\\frac{\\mathrm{OLR}}{\\sigma_\\mathrm{sb}}\\right)^{1\/4}, \\label{eq:tp}\n\\end{eqnarray}\n\n\\noindent with $\\sigma_\\mathrm{sb}$ the Stefan-Boltzmann constant. In order to quantify the irradiation of the planet by its host star, we define the irradiance temperature\n\n\\begin{eqnarray}\n\tT_\\mathrm{irr} = T_\\mathrm{eff} \\sqrt{\\frac{R_\\star}{2a}}, \\label{eq:tirr_obs}\n\\end{eqnarray}\n\n\\noindent where $T_\\mathrm{eff}$ and $R_\\star$ are the host star effective temperature and radius, respectively, and $a$ is the semi-major axis of the planet. The atmospheric model computes the Bond albedo from the atmosphere's reflectance \\citep[][using the method presented in]{Pl19}  assuming a G-type star linking both temperatures:\n\n\\begin{eqnarray}\n\tT_\\mathrm{irr} = \\left(\\frac{\\mathrm{OLR}}{(1-A)\\sigma_\\mathrm{sb}}\\right)^{1\/4} = \\frac{T_\\mathrm{p}}{(1-A)^{1\/4}}. \\label{eq:teq}\n\\end{eqnarray}\n\nThe literature often approximates $T_\\mathrm{irr}$ to the equilibrium temperature $T_\\mathrm{eq}$, which is the temperature the planet would have for an albedo $A=0$ (all the incoming heat is absorbed and re-emitted by the planet). Since it is the observable quantity, our results will be presented in term of $T_\\mathrm{irr}$. Equation \\ref{eq:teq} assumes that the planet is in radiative equilibrium with its host star. Any heating source in the planet interior would add an additional term in the radiative equilibrium of the planet with its host star, increasing the effective temperature of the planet for the same received irradiation \\citep{Ne11}. In this work, we model the structure of planets that have either no interior heating source, or that had time to cool off.\n\nFor a given planet mass, boundary radius and irradiation temperature, the atmosphere thickness and boundary temperature are retrieved from the atmospheric model. The latter is then used to compute the interior structure, and the former is taken into account to compute the total (transiting) radius.\n\n\\begin{figure*}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{profiles.pdf}}\n\t\\caption{$(P,T)$ profiles for 100\\% H$_2$O planets of masses $M_\\mathrm{p}=$1--20 $M_\\Earth$, and irradiation temperatures $T_\\mathrm{irr}=$400--1300 K with the Ma19+ parametrization (see section \\ref{sec:gruneisen}). Cases corresponding to smallest masses and highest temperatures are not shown, as their surface gravities are below the limit fixed in Sec. \\ref{sec:connect}. Phase transitions of H$_2$O are taken from \\cite{Wa11} at low temperatures (solid turquoise lines) and from \\cite{Ne11} at high temperatures (dashed turquoise lines), with labels IF (ionic fluid), SI (super ionic), P (plasma), and iN for the ice N.}\n\t\\label{fig:profiles}\n\\end{figure*}\n\n\\section{Model update} \\label{sec:improvements}\n\nThis section presents the improvements made on the existing model to push further its physical limitations. Since we are interested in planets with substantial amounts of water, we define a specfic CMF, which is only related to the mass budget of the rocky part:\n\n\\begin{eqnarray}\nx_\\mathrm{core}^\\prime = \\frac{x_\\mathrm{core}}{1-x_{\\mathrm{H}_2 \\mathrm{O}}}.\n\\end{eqnarray}\n\n\\noindent where $x_\\mathrm{core}$ is  the ``true'' CMF. $x_\\mathrm{core}^\\prime$ will be used to compare planets that have different WMF, but with similar refractory contents. For example, $x_\\mathrm{core}^\\prime=0.325$ corresponds to an Earth-like CMF, regardless the amount of water present in the planet.\n\n\\subsection{Used EoSs}\n\nThe choice of the EoS is critical, as it strongly impacts the estimate of the mass-radius relationships. Three EoS are then considered in this study:\n\n\\begin{itemize}\n\t\\item EoS from the latest revision of the IAPWS-95 formulation from \\cite{Wa02}\\footnote{http:\/\/iapws.org} (hereafter WP02). This reference EoS gives an analytical expression of the specific Helmholtz free energy $f(\\rho,T)$. Any thermodynamic quantity (pressure, heat capacity, internal energy, entropy etc.) can be computed by taking the right derivative of $f$, and those quantities have analytical expressions.\n\t\\item EoS from \\cite{Du06} (hereafter DZ06). This EoS is corrected around the critical point, and gives an analytical expression for pressure as function of density and temperature $P(\\rho,T)$.\n\t\\item EoS from \\cite{Ma19} (hereafter Ma19). This formulation was developed for planetary interiors by extending the IAPWS-95 EoS with ingredients from statistical physics allowing transition to plasma and superionic states. The authors created a fortran implementation\\footnote{http:\/\/cdsarc.u-strasbg.fr\/viz-bin\/qcat?J\/A+A\/621\/A128} that computes pressure, specific Helmholtz free energy, specific internal energy and specific heat capacity for a given couple $(\\rho,T)$.\n\\end{itemize}\n\n\\begin{figure*}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{aux_EOS_650.pdf}\\includegraphics[angle=0,width=5cm]{aux_EOS_2000.pdf} }\n\t\\caption{Pressure as function of density calculated with WP02 (blue), DZ06 (red) and Ma19 (green) in the cases of two different temperatures. The solid horizontal lines indicate the range of validity for WP02 and DZ06, and the dashed horizontal lines give the extended range. The black dotted line corresponds to the ideal gas law for water steam.}\n\t\\label{fig:comp_eos}\n\\end{figure*}\n\nThe validity ranges of the different EoSs, which rely on the availability of experimental data, are given in Table \\ref{tab:range}. Extended ranges proposed by \\cite{Wa02} and \\cite{Du06} are also indicated because the mathematical expressions of their EoSs allow for extrapolations beyond the corresponding validity ranges. However, they become invalid when phase transition occurs (e.g. dissociation of water). Other EoSs exist in the literature, covering various regions of the phase diagram of water, or being used for specific purposes. Our choice of EoSs among others is discussed in Sec. \\ref{sec:ccls}.\n\n\\begin{table}\n\t\\centering\n\t\\caption{Validity ranges of the different EoSs.} \n\t\\label{tab:range}\n\t\\begin{tabular}{lll}\n\t\t\\tablewidth{0pt}\n\t\t\\hline\n\t\t\\hline\n\t\tEoS \t& Valid \t\t& Extended \t\\\\\t\n\t\t\\hline\n\t\tWP02 & $P<1$ GPa & $P<100$ GPa   \\\\ \t\n\t\t&$T<1~273$ K & $T<5~000$ K \\\\\n\t\tDZ06  \t& $P<10$ GPa & $P<35$ GPa\\footnote{Limit given by \\cite{Du96}.\\label{limdz}}  \\\\\n\t\t&$T<2~573.15$ K& $T<2~800$ K\\footref{limdz} \t \\\\\n\t\tMa19   \t&$\\rho<100\\times 10^3$ kg.m$^{-3}$\t&  not specified \\\\\n\t\t&$T<100~000$ K\t&  not specified  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nFigure \\ref{fig:comp_eos} shows the $P(\\rho)$ profiles derived from the considered EoSs at different temperatures. All EoSs present minor differences in their validity range, regardless the considered temperature. \\cite{Ma19} find that the WP02 overestimates the pressure beyond its extended range. For a given pressure in a planet's interior, this would underestimate the density, and then overestimate the total radius of the planet. A more pronounced deviation is visible for DZ06 above its validity range. Around the critical point ($\\rho\\sim350$ kg.m$^{-3}$, mostly visible at 650 K), WP02 is closer to DZ06, compared to Ma19, as expected. In the low density limit, all EoSs behave following the ideal gas law $P\\propto \\rho T$, which has a characteristic slope of 1 in log-log scale. \n\n\n\\subsection{Gr\\\"uneisen parameter for fluids} \\label{sec:gruneisen}\n\nThe Gr\\\"uneisen parameter $\\gamma$, already introduced in Eq. (\\ref{eq:temp_grad}), has many definitions. For solids, it gives the rate of change in phonon frequencies $\\omega_i$ relative to a change in volume $V$ \\citep{Gr12}:\n\n\\begin{eqnarray}\n\\gamma_i = -\\left(\\frac{\\partial \\ln \\omega_i}{\\partial \\ln V}\\right)_T.\n\\end{eqnarray}\n\nBy averaging over all lattice frequencies, it is possible to obtain a thermodynamic definition (using the internal energy $U$ and entropy $S$) of the Gr\\\"uneisen parameter \\citep{Ar84} via the following expression:\n\n\\begin{eqnarray}\n\\gamma = V \\left(\\frac{\\partial P}{\\partial U}\\right)_V = \\frac{V}{C_V} \\left(\\frac{\\partial P}{\\partial T}\\right)_V = \\frac{\\rho}{T} \\left(\\frac{\\partial T}{\\partial \\rho}\\right)_S. \\label{eq:gamma-therm-id}\n\\end{eqnarray}\n$\\gamma$ relates a pressure (or density) variation to a temperature change. Although initialy defined for solids, the meaning of $\\gamma$ holds for fluids. In planetary interiors, adiabatic heat exchange is mostly driven by convective heat transfer \\citep{St19}. At planetary scales, the Gr\\\"uneisen parameter can thus be used for both solids and fluids. From identities in Eq. (\\ref{eq:gamma-therm-id}), $\\gamma$ can be expressed using other thermodynamic constants such as the thermal expansion coefficient $\\alpha$, the isothermal bulk modulus $K_T$, and the specific isochoric heat capacity $c_V$\n\\begin{eqnarray}\n\\gamma = \\frac{\\alpha K_T }{\\rho c_V}. \\label{eq:gruneisen-expression}\n\\end{eqnarray}\n\n\\noindent  $\\gamma$ is assumed to be temperature-independent in solid phase, and its value is fitted from experimental data, taking into account small density variations. In this study, we use the Helmholtz free energy $F$ given in \\cite{Wa02} and \\cite{Ma19}. In the IAPWS95 release, the specific Helmholtz free energy $f$ in its dimensionless form $\\phi$ is divided into its ideal part (superscript $\\circ$) and a residual (superscript ``r\") via the following expression:\n\n\\begin{eqnarray}\n\\frac{f(\\rho,T)}{RT} = \\phi(\\delta,\\tau) = \\phi^\\circ(\\delta,\\tau)+\\phi^\\mathrm{r}(\\delta,\\tau),\n\\end{eqnarray}\n\n\\noindent with $\\delta=\\rho\/\\rho_c$ and $\\tau=T_c\/T$, $\\rho_c$ and $T_c$ being the supercritical density and temperature, respectively. After defining the derivatives of the ideal and residual part:\n\n\\begin{eqnarray}\n\\phi_{m n}^{\\circ}=\\frac{\\partial^{m+n} \\phi^{\\circ}(\\tau, \\delta)}{\\partial \\tau^{m} \\partial \\delta^{n}}, \\\\\n\\phi_{m n}^{\\mathrm{r}}=\\frac{\\partial^{m+n} \\phi^{\\mathrm{r}}(\\tau, \\delta)}{\\partial \\tau^{m} \\partial \\delta^{n}},\n\\end{eqnarray}\n\n\\noindent where integers $m$ and $n$ define the order of the derivative with respect to $\\tau$ and $\\delta$, respectively. From these expressions, \none can derive:\n\n\\begin{eqnarray}\n\\gamma_- = -\\frac{1+\\delta \\phi^\\mathrm{r}_{01}-\\delta \\tau \\phi^\\mathrm{r}_{11}}{\\tau^2 \\left(\\phi^\\circ_{20}+\\phi^\\mathrm{r}_{20}\\right)}, \\label{eq:grun-iapws}\n\\end{eqnarray}\n\n\\noindent where $\\gamma_-$ is the formulation of the Gr\\\"uneisen parameter computed following the approach of  \\cite{Wa02}.\n\nThe fortran implementation of \\cite{Ma19} computes $F(\\rho,T)$, along with other useful quantities such as $\\chi_T = \\left(\\frac{\\partial \\ln P}{\\partial \\ln T}\\right)_V$ and the specific isochoric heat capacity $c_V$. In this case, the Gr\\\"uneisen parameter is expressed as:\n\\begin{eqnarray}\n\\gamma_+ = \\frac{P(\\rho,T) \\chi_T (\\rho,T)}{\\rho c_V T}, \\label{eq:grun-mazevet}\n\\end{eqnarray}\n\n\\noindent where $\\gamma_+$ is the formulation of the Gr\\\"uneisen parameter derived from the quantities calculated via the approach of  \\cite{Ma19}.\n\nIn the case of an ideal gas, one can derive the theoretical value $\\gamma=\\frac{2}{l}$, where $l$ is the number of degrees of freedom for a given molecule. For H$_2$O,  $\\gamma \\simeq \\frac{1}{3}$, since $l=6$ (3 rotational and 3 vibrational degrees of freedom).\n\nThe Gr\\\"uneisen parameter is crucial to compute the adiabatic temperature gradient inside a planet's interior.  However, because temperature has low impact on EoSs used in solid phase, it is possible to assume isothermal layers in interior models when thermodynamic data are lacking, and generate internal structures close to reality \\citep{Ze19}. In the case of fluids (here H$_2$O), temperature rises sharply with depth. This strongly impacts the EoS and leads to different phase changes that are not visible in the case of isothermal profiles.\n\nEach computation for the interior model can be performed by using any of the three EoS (WP02, DZ06, Ma19) to solve Eq. (\\ref{eq:solve_eos}), and WP02 or Ma19 EoS to solve Eq. (\\ref{eq:temp_grad}) (i.e. computing $\\gamma$ with EoS WP02 or Ma19). In the following, we will use the name of the EoS used to solve Eq. (\\ref{eq:solve_eos}), and add + or - depending on the EoS used to compute the Gr\\\"uneisen parameter, $\\gamma_+$ (Ma19) or $\\gamma_-$ (WP02) respectively. For example, Ma19- indicates that the Ma19 EoS was used to solve Eq. (\\ref{eq:solve_eos}), and that the WP02 approach was used to solve Eq. (\\ref{eq:temp_grad}).\n\nFigure \\ref{fig:gruneisen_val} shows the values of $\\gamma_+$ and $\\gamma_-$ in the H$_2$O phase diagram. Since $\\gamma$ is integrated to obtain the temperature gradient, a small difference leads to different paths in the $(P,T)$ plane. The indiscernability between the WP02- and the Ma19- profiles shows that the internal structure (and thus mass-radius relationships) is more impacted by the temperature profile than the difference in the EoS in the case of a water layer. The difference in temperature between Ma19+ and Ma19-\/WP02- profiles is as high as $\\sim$2000 K, which also results in a difference of $\\sim 200$ kg.m$^{-3}$ in density at the center of the planet.\n\n\\begin{figure}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{grunmap+.png}}\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{grunmap-.png}}\n\t\\caption{Color maps showing $\\gamma_+$ (top panel) and $\\gamma_-$ (bottom panel) in the H$_2$O phase diagram. The phase diagram of water is  identical to the one shown in Figure \\ref{fig:profiles}. Ma19+, Ma19- and WP02- are the $(P,T)$ profiles defined in the case of a 1 M$_\\Earth$ planet fully made of H$_2$O, with the atmosphere part shown with short dashes. Ma19- and WP02- interiors are almost indistinguishable, hence represented by the same color, and all atmospheric profiles are identical, although the  gravity at the boundary is different for each case.}\n\t\\label{fig:gruneisen_val}\n\\end{figure}\n\n\\subsection{Connection between interior and atmospheric models} \\label{sec:connect}\n\nAtmospheric properties (OLR, albedo, mass and thickness) are all quantities that evolve smoothly. To enable a smooth connection between the two models, we implemented a trilinear interpolation module that can estimate atmospheric properties for a planet whose physical parameters $g_\\mathrm{b}$, $M_\\mathrm{b}$, and  $T_\\mathrm{b}$ are in the 3--30 m.s$^{-2}$,  0.2--20 $M_\\Earth$, and 750--4500 K ranges, respectively. This allows us to correct the slight deviations from nods of the grid, and trilinear interpolation ensures that properties computed at a nod are exactly those at the nod, which would not be the case if a polynomial fit was performed on data. Details of the connection between the two models are given in Appendix \\ref{sec:connection}.\n\nFigure \\ref{fig:teq_tp} shows $T_\\mathrm{irr}$ as a function of $T_\\mathrm{p}$ for a set of fixed $g_\\mathrm{b}$ and $M_\\mathrm{b}$. Due to a strong greenhouse (or blanketing) effect from the steam atmosphere, most cases lead to $T_\\mathrm{b} > 2000$ K. As previously stated, this consequence discards any EoS that does not hold for such high temperatures. A second observation is that at low temperatures, one input irradiation temperature $T_\\mathrm{irr}$ can correspond to two different planet temperatures $T_\\mathrm{p}$ (and atmospheric properties). Since our work focuses on highly irradiated exoplanets, we will only investigate cases with $T_\\mathrm{irr}~>~400$ K to bypass this degeneracy.\n\n\\begin{figure}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{tp_teq.pdf}}\n\t\\caption{Irradiation temperature $T_\\mathrm{irr}$ as a function of the planet's temperature $T_\\mathrm{p}$. Several curves are obtained due to different values of $g_\\mathrm{b}$ and $M_\\mathrm{b}$ in the available parameter range. Coldest planets exhibit a degeneracy, as the same amount of irradiation is consistent with two different atmospheric structures. As shown by the color bar, when taking the ``hot\" solution for $T_\\mathrm{p}$, the temperature at the bottom of the atmosphere $T_\\mathrm{b}$ is $>2000$ K.}\n\t\\label{fig:teq_tp}\n\\end{figure}\n\n\\section{Atmospheric escape} \\label{sec:atmos-escape}\n\nPlanetary atmospheres are subject to two types of instabilities : hydrostatic and thermal escape. The former is encountered when the gravity at a given height is insufficient to retain the gas. In this case, the atmosphere cannot exist in hydrostatic equilibrium and atmospheric models fail to produce static $(P,\\rho)$ profiles. The choice of $g_\\mathrm{b}>3$ m.s$^{-2}$ is arbitrary, but allows to avoid these cases. The latter occurs when the thermal energy of gas molecules exceeds the gravitational potential, allowing their escape. Escape rates are then computed, indicating which molecules can remain in an atmosphere. Several mechanisms of non-thermal escape exist as well, involving collisions between atoms and ions producing kinetic energy that leads to knock-off \\citep{Hu82}, but they rely on processes that are beyond the scope of this study.\n\n\\subsection{Jeans' escape}\n\nOne widely known process of atmospheric escape is the Jeans escape. Gas molecules have a velocity distribution given by the Maxwell-Boltzmann distribution, which displays an infinite extension in the velocity space, meaning that \\textit{some} particles have velocities greater than the escape velocity. By integrating this distribution, one can derive the Jeans' particle flux (particles per time unit per surface unit) escaping the atmosphere at the exobase \\citep{Je25}:\n\n\\begin{eqnarray}\n\t\\Phi_J = \\frac{n_\\mathrm{e} v_\\mathrm{esc}}{2 \\sqrt{\\pi}} \\frac{1}{\\sqrt{\\lambda}} \\left(1+\\lambda\\right) \\mathrm{e}^{-\\lambda}, \\label{eq:jflux1}\n\\end{eqnarray}\n\n\\noindent where $n_\\mathrm{e}$ is the particle number density at the top of the atmosphere (exobase), $v_\\mathrm{esc} = \\sqrt{2 g_\\mathrm{b} R_\\mathrm{p}}$ is the escape velocity (we assume $R_\\mathrm{p}\\simeq R_\\mathrm{b}$ and $M_\\mathrm{p}\\simeq M_\\mathrm{b}$). $\\lambda = \\left(\\frac{v_\\mathrm{esc}}{v_\\mathrm{th}}\\right)^2$ is the escape parameter, with $v_\\mathrm{th}=\\sqrt{2R_g T_\\mathrm{e}\/\\mu}$ the average thermal velocity of molecules of mean molar mass $\\mu$ at the exobase temperature $T_\\mathrm{exo}$, and $R_g$ is the ideal gas constant. \n\nWe wish to provide an estimate of the physical characteristics of the planets that would lose more than a fraction $x_\\mathrm{lost}=0.1$ of water content over a typical timescale of $\\Delta t= 1$ Gyr. This condition is met when\n\n\\begin{eqnarray}\n4\\pi R_\\mathrm{p}^2 \\frac{\\mu}{\\mathcal{N}_\\mathrm{A}} \\Phi_J \\ge \\frac{x_\\mathrm{lost} M_\\mathrm{p}}{\\Delta t}, \\label{eq:Jescape0}\n\\end{eqnarray}\n\n\\noindent with $\\mathcal{N}_\\mathrm{A}$ the Avogadro number. Solving Eq. (\\ref{eq:Jescape0}) with Earth's properties ($n_\\mathrm{e} = \\frac{P_\\mathrm{top} \\mathcal{N}_\\mathrm{A}}{Rg T_\\mathrm{exo}} \\sim 10^{19}$, $R_\\mathrm{p}=R_\\Earth$,  $M_\\mathrm{p}=M_\\Earth$) yields $\\lambda \\le 100$. Due to the exponential term, the result is poorly sensitive to changes in parameters, including the exact location of the exobase. Assuming $T_\\mathrm{exo}=T_\\mathrm{irr}$, this condition can be rewritten as\n\n\\begin{eqnarray}\n\tR_\\mathrm{p} > \\frac{1}{\\lambda} \\frac{G \\mu}{R_g T_\\mathrm{irr}} M_\\mathrm{p}, \\label{eq:Jescape}\n\\end{eqnarray}\n\n\\noindent with $G$ the gravitational constant. This estimate is consistent with today's composition of planets of the solar system (see Fig. \\ref{fig:final}). Equation \\ref{eq:Jescape} gives an indication of the properties of the planets that are subject to H$_2$ or H$_2$O escape, implying that their atmospheres should be dominated by heavier molecules (H$_2$O, CO$_2$, O$_2$, CH$_4$, etc) or be rocky planets, respectively.\n\n\\subsection{Hydrodynamic escape}\n\nHydrodynamic escape, also referred to as hydrodynamic blowoff, occurs when upper layers of the atmosphere are heated by intercepting the high energy irradiation (Far UV, Extreme UV and X-ray fluxes, the sum of which is often called XUV flux) from the host star. This heating induces an upward flow of gas, leading to mass-loss at a rate \\citep{Er07,Ow13}\n\n\\begin{eqnarray}\n\\dot{M} = \\epsilon \\frac{L_\\mathrm{XUV} R_\\mathrm{p}^3}{G M_\\mathrm{p} (2a)^2}, \\label{eq:Hescape}\n\\end{eqnarray}\n\n\\noindent where $L_\\mathrm{XUV}$ is the host star XUV luminosity, $a$ is the planet's orbital distance and $\\epsilon$ is a conversion factor between incident irradiation energy and mechanical blowoff energy. Note that Eq. (\\ref{eq:Hescape}) is only true in the energy-limited case. Heating occurs by absorption of high-energy photons by molecules which are dissociated in the upper atmosphere, meaning that blowoff can be limited by i) the number of photons as 1 photon breaks 1 molecule, and ii) recombination time as a dissociated molecule may recombine before being able to absorb the XUV irradiation again. Boundaries between these regimes have been explored by \\cite{Ow16}, who showed that the sub-Neptune population undergoes mostly energy-limited mass-loss, validating the use of Eq. \\ref{eq:Hescape} in our case.\n\nFor our estimate, we use the X-ray and UV luminosities obtained by fits on observational data for M to F type stars by \\cite{Sa11}:\n\n\\begin{eqnarray}\n&L_\\mathrm{EUV} = 10^{3.8} L_\\mathrm{X}^{0.86},&\\\\\n&L_\\mathrm{X} = 6.3\\times 10^{-4} L_\\star,& \\qquad \\tau<\\tau_\\mathrm{sat} \\\\\n&\\phantom{L_\\mathrm{X}xxx} = 1.89\\times 10^{21} \\tau^{-1.55},& \\qquad \\tau>\\tau_\\mathrm{sat} \\nonumber\n\\end{eqnarray}\n\n\\noindent where $\\tau$ is the host star age in Gyr and $\\tau_\\mathrm{sat} \n= 5.72\\times 10^{15} L_\\star^{-0.65}$ \\citep{Sa11}. To estimate the XUV luminosity, the star's bolometric luminosity is assumed constant, a hypothesis supported by the stellar evolution tracks of \\cite{Ba15}. Integrating the XUV luminosity in the saturation regime ($0<\\tau < \\tau_\\mathrm{sat}$) and beyond, gives the finite quantity $E_\\mathrm{XUV}=\\int_{0}^{+\\infty} L_\\mathrm{XUV}~dt = 1.8 \\times 10^{39} $ W for a solar type star.\n\nAgain, we look for planets that could lose more than 10\\% of their mass over a 1 Gyr period, due to atmospheric blowoff:\n\n\\begin{eqnarray}\n\\epsilon \\frac{E_\\mathrm{XUV} R_\\mathrm{p}^3}{G M_\\mathrm{p} (2a)^2} \\ge x_\\mathrm{lost} M_\\mathrm{p}. \\label{eq:Hescape1}\n\\end{eqnarray}\n\n\\noindent Combining Eq. (\\ref{eq:tirr_obs}) and Stefan-Boltzmann's law $L_\\star~=~4\\pi R_\\star^2 \\sigma_\\mathrm{sb} T_\\mathrm{eff}^4$ gives\n\\begin{eqnarray}\n\t(2a)^2 = \\frac{1}{T_\\mathrm{irr}^4} \\frac{L_\\star}{4 \\pi \\sigma_\\mathrm{sb}}.\n\\end{eqnarray}\nSubstituting this expression in Eq. (\\ref{eq:Hescape1}) yields to the condition:\n\n\\begin{eqnarray}\nR_\\mathrm{p} \\ge M_\\mathrm{p}^{\\frac{2}{3}} \\left(\\frac{x_\\mathrm{lost} G}{\\epsilon 4\\pi \\sigma_\\mathrm{sb} T_\\mathrm{irr}^4 E_\\mathrm{XUV}}\\right)^{\\frac{1}{3}}. \\label{eq:Hescape2}\n\\end{eqnarray}\n\n\\noindent This condition only gives an indication of the planets that are subject to substantial hydrodynamic escape. All arbitrary quantities such as $\\epsilon\\simeq1$ \\citep{Ow12,Bo17} and $E_\\mathrm{XUV}$, are affected by a power of $1\/3$, resulting in a low dependency on the chosen values.\n\nThe nature of escaping particles is not considered in Eq. (\\ref{eq:Hescape2}), meaning the computed quantity is the total lost mass. \\cite{Bo17} developed a method to quantify the hydrodynamic outflow $r_\\mathrm{F}$ (how many atoms of oxygen leave for each hydrogen atom). Based on their work, we compute $r_\\mathrm{F}\\sim 0.2$, indicating substantial loss of both H and O, with an accumulation of O$_2$. Mass loss of water content and accumulation of O$_2$ have several implications for the habitability of exoplanets \\citep{Ri16,Sc16}. The power laws for mass is 1 and $\\frac{2}{3}$ in the cases of Eqs. (\\ref{eq:Jescape}) and (\\ref{eq:Hescape2}), respectively. This implies that hydrodynamic escape is more efficient for less dense planets. In contrast, Jeans escape is dominant in the case of denser planets. The power-law for $T_\\mathrm{irr}$ is $-1$ and $-\\frac{4}{3}$ in the cases of Eqs. \\ref{eq:Jescape} and \\ref{eq:Hescape2}, respectively, implying that hydrodynamic escape will take over Jeans escape at higher irradiation temperatures. As shown in Fig. \\ref{fig:mr+}, Eqs. \\ref{eq:Jescape} and \\ref{eq:Hescape2} leave a window for planets that lost their H$_2$ reservoir but kept heavier volatiles from which they formed \\citep{Ze19}. This result highlights the consistency between the possible existence of irradiated ocean planets and atmospheric escape.\n\n\\section{Results} \\label{sec:results}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{mr400+.pdf} \\includegraphics[angle=0,width=5cm]{mr600+.pdf}}\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm,]{mr800+.pdf} \\includegraphics[angle=0,width=5cm]{mr1000+.pdf}}\n\t\\caption{Mass-radius relationships for planets with Earth-like properties regarding their rocky part (see Table \\ref{tab:parameters}), and computed with $\\gamma_+$, for multiple temperatures and water contents. Colors correspond to the three used EoSs: DZ06 (red), WP02 (blue) and Ma19 (green). Dashed lines correspond to regions where the atmosphere model is extrapolated beyond the available grid (see Appendix \\ref{sec:trilinear}). Filled circles correspond to cases where both $P$ and $\\gamma$ remain in the range of validity of used EoS. Open circles correspond to cases where $P$ or $\\gamma$ are computed in the extended range. Crosses correspond to cases where $P$ or $\\gamma$ are in the extrapolated range. Shaded areas correspond to H$_2$ (gray), H$_2$O (pink) and hydrodynamic escape (shaded) (see Sec. \\ref{sec:atmos-escape}).}\n\t\\label{fig:mr+}\n\\end{figure*}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{mdr400.pdf} \\includegraphics[angle=0,width=5cm]{mdr1000.pdf}}\n\t\\caption{Relative difference on radius between Ma19+ and Ma19- parametrizations showing the large impact of the temperature profile on mass-radius relationships.}\n\t\\label{fig:mdr}\n\\end{figure*}\n\nThe aim of this paper is to quantify the impact of the choice of EoS and $\\gamma$ computation on mass-radius relationships. With three EoSs and two $\\gamma$ parametrizations, 6 cases are considered : WP02$\\pm$, DZ06$\\pm$ and Ma19$\\pm$, with the +\/- sign standing for $\\gamma_+$ and $\\gamma_-$, respectively. For each case, three validity domains are explored: true validity range, extended range, or extrapolated range. A case is valid when the $(P,T)$ profile remains strictly in the true validity range of the used EoS and $\\gamma$ computation. It is extended if the EoS and\/or $\\gamma$ computation reaches the extended range. If either the EoS or $\\gamma$ reaches the extrapolated region, the whole case is considered extrapolated. For example, the Ma19+ parametrization (see Fig. \\ref{fig:mr+}) is always valid due to the important validity range of Ma19 EoS and $\\gamma_+$ computation. On the other hand, the Ma19- parametrization is always extrapolated because the computed $\\gamma_-$ is out of its validity and extended range.\n\n\\subsection{Mass-radius relationships and choice of EoS}\n\nFigure \\ref{fig:mr+} presents computed mass-radius relationships for the $\\gamma_+$ parametrization, and assuming Earth-like properties for the rocky part (see Table \\ref{tab:parameters}). As predicted from the shape of EoSs curves, WP02 and DZ06 EoSs underestimate the density and thus produce larger planets. This effect is accentuated for more massive planets with a larger amount of water, corresponding to cases where water pressure reaches the highest values. The radius is also overestimated for low-mass planets, because the hydrosphere becomes extended due to the low gravity, implying that a slight underestimation of the density can still lead to a substantial difference in radius. These results show the incontestable asset of the EoS developed by \\cite{Ma19}, and rule out the possibility of using WP02 or DZ06 EoSs to produce reliable mass-radius relationships for planets with substantial amounts of water. To remain in the true validity ranges of WP02 or DZ06 EoS, one should consider a few \\% of water content at most in the planet.\n\nAs discussed in Sec. \\ref{sec:gruneisen}, $\\gamma_+$ is always lower than $\\gamma_-$ in Earth-sized planets fully made of water. As a result $(P,T)$ profiles for $\\gamma_+$ parametrizations are steeper than for $\\gamma_-$ parametrizations (see Fig. \\ref{fig:gruneisen_val}), meaning the interior is colder for $\\gamma_+$. In turn, colder planets will be denser and thus smaller. The impact of the choice between $\\gamma_+$ and $\\gamma_-$ is shown in Fig. \\ref{fig:mdr}, where the relative difference on the radius between Ma19+ and Ma19- parametrizations is presented. In all cases, the relative difference between the models is 10\\%  at most.\n\nAs the mass of a planet increases, its gravity becomes more important, and its hydrosphere (interior structure and atmosphere) consequently thinner. Thinner hydrospheres, especially in the case of massive planets, lead to smaller relative differences in radii. Moreover, values of $\\gamma_+ $ and $ \\gamma_-$ become closer (and even equal) in the 10$^1$--10$^2$ GPa pressure range (see Fig. \\ref{fig:gruneisen_val}), thus reducing even more significantly the radii differences between the Ma19+ and Ma19- parametrizations.\n\nThe value of $\\gamma$ increases when the ($P$,$T$) curves of a hydrosphere approaches the liquid--Ice VII transition, which leads to a more important temperature gradient that prevents the formation of high pressure ices. This observation is in major disagreement with models assuming isothermal hydrospheres \\citep{Va06,Va07,Se07,Ze13,Br17,Ze19}, an hypothesis often justified by assuming that temperature has a secondary impact on EoSs, \nwhich remains a valid statement  for solid phases but not in the case of the hydrosphere. A correct treatment of the temperature gradient \\citep{Mo20} leads to the presence of high-temperature phases for H$_2$O (ionic, super ionic, plasma), which are more dilated, impacting significantly the mass-radius relationships.\n\nIn the following, we use $\\gamma_+$ and Ma19 to compute the mass-radius relationships. Indeed, the pressure and temperature ranges in the hydrospheres of sub-Neptunes-like planets lie well in the region for which the Ma19 formulation was developed. Also, due to the blanketing effect of the atmosphere, even the coldest planets irradiated at $T_\\mathrm{irr}~=~400$ K have a temperature of more than 2000 K at the 300 bar interface (see Fig. \\ref{fig:teq_tp}), which corresponds to the pressure at which the atmospheric and the internal model are connected. This interface is already located well above the range of validity of $\\gamma_-$.\n\n\\subsection{Planetary composition}\n\n\\begin{figure}[!ht]\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjb.pdf}}\\\\\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjc.pdf}}\\\\\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjd.pdf}}\n\t\\caption{From top to bottom: possible compositions of planets b, c and d of the GJ 9827 system \\citep{Ri19} in the forms of compositional ternary diagrams. Ternary diagrams were computed for the central masses of the planets, and contours are plotted for the measured radius and 1$\\sigma$ error bar.}\n\t\\label{fig:ternary}\n\\end{figure}\n\nMass-radius relationships only provide an order of estimate of the possible exoplanet composition. A more precise assessment is achieved via the use of compositional ternary diagrams. For a given planet mass and irradiation temperature, such a diagram shows the radius as a function of the planet's WMF and CMF. Possible compositions as thus retrieved from the contour at the level of the planet's measured radius. Computations presented here use only the central value of the mass of each planet, thus not taking into account the measurement error on the planet's mass.\n\nPossible compositions of the three planets of the GJ 9827 system are shown in Fig. \\ref{fig:ternary}, based on the planets parameters measurements made by \\cite{Ri19}. Planet b exhibits an Earth-like interior without the need of invoking a significant steam atmosphere. The presence of a thick steam atmosphere is rather consistent with the low-density measurements made for planet c, with a water content ranging from 1 to 8$\\%$. Physical properties (mass, radius and temperature) of planet c lead to important Jeans' escape (with our criterion in Eq. \\ref{eq:Jescape0}, see Fig. \\ref{fig:final}), suggesting the absence of H$_2$ and He in the atmosphere. Moreover, planet c is unlikely to accrete substantial amount of H$_2$ and He due to its low mass. Although planet d is consistent with a Jupiter-like interior due to their similar bulk densities, again, its high irradiation temperature suggests the presence of a H$_2$-He free atmosphere. Isochrones used by \\cite{Ri19} fix a lower limit on the age of 5 Gyr on the age of the system, which makes an H$_2$-He atmosphere less likely as Jeans' escape would remove them. Applying our model to the current measurements yields a WMF in the 5--30\\% range for planet d. These results are summarized in Table \\ref{tab:gj}.\n\n\\begin{table}\n\t\\centering\n\t\\caption{Planetary parameters of the GJ 9827 system used as input for the model, and estimated WMF using ternary diagrams (Fig. \\ref{fig:ternary}).} \n\t\\label{tab:gj}\n\t\\begin{tabular}{llll}\n\t\t\\tablewidth{0pt}\n\t\t\\hline\n\t\t\\hline\n\t\tPlanet \t& b \t\t& c & d \t\\\\\t\n\t\t\\hline\n\t\t$M_\\mathrm{p} ~(M_\\Earth)$ & $4.91\\pm0.49$ & $0.84\\pm0.66$ & $4.04\\pm0.83$ \\\\ \t\n\t\t$R_\\mathrm{p}~(R_\\Earth)$ & $1.58\\pm0.03$ & $1.24\\pm0.03$ & $2.02\\pm0.05$ \\\\\n\t\t$T_\\mathrm{irr}$ (K) & 1184 K & 820 K & 686 K \\\\ \\hline\n\t\tWMF (\\%) & 0--5 & 1--5 & 5--30 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nTernary diagrams presented here do not take into account the uncertainty on each planet's mass, and were computed for the central value only. If a planet's mass is slightly higher (resp. lower), its density increases (resp. decreases), while the estimated WMF diminishes (resp. grows). This implies that the mass and radius of a planet must be measured with extreme accuracy to constrain the WMF properly. Additional constraints can be applied from observational data such as the stellar elemental ratios (Fe\/Si, Mg\/Si) that could help constraining the core to mantle mass ratio \\citep{Br17}, and methods such as MCMC can be performed to simultaneously determine all parameters \\citep{Ac21}.\n\nFigure \\ref{fig:final} represents the computed mass-radius relationships for WMF of 0.2, 0.5 and 1. In this figure, the condition for substantial atmospheric loss due to Jeans' escape is derived by solving equation (\\ref{eq:Jescape0}) for each planet. One already known effect is that steam atmospheres are very extended \\citep{Mo20}, allowing to compute compositions without invoking small H$_2$-He enveloppes (1-5\\% by mass). The second effect is heating due to the adiabatic gradient, which decreases the density, and then increases the radius. In the 10--20 M$_\\Earth$ range, the radius of a planet with a WMF of 50\\% made of liquid H$_2$O is equal to that of a planet with a WMF of 20\\% constitued of supercritical H$_2$O. Also, the radius of a planet fully made of liquid H$_2$O is equivalent to that of a planet with half of its mass constituted of supercritical H$_2$O. This shows how important the error on the computation of WMF can be, depending on the physical assumptions made. In the figures presented in \\cite{Mo20}, where the DZ06 EoS was used, the model was able to match Neptune's mass (17 $M_\\Earth$) and radius (3.88 $R_\\Earth$) with a $95\\%$ H$_2$O interior at 300 K. With the Ma19 EoS, a 100\\% water planet presents a radius of 3.25 $R_\\Earth$ at $T_\\mathrm{irr}=400$ K and 3.6 $R_\\Earth$ at $T_\\mathrm{irr}=1300$ K.\n\n\\subsection{Analytical expression of mass-radius relationships}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=10cm]{mazevet.pdf}}\n\t\\caption{Comparison between mass-radius relationships computed with the Ma19+ model and those existing in literature. Our mass-radius relationships were computed for WMF of 20\\%, 50\\% and 100\\% with no metallic core, and temperatures of 400, 600, 800 and 1000 K. Thin solid lines and thin dashed lines are from  \\cite{Ze16} and \\cite{Br17}, respectively. Empty triangles, solid circles and stars correspond to planets subject to no atmospheric escape, to escape of H$_2$ only and to escape of both H$_2$ and H$_2$O (Jeans or blowoff), respectively. Planetary data are taken from the NASA exoplanet archive and updated to July 2020.}\n\t\\label{fig:final}\n\\end{figure*}\n\nAll produced mass-radius relationships are very well approximated by an equation of the form\n\\begin{eqnarray}\n\t\\log R_\\mathrm{p} = a \\log M_\\mathrm{p}+b\n\t+\\exp \\left(-d(\\log M_\\mathrm{p}+c)\\right), \\label{eq:fit1}\n\\end{eqnarray}\n\n\\noindent where log denotes the decimal logarithm, and $R_\\mathrm{p}$ and $M_\\mathrm{p}$ are normalized to Earth units. $a$, $b$, $c$ and $d$ are coefficients obtained by fits, and have one value for each composition $(x_\\mathrm{core},x_{\\mathrm{H}_2 \\mathrm{O}})$ and each temperature $T_\\mathrm{irr}$. For each fitted curve, we define the mean absolute error between data and fit as\n\\begin{eqnarray}\n\t\\mathrm{MAE} = \\frac{1}{N}\\sum_{i=1}^{N} \\left|\\frac{R_\\mathrm{p,model}-R_\\mathrm{p,fit}}{R_\\mathrm{p,model}}\\right|,\n\\end{eqnarray}\nValues of the MAE are 0.01--1\\% for all fits, indicating a good accuracy. The largest deviation between one point $(M_\\mathrm{p},R_\\mathrm{p})$ and the fitted curve is of $2.3\\%$, meaning the deviation between data and fit can be neglected. Fitted coefficients vary smoothly with respect to the three parameters $(x_\\mathrm{core},x_{\\mathrm{H}_2 \\mathrm{O}},T_\\mathrm{irr})$, allowing a good interpolation of the intermediate values. The produced grid uses the compositional parameters for the core and mantle calibrated for Earth (see Table \\ref{tab:parameters}), and data may be different if Fe\/Si or Mg\/si ratios are different.\n\n\\section{Discussion and conclusion} \n\\label{sec:ccls}\n\nThis work aimed at describing a model that computes a realistic structure for water-rich planets. This was achieved by combining an interior model with an updated EoS for water, and an atmospheric model that takes into account radiative transfer.\n\nVarious EoSs were investigated, and we find that results are identical when all of them are used within their validity range. However, the pressure profile rises sharply for planets with substantial amounts of water, invalidating the use of WP02 and DZ06 EoSs for WMF $>5\\%$. The blanketing effect due to the presence of the atmosphere leads to boundary temperatures greater than $2000$ K, leaving even less room for the DZ06 EoS to work properly. Both non-valid EoSs lead to the common result of overestimating the planetary radius by up to $\\sim$10\\%. Inexact computation of the Gr\\\"uneisen parameter yields another $\\sim$10\\% of error on the radius, at most. This requires to use an EoS that holds for pressures up to a few TPa and temperatures of $10^4$ K (conditions at the center of a pure water sphere of 1 Jupiter mass), such as \\cite{Ma19}.\n\nOther EoSs exist in the literature, such as those proposed by \\cite{Br18} and \\cite{Ha20}, which are functions either fitted or derived from the Gibbs or Helmholtz free energy. The range of validity for the EoS of \\cite{Br18} is less extended than that of \\cite{Ma19}, justifying our choice of EoS. \\cite{Ha20} presents a unified EoS for water from the connexion of already existing EoSs in their validity range, incuding \\cite{Ma19}. This EOS is then consistent with ours in the range of temperature and pressure explored here. The implementation of such an EoS is interesting for future works, especially when combining high pressure ices. \n\nIt should be noted that the most accurate EoS possible is not sufficient to produce precise mass-radius relationships for such planets. Assuming an adiabatic profile for the atmosphere (i.e. not taking into account radiative transfer) results in more extended atmospheres, as heat is transported solely by convection. Isothermal water layers seem closer to reality, but they produce the same mass-radius relationships as for liquid water \\citep{Ze16,Br17,Ha20}. Atmospheric models are essential for computing the atmosphere thickness and the energy that is transported to the interior.\n\nDerived MR relationships produce radii that match well those of the population of sub-Neptunes (1.75--3.5 $R_\\Earth$). This population corresponds to the second peak of the bimodal distribution of planet radii highlighted by \\cite{Fu17}, thus suggesting that irradiated ocean planets are good candidates to represent such planets \\citep{Mo20}. This bimodal distribution in planet radii has been predicted by \\cite{Ow13} and \\cite{Lo13} who investigated the atmospheric mass loss for Jupiter-like planets. However, the authors focused mainly on the loss of the enveloppe of a H\/He rich atmosphere. More recently, \\cite{Ow19} pointed out the need to extend this work to steam atmospheres. Our calculations aimed to do so in a very simplistic manner. Due to its greater density, we find that water is much less subject to atmospheric escape than H\/He. This suggests that highly irradiated planets could have lost their H\/He content through atmospheric loss processes, and the remaining matter led to either super-Earths ($R_\\mathrm{p} =$ 1--1.75 $R_\\Earth$) or a sub-Neptunes ($R_\\mathrm{p}=$ 1.75--3.5 $R_\\Earth$), depending on the final WMF.\n\nThe data grid can be used to assess a planet's composition once its mass and radius are known. Interpolating between the values can provide better precision. For a very precise computation, the full model is required since compositional parameters such as Fe\/Si and Mg\/Si ratios are required as well and depend on the star spectral analysis.\n\nTabulated mass-radius relationships and the coefficients obtained by fit for analytical curves can be found at \\url{https:\/\/doi.org\/10.5281\/zenodo.4552188} or \\url{https:\/\/archive.lam.fr\/GSP\/MSEI\/IOPmodel}. Explored parameter ranges are large enough to constrain planetary compositions for any WMF and CMF, and interpolate between given values without the need for the full model. We used the GJ 9827 system as a test case for our new relationships. Measured masses and radii of planets b and c of the GJ 9827 system indicate Earth-like or Venus-like interiors. We find that planet d could be an irradiated ocean planet with a WMF of $20\\pm10\\%$.\n\nIn the present model, only H$_2$O as a volatile is considered. Other volatiles such as CO$_2$, CH$_4$ or N$_2$ are expected to have similar densities as H$_2$O, thus producing similar mass-radius relationships. However, using a different gas will highly impact radiative transfer. Efficient radiative transfer for gases such as N$_2$ could keep the interior cold enough for maintaining a liquid water ocean, as it is the case for the Earth. An atmosphere dominated by gases such as H$_2$O or CO$_2$ lead to important blanketing, resulting in a Venus-like case.\n\nAtmospheric escape has motivated our focus on H\/He-free atmospheres. The addition of H$_2$ to the atmosphere is the scope of future work. The addition of O$_2$ as the product of water photodissociation will be considered as well.\n\n\\section*{Aknowledgements}\nOM and MD acknowledge support from CNES. We thank the anonymous referee for useful comments that helped improving the clarity of our paper and added important discussion.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nAntihydrogen, the bound state of an antiproton and a positron, is the simplest pure antimatter atomic system.\nThe first cold (non-relativistic) antihydrogen atoms were synthesised by the ATHENA experiment in 2002 by combining antiprotons and positrons under cryogenic conditions in a Penning trap \\cite{ATHENA_Nature}.\nThe neutral antihydrogen atoms formed were not confined by the electric and magnetic fields used to hold the antiprotons and positrons as non-neutral plasmas, but escaped to strike the matter of the surrounding apparatus and annihilate.\nDetection of the coincident antiproton and positron annihilation signals was used to identify antihydrogen in these experiments.\nHowever, before performing high-precision spectroscopy, it is highly desirable, perhaps even necessary, to confine the antihydrogen in an atomic trap.\n\n\\section{Atom Trap}\n\nAtoms with a permanent magnetic dipole moment $\\vec{\\mu}$ can be trapped by exploiting the interaction of the dipole moment with an inhomogeneous magnetic field.\nA three-dimensional maximum of magnetic field is not compatible with Maxwell's equations, but a minimum is.\nThus, only atoms with $\\mu$ aligned antiparallel to the magnetic field (so-called `low-field seekers') can be trapped.\n\nALPHA creates a magnetic minimum using a variation of the Ioffe-Pritchard configuration \\cite{Ioffe_Pritchard}, replacing the transverse quadrupole magnet with an octupole \\cite{ALPHA_magnet}.\nThe octupole and the `mirror coils' that complete the trap are superconducting and are cooled to 4~K by immersing them in liquid helium.\nThe depth of the magnetic minimum produced is approximately 0.8~T, equivalent to a trap depth of $0.6~\\mathrm{K}\\times k_\\mathrm{B}$ for ground state antihydrogen.\n\nALPHA's scheme to detect trapped antihydrogen is to quickly release trapped atoms from the atomic trap and detect their annihilation as they strike the apparatus.\nHaving the antihydrogen atoms escape over a short time minimises the background from cosmic rays that can mimic antihydrogen annihilations (see section \\ref{sec:detector}), so the magnet systems have been designed to remove the stored energy in as short a time as possible.\nThe current has been measured to decay with a time constant of 9~ms for the octupole and 8.5~ms for the mirror coils.\n\nThe atom trap is superimposed on a Penning trap, which is used to confine the charged particles used in antihydrogen production.\nThe Penning trap electrodes are also cooled by a liquid helium reservoir and reach a temperature of approximately 7~K.\nIn the absence of external heating sources, the stored non-neutral plasmas should come into thermal equilibrium at this temperature.\n\nIntroduction of the multipolar transverse magnetic field modifies the confinement properties of the Penning trap.\nIn the most extreme case, this manifests as a `critical radius' \\cite{CriticalRadiusTheory}, outside which particles can be lost from the trap simply because the magnetic field lines along which the particles move intersect the electrode walls.\nEven if particles are not lost, the transverse field results in a higher rate of plasma diffusion \\cite{GilsonFajans_Diffusion}.\nAs the plasma diffuses and expands, electrostatic potential energy is converted to thermal energy, resulting in a temperature higher than would be otherwise expected.\n\nALPHA chose to use an octupole instead of the prototypical quadrupole in its Ioffe trap to reduce the transverse fields close to the axis of the Penning trap, where the non-neutral plasmas are stored.\nThough this choice can significantly ameliorate the undesirable effects, it does not eliminate them entirely.\nOther sources of heating, notably the coupling of the particles to electronic noise \\cite{NoiseTemperature}, will also increase the temperature.\nThis highlights the importance of direct, absolute measurements of the particle temperature to accurately determine the experimental conditions.\n\n\\section{Cooling and temperature measurements of antiprotons}\n\nThe temperature of a plasma can be determined by measuring the distribution of particles in the tail of a Boltzmann distribution - a technique common-place in non-neutral plasma physics \\cite{TemperatureMeasurement}.\nThis measurement has the advantage of yielding the absolute temperature of the particles without recourse to supporting measurements (for example, of the density distribution), unlike measurements of the frequencies of the normal plasma modes \\cite{modes}, which can only give a relative temperature change.\nThe plasmas typical in ALPHA have densities in the range $10^6$ to $10^8~\\mathrm{cm^{-3}}$, with collision rates high enough to ensure that the plasma comes to equilibrium in a few seconds.\nIn equilibrium, the energy of the particles conforms to a Boltzmann distribution.\n\nTo measure the temperature, the particles are released from a confining well by slowly (compared to the axial oscillation frequency) reducing the voltage on one side of the well.\nAs the well depth is reduced, particles escape according to their energy; the first (highest-energy) particles to be released will be drawn from the tail of a Boltzmann distribution.\nAs the dump progresses, the loss of particles causes redistribution of energy and, at later times, the measured distribution deviates from the expected Boltzmann distribution.\nThe escaping particles can be detected using a micro-channel plate as a charge amplifier, or for antimatter particles, by detecting their annihilation.\nThe temperature is determined by fitting an exponential curve to the number of particles released as a function of energy, such as in the example measurement shown in Fig. \\ref{fig:temperature}.\n\n\n\\begin{SCfigure}[1.0][h]\n\\centering\n\\input{prettyTemp}\n\\caption{An example temperature measurement of approximately 45,000 antiprotons, after separation from the cooling electrons and with the inhomogeneous trapping fields energised. The straight line shows an exponential fit to determine the temperature, which in this case, is $\\left(310~\\pm~20\\right)~\\mathrm{K}$}\n\\label{fig:temperature}\n\\end{SCfigure}\n\nThe actual process of manipulating the trap potentials can change the temperature of the particles as the measurement takes place.\nParticle-in-cell (PIC) simulations of the measurement process have predicted that the temperature obtained from the fit is around 15\\% higher than the initial temperature for a typical antiproton cloud.\nFor the denser electron and positron plasmas, the measured temperature can be as much as factor of two higher than the initial temperature.\nWe can apply the corrections determined from these simulations to the measured temperature to find the true temperature.\nThis temperature diagnostic has been applied to all three particle species used in ALPHA - antiprotons, positrons and electrons.\nThe lowest temperatures measured for electron or positron plasmas at experimentally relevant densities $\\left(10^6~\\mathrm{cm^{-3}} \\text{or more}\\right)$ is of the order of 40~K.\n\nElectrons are used to collisionally cool the antiprotons, which, due to their larger mass, do not effectively self-cool via synchrotron radiation.\nBefore mixing the antiprotons with positrons to produce antihydrogen, the electrons must be removed.\nIf the electrons were allowed to remain, they could potentially deplete the positron plasma by forming positronium, destroy antihydrogen atoms through charge exchange, or destabilise the positron plasma by partially neutralising it.\n\nElectron removal is accomplished through the application of electric field pulses.\nThese pulses remove the confining potential on one side of the well holding the antiproton\/electron two-component plasma, typically for 100-300~ns.\nThe electrons, moving faster than the antiprotons, escape the well.\nThe well is restored before the antiprotons can escape, so they remain trapped.\nHowever, the process does not avoid disturbing the antiprotons.\nThe electron removal process has been the focus of a significant portion of experimental effort at ALPHA, and the coldest antiproton temperatures obtained have been around 200-300~K.\n\n\\section{Evaporative Cooling}\n\nAntiprotons at a few hundred Kelvin will have a very small probability of forming low-energy, trappable, antihydrogen atoms.\nTo further cool the antiprotons, ALPHA has implemented a technique of forced evaporative cooling.\nEvaporative cooling is a common-place technique in neutral particle trapping, and has been instrumental in the production of Bose-Einstein condensates \\cite{EVC_in_atoms}.\nHowever, evaporative cooling has found limited application to charged particles.\n\nBefore evaporative cooling, a cloud of antiprotons, containing 45,000 particles, with a radius of 0.6~mm, density $7.6\\times10^6~\\mathrm{cm^{-3}}$, and initial temperature of $\\left(1040~\\pm~45\\right)~\\mathrm{K}$ was prepared in a 1.5~V deep potential well.\nThe collision rate between antiprotons was of order 200~$\\mathrm{s}^{-1}$, high enough to ensure that the temperatures in the parallel and perpendicular degrees of freedom had equilibrated before evaporative cooling commenced.\n\nTo perform evaporative cooling, the confining potential on one side of the well is slowly (with respect to the equilibration rate) lowered.\nParticles with kinetic energy higher than the instantaneous well depth escape the trap, carrying with them energy in excess of the mean thermal energy.\nThe distribution then evolves towards a Boltzmann distribution with lower temperature, and the process continues.\n\nStarting with $45,000$ antiprotons at 1040~K, we have obtained temperatures as low as (9~$\\pm$~4)~K with $\\left(6\\pm1\\right)\\%$ of the particles stored in a 10~mV deep well.\nMeasurements of the temperature, number of particles and transverse size of the clouds were made at a number of points between the most extreme well depths.\nThe temperatures and number of particles remaining at each measurement point are shown in Fig. \\ref{fig:EVC_data}.\n\n\\captionsetup[subfloat]{position=top,captionskip=-10pt, justification=raggedright, singlelinecheck=false, margin=20pt}\n\n\\vspace{-0.5cm}\n\\begin{figure}[h]\n\\centering\n\\subfloat[]{\\input{evcTempCor}}\n\\subfloat[]{\\input{evcEff}}\n\\caption{The temperature (a) and the fraction of the initial number of particles (b) after evaporative cooling to a series of well depths. The minimum temperature is (9 $\\pm$ 4)~K}\n\\label{fig:EVC_data}\n\\end{figure}\n\nThe evaporation process can be described using simple rate equations for the number of particles $N$ and the temperature $T$;\n\n\\vspace{-0.2cm}\n\\begin{subequations}\n\\begin{center}\n\\begin{tabular}{p{0.4\\textwidth} p{0.4\\textwidth}}\n\t\\begin{equation}\n\t\t\\frac{\\mathrm{d}N}{\\mathrm{d}t} = - \\frac{N}{\\tau_{ev}}, \n\t\\end{equation} \t&\n\t\\begin{equation}\n\t\t\\frac{\\mathrm{d}T}{\\mathrm{d}t} = - \\alpha \\frac{T}{\\tau_{ev}} .\n\t\\end{equation}\n\\end{tabular}\n\\end{center}\n\\end{subequations}\n\\vspace{-0.5cm}\n\n\\noindent Here, $\\tau_{ev}$ is the characteristic evaporation timescale and $\\alpha$ is the excess energy carried away by an evaporating particle, in multiples of $k_\\mathrm{B} T$.\nAt a given time, the distribution of energies can be thought of as a truncated Boltzmann distribution, characterised by a temperature $T$, and the well depth $U$.\n$\\tau_{ev}$ is linked to the mean time between collisions, $\\tau_{col}$ as \\cite{EVC_Theory}\n\\begin{equation}\n\t\\frac{\\tau_{ev}}{\\tau_{col}} = \\frac{\\sqrt{2}}{3} \\eta e^\\eta,\n\t\\label{eqn:tau}\n\\end{equation}\nwhere $\\eta = U\/{k_\\mathrm{B}T}$ is the rescaled well depth.\nWe note the strong dependence of $\\tau_{ev}$ on $\\eta$, indicating that this is the primary factor determining the temperature in a given well. \nWe find values of $\\eta$ between 10 and 20 over the range of our measurements.\nThe value of $\\alpha$ can be calculated using the treatment in reference \\cite{ketterleReview}.\nWe have numerically modelled evaporative cooling in our experiment using these equations and have found very good agreement between our measurements and the model \\cite{ALPHA_EVC}.\n\n\nMeasurements of the transverse density profile were made by ejecting the particles onto an MCP\/phosphor\/CCD imaging device \\cite{ALPHA_MCP}.\nIt was seen that, as evaporation progressed, the cloud radius increased dramatically - see Fig. \\ref{fig:radius}.\nWe interpret this effect to be due to escape of the evaporating particles principally from the radial centre of the cloud, and the conservation of the total canonical angular momentum during the subsequent redistribution process.\nInside the cloud, the space charge reduces the depth of the confining well.\nThis effect is accentuated closer to the trap axis, with the result that the well depth close to the axis can be significantly lower than further away.\nThe evaporation rate is exponentially suppressed at higher well depths (eqn. \\ref{eqn:tau}), so evaporation is confined to a small region close to the axis, causing the on-axis density to become depleted.\nThis is a non-equilibrium configuration, and the particles will redistribute to replace the lost density.\nIn doing so, some particles will move inwards, and to conserve the canonical angular momentum, some particles must also move to higher radii \\cite{confinementTheorem}.\nAssuming that all loss occurs at $r=0$, the mean squared radius of the particles, $\\left< r^2 \\right>$, will obey the relationship\n\\vspace{-0.5cm}\n\\begin{equation}\n\t\\label{eq:expansion}\n\tN_0\\left< r_0^2\\right> = N \\left< r^2 \\right>,\n\t\\vspace{-0.5cm}\n\\end{equation}\nwhere N is the number of particles, and the zero subscript indicates the initial conditions.\n\nAs seen in Fig. \\ref{fig:radius}, this model agrees very well with the measurements.\nThis radial expansion can be problematic when attempting to prepare low kinetic energy antiprotons to produce trappable antihydrogen atoms, as the energy associated with the magnetron motion grows with the distance from the axis, and the electrostatic potential energy released as the radius expands can reheat the particles.\nThe effect can be countered somewhat by taking a longer time to cool the particles, resulting in a higher efficiency and, thus, a smaller expansion, but we find that the efficiency depends very weakly on the cooling time.\n\n\\begin{SCfigure}[1.0][h]\n\n\t\\input{EVCSize}\n\t\\caption{The measured size of the antiproton cloud using a MCP\/phosphor\/CCD device as a function of the number of particles lost. This is compared to the size predicted from eqn \\ref{eq:expansion}}\n\t\\label{fig:radius}\n\\end{SCfigure}\n\nColder antiprotons are of great utility in the effort to produce cold antihydrogen atoms.\nAntihydrogen production techniques can be broadly categorised as `static' - in which a cloud of antiprotons is held stationary and positrons, perhaps in the form of positronium atoms are introduced \\cite{positronium}, or `dynamic' - where antiprotons are passed through a positron plasma \\cite{Nested}.\nIn the first case, the advantages of cold antiprotons are obvious, as the lower kinetic energy translates directly into lower-energy antihydrogen atoms.\nIn the second case, the colder temperature allows the manipulations used to `inject' the antiprotons into the positrons to produce much more precisely defined antiproton energies.\nIndirectly, this will also permit these schemes to produce more trappable antihydrogen.\n\n\\section{Annihilation vertex detector}\\label{sec:detector}\n\nAmong the most powerful diagnostic tools available to experiments working with antimatter are detectors capable of detecting matter-antimatter annihilations.\nAntiproton annihilations produce an average of three charged pions, which can be detected by scintillating material placed around the trap.\nThe passage of a pion through the scintillator produces photons, which trigger a cascade in a photo-multiplier tube to produce a voltage pulse.\nIndividual voltage pulses can be counted to determine the number of annihilations.\n\nA further technique uses a position-sensitive detector to reconstruct the trajectories of the pions and find the point where the antiproton annihilated (usually called the `vertex').\nThe ALPHA annihilation vertex detector comprises sixty double-sided silicon wafers, arranged in three layers in a cylindrical fashion around the antihydrogen production and trapping region.\nEach wafer is divided into 256 strips, oriented in orthogonal directions on the p- and n- sides.\nCharged particles passing through the silicon result in charge deposits, and the intersection of perpendicular strips with charge above a defined threshold marks the location a particle passed through the silicon.\n\nEach module is controlled by a circuit that produces a digital signal when a charge is detected on the silicon.\nIf a coincidence of modules is satisfied in a 400~ns time window, the charge profile is `read-out' and digitised for further analysis.\nEach readout and associated trigger and timing information comprises an `event'.\nThe pion trajectories are reconstructed by fitting helices to sets of three hits, one from each layer of the detector.\nThe point that minimises the distance to the helices is then identified as the annihilation vertex.\nAn example of an annihilation event is shown in Fig. \\ref{fig:vertex}(a).\n\n\\vspace{-0.5cm}\n\\begin{SCfigure}[1.0][h]\n\\includegraphics[width=0.6\\textwidth]{Verticesdrawing}\n\\caption{(a) an example reconstruction of an antihydrogen annihilation and (b) a cosmic ray event. The diamond indicates the position of the vertex identified by the reconstruction algorithm, the polygonal structure shows the locations of the silicon wafers, the dots are the positions of the detected hits, and the inner circle shows the radius of the Penning trap electrodes. Also shown are annihilation density distributions associated with antihydrogen production (c, e) and deliberately induced antiproton loss (d, f). (c) and (d) are projected along the cylindrical axis, with the inner radius of the electrodes marked with a white circle, while (e) and (f) show the azimuthal angle $\\phi$ against the axial position $z$}\n\\label{fig:vertex}\n\\end{SCfigure}\n\\vspace{-0.5cm}\n\nExamination of the spatial distributions of annihilations can yield much insight into the physical processes at work.\nATHENA established that antihydrogen production resulted in a characteristic `ring' structure - an azimuthally smooth distribution concentrated at the radius of the trap electrodes \\cite{ATHENA_imaging}, shown in \\ref{fig:vertex}(c) and (e).\nIn contrast, the loss of bare antiprotons occurred in spatially well-defined locations, called `hot-spots', examples of which are shown in \\ref{fig:vertex}(d) and (f).\nThis was interpreted to be due to microscopic imperfections in the trap elements.\nThese produce electric fields that break the symmetry of the trap and give rise to preferred locations for charged particle loss.\nWhen antihydrogen is produced in a multipole field, antiprotons generated by ionisation of weakly-bound antihydrogen also contribute small asymmetries \\cite{ALPHA_HbarOct}.\nThese features are present in Fig. \\ref{fig:vertex}(c) and (e).\n\nThe vertex detector is also sensitive to charged particles in cosmic rays.\nWhen passing through the detector, they are typically identified as a pair of almost co-linear tracks (Fig. \\ref{fig:vertex}(b)), and can be misidentified as an annihilation.\nCosmic-ray events when searching for the release of trapped antihydrogen thus present a background.\n\nTo develop a method to reject cosmic ray events, while retaining annihilations, we compared samples of the events using three parameters, shown in Fig. \\ref{fig:distributions}.\nCosmic rays have predominantly two tracks, while antiproton annihilations typically have more. 95\\% of cosmic events have two or fewer identified tracks, while 58\\% of antiproton annihilations have at least three.\nA significant number of antiproton annihilations can have only two tracks, so it is not desirable to reject all these events as background.\n\n\\vspace{-0.2cm}\n\\begin{SCfigure}[1.0][h]\n\\includegraphics[width=0.6\\textwidth]{vertexDistributions}\n\\caption{Comparison of the distributions of event parameters for antiproton annihilations (solid line) and cosmic rays (dashed line). Shown are (a) the number of identified charged particle tracks, (b) the radial coordinate of the vertex, and the squared residual from a linear fit to the identified positions for the events with (c) two tracks and (d) more than two tracks. The shaded regions indicate the range of parameters that are rejected to minimise the p-value as discussed in the text}\n\\label{fig:distributions}\n\\end{SCfigure}\n\\vspace{-0.4cm}\n\nWe determine if the tracks form a straight line by fitting a line to the hits from each pair of tracks, and calculating the squared residual value.\nAs seen in Fig. \\ref{fig:distributions}(c) and (d), cosmic events have much lower squared residual values than annihilations.\nThis is to be expected, since particles from cosmic rays have high momentum and pass through the apparatus and the magnetic field essentially undeflected, while the particles produced in an annihilation will, in general, move in all directions.\nIn addition, annihilations occur on the inner wall of the Penning trap, at a radius of $\\sim$2.2~cm, and as shown in Fig. \\ref{fig:distributions}(b), reconstructed annihilation vertices are concentrated here, whereas cosmic rays pass through at a random radius.\n\nBy varying the ranges of parameters for which events are accepted, we could optimise the annihilation detection strategy.\nThe point where the `p-value' -- the probability that an observed signal is due to statistical fluctuations in the background \\cite{PDG} --  was minimised requiring the vertex to lie within 4~cm of the trap axis, and the squared residual value to be at least 2~$\\mathrm{cm}^2$ or 0.05~$\\mathrm{cm}^2$ for events with two tracks and more than two tracks, respectively.\n\nThese thresholds reject more than 99\\% of the cosmic background, reducing the absolute rate of background events to 22~mHz, while still retaining the ability of identify $\\sim 40\\%$ of antiproton annihilations.\nWhile this method effectively removes cosmic rays as a source of concern, other background processes, including mirror-trapped antiprotons must also be considered when searching for trapped antihydrogen.\nOur cosmic-ray rejection method has been applied to data taken from the 2009 ALPHA antihydrogen trapping run, and a full discussion of the results obtained will be made in a forthcoming publication.\n\n\\section{Conclusions and outlook}\nIn this paper we have described two of the most recent techniques developed by the ALPHA collaboration in our search for trapped antihydrogen.\nEvaporative cooling of antiprotons has the potential to greatly increase the number of low-energy, trappable atoms produced in our experiment.\nThe use of our unique annihilation vertex imaging detector to discriminate with high power between annihilations and cosmic rays will be a vital tool to identify the first trapped antihydrogen atoms.\nWe have integrated both of these techniques into our experiment and are hopeful of soon being able to report detection of trapped antihydrogen.\n\n\\begin{acknowledgements}\nThis work was supported by CNPq, FINEP\/RENAFAE (Brazil), ISF (Israel), MEXT (Japan), FNU (Denmark), VR (Sweden), NSERC, NRC\/TRIUMF, AIF (Canada), DOE, NSF (USA), EPSRC and the Leverhulme Trust (UK).\nWe are also grateful to the AD team for the delivery of a high-quality antiproton beam, and to CERN for its technical support.\n\\end{acknowledgements}\n\n\\subsection*{Note added in proof}: Since the preparation of this article, trapping of antihydrogen atoms has been\nachieved by the ALPHA collaboration \\cite{ALPHA_Nature}\n\n\n\t\\bibliographystyle{aipnum4-1}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\n\\label{intro}\n\nThe recent developments in our understanding of the nuclear regions of\nnearby galaxies  provide us with a  new framework in  which to explore\nthe classical issue of the connection between host galaxies and AGN.\n\nAll evidence now points to the idea that most galaxies\nhost a supermassive black hole (SMBH) in their centers\n\\citep[e.g.][]{kormendy95} and that its mass is closely linked to the\nhost galaxies properties, such as the stellar velocity dispersion\n\\citep{ferrarese00,gebhardt00}.  This is clearly indicative of a\ncoevolution of the galaxy\/SMBH system and it also provides us with\nindirect, but robust, SMBH mass estimates for large sample of objects.\nFurthermore, the innermost structure of nearby galaxies have been\nrevealed by HST imaging, showing the ubiquitous presence of singular\nstarlight distributions with surface brightness diverging as\n$\\Sigma(r)\\sim r^{-\\gamma}$ with $\\gamma>0$ \\citep[e.g.][]{lauer95}.\nThe distribution of cusp slopes \\citep{faber97} is bimodal, with a\npaucity of objects with $0.3<\\gamma<0.5$.  Galaxies can then be\nseparated on the basis of their brightness profiles in the two classes\nof ``core'' ($\\gamma \\leq 0.3$) and ``power-law'' ($\\gamma \\geq 0.5$)\ngalaxies, in close correspondence to the revision of the Hubble\nsequence proposed by \\citet{kormendy96}.\n\nBut despite these fundamental breakthroughs we still lack a clear\npicture of the precise relationship between AGN and host galaxies.\nFor example, while spiral galaxies preferentially harbour radio-quiet\nAGN, early-type galaxies host both radio-loud and radio-quiet\nAGN. Similarly, radio-loud AGN are generally associated with the most\nmassive SMBH as there is a median shift between the radio-quiet and\nradio-loud distribution, but both distributions are broad and overlap\nconsiderably \\citep[e.g.][]{dunlop03}.\n\nIn this framework, in two senses\nearly-type galaxies appear to be the critical class of\nobjects, where the transition between the two profiles classes occurs\n(i.e. in which core and power-law galaxies coexist) and \nin which they can host  \neither radio-loud and radio-quiet AGN. We thus started a\ncomprehensive study of a sample of early-type galaxies (see below for\nthe sample definition) to \nexplore the connection between  the multiwavelength \nproperties of AGN and the characteristics of their hosts. \nSince the 'Nuker' classification can only be obtained when the \nnuclear region, potentially associated with a shallow cusp,\ncan be well resolved, such a study must be limited to nearby galaxies.\nThe most compact cores will be barely resolved at a\ndistance of 40 Mpc (where 10 pc subtend 0\\farcs05)\neven in the HST images. Furthermore, high quality radio-images \nare required for an initial selection of AGN candidates. \n\nWe then examined two samples of  nearby objects for\nwhich  radio observations combining  relatively high  resolution, high\nfrequency and sensitivity  are available,  in order  to  minimize the\ncontribution from radio emission not related to the galaxy's nucleus\nand confusion from background sources.\nMore specifically we focus on the samples of early-type galaxies \nstudied by \\citet{wrobel91b}  and \\citet{sadler89} both observed with\nthe VLA at  5 GHz with a  flux limit  of\n$\\sim$  1 mJy.   The two  samples were  selected with  a  very similar\nstrategy. \\citet{wrobel91a} extracted  a northern\nsample  of  galaxies from  the  CfA  redshift survey  \\citep{huchra83}\nsatisfying  the following  criteria: (1)  $\\delta_{1950} \\geq  0$, (2)\nphotometric magnitude  B $\\leq$ 14; (3) heliocentric  velocity $\\leq$ 3000\nkm s$^{-1}$, and (4) morphological  Hubble type T$\\leq$-1, for a total\nnumber of  216 galaxies.  \\citet{sadler89} selected  a similar southern sample\nof 116 E and S0 with  $-45 \\leq \\delta \\leq -32$. \nThe only difference between\nthe  two  samples  is  that  \\citeauthor{sadler89} did  not  impose  a\ndistance limit.   Nonetheless,  the  threshold in  optical  magnitude\neffectively limits the  sample to a recession velocity  of $\\sim$ 6000\nkm s$^{-1}$. \n\nIn \\citet[ hereafter Paper I]{capetti05}, we focused on the 116 \ngalaxies detected in these VLA surveys to boost the fraction\nof AGN with respect to a purely optically selected sample.\nWe used archival HST observations, available for 65 objects, to study their \nsurface brightness profiles and to separate these early-type \ngalaxies into core and power-law galaxies following the Nukers scheme, \nrather than on the traditional morphological classification (i.e. into E and\nS0 galaxies). \nHere we focus on the sub-sample formed by the 29 ``core'' galaxies. \n\nWe adopt a Hubble constant H$_{\\rm o}=75$ km  s$^{-1}$ Mpc$^{-1}$.\n\n\\section{A critical analysis of the classification as core galaxies.}\n\\label{sersic}\n\nIn \\citetalias{capetti05} we adopted the classification into power-law and\ncore galaxies following the scheme proposed by \\citet{lauer95}.\nWe then separated early-type galaxies on the  basis of the \nslope of their  nuclear  brightness\nprofiles obtained using the Nukers law (i.e. a double power-law \nwith innermost slope $\\gamma$)\ndefining as core-galaxies all objects with $\\gamma \\leq 0.3$.\nSince this strategy has been subsequently challenged by\n\\citet{graham03}, who introduced a different definition of\ncore-galaxies,\nit is clearly important to assess whether the identification\nof an object as a core galaxy is dependent on the fitting scheme adopted.\n\n\\citeauthor{graham03} argued that a S\\'ersic model\n\\citep{sersic68}\nprovides a better\ncharacterization of the brightness profiles of early-type galaxies.\nIn particular they pointed out that, among other issues, \ni) the values of the Nukers law\nparameters depend on the radial region used for the\nfit, ii) the Nukers fit is unable to reproduce the large scale\nbehaviour of early-type galaxies \nand, most importantly for our purposes, iii) the identification of\na core galaxy from a Nuker fit might not be recovered by a S\\'ersic\nfit. Conversely, they were able to fit power-law galaxies (in the\nNukers scheme), as well\nas dwarf ellipticals \\citep{graham03b}, with\na single S\\'ersic law over the whole range of radii. \nThey also suggested a new definition of \ncore-galaxy as the class of objects \nshowing a light deficit toward the center with respect to the S\\'ersic\nlaw \\citep{trujillo04}.\n\nIn this context,\nwe discuss in detail  here the behaviour of the most critical\nobjects, i.e. the two core galaxies for which the Nuker\nlaw returns the smallest values for the break radius,\nnamely UGC~7760 and UGC~7797 for which $r_b = 0\\farcs49$\nand $r_b = 0\\farcs21$ respectively. We fit both objects with\na S\\'ersic law. The final fits, shown in Fig. \\ref{sersicfig},\nwere obtained iteratively, fitting the external regions\nwhile flagging the innermost points\nout to a radius at which the residual from the S\\'ersic law exceeded\na threshold of 5 \\%. The S\\'ersic law  in general provides a remarkably good\nfit to the outer regions, with typical residuals of $\\sim$ 1\\%, but \na substantial central light deficit is clearly\npresent in both objects.\nThis indicates that both objects can be \nclassified as core-galaxies in the Graham et al. scheme.\n\nUsing the brightness profiles for the core-galaxies for which\nwe obtained Nuker fits in \\citetalias{capetti05} (14 additional objects) \nwe obtained similar results. Very\nsatisfactory fits can be obtained with a S\\'ersic law on the external\nregions of these galaxies, but \nthey all show an even clearer central light\ndeficit, as expected given the presence of well resolved shallow cores.\n\nWe conclude that, for the galaxies of our sample,\nthe objects classified as core-galaxies in the Nuker scheme \nare recovered as such with the Graham et al. \ndefinition.\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f1a.ps,width=0.50\\linewidth}\n\\psfig{figure=f1b.ps,width=0.50\\linewidth}\n}\n\\caption{S\\'ersic fits for the two core galaxies of our sample\nwith the smallest values for the core radius. A substantial central light \ndeficit is clearly present in both objects, conforming to\nthe ``core'' classification in the Graham et al. scheme.}\n\\label{sersicfig}\n\\end{figure*}\n\n\\section{Basic data and nuclear luminosities}\n\\label{nuc}\n\n\\input{tab1.tex}\n\nThe basic\ndata for  the selected galaxies, namely the recession velocity  \n(corrected for Local Group infall onto Virgo), the K\nband  magnitude from  the Two  Micron  All Sky  Survey (2MASS),\nthe galactic extinction and the total and\ncore radio fluxes were given in \\citetalias{capetti05}.\n \nIn the following three subsections, \nwe derive and discuss the measurements for the\nnuclear sources in the optical, X-ray and radio bands.\n\n\\subsection{Optical nuclei.}\n\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f2a.ps,width=0.33\\linewidth}\n\\psfig{figure=f2b.ps,width=0.33\\linewidth}\n\\psfig{figure=f2c.ps,width=0.33\\linewidth}\n}\n\\caption{\\label{hstnuc} Brightness profile and its derivative for \nthree objects of the sample, namely IC~4296, NGC~4373 and NGC~1316.\nThe first two galaxies show, at decreasing significance level, the\ncharacteristic up-turn in the profile associated with a nuclear source.\nThis is not seen in the third object which is then considered as a\nnon-detection.}\n\\end{figure*}\n\nThe detection and measurement of an optical nuclear source at the \ncenter of a galaxy is a challenging task particularly  \nwhen it represents only a small \ncontribution with respect to the host emission, \nas is likely often to be the case of the weakly active galaxies\nmaking up our sample.\n\nDifferent approaches have been employed in the literature.\nThe most widely used method is to fit the overall brightness \nprofile of a galaxy\nwith an empirical functional form and to define a galaxy as\n``nucleated'' when it shows a light excess in its central region\nwith respect to the model \\citep[e.g.][]{lauer04,ravi01}.\nThe drawback of this ``global'' approach is that it \nassumes that the model can be extrapolated inwards\nfrom the radial domain over which the fit was performed.\nFurthermore, the measurement and identification of the nuclear\ncomponent are coupled with the behaviour of the brightness profile at all radii\nand with the specific choice of an analytic form. \nAlthough this is not a significant issue for bright\npoint sources, it is particularly worrisome for the faint nuclei\nwe are dealing with.\nNonetheless, \\citet{rest01} pointed out that in general nuclear light excesses\nare associated with a steepening of the profile as the HST resolution\nlimit is approached. Indeed this is expected in the presence of\na nuclear point source, since the convolution with the Point Spread \nFunction produces a smooth decrease of the slope toward the center\nwhen only a diffuse galactic component is present.\nWe therefore preferred to adopt a ``local'' approach\nto identify nuclear sources, based on the characteristic up-turn\nthey cause in the nuclear brightness profile.\n\nMore specifically, we  evaluated the  derivative  of  the\nbrightness profile in a log-log representation \nfor the sources of our sample. In order to increase the \nstability of the slope measurement this has been estimated \nby combining the brightness \nmeasured over two adjacent points on each side of the radius of interest,\nyielding a second order accuracy. We then look for  \nthe presence of a nuclear up-turn in the derivative \nrequiring for a nuclear detection a difference larger than\n3 $\\sigma$ from the slopes at the local minimum and maximum.\nThis a rather conservative definition since \nthe region over which the up-turn is detected \nextends over several pixels while we only consider the\npeak-to-peak difference. \n\nTo illustrate this we focus on three\ncases. In the HST image as well as in the brightness profile of IC~4296 \na nucleus clearly stands out against the underlying background and\nthe central  steepening at about $r=0\\farcs15$ is highly significant. \nNGC~4373 is the detection with the least significance of our sample,\nin  which the presence of a  nucleus is uncertain\nfrom just the visual inspection of \nthe optical image,  but the\nderivative of the brightness profile reveals the effect  of the point\nsource with an increase of 0.013 $\\pm$ 0.004 from\n$r=0\\farcs1$ and $r=0\\farcs07$.  \nInstead  in NGC~1316 we do not  have evidence for any\ncompact point source, both in the image and in the brightness profile\nderivative, and it is considered as a non-detection.\n\nAdopting this strategy in 18 out of 29 objects we identify an optical nucleus,\nwith a  percentage of $\\sim$ 60\\% of the total sample. \nIn seven objects we did not find any upturn and these are \nconsidered upper limits.\nNote that this is again a conservative approach, since \na nuclear source can still be\npresent but its intensity might not be sufficient to compensate the\ndownward trend of the derivative sets by the host galaxy. \n\nIn the remaining 4 objects the central regions have a complex \nstructure and no estimate\nof the optical nucleus intensity can be obtained. \nIn two cases (UGC~8745 and UGC~9723) \nthe central regions are completely hidden by a kpc scale edge-on disk, \nwhile in NGC~3557 the study of its nuclear regions \nis hampered by the presence \nof a circumnuclear dusty disk. In UGC~9655, the innermost region\n($r<0\\farcs1$) has a lower brightness than its surrounding; since\nonly a single band image is available we cannot assess \nif this is due to dust absorption or\nto a genuine central brightness minimum \nas in the cases discussed by \\citet{lauer02}.\n\nWe measured the  nuclear luminosity with the task RADPROF  in IRAF,\nchoosing as the  extraction region a circle centered on the  nucleus with\nradius set at the location of the up-turn and as the background region \na circumnuclear annulus, 0.1\\arcsec\\ in width. \nFor the undetected nuclei we set as upper limits the light excess\nwith respect to the starlight background\nwithin a circular aperture 0.1\\arcsec\\ in diameter.\nThen  we use  the PHOTFLAM and  EXPTIME keyword in  the image\nheader to convert  the total counts to fluxes. \nErrors on the measurements of the optical nuclei are dominated\nby the uncertainty in the behaviour of the host's profile, \nwhile the statistical and absolute calibration errors amount\nto less than 10 \\%.h\nThe very presence of the nucleus prevents\nus from determining accurately the host contribution within\nthe central aperture. Our strategy is to remove the background measured\nas close as possible to the nucleus, i.e. effectively we adopted\na constant starlight distribution in the innermost regions. \nAn alternative approach would be to extrapolate the profile with\na constant slope instead. Our definition of\nnuclear sources (an increase in the profile's derivative) \nimplicitly requires that the observed profile lies\nabove this extrapolation, but the resulting flux is reduced\nby at most a factor of 2 (with respect to the case of constant\nbackground) for the nuclei with the smallest contrast\nagainst the galaxy. As will become clear in the next sections,\nerrors of this magnitude only have a marginal impact on our conclusions.\nThe resulting fluxes are reported\nin Table \\ref{tabsample2}.\nWe finally derived all the\nluminosities referred to 8140 \\AA\\ (see Table \\ref{lum}),\nafter correcting for the Galactic extinction as tabulated in \n\\citetalias{capetti05} and\nadopting an optical spectral \nindex\\footnote{We define the spectral index $\\alpha$ with the spectrum\nin the  form F$_\\nu  \\propto \\nu^{-\\alpha}$} $\\alpha_o = 1$.\n\n\\subsection{X-ray nuclei}\n\nFor the measurements of the X-ray nuclei we concentrate \nonly on the Chandra measurements, as this telescope provides \nthe best  combination of\nsensitivity and resolution necessary to detect the faint nuclei expected in\nthese weakly active galaxies.\nData for 21 core galaxies are available in the Chandra public archive.\n\nWhen available, we used the results of the analysis of the X-ray\ndata from the literature. \nWe find estimates of the luminosities of the nuclear sources (usually defined\nas the detection of a high energy power-law component)\nbased on  Chandra data  for 16\nobjects of our sample (12 of which are detections and 4 are upper limits) \nwhich we rescaled to our adopted distance \nand converted to the 2-10 keV  band, using  the\npublished power law index. In\nTab \\ref{tabsample2} we give a summary of the available\nChandra data, while references and details on the X-ray \nobservations and analysis are presented in Appendix \\ref{notes}. \n\nWe also considered the Chandra archival data for the 5 unpublished objects,\nnamely UGC~5902, UGC~6297, UGC~7203, NGC~3557 and NGC~5419. We analyzed these\nobservations  using the Chandra  data analysis  CIAO v3.0.2,  with the\nCALDB version 2.25, using the same strategy as in\n\\citet{balmaverde05}. \nWe reprocessed all the data from level 1 to level 2,\nsubtracting the bad pixels,  applying ACIS CTI correction, coordinates\nand pha randomization.  We searched for background flares and \nexcluded some period of bad aspect.\n\nWe then extracted the spectrum in a circle region centered on the nucleus \nwith a radius of 2\\arcsec\\ and we take the background in an annulus\nof 4\\arcsec. We grouped the spectrum to have at least 10 counts \nper bin and applied Poisson statistics. \n     \nFor two  objects  (NGC~3557 and NGC~5419) we obtain  a detection of a\nnuclear power-law source by\nfitting the spectrum using an absorbed power-law plus a thermal\nmodel,  with  the  hydrogen  column  density  fixed  at  the  Galactic\nvalue. Details of the results are given in Appendix \\ref{notes}.\nFor the remaining 3 galaxies we \nset an  upper  limit to  any  nuclear  emission, with  the\nconservative   hypothesis   that   all   flux  that   we   measure   is\nnon-thermal. We then fit the  spectrum with an  absorbed (to  the galactic\nvalue)  power law  model with  photon index  $\\Gamma=2$.  \n\nThe X-ray luminosities for all objects are given in Table \\ref{lum}.\n\n\\subsection{Radio nuclei.}\n\nThe radio data available for all objects of our sample \nare drawn from the surveys by \\citet{wrobel91b} \nand \\citet{sadler89}, performed with the VLA at 5 GHz with a resolution of\n$\\sim$ 5\\arcsec. Although these represent the most uniform\nand comprehensive studies of radio emission in early-type galaxies,\nthey do not always have a resolution sufficient to separate the core\nemission from any extended structure. \\citet{sadler89} \nargued that at decreasing radio luminosity there is a corresponding\nincrease of the fractional contribution of the radio core.\n\nTo verify whether the VLA data overestimate the core flux, we searched the literature  for radio  core measurements  obtained at\nhigher  resolution  (and\/or  higher  frequency) than  our  data.  This\nwould  improve  the  estimate  of the  core  flux  density,\navoiding the contribution of  extended emission or spurious sources to\nthe nuclear  flux as  well as revealing  any radio  structure.  Better\nmeasurements, from  VLBI data  or from  higher  frequency\/resolution VLA\ndata, are available for most CoreG  (23 out of 29) and compact cores\nwere detected in all but 2 objects. The radio core\nfluxes are taken from \\citet{nagar02} (15 GHz VLA data and 5 GHz VLBI\ndata), \\citet{filho02} and \\citet{krajnovic02} (8.4 GHz at the VLA),\n\\citet{jones97} (8.4 GHz VLBI data) and\n\\citet{slee94} (PTI 5 GHz interpolated data).\n\nIn Fig. \\ref{cfr} we\ncompare the radio  core flux density used in our  analysis \nagainst observations made at higher resolution.\nOverall there  is a  substantial agreement between the two datasets,\nwith a  median difference  of only\n$\\sim$0.25 dex (a factor  $\\sim$1.6), with only two objects\nsubstantially offset (by a factor of $\\sim$ 10).\nHowever, since  these data are  highly inhomogeneous and  given the\ngeneral  agreement with the  5 GHz VLA measurements, we\nprefer to retain the values of \\citeauthor{wrobel91b} and \\citeauthor{sadler89}.\nNonetheless, we always checked that using these higher resolution \ncore fluxes our main results are not significantly affected \n(see Appendix \\ref{radionuc} for a specific example).\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f3.ps,width=1.00\\linewidth}\n}\n\\caption{Radio core flux density for CoreG obtained at \n5 GHz with VLA (used in this work) compared to higher resolution data from\n\\citet{slee94,nagar02,filho02,\nkrajnovic02,jones97}.\nThe dotted line is the bisectrix of the plane.}\n\\label{cfr}\n\\end{figure}\n\n\n\\section{The multiwavelength properties of nuclei of core galaxies.}\n\\label{nuclei}\n\n\\input{tab2.tex}\n\nHaving collected the multiwavelength  information for the nuclei of our\ncore  galaxies we  can compare  the emission  in the  different bands.\nFirst of all, we can estimate the ratio between the radio, optical and\nX-ray  luminosities: the  median values  are Log$(\\nu L_r\/\\nu L_o)  \\sim -1.5$\n(equivalent to  a standard radio-loudness parameter Log  R $\\sim$ 3.6)\n\\footnote{R =  L$_{5\\rm {GHz}}$ \/ L$_{\\rm B}$. As in\n  Sect. \\ref{nuc} we transformed the optical fluxes to the B band\nadopting an optical spectral index $\\alpha_o = 1$.}\nand Log R$_{\\rm X}$  = Log$(\\nu L_{\\rm r}\/L_{\\rm X})  \\sim -1.3$,  \nboth with a  dispersion of  $\\sim$ 0.5\ndex.  These  ratios are clearly  indicative of a radio-loud  nature for\nthese  nuclei when compared  to both  the traditional  separation into\nradio-loud  and radio-quiet  AGN \n\\citep[Log R =  1, e.g.][]{kellermann94}, as well  as with  the\nradio-loudness threshold introduced by \\citet{terashima03}\nbased on  the X-ray to radio luminosity  ratio  (Log R$_{\\rm X}$ = -4.5).  \nFurthermore, the nuclear\nluminosities  in   all  three   bands  are clearly correlated (see\nFig. \\ref{corr1} and Table \\ref{tab0} for a summary of the results of\nthe statistical analysis): the generalized (including the presence\nof upper limits) Spearman rank correlation coefficient\n$\\rho$ is 0.63 and 0.89 for $L_{\\rm r}$ vs. $L_{\\rm o}$ \nand $L_{\\rm r}$ vs. $L_{\\rm X}$ respectively, with probabilities that\nthe correlations are not present of only 0.002 and 0.0001.\n\nBoth results are reminiscent of what is  observed for the\nradio-loud nuclei \nof low luminosity radio-galaxies (LLRG). \\citet{chiaberge:ccc} \nand \\citet{balmaverde05} reported on similar multiwavelength\nluminosity trends for the sample of\nLLRG formed  by the  3C sources with  FR~I morphology.\nThe connection between the CoreG and LLRG becomes \nmore evident if we add LLRG  in the diagnostic\nplanes (see  Fig. \\ref{corr} and Table \\ref{lumfr1}). \nThe early-type core  galaxies \nfollow the same behaviour of the stronger radio galaxies,\nextending it downward by 3 orders  of magnitude in radio-core\nluminosity as they reach levels\nas low as $L_{\\rm r} \\sim 10^{36}$ erg s$^{-1}$.\n\nWe estimated the best linear fit for the combined CoreG\/LLRG sample\nin both the \n$L_{\\rm r}$ vs. $L_{\\rm o}$ and $L_{\\rm r}$ vs. $L_{\\rm X}$ planes. \nThe best  fits were  derived  as the\nbisectrix of the  linear fits using the two  quantities as independent\nvariables following the suggestion by \\citet{isobe90} that this is\npreferable for problems needing symmetrical treatment of the variables. \nThe presence of upper limits in the independent variable\nsuggests that we could take advantage of the \nmethods of survival analysis proposed by e.g. \\citet{schmitt85}.\nHowever, the drawbacks discussed by \\citet{sadler89}\nand, in our specific case, \nthe non-random distribution of upper limits, argue against this approach. \nWe therefore preferred to exclude  upper limits from the  linear\nregression analysis. Nonetheless, a posteriori, \n1) the  objects  with an undetected nuclear\ncomponent in  the optical or X-ray are\nconsistent  with the correlation  defined by  the detections  only;\n2) the application of the Schmidt methods provides correlation\nslopes that agree, within the errors, with our estimates.\n \nWe obtained (indicating the Pearson \ncorrelation coefficient with $r$ and slope with $m$)\n$r_{ro}$=0.90 and $m_{ro}=0.89  \\pm 0.07 $, \n$r_{rx}$=0.89 $m_{rx} = 1.02\\pm 0.10 $ for the radio\/optical and\nradio\/X-ray correlations respectively.\nThe slopes and normalizations derived for CoreG, LLRG and the\ncombined CoreG+LLRG sample \n(see Table \\ref{tab0}) are consistent within the errors\nand this indicates that there is no significant change in the\nbehaviour between the two samples. \nOnly the dispersion is slightly larger for the CoreG nuclei \nbeing a factor of $\\sim 4$ rather than $\\sim 2$ for the LLRG sample alone. \n\n\\citet{chiaberge:ccc} first reported the presence of a correlation\nbetween radio and optical emission in the LLRG and they concluded that\nthis is most likely due to a common non-thermal jet origin for the\nradio and optical cores. \nRecently \\citet{balmaverde05} extended the analysis to the X-ray cores;\nthe nuclear X-ray luminosity also correlates with those of the radio\ncores and with a much smaller dispersion \n($\\sim$ 0.3 dex) when compared to similar\ntrends found for other classes of AGN \\citep[see e.g.][]{falcke95}, \nagain pointing to a common origin for the emission in the three bands.\nFurthermore, the broad band spectral indices of the 3C\/FR~I cores \nare very similar to those measured in BL Lacs objects (for which a jet\norigin is well established) in accord with the\nFR~I\/BL~Lacs unified model \n(we will return to this issue in Section \\ref{bllac}).\n\nThe core galaxies of our sample thus appear to smoothly extend the results\nobtained for LLRG to much lower radio luminosity, expanding the\nmultiwavelenght nuclear correlations to a total of 6 orders of\nmagnitude. This strongly argues in favour of a jet\norigin for the nuclear emission also in the core galaxies and that they \nsimply represent the scaled down versions of these already low luminosity AGN.\n\n\\begin{table}\n\\caption{Correlations summary}\n\\begin{tabular}{l l c c c l } \\hline \\hline\nSample  & Var. A    & Var. B & r$_{AB}$& Slope & rms     \\\\\n\\hline                                       \nCoreG     & L$_{O}$ & L$_{r}$   & 0.59 & 0.76$\\pm$0.21  & 0.62  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.78 & 1.36$\\pm$0.20  & 0.59  \\\\\nLLRG      & L$_{O}$ & L$_{r}$   & 0.94 & 0.82$\\pm$0.11  & 0.32  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.95 & 0.99$\\pm$0.11  & 0.33  \\\\\nLLRG+CoreG& L$_{O}$ & L$_{r}$   & 0.90 & 0.89$\\pm$0.07  & 0.56  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.89 & 1.02$\\pm$0.10  & 0.58  \\\\\n\\hline\n\\end{tabular}\n\\label{tab0}\n\\end{table}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f4a.ps,width=0.50\\linewidth}\n\\psfig{figure=f4b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{corr1} Radio core luminosity for the early-type galaxies\nwith a ``core'' profile versus the optical (left) \nand X-ray (right) nuclear luminosities.}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f5a.ps,width=0.50\\linewidth}\n\\psfig{figure=f5b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{corr} Comparison of radio and optical \n(left) and X-ray (right) nuclear luminosity\nfor the sample of core-galaxies (filled circles) and for the \nreference 3C\/FR~I sample of low luminosity radio-galaxies (empty\ncircles). \nThe three sources in common are marked\nwith a filled square. The solid lines reproduce the best linear fits.\n}\n\\end{figure*}\n\n\\subsection{Core galaxies vs. low luminosity radio-galaxies.}\n\\label{cg-fri}\n\n\\input{tab4.tex}\n\nThe results presented above indicate that the nuclei of the CoreG\nshow a very similar behaviour to those of LLRG.\nHere we explore in more detail how CoreG and LLRG\ncompare in their other properties, such\nas the structure of the host, black hole mass, radio-morphology\nand optical spectra.\n\nOur sample was selected to include only\nearly-type galaxies with a core profile, e.g. with an asymptotic slope \n(toward the nucleus) of their surface brightness profiles $\\gamma < 0.3$. \nRecently \\citet{deruiter05} showed, from the analysis of a combined\nsample of B2 and 3C sources, that they are all hosted\nby early-type galaxies and that the presence of a flat core\nis a characteristic of the host galaxies of all nearby radio-galaxies.\n\nA strong similarity between CoreG and LLRG emerges when \ncomparing the mass of their\nsupermassive black holes. \nWhen no direct measurement \n\\citep[taken from the compilation by][]{marconi03} \nwas available, we estimated $M_{BH}$ using\nthe relationship with the stellar velocity dispersion\n(taken from the LEDA database) \nin the form given by \\citet{tremaine02}.\nThe distributions of $M_{BH}$ (see Fig. \\ref{mbhhis}) of\nthe two samples are almost indistinguishable\\footnote{The \nprobability that the two samples are drawn \nfrom the same parent distribution is 0.32, \naccording to the Kolmogorov-Smirnoff test.}, as they \nhave median values of Log $M_{BH} = 8.54 $ and Log $M_{BH} = 8.70 $,\nfor CoreG and LLRG respectively,\nand they also cover  \nthe same range, with most objects with Log $M_{BH} = 8 - 9.5$.\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f6b.ps,width=0.5\\linewidth}\n\\psfig{figure=f6a.ps,width=0.5\\linewidth}\n}\n\\caption{\\label{mbhhis} Distributions for CoreG (shaded histograms) \nand for LLRG (empty histogram) of \n(left panel) black hole mass M$_{BH}$ and \n(right panel) absolute magnitude M$_K$.  \nThe LLRG histograms have been re-normalized multiplying \nby a factor 29\/19 for M$_{BH}$ and 29\/17 for M$_K$ respectively, \ni.e. the number of objects\nin the two samples for which estimates of these parameters are available.}\n\\end{figure*}\n\nFurther indications  of the  nature of CoreG  cores and  their connection\nwith LLRG come from  the emission lines in their optical spectra.\nLLRG are characterized as a class by their LINER spectra\n\\citep[e.g.][]{lewis03} and this\nis the case also for the CoreG of our sample.\nIn the NED  database, although about half of the CoreG do not have\na spectral  classification, 13 objects are classified \nas LINERs\\footnote{This result provides further support to the \nsuggestion by \\citet{chiaberge05}\nthat a dual population is associated with galaxies with a LINER\nspectrum, being formed by both radio-quiet and by radio-loud objects.\nThe CoreG are part of this latter sub-population of radio-loud LINER.}.  \nThe only exception is UGC~7203, with a Seyfert spectrum, but its\ndiagnostic line  ratios are borderline  with those of LINERs \\citep{ho97}.\nConcerning the emission line luminosity, \\citet{capetti:cccriga} found a\ntight relationship  between radio core  and line luminosity  studying a\ngroup of LLRG formed by the 3C\/FR~I complemented by\nthe sample of 21 radio-bright ($F_r > 150$ mJy) \nUGC galaxies defined by \\citet{noelstorr03}. Line luminosity for \nour CoreG clearly follow the same trend defined by LLRG, \nalthough with a substantially larger  dispersion, not unexpected given their\nlow line luminosity and the non uniformity of the data used for this\nanalysis. \n\nConsidering the radio structure,\nseveral objects of our CoreG sample have a radio-morphology with well developed \njets and lobes: UGC~7360, UGC~7494 and UGC~7654 are FR~I\nradio-galaxies part of the 3C sample (3C~270, 3C~272.1 and 3C~274), \nwhile in the Southern sample we\nhave the well studied radio-galaxies NGC~1316 (Fornax A), a FR~II\nsource, NGC~5128 (Cen A) and IC~4296.\nA literature search shows that at least another 11 sources \nhave extended radio-structure indicative of a collimated outflow,\nalthough in several cases this can only be seen in high\nresolution VLBI images, such as\nthe mas scale double-lobes in UGC~7760 or the one-sided jet of \nUGC~7386 \\citep{nagar02,falcke00}. \n\nConversely, hosts of 3C\/FR~I radio-sources are on average more \nluminous than core-galaxies\n(see Fig. \\ref{mbhhis}, left panel) although there is\na substantial overlap between the two groups: the median values are\n$M_K=-24.8$ and $M_K=-25.7$ for CoreG and 3C\/FR~I respectively,\nwith a KS probability of only 0.003 of being drawn from the same population. \nThis reflects the well known trend,\nalready noted by \\citet{auriemma77}, for which \na brighter galaxy has a higher probability of being a stronger\nradio emitter, and which is present also in our sample \\citepalias{capetti05}. \nThe selection of \nrelatively bright radio sources, such as the 3C\/FR~I, corresponds\nto a bias toward more luminous galaxies. Indeed, within our sample,\nimposing a threshold in total radio-luminosity of \n$L_{\\rm tot} > 10^{39}$ erg\/s,\\footnote{The 5 GHz luminosity was\nconverted to 178 MHz for consistency with the 3C\/FR~I values adopting\na spectral index of 0.7.} the low end for LLRG,  \ndecreases the median magnitude to -25.1,\nin closer agreement with the 3C\/FR~I value.\n\nWe conclude that the properties of our low radio luminosity CoreG show\na remarkable similarity to those of classical LLRG, in particular,\nthey share the presence of a flat core in their host's brightness profiles,\nthey have the same distribution in black hole\nmasses, as well as analogous properties concerning their optical\nemission lines and radio-morphology. \nThese results indicate that core galaxies and LLRG\ncan be considered, from these different point of view, as being drawn\nfrom the same population of early-type galaxies. They can only be\nseparated on the basis of their different level of nuclear activity,\nwith the LLRG forming the tip of the iceberg of (relatively) \nhigh luminosity objects. Furthermore, the emission processes\nassociated to\ntheir activity scale almost linearly over 6 orders of magnitude in\nall bands for which data are available.\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f7.ps,width=1.00\\linewidth}\n}\n\\caption{Emission line vs. radio core luminosity for CoreG galaxies (filled circles)\nand for the LLRG 3C\/FR~I sample (empty circles), from Capetti et al. (2005).}\n\\label{line}\n\\end{figure}\n\n\\section{Black hole mass and radio luminosity.}\n\nThe issue of the relationship between\nthe black hole mass and the radio-luminosity has been discussed by\nseveral authors, taking advantage of the recent possibility to measure\n(or at least estimate) $M_{BH}$.\n\\citet{franceschini98} pioneered this field showing, \nfrom a compilation of objects with\navailable black hole estimates, that the radio-luminosity tightly\ncorrelates with the black hole mass, with a logarithmic index of\n$\\sim$ 2.5. This result was subsequently challenged, by\ne.g. \\citet{ho02}.\nWe here re-explore this issue limiting ourselves to the sample\nof core early-type galaxies; while this substantially restricts\nthe accessible range in $M_{BH}$ and it applies only to radio-loud\nnuclei, it has the\nsubstantial advantage of performing the analysis on a complete sample\nwith well defined selection criteria and\ncovering a large range of radio-luminosity.\n\nThe comparison of  the radio-core luminosity with the  black hole mass\nis presented in  Fig.  \\ref{mbh}. Apparently,\na dependence of $L_r$ on  $M_{BH}$ is present,\nalthough   with  a  substantial   scatter. However, \nthe radio flux limit of the samples exclude objects with lower radio\nluminosity, potentially populating the lower part of the \n$L_R$ vs. $M_{BH}$ plane.\nFurthermore, the\ninclusion of LLRG (which, as discussed above, represent the high\nactivity end of the early-type population) radically changes the picture, as they populate the\nwhole upper portion of this plane. This indicates that  a very large\nrange (at least 4 orders  of magnitude) of radio-power can correspond to\na given $M_{BH}$ \n\\footnote{With respect to previous studies we report the\nnuclear   radio   emission   only,   instead  of   the   total   radio\nluminosity.  However, since  the fraction  of extended  emission grows\nwith radio luminosity, using the  total power would just move the LLRG\nupward, further increasing  the spread.}. \nThis is a clear\nindication that, not unexpectedly, \nparameters other than the black hole mass play a\nfundamental role in determining the radio luminosity of a galaxy.\n\nMore notable is the lack of sources with \n$M_{BH} < 10^8 M_{\\sun}$ \n(with only one exception). \nThe  effects  produced  by  our\nselection  criteria must be  considered before  any conclusion  can be\ndrawn.  In particular the correlation between the black hole mass and\nthe  spheroidal   galactic  component,  combined   with  the  limiting\nmagnitude,  translates into  a threshold  in the  accessible  range of\nblack  hole masses.  Using the limit in apparent  magnitude of our\nsample ($m_B <  14$), an\naverage color  of B-K =  4.25 \\citep{mannucci01}\nand the best fit to the relationship between\n$M_{BH}$ and $M_K$  from \\citet{marconi03} we  obtain that at distances\nlarger  than  20 Mpc  (corresponding  to 7\/8 of  the  volume covered\nin the Wrobel's sample) we  do include galaxies with expected\nblack hole masses $M_{BH} < 10^8 M_{\\sun}$. This represents\na severe bias against the inclusion of galaxies with low values\nof $M_{BH}$, regardless of their radio emission.\nThe lack of low black hole mass LLRG seems to favour  \nthe reality of this effect, as they\nare  not directly selected imposing an optical threshold; \nhowever, the already discussed statistical trend linking\nradio and optical luminosity might represent a more subtle\nbias leading to the same effect.\nThe existence of a minimum black hole mass\nto produce a radio-loud nucleus must be properly\ntested extending the analysis to a  sample  of less luminous\ngalaxies, likely to harbour less massive black holes.\n\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f8.ps,width=1.00\\linewidth}\n}\n\\caption{Nuclear radio-luminosity vs. black hole mass M$_{BH}$ for CoreG galaxies (filled circles)\nand for the LLRG 3C\/FR~I sample (empty circles).}\n\\label{mbh}\n\\end{figure}\n\n\\section{Constraints on the radiative manifestation of the accretion process}\n\\label{adaf}\n\nTaking advantage of the estimates of black hole mass we \ncan convert the measurements of\nthe nuclear luminosities to units of the Eddington luminosity. \nAll CoreG nuclei are associated with a low fraction of $L_{\\rm {Edd}}$, \nbeing confined to the range $L\/L_{\\rm {Edd}} \\sim 10^{-6} - 10^{-9}$\nin both the X-ray and optical bands (with only one X-ray exception), \nsee Fig. \\ref{eddihis}. \nFurthermore, as discussed in Sect. \\ref{nuclei},\nthe tight correlations between radio, optical and X-ray nuclear \nluminosities extending across LLRG and CoreG strongly argue in favour of a jet\norigin for the nuclear emission also in the core galaxies.\nIf this is indeed the case, the observed nuclear emission \ndoes not originate in the accretion process and the\nvalues reported above should be considered\nas upper limits.\n\nOur results add to the already vast literature reporting \nemission corresponding to a very low Eddington fraction associated with \naccretion onto supermassive black holes. \nThese results prompted the idea that in these objects accretion occurs\nnot only at a low rate but also at a low radiative efficiency,\nsuch as in the Advection Dominated Accretion Flows \n\\citep[ADAF,][]{narayan95} in which most of the\ngravitational energy of the accreting gas is advected into the\nblack hole before it can be dissipated radiatively, thus reducing the\nefficiency of the process with respect to the standard models of\ngeometrically thin, optically thick, accretion disks.\nThe ADAF models have been rather successful in modeling the observed\nnuclear spectrum in several galaxies, such as e.g. the\nGalactic Center and NGC 4258 \\citep{narayan95,lasota96}. \nConversely, ADAF models substantially \nover-predict the observed emission in the nuclei of nearby bright elliptical\ngalaxies \\citep{dimatteo00,loewenstein01}.\n\nThis suggested the possibility that a substantial fraction\nof the mass included within the Bondi's accretion radius\n\\citep{bondi52} might not\nactually reach the central object,\nthus further reducing the radiative emission from the accretion\nprocess with respect to the ADAF models. \nThis may be the case in the presence of an outflow\n\\citep[Advection Dominated Inflow\/Outflow Solutions, or ADIOS,][]{blandford99}\nor strong convection \\citep[Convection Dominated Accretion Flows,\n  or CDAF,][]{quataert00} in which most gas circulates in \nconvection eddies rather than accreting onto the black hole. \n\nUnfortunately, in the case of the galaxies under investigation,\nthe comparison of the theoretical predictions \nwith the observations so as to get constraints\non the properties of the accretion process is quite difficult. \nThis is due to the presence of different competing models, all\nof these with several free parameters, and to the observational data, in particular to\nthe scarce multiwavelength coverage of the nuclear emission\nmeasurements which\nprevents us from deriving a detailed Spectral Energy Distribution of these objects.   \nAs discussed above, this is more complicated for our radio-loud\nnuclei in which  the emission is most likely dominated by the\nnon-thermal radiation from their jets.\n\nNonetheless, \\citet{pellegrini05} recently studied in detail a sample\nof nearby galaxies for which the Chandra observations provide an\nestimate of the temperature and density of the gas in the nuclear\nregions, thus enabling one to derive the expected Bondi \naccretion rate, $\\dot{M}_B$. It is interesting to note that the estimates \nof $\\dot{M}_B$ for three sources common to both samples\nwith the lowest X-ray luminosity (namely NGC~1399, UGC~7629 (AKA\nNGC~4472) and UGC~7898 (AKA NGC~4649)) are relatively large, \n$\\dot{M}_B\/\\dot{M}_{\\rm Edd} = 10^{-2} - 10^{-4}$,\nwhile their X-ray luminosities are $L_X\/L_{Edd} = 10^{-8} - 10^{-10}$\n(see her Fig. 3). These luminosities are between\n3 and 5 orders of magnitude lower than expected from an\nADAF model, and they should be considered\nonly as upper limits. \nThese results argue in favour of an effective accretion rate\nsubstantially smaller than expected in the case\nof spherical accretion, suggesting that \nan important role is played by mass loss due to an outflow or \nby convection.\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f9a.ps,width=0.50\\linewidth}\n\\psfig{figure=f9b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{eddihis} Distributions of the nuclear luminosities \nmeasured as fraction\nof the Eddington luminosity in the X-ray (left) and optical (right)\nbands. }\n\\end{figure*}\n\n\\section{CoreG and the BL~Lacs\/LLRG unifying model}\n\\label{bllac}\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f10.ps,width=1.00\\linewidth}\n}\n\\caption{Broad band spectral indices, calculated \nbetween 5 GHz, 5500 \\AA\\ and 1 keV, for core galaxies (filled\ncircles), low luminosity 3C\/FR~I radio-galaxies (empty circles), \nLow energy peaked BL Lacs (stars) and High energy peaked BL Lacs\n(squares).\nSolid lines mark the regions within 2\n$\\sigma$ from the mean $\\alpha_{ro}$ and $\\alpha_{ox}$ for BL Lacs \ndrawn from the DXRB and RGB surveys.\nThe dashed lines\nrepresent constant values for the third index, $\\alpha_{rx}$.}\n\\label{roox}\n\\end{figure}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f11a.ps,width=0.50\\linewidth}\n\\psfig{figure=f11b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{spix} \nBroad band spectral indices vs. extended radio luminosity \nfor core galaxies (filled\ncircles), LLRG (empty circles), \nLBL (stars) and HBL (squares). The CoreG luminosity has been extrapolated\nto 178 MHz adopting a spectral index of 0.7.}\n\\end{figure*}\n\nIn Section \\ref{cg-fri} we presented evidence that \n``core'' galaxies and LLRG are drawn from the same\npopulation of early-type galaxies. They can only be\nseparated on the basis of their different level of nuclear activity,\nwith CoreG representing the low luminosity extension of LLRG. \nThe CoreG nuclei appear to be the scaled down versions of those of LLRG\nwhen their multiwavelength nuclear properties are considered. \nThus here we are sampling a new \nregime for radio-galaxies in terms of nuclear power and\nit is important to explore the implications of this result \nfor the model unifying BL Lac objects and radiogalaxies.\n\nUnification models ascribe the differences between the observed\nproperties of different classes of AGN to the anisotropy of the\nnuclear radiation \n\\citep[see e.g.][ for reviews]{antonucci93,urry95}.\nIn particular, for low\nluminosity radio-loud objects, it is believed \nthat BL Lac objects are the pole-on counterparts of radio-galaxies,\ni.e. their emission is dominated by the radiation \nfrom the inner regions of a relativistic jet seen \nat a small angle from its axis which is thus strongly\namplified by relativistic Doppler beaming. \nIn FR~I, whose jets are\nobserved at larger angles with respect to the line of sight,\nthe nuclear component is strongly de-amplified.  \nContrary to other classes of AGN there is growing evidence that\nobscuration does not play a significant role in\nthese objects \\citep{henkel98,chiaberge:ccc,donato04,balmaverde05}.\n\n\\citet{balmaverde05} found that there is a close similarity of the\nbroad band spectral indices between LLRG and the \nsub-class of the BL~Lacs, the Low energy peaked BL~Lac \n\\citep[LBL, ][]{padovani95}, in agreement with the\nunified model\\footnote{The small offsets between the two classes\ncan be quantitatively \naccounted for by the effects of beaming since\nDoppler beaming not only affects the angular pattern\nof the jet emission, but it also causes a shift in frequency of the\nspectral energy distribution \\citep[see ][]{marco3,trussoni03}}.\nWe performed the same comparison \n\\footnote{We used the standard definition of spectral indices,\nmeasured between 5 GHz, 5500 \\AA\\ and 1 keV. Optical fluxes have been\nconverted from 8140 \\AA\\ to 5500 \\AA\\ using a local slope of $\\alpha =\n1$; 1 keV fluxes are\ndirectly derived from the spectral fit.} including CoreG,\nsee Fig. \\ref{roox}. \nWe considered the radio selected BL Lacs sample\nderived from the 1Jy catalog \\citep{stickel91}\nand the BL Lac sample selected from the\n{\\it Einstein} Slew survey \\citep{elvis92,perlman96}. \nWe used the classification into \nHigh and Low energy peaked BL Lacs (HBL and LBL respectively), as well\nas their multiwavelength data \ngiven by \\citet{fossati98}. We also report the regions \n(solid lines) of the plane within 2\n$\\sigma$ from the mean $\\alpha_{ro}$ and $\\alpha_{ox}$ for the BL Lacs \ndrawn from the Deep X-Ray Radio Blazar Survey (DXRBS) \nand the ROSAT All-Sky Survey-Green Bank Survey (RGB) \\citep{padovani03}.\n\nCore galaxies are found to be located in  the same region covered by  LLRG. \nThis   is  not  surprising  since  they extend\nthe behaviour of LLRG in  the radio\/optical and radio\/X-ray planes,\nfollowing Log-Log linear correlations whose slope is close to\nunity, implying only a small dependence of spectral indices on\nluminosity. More importantly, they populate the same area\nin which LBL are found.\n\nWe also compared \nthe spectral indices of the different groups taking into\naccount the extended radio-luminosity L$_{ext}$\n(see Fig. \\ref{spix}) which does not depend on orientation. This enables us \nto properly relate objects from the same region of the luminosity\nfunction of the parent population. \nIndeed, the strongest evidence in favour of \nthe FR~I\/BL~Lac unifying model comes from the similarity in\nthe power and morphology of the\nextended radio emission of BL Lacs and FR~I \n\\citep[see e.g.][]{antonucci85,kollgaard92,murphy93}.\n\nThe CoreG reach radio-luminosities $\\sim$ 100 smaller \nthan in LLRG and the 1 Jy LBL.\nIn addition, in 13 CoreG the available radio-maps do not allow us to separate\ncore and extended radio-emission and $L_{ext}$ must be considered as\nan upper limit.\nThis suggests that the CoreG represent the counterparts of \nthe large low luminosity population of BL~Lac of LBL type which is\nnow emerging from the low radio flux limit surveys such as the DXRBS \n\\citep{landt01}. Clearly, this still requires measurements of the\nextended radio-luminosity of these low power BL~Lac.\nA ramification of this possible extension of the unified model\ntoward lower luminosities \nwould be the presence of relativistic jets also in our \nsample of quasi-quiescent early-type galaxies, as this is a prerequisite \nto produce a substantial dependence of the luminosity on the viewing angle.\n\nWe did not find any CoreG with spectral properties similar\nto those of the High energy peaked BL~Lac (HBL), even though HBL have\nextended radio-emission values of $L_{ext}$ similar to CoreG.\nThe spectral indices of CoreG imply \na difference  in both the radio-to-optical and\nradio-to-X-ray flux ratios of an average factor \nof  $\\sim$100 with respect to HBL.\nThe same result applies to LLRG, as all have a LBL-type SED, with the\nonly exception of 3C~264 \\citep{fr1sed}.\nOur optical selection criteria did not exclude the  parent population of  HBL \nsince their host galaxies are early-type  sufficiently luminous \n\\citep[$M_R < -22.5$,][]{scarpa00} to be included  in our\nsample. Most likely, the dearth of HBL-like CoreG is induced by the radio\nthreshold. Purely radio selected\nsamples of BL~Lacs are known to strongly favour the inclusion of LBL; \ne.g. in the 1 Jy sample\nthere are only 2 HBL out of 34 objects \\citep{giommi94}. \n\n\n\\section{Summary and conclusions}\n\nThe aim of this series of papers is to explore\nthe classical issue of the connection between host galaxies and AGN,\nin the new light shed by the recent developments \nin our understanding of the nuclear regions of\nnearby galaxies. \n\nWe thus selected a samples of nearby early-type \ngalaxies comprising 332 objects. We performed an initial\nselection of AGN candidates requiring a radio detection \nabove $\\sim$1 mJy leading to a sub-sample of 112 sources.\nArchival HST images enabled us to classify 51 of them\ninto core and power-law galaxies on the basis\nof their nuclear brightness profile.\nWe here focused on the 29 core galaxies.\n\nWe used HST and Chandra archival data to isolate their nuclear\nemission in the optical and X-ray bands, \nthus enabling us (once\ncombined with the radio data) to study the\nmultiwavelength behaviour of their nuclei.\nThe detection rate of nuclear sources is 18\/29 in\nthe optical (62 \\%, increasing to 72\\% if the sources affected by large\nscale dust are not considered) and 14 in the X-ray, out \nof the 21 objects with available Chandra data (67 \\%).\nOur selection criteria required a radio detection \nin order to select AGN candidates;\n26 CoreG are confirmed as genuine active galaxies \nbased on the presence of i) an optical (or X-ray) core, ii) a AGN-like\noptical spectrum, or iii) radio-jets, with only 3 exceptions,\nnamely UGC~968, UGC~7898 and NGC~3268.\n\nThe most important result of this analysis is that\n``core'' galaxies invariably host a radio-loud nucleus.\nThe radio-loudness parameter $R$ for the nuclei\nin these sources is on average Log R $\\sim$ 3.6, a factor \nof 400 above the classical threshold between radio-loud and\nradio-quiet nuclei. The X-ray data provide a completely independent\nview of their multiwavelength behaviour leading to the same result,\ni.e. a large X-ray deficit, at the same radio luminosity, \nwhen compared to radio-quiet nuclei.\n\nConsidering the multiwavelength nuclear diagnostic planes, \nwe found that optical and X-ray nuclear luminosities are \ncorrelated with the\nradio-core power, reminiscent of the behaviour\nof low luminosity radio-galaxies. The inclusion of CoreG\nindeed extends the correlations reported for LLRG toward much lower\npower, by a factor of $\\sim 1000$.\n\nThe available radio maps show that in 17 CoreG\nthe extended radio morphology is clearly indicative of a collimated outflow,\nin the form of either double-lobed structures or jets, \nalthough in several cases this can only be seen in high\nresolution VLBI images. This finding, combined with the \nanalogy of the nuclear properties, leads us to the conclusion that\nminiature radio-galaxies are associated with all core galaxies\nof our sample. \n\nThe similarity between CoreG and classical low luminosity\nradio-galaxies extends to other properties. Recent results show that\nLLRG are always hosted by early-type galaxies with a shallow cusp in their\nnuclear profile, and this is the case, by definition, for our CoreG.\nWhile the distributions of black hole masses, $M_{BH}$,\nof the two classes are indistinguishable,\nhosts of 3C\/FR~I radio-sources are on average slightly more \nluminous than CoreG but there is\na substantial overlap between the two groups.\nCoreG and LLRG also share similar properties from the point of view of\ntheir emission lines, as all sources with available data conform\nto the definition of a LINER on the basis of the optical line ratios\nand they follow a common dependence of line luminosity\nwith radio core power.\nCoreG and LLRG thus appear to be drawn\nfrom the same population of early-type ``core'' galaxies. They host\nactive nuclei with the same multiwavelength characteristics\ndespite covering a range of 6 orders of magnitude in\nluminosity. Thus LLRG represent the tip of the iceberg of (relatively) \nhigh luminosity objects.\n\nIt is unclear what mechanism is driving the level of nuclear\nactivity. As noted above, there is a marginal difference (less than 1\nmag) in the host\ngalaxies of CoreG and LLRG; this reflects the well known (but as yet\nunexplained) trend for which \na brighter galaxy has a higher probability of being a stronger\nradio emitter. As described in \\citetalias{capetti05},\nthis effect is present also within our sample of CoreG\nbut it cannot\nbe simply described as a correlation between  L$_r$ and  M$_K$.\n\nWe explored if there is a relationship between the black\nhole mass and the radio-luminosity.\nAgain, a  very large  range of\nradio-power corresponds  to a  given $M_{BH}$. We do not  find any\nrelationship   between  radio-power  and   black  hole   mass, clearly\nindicating  that parameters  other than  the  black hole  mass play  a\nfundamental role in determining the  radio luminosity of a galaxy.  No\nsources with $M_{BH} < 10^8  M_{\\sun}$ are found.  However, this might\nbe due to a  bias induced by the  sample's selection criteria. \nThe limit in optical  magnitude translates  into a\nthreshold of  accessible black  hole masses.  \nOnly by extending  this study to a sample of less\nluminous  galaxies (harbouring,  on average,  smaller black  holes) \nwill it be possible  to test the reality of a minimum  black hole mass to\nproduce a radio-loud nucleus.\n\nOur data can also be used to set constraints \non the radiative manifestation of the accretion process.\nThe nuclear luminosities of CoreG correspond, in units of the\nEddington luminosity, to the range $L\/L_{\\rm {Edd}} \\sim 10^{-6} -\n10^{-9}$ in both the optical and X-ray bands. \nIn analogy with the scenario\nproposed for LLRG, the available data support a common\njet origin for the nuclear emission in these observing bands also for CoreG.\nThus, the above values should be considered as upper limits to\nthe radiative manifestation of the accretion process, suggesting\nthat accretion occurs both at a low accretion level and at a low efficiency.\nIt is difficult to derive from these results clear \nconstraints on the properties of the accretion flow. In part this is due to\nthe limited information on the Spectral Energy Distribution of the CoreG\nnuclei and by the fact that in these\nradio-loud nuclei the observed emission is most likely dominated by \nradiation from their jets rather than from the accretion.\nThis is further complicated by \nthe presence of several competing accretion models\nwhose predictions of the emitted spectra depend on parameters that\nare not well constrained by the observations.\nNonetheless, in the galaxies with the least luminous nuclei, \nthe estimates of the accretion rate  from the literature\n(derived for the case of spherical accretion), combined with\nthe very low level of X-ray emission, suggest that \nan important role is played by outflows (or \nby convection) in order to substantially suppress\nthe amount of gas actually reaching the central object.\n\nAs reported above, the CoreG can be effectively considered\nas miniature radio-galaxies, in terms of nuclear luminosity,\nthus we are sampling a new \nregion in terms of luminosity for radio-loud AGN.\nIt is interesting to explore the implications of this result \nalso for the model unifying BL Lac objects and radio-galaxies.\nThe broad band spectral indices of CoreG present a very close similarity\nto those of Low Energy Peaked BL Lac, suggesting the extension\nof the unified models to these lower luminosities. The CoreG might \nrepresent the mis-aligned counterpart of the large population of low\nluminosity BL Lac emerging from the recent\nsurveys at low radio flux limits. Clearly, a more detailed comparison,\ntaking into account e.g. the (as yet not available) information on the\nextended radio power and morphology, is needed before this result\ncan be confirmed. An important ramification \nof this possible extension of the unifying model\ntoward lower luminosities would be the presence \nof relativistic jets, \nthe essential ingredient of this model, \nalso in our quasi-quiescent early-type galaxies.\n\nIn the third paper of the series we will explore the properties\nof the AGN hosted by galaxies with a power-law brightness profile.\n\n\\begin{acknowledgements}\nThis work was partly supported by the Italian MIUR under \ngrant Cofin 2003\/2003027534\\_002.\nThis research has made use of the NASA\/IPAC Extragalactic Database (NED)\n(which is operated by the Jet Propulsion Laboratory, California Institute of\nTechnology, under contract with the National Aeronautics and Space\nAdministration), of the NASA\/ IPAC Infrared Science Archive\n(which is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under contract with the National Aeronautics\nand Space Administration) and of the LEDA database.\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn two dimensions, the configuration space\nof $n$ point-like particles ${\\cal C}_n^{2D}$\nis multiply-connected. Its first homotopy\ngroup, or {\\it fundamental group},\nis the $n$-particle braid group,\n${\\pi_1}({\\cal C}_n^{2D})={\\cal B}_n$.\nThe braid group ${\\cal B}_n$ is generated\nby counter-clockwise exchanges $\\sigma_i$\nof the $i^\\text{th}$ and $(i+1)^\\text{th}$ particles\nsatisfying the defining relations:\n\\begin{eqnarray}\n{\\sigma_i} {\\sigma_j} &=& {\\sigma_j} {\\sigma_i} \\hskip 0.5 cm\n\\mbox{for } |i-j|\\geq 2\\cr {\\sigma_i} \\sigma_{i+1} {\\sigma_i} &=&\n\\sigma_{i+1} {\\sigma_i}\\, \\sigma_{i+1} \\hskip 0.5 cm \\mbox{for }\n1\\leq i \\leq n-2\n\\label{eq:braidrelation1}\n\\end{eqnarray}\nThis is an infinite group, even for only two particles,\nsince $(\\sigma_i)^m$ is a non-trivial element of the\ngroup for any $m>0$. In fact, even if we consider\ndistinguishable particles, the resulting group,\ncalled the `pure Braid group' is non-trivial.\n(For two particles, the pure braid group\nconsists of all even powers\nof $\\sigma_1$.)\n\nIn quantum mechanics, the equation\n${\\pi_1}({\\cal C}_n^{2D})={\\cal B}_n$\nopens the door to the possibility of anyons\\cite{Leinaas77,Wilczek82a}.\nHigher-dimensional representations\nof the braid group give rise to\nnon-Abelian anyons \\cite{Bais80,Goldin85,Frohlich90}.\nThere has recently been intense effort directed towards\nobserving non-Abelian anyons due, in part,\nto their potential use for fault-tolerant quantum\ncomputation \\cite{Kitaev97,Nayak08}.\nOne of the simplest models of non-Abelian\nanyons is called {\\it Ising anyons}. They arise\nin theoretical models of the $\\nu=5\/2$ fractional quantum\nHall state \\cite{Moore91,Nayak96c,LeeSS07,Levin07}\n(see also Ref. \\onlinecite{Bonderson08}),\nchiral $p$-wave superconductors \\cite{Read00,Ivanov01},\na solvable model of spins on the honeycomb lattice\n\\cite{Kitaev06a}, and interfaces between\nsuperconductors and either 3D topological\ninsulators \\cite{Fu08} or spin-polarized\nsemiconductors with strong spin-obrit coupling \\cite{Sau09}.\nA special feature of Ising anyons, which makes\nthem relatively simple and connects them to BCS\nsuperconductivity, is that they can be understood\nin a free fermion picture.\n\nA collection of $2n$ Ising anyons has a $2^{n-1}$-dimensional\nHilbert space (assuming fixed boundary condition).\nThis can be understood in terms of $2n$ Majorana fermion\noperators ${\\gamma^{}_i}={\\gamma_i^\\dagger}$,\n$i=1,2,\\ldots,n$, one associated to\neach Ising anyon, satisfying the anticommutation rules\n\\begin{equation}\n\\label{eqn:clifford}\n\\{{\\gamma^{}_i},{\\gamma^{}_j}\\}=2\\delta_{ij}\\,.\n\\end{equation}\nThe Hilbert space of $2n$ Ising anyons with fixed boundary condition furnishes a representation\nof this Clifford algebra; by restricting to fixed boundary condition, we obtain\na representation\nof products of an even number of $\\gamma$ matrices, which has minimal dimension $2^{n-1}$.\nWhen the $i^\\text{th}$ and $(i+1)^\\text{th}$ anyons\nare exchanged in a counter-clockwise manner,\na state of the system is transformed according to the action of\n\\begin{equation}\n\\label{eqn:Ising-braid}\n\\rho({\\sigma_i})=e^{i\\pi\/8}\\,e^{-\\pi{\\gamma^{}_i}\\gamma^{}_{i+1}\/4}\\,.\n\\end{equation}\n(There is a variant of Ising anyons, associated with\nSU(2)$_2$ Chern-Simons theory,\nfor which the phase factor $e^{i\\pi\/8}$\nis replaced by $e^{-i\\pi\/8}$. In the fractional quantum\nHall effect, Ising anyons are tensored with\nAbelian anyons to form more complicated models\nwith more particle species; the phase factor depends\non the model.) A key property, essential for applications\nto quantum computing, is that {\\it a pair} of Ising anyons\nforms a two-state system. The two states\ncorrespond to the two eigenvalues $\\pm 1$\nof ${\\gamma^{}_i}\\gamma^{}_{j}$. No local degree of freedom\ncan be associated with each anyon; if we insisted on doing so,\nwe would have to say that each Ising anyon has $\\sqrt{2}$\ninternal states. In superconducting contexts,\nthe ${\\gamma^{}_i}$s are the Bogoliubov-de Gennes operators\nfor zero-energy modes (or, simply, `zero modes')\nin vortex cores; the vortices\nthemselves are Ising anyons if they possess a single\nsuch zero mode ${\\gamma^{}_i}$.\nAlthough the Hilbert\nspace is non-local in the sense that it cannot be decomposed\ninto the tensor product of local Hilbert spaces associated\nwith each anyon, the system is perfectly compatible\nwith locality and arises in local lattice models and\nquantum field theories.\n\nIn three or more dimensions,\nthe configuration space of $n$ point-like particles\nis simply-connected if the particles are distinguishable.\nIf the particles are indistinguishable, it\nis multiply-connected,\n${\\pi_1}({\\cal C}_n^{3D})=S_n$.\nThe generators of the permutation group\nsatisfy the relations (\\ref{eq:braidrelation1})\nand one more, ${\\sigma_i^2}=1$. As a result\nof this last relation, the permutation group\nis finite. The one-dimensional representations\nof $S_n$ correspond to bosons and fermions.\nOne might have hoped that higher-dimensional\nrepresentations of $S_n$ would give rise to\ninteresting 3D analogues of non-Abelian anyons.\nHowever, this is not the case, as shown in\nRef. \\onlinecite{Doplicher71a,Doplicher71b}: any higher-dimensional\nrepresentation of $S_n$ which is compatible with\nlocality can be decomposed into the tensor product\nof local Hilbert spaces associated\nwith each particle. For instance, suppose we\nhad $2n$ spin-$1\/2$ particles but ignored\ntheir spin values. Then we would have $2^{2n}$\nstates which would transform into each other\nunder permutations. Clearly, if we discovered such a system,\nwe would simply conclude that we were missing\nsome quantum number and set about trying to\nmeasure it. This would simply lead us back\nto bosons and fermions with additional\nquantum numbers. (The color quantum number of quarks\nwas conjectured by essentially this kind of reasoning.)\nThe quantum information contained in these\n$2^{2n}$ states would not have any special protection.\n\nThe preceding considerations point to\nthe following tension. The Clifford algebra\n(\\ref{eqn:clifford}) of Majorana fermion zero modes\nis not special to two dimensions. One could imagine\na three (or higher) dimensional system with topological defects supporting such zero modes. But the Hilbert space of these\ntopological defects would be $2^{n-1}$-dimensional, which\nmanifestly cannot be decomposed into the tensor product\nof local Hilbert spaces associated\nwith each particle, seemingly in contradiction with the results of \nRefs. \\onlinecite{Doplicher71a,Doplicher71b} on\nhigher-dimensional representations of the permutation group\ndescribed above. However, as long as\nno one had a three or higher dimensional system\nin hand with topological defects supporting\nMajorana fermion zero modes,\none could, perhaps, sweep this worry under the rug.\nRecently, however, Teo and Kane \\cite{Teo10}\nhave shown that a 3D system which is simultaneously\na superconductor and a topological insulator \\cite{Moore07,Fu07,Roy09,Qi08}\n(which, in many but not all examples, is\narranged by forming superconductor-topological insulator\nheterostructures) supports Majorana zero modes at\npoint-like topological defects.\n\nTo make matters worse, Teo and Kane \\cite{Teo10}\nfurther showed that exchanging these\ndefects enacts unitary operations on this\n$2^{n-1}$-dimensional Hilbert space which are\nessentially equal to (\\ref{eqn:Ising-braid}).\nBut we know that these unitary matrices\nform a representation of the braid group,\nwhich is not the relevant group in 3D.\nOne would naively expect that the relevant group is\nthe permutation group, but $S_n$ has no such\nrepresentation (and even if it did, its use in this\ncontext would contradict locality, according to Ref. \\onlinecite{Doplicher71a,Doplicher71b}\nand arguments in Ref. \\onlinecite{Read03}).\nSo this begs the question: what is the group ${\\cal T}_{2n}$\nfor which Teo and Kane's unitary\ntransformations form a representation?\n\nWith the answer to this question in hand,\nwe could address questions such as the following.\nWe know that a 3D incarnation of Ising anyons\nis one possible representation of ${\\cal T}_{2n}$;\nis a 3D version of other anyons another\nrepresentation of ${\\cal T}_{2n}$?\n\nAttempts to generalize the braiding of anyons\nto higher dimensions sometimes start with\nextended objects, whose configuration space\nmay have fundamental group which is\nricher than the permutation group.\nObviously, if one has line-like defects\nin 3D which are all oriented in the same direction,\nthen one is essentially back to the 2D situation\ngoverned by the braid group. This is too\ntrivial, but it is not clear what kind of extended\nobjects in higher dimensions would be the best starting point.\nWhat is clear, however, is that Teo and Kane's topological\ndefects must really be some sort of extended objects.\nThis is clear from the above-noted contradiction\nwith the permutation group. It also follows from\nthe `order parameter' fields which must deform\nas the defects are moved, as we will discuss.\n\nIn this paper, we show that Teo and Kane's\ndefects are properly viewed as point-like\ndefects connected pair-wise by ribbons.\nWe call the resulting $2n$-particle configuration\nspace $K_{2n}$. We compute\nits fundamental group ${\\pi_1}(K_{2n})$, which we denote by\n${\\cal T}_{2n}$ and find that\n${\\cal T}_{2n}={\\mathbb Z} \\times {\\cal T}^r_{2n}$.\nHere, ${\\cal T}^r_{2n}$ is the `ribbon permutation group',\ndefined by ${\\cal T}^r_{2n} \\equiv {\\mathbb Z}_2\n\\times E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$.\nThe group $E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$\nis a non-split extension of the permutation group $S_{2n}$ by ${\\mathbb Z}_2^{2n-1}$\nwhich is defined as follows: it is the\nsubgroup of $(\\mathbb{Z}_2)^{2n} \\rtimes S_{2n}$\ncomposed of those elements for which the total parity of\nthe element in $(\\mathbb{Z}_2)^{2n}$ added to\nthe parity of the permutation is even.\nThe `ribbon permutation group' for $2n$ particles,\n by ${\\cal T}^r_{2n}$, is the fundamental group of the\n reduced space of $2n$-particle configurations.\n\nOur analysis relies on the topological\nclassification of gapped free fermion Hamiltonians\n\\cite{Ryu08,Kitaev09} -- band insulators\nand superconductors -- which\nis the setting in which Teo and Kane's 3D defects\nand their motions are defined.\nThe starting point for this classification is reducing the problem\nfrom classifying gapped Hamiltonians defined on a lattice to\nclassifying Dirac equations with a spatially varying mass term.\nOne can motivate the reduction to a Dirac equation as\nTeo and Kane do: they\nstart from a lattice Hamiltonian and assume\nthat the parameters in the Hamiltonian vary smoothly in space, so that\nthe Hamiltonian can be described as a function of both the momentum $k$\nand the position $r$.  Near the minimum of the band gap, the\nHamiltonian can be expanded in a Dirac equation, with a position-dependent\nmass term.  In fact, Kitaev\\cite{Kitaev09} has shown that the reduction\nto the Dirac equation with a spatially varying mass term can be derived\nmuch more generally:\ngapped lattice Hamiltonians, even if the parameters\nin the Hamiltonian do not vary smoothly in space, are {\\it stably equivalent}\nto Dirac Hamiltonians with a spatially varying mass term.  Here, equivalence\nof two Hamiltonians means that one can be smoothly deformed into the other\nwhile preserving locality of interactions and the spectral gap, while\nstable equivalence means that one can add additional ``trivial\" degrees of freedom (additional\nsites which have vanishing hopping matrix elements) to the original lattice\nHamiltonian to obtain a Hamiltonian which is equivalent to\na lattice discretization of the Dirac Hamiltonian.\n\nSince this classification of Dirac Hamiltonians\nis essential for the definition of $K_{2n}$, we give\na self-contained review, following Kitaev's\nanalysis \\cite{Kitaev09}. Our exposition parallels\nthe discussion of Bott periodicity in Milnor's book\n\\cite{Milnor63}. The basic idea is that each additional\ndiscrete symmetry which squares to $-1$ which we\nimpose on the system is encapsulated by an\nanti-symmetric matrix which defines a complex structure\non $\\mathbb{R}^N$, where $N\/2$ is the number of\nbands (or, equivalently, $N$ is the number of bands\nof Majorana fermions). For any given system,\nthese are chosen and fixed. This leads to\na progression of symmetric spaces $\\text{O}(N)\\rightarrow\n\\text{O}(N)\/\\text{U}(N\/2) \\rightarrow \\text{U}(N\/2)\/\\text{Sp}(N\/4)\n\\rightarrow \\ldots$ as the number of such symmetries is increased.\nFollowing Kitaev \\cite{Kitaev09}, we view the Hamiltonian\nas a final anti-symmetric matrix which must be chosen (and, thus,\nput almost on the same footing as the symmetries); it is defined by a choice\nof an arbitrary point in the next symmetric space in the progression.\nThe space of such Hamiltonians is topologically-equivalent\nto that symmetric space.\nHowever, as the spatial dimension is increased, $\\gamma$-matrices\nsquaring to $+1$ must be chosen in order to expand\nthe Hamiltonian in the form of the Dirac equation\nin the vicinity of a minimum of the band gap. These halve the dimension\nof subspaces of $\\mathbb{R}^N$ by separating it\ninto their $+1$ and $-1$ eigenspaces and thereby\nlead to the opposite progression of symmetric spaces. Thus,\ntaking into account both the symmetries of the system and\nthe spatial dimension, we conclude that the space of gapped\nHamiltonians with no symmetries in $d=3$ is topologically\nequivalent to $\\text{U}(N)\/\\text{O}(N)$. (However, by the preceding\nconsiderations, the same symmetric space also, for instance, classifies\nsystems with time-reversal symmetry in $d=4$.)\nAll such Hamiltonians can be continuously deformed into each\nother without closing the gap, $\\pi_{0}(\\text{U}(N)\/\\text{O}(N))=0$.\nHowever, there are topologically-stable point-like defects\nclassified by $\\pi_{2}(\\text{U}(N)\/\\text{O}(N))=\\mathbb{Z}_2$.\nThese are the defects whose multi-defect configuration space\nwe study in order to see what happens when they are exchanged.\n\nThe second key ingredient in our analysis\nis 1950's-vintage homotopy theory, which we use to compute\n${\\pi_1}(K_{2n})$.  We apply the Pontryagin-Thom\nconstruction to show that $K_{2n}$, which\nincludes not only the particle locations but also\nthe full field configuration  around the particles\n(i.e. the way in which the gapped free fermion\nHamiltonian of the system explores $\\text{U}(N)\/\\text{O}(N)$),\nis topologically-equivalent to a much simpler\nspace, namely point-like defects connected\npair-wise by ribbons. In order to then\ncalculate ${\\pi_1}(K_{2n})$, we rely on the long\nexact sequence of homotopy groups\n\\begin{equation}\n\\label{eqn:long-exact-sequence}\n\\ldots \\rightarrow \\pi_{i}(E)\\rightarrow\n{\\pi_i}(B)\\rightarrow\\pi_{i-1}(F)\\rightarrow\\pi_{i-1}(E)\n\\rightarrow ...\n\\end{equation}\nassociated to a fibration defined by\n$F \\rightarrow E\\rightarrow B$.\n(In an exact sequence, the kernel of each map\nis equal to the image of the previous map.)\nThis exact sequence may be familiar to some readers\nfrom Mermin's review of the topological theory of\ndefects \\cite{Mermin79}, where a symmetry associated\nwith the group $G$ is spontaneously broken to $H$,\nthereby leading to topological\ndefects classified by homotopy groups ${\\pi_n}(G\/H)$.\nThese can be computed by (\\ref{eqn:long-exact-sequence})\nwith $E=G$, $F=H$, $B=G\/H$, e.g.\nif $\\pi_{1}(G)=\\pi_{0}(G)=0$,\nthen $\\pi_{1}(G\/H)=\\pi_{0}(H)$.\n\nThe ribbon permutation group is a rather weak enhancement\nof the permutation group and, indeed, we conclude\nthat Teo and Kane's unitary operations are\n{\\it not} a representation of the ribbon permutation\ngroup. However, they are a {\\it projective} representation\nof the ribbon permutation group. In a\n{\\it projective} representation, the group\nmultiplication rule is only respected up to\na phase, a possibility allowed in quantum mechanics.\nA representation $\\rho$ (sometimes called a linear\nrepresentation) of some group $G$ is\na map from the group to the group of linear transformations\nof some vector space such that\nthe group multiplication law is reproduced:\n\\begin{equation}\n\\rho(gh)=\\rho(g)\\cdot\\rho(h)\n\\end{equation}\nif $g,h\\in G$. Particle statistics arising as a projective\nrepresentation of some group\nrealizes a proposal of Wilczek's \\cite{Wilczek98},\nalbeit for the ribbon permutation group rather than\nthe permutation group itself. This difference\nallows us to sidestep a criticism of Read \\cite{Read03}\nbased on locality, which Teo and Kane's\nprojective representation respects.\nThe group $(\\mathbb{Z}_2)^{2n-1}$ is generated\nby $2n-1$ generators ${x_1}$, ${x_2}$,\n\\ldots, $x_{2n-1}$ satisfying\n\\begin{eqnarray}\n{x_i^2} &=& 1\\cr\n{x_i} {x_j} &=& {x_j} {x_i}\n\\label{eq:Z_2-def}\n\\end{eqnarray}\nHowever, the projective representation of\n$(\\mathbb{Z}_2)^{2n-1}$, which gives a subgroup of\nTeo and Kane's transformations, is an ordinary\nlinear representation of a ${\\mathbb Z}_2$-central extension,\ncalled the extra special group $E^1_{2n-1}$:\n\\begin{eqnarray}\n{x_i^2} &=& 1\\cr\n{x_i} {x_j} &=& {x_j} {x_i} \\hskip 0.5 cm\n\\mbox{for } |i-j|\\geq 2\\cr\n{x_i} x_{i+1} &=& z \\,x_{i+1} {x_i}\\cr\n{z^2}&=&1\n\\label{eq:extra-special}\n\\end{eqnarray}\nHere, $z$ generates the central extension, which we may\ntake to be $z=-1$. The operations generated\nby the ${x_i}$s were dubbed `braidless operations'\nby Teo and Kane \\cite{Teo10} because they could\nbe enacted without moving the defects. While these\noperations form an Abelian subgroup of ${\\cal T}_{2n}$,\ntheir representation on the Majorana zero mode\nHilbert space is {\\it not} Abelian -- two such operations\nwhich twist the same defect {\\it anti-commute} (e.g. $x_i$ and\n$x_{i+1}$).\n\nThe remaining sections of this paper will\nbe as follows. In Section \\ref{sec:strong-coupling},\nwe rederive Teo and Kane's zero modes and unitary transformations\nby simple pictorial and counting arguments in a `strong-coupling'\nlimit of their model. In Section \\ref{sec:free-fermion},\nwe review the topological classification of free-fermion\nHamiltonians, including topological insulators and\nsuperconductors. From this classification, we obtain\nthe classifying space relevant to Teo and Kane's model\nand, in turn, the topological classification of\ndefects and their configuration space.\nIn Section \\ref{sec:tethered}, we use a toy\nmodel to motivate a simple picture for the defects\nused by Teo and Kane and give a heuristic\nconstruction of the ribbon permutation group.\nIn Section \\ref{sec:Kane_space}, we give a full\nhomotopy theory calculation.\nIn Section \\ref{sec:projective}, we compare\nthe ribbon permutation group to\nTeo and Kane's unitary transformations and conclude that\nthe latter form a projective, rather than a linear,\nrepresentation of the former. Finally, in Section\n\\ref{sec:discussion}, we review and discuss our results.\nSeveral appendices contain technical details.\n\n\n\n\n\\section{Strong-coupling limit of the Teo-Kane Model}\n\\label{sec:strong-coupling}\n\nIn this section, we present a lattice model\nin $d$ dimensions which has,\nas its continuum limit in $d=3$, the model discussed by\nTeo and Kane \\cite{Teo10}. In the limit that the mass terms\nin this model are large (which can be viewed as\na `strong-coupling' limit), a simple picture of\ntopological defects (`hedgehogs') emerges.\nWe show by a counting argument that hedgehogs\npossess Majorana zero modes which evolve as the\nhedgehogs are adiabatically moved. This adiabatic evolution\nis the 3D non-Abelian statistics which it is the main purpose\nof this paper to explain.\n\nThe strong coupling limit which we describe is the\nsimplest way to derive the existence of Majorana zero\nmodes and the unitary transformations of their Hilbert space\nwhich results from exchanging them. This section does\nnot require the reader to be {\\it au courant} with the\ntopological classification of insulators and superconductors\n\\cite{Ryu08,Kitaev09}. (In the next section, we will\nreview that classification in order to make our exposition\nself-contained.)\n\nWe use a hypercubic lattice in $d$-dimensions, with a single Majorana degree of freedom at each site.\nThat is,\nfor $d=1$, we use a chain, in $d=2$ we use a square lattice,\nin $d=3$ we use\na cubic lattice, and so on.\nWe first construct a lattice model whose continuum limit is the Dirac equation with $2^d$-dimensional\n$\\gamma$-matrices to reproduce the Dirac equation considered by Teo\nand Kane; we then show how to perturb this model to open a mass gap.\nWe begin by considering only nearest neighbor couplings.  The Hamiltonian $H$ is an anti-symmetric\nHermitian matrix.  In $d=1$, we can take the linear chain to give a lattice model with the Dirac equation as its continuum\nlimit.  That is, $H_{j,j+1}=i$ and $H_{j+1,j}=-i$.  To describe this state in\npictures, we draw these bonds as oriented lines,\nas shown in Fig.~(\\ref{figDirac}a), with the orientation indicating the\nsign of the bond.  The continuum limit of this Hamiltonian is described by a Dirac equation with $2$-dimensional $\\gamma$ matrices.\nWhile this system can be described by a unit cell of a single site, we instead choose to describe it by a unit\ncell of two sites for convenience when adding mass terms later.\nIn $d=2$, we can take a $\\pi$-flux\nstate to obtain the Dirac equation in the continuum limit.\nA convenient gauge to take to describe the $\\pi$-flux state is shown in Fig.~(\\ref{figDirac}b), with all the vertical bonds\nhaving the same orientation, and the orientation of the horizontal bonds alternating from row to row.  The continuum limit\nhere has $4$-dimensional $\\gamma$ matrices and we use a $4$-site unit cell.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{dirac.pdf}\n\\caption{(a) A lattice model giving the Dirac equation in $d=1$.\n(b) A lattice model in $d=2$.}\n\\label{figDirac}\n\\end{figure}\n\n\nIn general, in $d$ dimensions, we can obtain a Dirac equation with $2^d$-dimensional $\\gamma$ matrices\nby the following iterative\nprocedure.  Let the ``vertical\" direction refer to the direction of the $d$-th basis vector.\nHaving constructed the lattice Hamiltonian in $d-1$ dimensions, we stack these Hamiltonians vertically on top of each other,\nwith alternating signs in each layer.  Then, we take all the vertical bonds to be oriented in the\nsame direction.\nThis Hamiltonian is invariant under translation in the vertical direction by distance $2$.  Thus,\nif $H_{d-1}$ is the Hamiltonian in $d-1$ dimensions, the Hamiltonian $H_d$ is given by\n\\begin{equation}\nH_d=\\begin{pmatrix} H_{d-1} & 2\\sin(k\/2) I  \\\\ 2\\sin(k\/2) I& -H_{d-1} \\end{pmatrix},\n\\end{equation}\nwhere $I$ is the identity matrix and $k$ is the momentum in the vertical direction.\nNear $k=0$, this is\n\\begin{equation}\n\\label{Hdcontinuum}\nH_d \\approx H_{d-1} \\otimes \\sigma_z + k \\otimes \\sigma_x.\n\\end{equation}\n\nThis iterative construction corresponds to an iterative construction of $\\gamma$-matrices.  Having constructed $d-1$\ndifferent $2^{d-1}$-dimensional $\\gamma$-matrices $\\gamma^{}_1,...,\\gamma^{}_{d-1}$, we construct $d$ different $2^d$-dimensional $\\gamma$-matrices,\n$\\tilde \\gamma^{}_1,...,\\tilde \\gamma^{}_{d}$, by $\\tilde \\gamma^{}_i=\\gamma^{}_i \\otimes \\sigma_z$ for $i=1,...,d-1$, and $\\tilde \\gamma^{}_d=I\\otimes \\sigma_x$.\n\nIn one dimension, dimerization of bonds corresponds to alternately strengthening and weakening the bonds as shown in\nFig.~(\\ref{figDimer}).  In two dimensions, we can dimerize in either the horizontal or vertical directions.\nIn $d$-dimensions, we have $d$ different directions to dimerize.  Dimerizing in the ``vertical\" direction gives, instead\nof (\\ref{Hdcontinuum}), the result\n\\begin{equation}\n\\label{eqn:dimerizations}\nH_d \\approx H_{d-1} \\otimes \\sigma_z + k\\otimes \\sigma_x + m_d \\otimes \\sigma_y,\n\\end{equation}\nwhere $m_d$ is the dimerization strength.  This corresponds to an iterative construction of mass matrices, $M_i$, as follows.\nIn one dimension, we have $M_1=i\\sigma_y$.  Given $d-1$ different mass matrices in $d-1$ dimensions, $M_i$, we construct $\\tilde M_i$ in\n$d$-dimensions by $\\tilde M_i=M_i \\otimes \\sigma_z$, for $i=1...d-1$, and $\\tilde M_d=iI\\otimes \\sigma_y$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{dimer.pdf}\n\\caption{Dimerization in $d=1$.}\n\\label{figDimer}\n\\end{figure}\n\nIf the dimerization is non-zero, and constant, we can increase the dimerization strength without closing the gap until a strong\ncoupling limit is reached.  In one dimension, by increasing the dimerization strength, we eventually\nreach a fully dimerized configuration, in which each site has one non-vanishing bond\nconnected to it.\nIn two or more dimensions, the dimerization\ncan be a combination of dimerization in different directions.  However, if the dimerization is completely in one direction, for example\nthe vertical direction, we increase the dimerization strength until the vertical bonds are fully dimerized.  Simultaneously, we reduce the\nstrength of the other bonds to zero without closing the gap.\nThis is again a fully dimerized state, the columnar state, with each site having one non-vanishing bond.\nAny configuration with uniform, small dimerization can be deformed into this pattern without closing the gap by rotating the\ndirection of dimerization, increasing the strength of dimerization, and then setting the bonds in the other directions to zero.\n\nIt is important to understand that the ability to reach such a strong coupling\nlimits depends on the perturbation of the Dirac equation that we consider;\nfor dimerization, it is possible to reach a strong coupling limit, while if\nwe had instead chosen to open a mass gap by adding, for example, diagonal\nbonds with imaginary coupling to the two-dimensional Dirac equations, we\nwould open a mass gap by perturbing the Hamiltonian with the term $i\\gamma_1\\gamma_2$, and such a perturbation cannot be continued to the strong coupling\nlimit due to topological obstruction.\n\nFurther, if the dimerization is non-uniform then it may not be possible to reach a fully dimerized state without\nhaving defect sites.  Consider the configurations in\nFig.~(\\ref{figHedgehog}a) in $d=1$ and in Fig.~(\\ref{figHedgehog}b) in $d=2$.  These are the strong coupling limits of the\nhedgehog configuration,\nand each contains a zero mode, a single unpaired site.\nThis is one of the central results of the strong-coupling\nlimit: {\\it topological defects have unpaired sites which,\nin turn, support Majorana zero modes}.\n\nSuch strong-coupling hedgehog configurations can be constructed by the following iterative process in any dimension $d$.\nLet $x_d$ correspond to the coordinate in the vertical direction.  For $x_d\\geq 0$, stack $d-1$-dimensional hedgehog configurations.\nAlong the half-line given by $x_d>0$ and $x_i=0$ for $1\\leq i \\leq d-1$, arrange vertical bonds, oriented to connect\nthe site with $x_d=2k-1$ to\nthat with $x_d=2k$, for $k\\geq 1$.  Along the lower half plane, given by $x_d<0$, arrange vertical bonds oriented to connect\na site with $x_d=-(2k-1)$ to that with $-2k$, for $k\\geq 1$.  This procedure gives the $d=2$ hedgehog in\nFig.~(\\ref{figHedgehog}b) from the $d=1$ hedgehog in\nFig.~(\\ref{figHedgehog}a), and gives a strong coupling limit of the\nTeo-Kane hedgehog in $d=3$.  That is, the Teo-Kane\nhedgehog can be deformed into this configuration,  without closing the gap.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{hedgehog.pdf}\n\\caption{(a) A one-dimensional hedgehog.\n(b) A two-dimensional hedgehog.}\n\\label{figHedgehog}\n\\end{figure}\n\n\nSo long as we consider only nearest-neighbor bonds,\nthere is an integer index $\\nu$ describing different dimerization\npatterns in the strong-coupling limit.  This index, which\nis present in any dimension, arises from the sublattice symmetry\nof the system, and is closely-related to the U(1) symmetry\nof dimer models of spin systems\\cite{Rokhsar88}.  Label the two\nsublattices by $A$ and $B$.  Consider any set of sites, such that every site in that\nset has exactly one bond connected to it.  (Recall that,\nin the strong coupling\nlimit, every bond has strength $0$ or $1$ and every site\nhas exactly one bond connected to it, except for defect sites.)\nThen, the number of bonds going from $A$ sites in this set to $B$ sites outside the set\nis exactly equal to the number of bonds going from $B$ sites in this set to $A$ sites outside the set.  On the other hand,\nif there are defect sites in the set, then this rule is broken.\nConsider the region defined by the dashed line\nin Fig.~(\\ref{figU1}a).  We define the ``flux\" crossing the dashed line to be the number of bonds crossing that boundary which leave starting on an $A$ site, minus the number\nwhich leave starting on a $B$ site. The flux is non-zero in this case, but is unchanging as we increase the size of the region.\nThis flux is the index $\\nu$. By the argument given above for\nthe existence of zero modes, $\\nu$ computed for any region\nis equal to the number of Majorana zero modes contained\nwithin the region.\n\nThe index $\\nu$ can be defined beyond the strong-coupling\nlimit. Consider, for the sake of concreteness, $d=3$.\nThere are 3 possible dimerizations, one for each\ndimension, as we concluded in Eq. \\ref{eqn:dimerizations}.\nIn weak-coupling, the square of the gap is equal to the sum of\nthe squares of the dimerizations. Thus, if we assume\na fixed gap, we can model these dimerizations by a unit vector.\nThe integer index discussed above is simply the total winding\nnumber of this unit vector on the boundary of any region.\n\nHowever, once diagonal bonds are allowed,\nthe integer index $\\nu$ no longer counts zero modes.\nInstead, there is a $\\mathbb{Z}_2$ index,\nequal to $\\nu(\\text{mod} 2)$ which\ncounts zero modes modulo 2. To see this in\nthe strong-coupling limit,\nconsider the configuration in Fig.~(\\ref{figU1}b).\nThis is a configuration with $\\nu=2$\nbut no Majorana zero modes. However, a $\\nu=1$\nconfiguration must still have a zero mode and, thus,\nany configuration with odd $\\nu$ must have at least one zero mode.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{U1.pdf}\n\\caption{(a) Defect acting as source of $U(1)$-flux.  Bonds are oriented from $A$ to $B$ sublattice.  There is a net\nflux of one leaving the region defined by the dashed line.\n(b) Configuration with diagonal bond added, indicated by the undirected line\nconnecting the two circles; either orientation of this line, corresponding to different choices of the sign of the\nterm in the Hamiltonian, would lead to the same result.  There is a net of flux of two leaving the region\ndefined by the dashed line.}\n\\label{figU1}\n\\end{figure}\n\n\nIn Fig.~(\\ref{figU1}), we have chosen to orient the\nbonds from A to B sublattice\nto make it easier to compute $\\nu$.  However,\nthe $\\nu$ and its residue modulo 2,\ndefined above are independent of the orientation of\nthe bonds (which indicate the sign of terms in the Hamiltonian) and depend only on which sites are connected by bonds (which indicate which terms in the Hamiltonian are non-vanishing).\n\nThe $\\nu(\\text{mod} 2)$ with diagonal bonds\nis the same as Kitaev's ``Majorana number\"\\cite{Kitaev06a}.\nWe can use this to show the existence of\nzero modes in the Teo-Kane hedgehog even outside the\nstrong-coupling limit.\nConsider a hedgehog configuration.\nOutside some large distance $R$ from\nthe center of the hedgehog, deform to the\nstrong coupling limit without closing the gap.  Then, outside a distance $R$, we can count $\\nu(\\text{mod} 2)$ by counting bonds leaving\nthe region and we find a nonvanishing result relative to a reference configuration: if there are an even number of sites in the\nregion then there are an odd number of bonds leaving in a hedgehog configuration, and if there are an odd number of sites then there\nare an even number of bonds leaving.\nHowever, since this implies a nonvanishing\nMajorana number, there must be a zero mode inside the region, regardless of what the Hamiltonian inside is.\nWe note that this is a highly non-trivial result\nin the weak-coupling limit, where\nthe addition of weak diagonal bonds, all oriented\nthe same direction, to the configuration of Fig.~(\\ref{figDirac}b)\ncorresponds to adding the term\n$i\\gamma^{}_1 \\gamma^{}_2$ to the Hamiltonian in $d=2$.\nBy the argument given above, even this Hamiltonian\nhas a zero mode in the presence of a defect\nwith non-zero $\\nu(\\text{mod} 2)$.\n\nGiven any two zero modes, corresponding to defect sites in the strong coupling limit, we can identify a string of sites\nconnecting them.  If we have a pair of defect sites on opposite sublattices, corresponding to opposite hedgehogs, then one particular string\ncorresponds to the north pole of the order parameter, as in Fig.~(\\ref{figstring}a).  However, we\ncan simply choose {\\it any} arbitrary string.\nLet $\\gamma^{}_i,\\gamma^{}_j$ be the Majorana operators at the two defect sites.  The operation $\\gamma^{}_i\\rightarrow -\\gamma^{}_i,\\gamma^{}_j\\rightarrow\n-\\gamma^{}_j$ can be implemented as follows.  We begin with an adiabatic operation on one of the defect sites and the nearest $2$\nsites on the line.  The Hamiltonian on those three sites is an anti-symmetric, Hermitian matrix.  That is, it corresponds to\nan oriented plane in three dimensions.  We can adiabatically perform orthogonal rotations of this plane.  Thus, by rotating by $\\pi$\nin the plane corresponding to the defect site and the first site on the string,\nwe can change the sign of the mode on the defect and the orientation of the\nbond, as shown in Fig.~(\\ref{figstring}b).  This rotation is an adiabatic transformation\nof the three site Hamiltonian\n\\begin{equation}\n\\begin{pmatrix}\n0 & 0 & i\\sin(\\theta) \\\\\n0 & 0 & i\\cos(\\theta) \\\\\n-i\\sin(\\theta) & -i\\cos(\\theta) & 0\n\\end{pmatrix}\n\\end{equation}\nalong the path $\\theta=0\\rightarrow \\pi$.\nWe then perform rotations on consecutive triples of sites along the defect line, which changes\nthe orientation of pairs of neighboring bonds, arriving\nat the configuration in Fig.~(\\ref{figstring}c).  Finally, we rotate by $\\pi$\nin the plane containing the other defect site and the last site.\nThis returns the system to the original configuration,\nhaving effected the desired operation.\n\nSince we only consider adiabatic transformation, we can only perform orthogonal rotations with unit determinant.  Thus, any transformation which swaps two defects and returns the\nbonds to their original configuration, must change the sign of one of\nthe zero modes: $\\gamma^{}_i \\rightarrow \\gamma^{}_j, \\gamma^{}_j \\rightarrow -\\gamma^{}_i$.  Indeed, any orthogonal\ntransformation with determinant equal to minus one would change the sign of the fermion parity in the system, as the\nfermion parity operator is equal to the product of the $\\gamma_i$ operators.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{string.pdf}\n\\caption{(a) Pairs of defects connected by a string.\n(b) First rotation applied to the configuration in (a)  Open circle replaces filled circle to indicate sign change of the Majorana mode on the site.  (c)After rotating along the string. (d) Rotating the last site and restoring the string to its original configuration}\n\\label{figstring}\n\\end{figure}\n\nWe used the ability to change the orientation of a pair of bonds in this\nconstruction.\nThe fact that one can only change the\norientation of bonds in pairs, and not the\norientation of a single bond,\nis related to a global $Z_2$\ninvariant: the Hamiltonian is an anti-symmetric matrix and the sign\nof its Pfaffian cannot be changed without closing the gap.  Changing the direction of a single bond changes the sign of this Pfaffian and so is not possible.\n\nThe above discussion left open the question\nof which zero changes its sign, i.e. is the\neffect of the exchange  $\\gamma^{}_i \\rightarrow \\gamma^{}_j, \\gamma^{}_j \\rightarrow -\\gamma^{}_i$ or  $\\gamma^{}_i \\rightarrow -\\gamma^{}_j, \\gamma^{}_j \\rightarrow \\gamma^{}_i$? The answer is that it depends on how the bonds are returned to their original configuration\nafter the exchange is completed (which is a clue that\nthe defects must be understood as extended objects,\nnot point-like ones). For the bonds to be\nrestored, one of the defects must be rotated by\n$2\\pi$; the corresponding zero mode acquires a minus sign.\nWe will discuss this in greater detail in a later section.\nThe salient point here\nis that the effect of an exchange is a unitary transformation\ngenerated by the operator $e^{\\pm\\pi \\gamma^{}_i \\gamma^{}_j \/4}$.\nThis is reminiscent of the representation of braid group\ngenerators for non-Abelian quasiparticles in the quantum\nHall effect \\cite{Nayak96c} and vortices in chiral\n$p$-wave superconductors \\cite{Ivanov01},\nnamely the braid group representation realized\nby Ising anyons \\cite{Nayak08}.\nBut, of course, in 3D the braid group is not\nrelevant, and the permutation group, which is associated with\npoint-like particles in $d>2$, does not have non-trivial\nhigher-dimensional representations consistent with\nlocality \\cite{Doplicher71a,Doplicher71b}. As noted in the introduction,\nthis begs the question:\nwhat group are the unitary matrices\n$e^{\\pm\\pi \\gamma^{}_i \\gamma^{}_j \/4}$ representing?\n\n\n\n\\section{Topological Classification of Gapped Free\nFermion Hamiltonians}\n\\label{sec:free-fermion}\n\n\\subsection{Setup of the Problem}\n\nIn this section, we will briefly review the topological\nclassification of translationally-invariant or slowly\nspatially-varying free-fermion Hamiltonians following Kitaev's analysis\nin Ref. \\onlinecite{Kitaev09}. (For a different perspective,\nsee Schnyder {\\it et al.}'s approach in Ref. \\onlinecite{Ryu08}.)\nThe 3D Hamiltonian of the previous section is a specific example which\nfits within the general scheme and, by implication,\nthe 3D non-Abelian statistics which we derived at\nthe end of the previous section also holds for an entire\nclass of models into which it can be deformed without\nclosing the gap. Our discussion will\nfollow the logic of Milnor's treatment of Bott periodicity\nin Ref. \\onlinecite{Milnor63}.\n\nConsider a system of $N$ flavors of electrons\n${c_j}({\\bf k})$ in $d$ dimensions. The flavor index\n$j$ accounts for spin as well as the possibility of multiple\nbands. Since we will not be assuming charge conservation,\nit is convenient to express the complex fermion operators\n${c_j}({\\bf k})$ in terms of real fermionic operators\n(Majorana fermions),\n${c_j}({\\bf k})=(a_{2j-1}({\\bf k})+ia_{2j}({\\bf k}))\/2$\n(the index $j$ now runs from $1$ to $2N$).\nThe momentum ${\\bf k}$ takes values in\nthe Brillouin zone, which has the topology of the\n$d$-dimensional torus $T^d$. The Hamiltonian\nmay be written in the form\n\\begin{equation}\n\\label{eqn:basic}\nH = \\sum_{i,j,{\\bf p}} iA_{ij}({\\bf p}){a_i}({\\bf p}){a_j}(-{\\bf p})\n\\end{equation}\nwhere, by Fermi statistics, $A_{ij}({\\bf p})=-A_{ji}(-{\\bf p})$.\nLet us suppose that the Hamiltonian\n(\\ref{eqn:basic}) has an energy gap $2\\Delta$,\nby which we mean that its eigenvalues $E_{\\alpha}(p)$\n($\\alpha$ is an index labeling the eigenvalues of\n$H$) satisfy $|E_{\\alpha}(p)|\\geq \\Delta$.\nThe basic question which we address in this section\nis the following. What topological obstructions\nprevent us from continuously deforming one\nsuch gapped Hamiltonian into another?\n\nSuch an analysis can apply, as we will see,\nnot only to free fermion Hamiltonians, but also\nto those interacting fermion Hamiltonians which,\ndeep within ordered phases, are well-approximated\nby free-fermion Hamiltonians. (This can include\nrather non-trivial phases such as Ising anyons,\nbut not Fibonacci anyons.) In such settings, the\nfermions may be emergent fermionic quasiparticles;\nif the interactions between these quasiparticles\nare irrelevant in the renormalization-group sense,\nthen an analysis of free-fermion Hamiltonians\ncan shed light on the phase diagrams of\nsuch systems. Thus, the analysis of free fermion\nHamiltonians is equivalent to the analysis of\n{\\it interacting fermion ground states} whose\nlow-energy quasiparticle excitations are free fermions.\n\nLet us begin by considering a few concrete examples,\nin order of increasing complexity.\n\n\\subsection{Zero-Dimensional Systems}\n\nFirst, we analyze\na zero-dimensional system which we will not assume\nto have any special symmetry. The Hamiltonian\n(\\ref{eqn:basic}) takes the simpler form:\n\\begin{equation}\n\\label{eqn:zero-dim}\nH = \\sum_{i,j} iA_{ij}{a_i}{a_j}\n\\end{equation}\nwhere $A_{ij}$ is a $2N\\times 2N$ antisymmetric matrix,\n$A_{ij}=-A_{ji}$. Any real antisymmetric matrix can be written\nin the form\n\\begin{equation}\nA =  O^T \\left( \\begin{array}{rrrrr}\n0 & -{\\lambda_1} &   &  & \\\\\n{\\lambda_1} & 0  &    &  &  \\\\\n   &    & 0  & -{\\lambda_2}&  \\\\\n   &    & {\\lambda_2}  &  0&  \\\\\n   &    &     &    &  \\ddots\\end{array} \\right) O\n\\end{equation}\nwhere $O$ is an orthogonal matrix and\nthe $\\lambda_i$s are positive. The eigenvalues\nof $A$ come in pairs $\\pm i\\lambda_i$;\nthus, the {\\it absence of charge conservation} can\nalso be viewed as the {\\it presence of a particle-hole symmetry}.\nBy assumption, ${\\lambda_i}\\geq\\Delta$ for all $i$.\nClearly, we can continuously deform $A_{ij}$\nwithout closing the gap so that ${\\lambda_i}=\\Delta$ for all $i$.\n(This is usually called `spectrum flattening'.)\nThen, we can write:\n\\begin{equation}\nA = \\Delta\\,\\cdot \\,{O^T} J O\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:canonical-J}\nJ =  \\left( \\begin{array}{rrrrr}\n0 & -1 &   &  & \\\\\n1 & 0  &    &  &  \\\\\n   &    & 0  & -1&  \\\\\n   &    & 1  &  0&  \\\\\n   &    &     &    &  \\ddots\\end{array} \\right)\n\\end{equation}\nThe possible choices of $A_{ij}$\ncorrespond to the possible choices of $O\\in\\text{O}(2N)$,\nmodulo $O$ which commute with the matrix $J$.\nBut the set of $O\\in\\text{O}(2N)$ satisfying ${O^T} J O~=~J$\nis U($N$)$\\subset$O($2N$).\nThus, the space of all\npossible zero-dimensional free fermionic Hamiltonians\nwith $N$ single-particle energy levels\nis topologically-equivalent to the symmetric space O($2N$)\/U($N$).\n\nThis can be restated in more geometrical terms\nas follows. Let us here and henceforth\ntake units in which $\\Delta=1$.\nThen the eigenvalues of $A$ are $\\pm i$.\nIf we view the $2N\\times 2N$ matrix $A$\nas a linear transformation on $\\mathbb{R}^{2N}$, then\nit defines a complex structure.\nConsequently, we can view $\\mathbb{R}^{2N}$ as\n$\\mathbb{C}^{N}$ since multiplication of\n$\\vec{v}\\in \\mathbb{R}^{2N}$ by a\ncomplex scalar can be defined as\n$(a+ib) \\vec{v} \\equiv a\\vec{v} + b A\\vec{v}$.\nThe set of complex structures on $\\mathbb{R}^{2N}$\nis given by performing an arbitrary O($2N$) rotation\non a fixed complex structure, modulo the rotations\nof $\\mathbb{C}^{N}$ which respect the complex structure,\nnamely U($N$). Thus, once again, we conclude that\nthe desired space of Hamiltonians is topologically-equivalent\nto O($2N$)\/U($N$).\n\nWhat are the consequences of this equivalence?\nConsider the simplest case, $N=1$. Then, the\nspace of zero-dimensional Hamiltonians is topologically-equivalent to\nO($2$)\/U($1$)$=\\mathbb{Z}_2$: there are two classes\nof Hamiltonians, those in which the single fermionic level\nis unoccupied in the ground state,\n$c^\\dagger c = (1+i{a_1}{a_2})\/2=0$,\nand those in which it is occupied. For larger $N$,\nO($2N$)\/U($N$) is a more complicated space, but it\nstill has two connected components,\n${\\pi_0}(\\text{O}(2N)\/\\text{U}(N))=\\mathbb{Z}_2$,\nso that there are two classes of free fermion Hamiltonians,\ncorresponding to even or odd numbers of occupied\nfermionic levels in the ground state.\n\nSuppose, now, that we restrict ourselves to time-reversal\ninvariant systems and, furthermore, to those time-reversal\ninvariant systems which satisfy ${T^2}=-1$, where\n$T$ is the anti-unitary operator generating time-reversal.\nThen, following Ref. \\onlinecite{Kitaev09}, we\nwrite $T {a_i} T^{-1} = ({J_1})_{ij}{a_j}$. The matrix\n${J_1}$ is antisymmetric and satisfies ${J_1^2}=-1$.\n$T$-invariance of the Hamiltonian requires\n \\begin{equation}\n{J_1} A = -A {J_1}\n\\end{equation}\nSince ${J_1}$ is antisymmetric and satisfies ${J_1^2}=-1$,\nits eigenvalues are $\\pm i$.\nTherefore, ${J_1}$ defines a complex structure\non $\\mathbb{R}^{2N}$ which may, consequently,\nbe viewed as $\\mathbb{C}^{N}$.\nNow consider $A$, which is also\nantisymmetric and satisfies ${A^2}=-1$, in addition\nto anticommuting with $J_1$. It defines\na quaternionic structure on $\\mathbb{C}^{N}$\nwhich may, consequently, be viewed as\n$\\mathbb{H}^{N\/2}$.\nMultplication of $\\vec{v}\\in \\mathbb{R}^{2N}$\nby a quaternion can be defined as:\n$(a+bi+cj+dk)\\vec{v}\\equiv a\\vec{v}+b{J_1}\\vec{v}\n+cA\\vec{v}+d {J_1}A\\vec{v}$.\nThe possible choices of $A$\ncan be obtained from a fixed one\nby performing rotations of $\\mathbb{C}^{N}$,\nmodulo those rotations which respect the quaternionic structure,\nnamely Sp($N\/2$).\nThus, the set of time-reversal-invariant zero-dimensional free\nfermionic Hamiltonians with ${T^2}=-1$\nis topologically-equivalent to U($N$)\/Sp($N\/2$).\nSince ${\\pi_0}(\\text{U}(N)\/\\text{Sp}(N\/2))=0$,\nany such Hamiltonian can be continuously\ndeformed into any other. This can be understood\nas a result of Kramers\ndoubling: there must be an even number\nof fermions in the ground state so the division\ninto two classes of the previous case does not exist here.\n\n\n\\subsection{2D Systems: $T$-breaking\nsuperconductors}\n\nNow, let us consider systems in more than\nzero dimensions. Once again, we will assume\nthat charge is not conserved, and we will also\nassume that time-reversal symmetry is not\npreserved. For the sake of\nconcreteness, let us consider a single band of\nspin-polarized electrons on a two-dimensional lattice.\nLet us suppose that the electrons condense into\na (fully spin-polarized) $p_x$-wave superconductor.\nFor fixed superconducting order parameter,\nthe low-energy theory is a free fermion Hamiltonian\nfor gapless fermionic excitations at the nodal\npoints $\\pm\\vec{k}_{F}\\equiv (0,\\pm p_F)$. We now ask the question,\nwhat other order parameters could develop\nwhich would fully gap the fermions? For fixed values\nof these order parameters, we have a free fermion\nHamiltonian. Thus, these different possible order\nparameters correspond to different\npossible gapped free fermion Hamiltonians.\n\nThe low-energy Hamiltonian of a fully\nspin-polarized $p_x$-wave superconductor\ncan be written in the form:\n\\begin{equation}\nH = {\\psi^\\dagger}\\left( i{v_\\Delta}{\\partial_x}{\\tau_x}\n+ i{v_F}{\\partial_y}{\\tau_z}\\right)\\psi\n\\end{equation}\nwhere $v_F$, ${v_\\Delta}$ are, respectively, the\nFermi velocity and slope of the gap near the node.\nThe Pauli matrices $\\tau$ act in the particle-hole\nspace:\n\\begin{eqnarray}\n\\psi(k)\n\\equiv \\left(\n\\begin{array}{c}\nc_{\\vec{k}_{F}+\\vec{k}}\\\\\nc^{\\dagger}_{-\\vec{k}_{F}+\\vec{k}}\n\\end{array}\n\\right)\n\\end{eqnarray}\nThis Hamiltonian is invariant under the U(1):\n$\\psi\\rightarrow e^{i\\theta} \\psi$ which corresponds\nto conservation of momentum in the $p_y$ direction\n(not to charge conservation). Since we will be considering\nperturbations which do not respect this symmetry,\nit is convenient to introduce Majorana fermions\n${\\chi^{}_1}$, ${\\chi^{}_2}$ according to\n$\\psi={\\chi^{}_1}+i{\\chi^{}_2}$. Then\n\\begin{equation}\nH = i{\\chi^{}_a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right){\\chi^{}_a}\n\\end{equation}\nwith $a=1,2$. Note that we have suppressed the\nparticle-hole index on which the Pauli matrices\n$\\tau$ act. Since $\\chi^{}_1$, $\\chi^{}_2$ are each\na 2-component real spinor, this model has\n4 real Majorana fields.\n\nWe now consider the possible mass terms which\nwe could add to make this Hamiltonian fully gapped:\n\\begin{equation}\n\\label{eqn:Dirac+mass}\nH = i{\\chi^{}_a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right){\\chi^{}_a} + i{\\chi^{}_a}M_{ab}{\\chi^{}_b}\n\\end{equation}\nIf we consider the possible order parameters\nwhich could develop in this system, it is clear\nthat there are only two choices: an imaginary\nsuperconducting order parameter $ip_y$ (which\nbreaks time-reversal symmetry) and\ncharge density-wave order (CDW). These take the form:\n\\begin{equation}\nM^{ip_y}_{ab} = \\Delta_{ip_y}\\,i{\\tau^y} \\delta_{ab}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqn:CDW-mass-eg}\nM^{CDW}_{ab} = \\rho^{}_{2k_F}\n{\\tau^y} \\left(\\cos\\theta\\,  \\mu^z_{ab}+\n\\sin\\theta\\, \\mu^x_{ab}\\right)\n\\end{equation}\nwhere $\\mu^{x,z}$ are Pauli matrices and $\\theta$\nis an arbitrary angle.\nFor an analysis of the possible mass terms\nin the more complex situation\nof graphene-like systems, see, for instance,\nRef. \\onlinecite{Ryu09}.\n\nLet us consider the space of mass terms\nwith a fixed energy gap $\\Delta$ which is\nthe same for all 4 of the Majorana fermions in the model\n(i.e. a flattened mass spectrum).\nAn arbitrary gapped Hamiltonian can be continuously\ndeformed to one which satisfies this condition.\nThen we can have\n$\\Delta_{ip_y}=\\pm\\Delta$,\n$\\rho^{}_{2k_F}=0$\nor $\\rho^{}_{2k_F}=\\Delta$, $\\Delta_{ip_y}=0$\n(in the latter case, arbitrary $\\theta$ is allowed).\nIf both order\nparameters are present, then the energy gap\nis not the same for all fermions.\nIt's not that there's anything wrong with\nsuch a Hamiltonian -- indeed, one can imagine a\nsystem developing both kinds of order.\nRather, it is that such a Hamiltonian\ncan be continuously deformed to one with\neither $\\Delta_{ip_y}=0$ or $\\rho^{}_{2k_F}=0$\nwithout closing the gap. For instance, if\n$\\Delta_{ip_y}>\\rho^{}_{2k_F}$,\nthen the Hamiltonian can be continuously deformed to\none with $\\rho^{}_{2k_F}=0$. (However if we try to deform\nit to a Hamiltonian with $\\Delta_{ip_y}=0$, the gap will\nclose at $\\Delta_{ip_y}=\\rho^{}_{2k_F}$.)\nHence, we conclude that the space of possible mass terms is\ntopologically-equivalent to the disjoint union\nU($1$)$\\cup\\mathbb{Z}_2$: a single one-parameter family\nand two disjoint points.\n\nSince ${\\pi_0}(\\text{U}(1)\\cup\\mathbb{Z}_2) = \\mathbb{Z}_3$,\nthere are three distinct classes of quadratic Hamiltonians\nfor $4$ flavors of Majorana fermions in $2D$.\nThe one-parameter family of CDW-ordered\nHamiltonians counts as a single class since\nthey can be continuously deformed into each other.\nThe parameter $\\theta$ is the phase of the CDW,\nwhich determines whether the density is maximum\nat the sites, the midpoints of the bonds, or somewhere\nin between. It is important to keep in mind, however,\nthat, although there is no topological obstruction to\ncontinuously deforming one $\\theta$ into another,\nthere may be an energetic penalty which makes it costly.\nFor instance, the coupling of the system to the lattice may prefer\nsome particular value of $\\theta$.\nThe classification discussed here accounts only for topological\nobstructions; the possibility of energetic barriers must\nbe analyzed by different methods.\n\nWe can restate the preceding analysis in the following,\nmore abstract language. This formulation will\nmake it clear that we haven't overlooked a\npossible mass term and will generalize to more\ncomplicated free fermion models.\nLet us write\n${\\gamma^{}_1}={\\tau_x}\\delta_{ab}$,\n${\\gamma^{}_2}={\\tau_z}\\delta_{ab}$.\nThen\n\\begin{equation}\n\\{{\\gamma^{}_i},{\\gamma^{}_j}\\}=2\\delta_{ij}\n\\end{equation}\nThe Dirac Hamiltonian for $N=4$\nMajorana fermion fields takes the form\n\\begin{equation}\n\\label{eqn:Dirac-eqn-generic}\nH = i\\chi({\\gamma_i}{\\partial_i} + M)\\chi\n\\end{equation}\nThe matrix $M$ plays the role that the\nmatrix $A$ did in the zero-dimensional case.\nAs in that case, we assume a flattened spectrum\nwhich here means that each Majorana fermion\nfield has the same gap and that this gap is\nequal to $1$. (It does {\\it not} mean that the energy\nis independent of the momentum ${\\bf k}$.)\nIn order to satisfy this, we must require that\n\\begin{equation}\n\\{{\\gamma^{}_i},M\\}=0 {\\hskip 0.3 cm} \\text{and}\n{\\hskip 0.3 cm} {M^2}=-1\n\\end{equation}\n\nNote that it is customary to write the Dirac Hamiltonian\nin a slightly different form,\n\\begin{equation}\n\\label{eqn:Dirac-to-conventional}\nH = \\overline{\\psi}(i{\\gamma_i}{\\partial_i} + m)\\psi\n\\end{equation}\nwhich can be massaged into the form of (\\ref{eqn:Dirac-eqn-generic})\nusing $ \\overline{\\psi}=\\psi^\\dagger \\gamma_0$:\n\\begin{eqnarray}\nH &=& {\\psi^\\dagger}(i{\\gamma_0}{\\gamma_i}{\\partial_i} +\nm{\\gamma_0})\\psi\\cr\n&=&  {\\psi^\\dagger}(i{\\alpha_i}{\\partial_i} +\nm\\beta)\\psi\\cr\n&=&i{\\psi^\\dagger}({\\alpha_i}{\\partial_i} -\nim\\beta)\\psi\n\\end{eqnarray}\nwhere ${\\alpha_i}={\\gamma_0}{\\gamma_i}$ and\n$\\beta={\\gamma_0}$. Thus, if we write\n${\\gamma_i}\\equiv {\\alpha_i}$ and $M\\equiv\n-im\\beta$ and consider Majorana fermions\n(or decompose Dirac fermions into Majoranas),\nwe recover (\\ref{eqn:Dirac-eqn-generic}).\nWe have used the form (\\ref{eqn:Dirac-eqn-generic})\nso that it is analogous to (\\ref{eqn:zero-dim}), with\n$({\\gamma_i}{\\partial_i} + M)$ replacing $A_{ij}$\nand the $i$ pulled out front. Then, the matrix $M$\ndetermines the gaps of the various modes\nin the same way as $A$ does in the zero-dimensional\ncase. Similarly, assuming a `flattened' spectrum\nleads to the condition ${M^2}=-1$.\n\nHow many ways can we\nchoose such an $M$? Since ${\\gamma^2_2}=1$,\nits eigenvalues are $\\pm 1$. Hence, viewed as a\nlinear map from $\\mathbb{R}^4$ to itself, this matrix\ndivides $\\mathbb{R}^4$ into two 2D subspaces\n$\\mathbb{R}^4={X_+}\\oplus{X_-}$\nwith eigenvalue $\\pm 1$ under\n${\\gamma^{}_2}$, respectively.\nFor ${\\gamma^{}_2}={\\tau_z}\\delta_{ab}$,\nthis is trivial:\n\\begin{equation}\n{X_+}= \\text{span}\\left\\{\n\\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}} \\right) \\otimes  \\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}}\\right),\n \\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}} \\right) \\otimes  \\left(\\scriptstyle{\\begin{array}{c}\n0\\\\ 1 \\end{array}}\\right)\\right\\}\n\\end{equation}\nwhere $\\tau_z$ acts on the\nfirst spinor and the second spinor is indexed by $a=1,2$,\ni.e. is acted on by the Pauli matrices $\\mu^{x,z}$\nin (\\ref{eqn:CDW-mass-eg}).\nThis construction generalizes straightforwardly\nto arbitrary numbers $N$ of Majorana fermions, which is\nwhy we use it now.\n\nNow ${\\gamma^{}_1} M$ commutes with\n${\\gamma^{}_2}$ and satisfies ${({\\gamma^{}_1} M)^2}=1$.\nThus, it maps ${X_+}$ to itself and defines\nsubspaces ${X_+^1}, {X_+^2}$\nwith eigenvalue $\\pm 1$ under ${\\gamma^{}_1} M$\n(and equivalently for ${X_-}$). ${X_+}$ can decomposed\ninto ${X_+^1}\\oplus{X_+^2}={X_+}$.\nChoosing $M$ is thus equivalent to choosing\na linear subspace ${X_+^1}$ of ${X_+}$.\n\nThis can be divided into three cases.\nIf ${\\gamma^{}_1} M$\nhas one positive eigenvalue and one negative one\nwhen acting on ${X_+}$ then the space of possible choices\nof ${\\gamma^{}_1} M$ is equal to the space of 1D linear subspaces\nof $\\mathbb{R}^2$, which is simply U(1). If, on the other hand,\n${\\gamma^{}_1} M$ has two positive eigenvalues, then\nthere is a unique choice, which is simply\n$M={\\gamma^{}_1}{\\gamma^{}_2}$. If ${\\gamma^{}_1} M$\nhas two negative eigenvalues, then\nthere is again a unique choice,\n$M=-{\\gamma^{}_1}{\\gamma^{}_2}$.\nTherefore, the space of possible $M$s is topologically\nequivalent to $\\text{U}(1)\\cup\\mathbb{Z}_2$.\n\nNow, suppose that we have $2N$ Majorana fermions.\nThen $\\gamma^{}_2$ defines $N$-dimensional\neigenspaces\n${X_+},{X_-}$ such that\n$\\mathbb{R}^{2N}={X_+}\\oplus{X_-}$\nand ${\\gamma^{}_1} M$ defines eigenspaces\nof ${X_+}$: ${X_+^1}\\oplus{X_+^2}={X_+}$.\nIf ${\\gamma^{}_1} M$ has $k$ positive eigenvalues\nand $N-k$ negative ones, then the space\nof possible choices of ${\\gamma^{}_1}M$\nis O(N)\/O(k)$\\times$O(N-k), i.e we can\ntake the restriction of ${\\gamma^{}_1}M$\nto ${X_+}$ to be of the form\n\\begin{equation}\n{\\gamma^{}_1}M = O^T  \\left( \\begin{array}{cccccc}\n1 &  &   &  & & \\\\\n & \\ddots  &    &  &  &\\\\\n   &    & 1  & &  &\\\\\n   &    &   &  -1&  &\\\\\n   &    &     &    &\\ddots &\\\\\n      &    &     &    &  & -1  \\end{array} \\right) O\n\\end{equation}\nwith $k$ diagonal entries equal to $+1$\nand $N-k$ entries equal to $-1$. Thus, the space of Hamiltonians for\n$N$ flavors of free Majorana fermions\nis topologically equivalent to\n\\begin{equation}\n\\label{eqn:BO-def}\n{\\cal M}_{2N} = \\bigcup_{k=0}^{N} \\text{O}(N)\/(\\text{O}(k)\\times\n\\text{O}(N-k))\n\\end{equation}\nHowever, since ${\\pi_0}(\\text{O}(N)\/(\\text{O}(k)\\times\\text{O}(N-k)))\n=0$, independent of $k$ (note that $0$ is the group with a single\nelement, not the empty set $\\emptyset$),\n${\\pi_0}({\\cal M}_{2N})=\\mathbb{Z}_{N+1}$.\n\nIn the model analyzed above, we had\nonly a single spin-polarized band of electrons.\nBy increasing the number of bands and allowing\nboth spins, we can increase the number of\nflavors of Majorana fermions. In principle, the number of\nbands in a solid is infinity. Usually, we can introduce\na cutoff and restrict attention to a few bands\nnear the Fermi energy. However, for a purely\ntopological classification, we can ignore energetics\nand consider all bands on equal footing.\nThen we can take $N\\rightarrow\\infty$,\nso that ${\\pi_0}({\\cal M}_\\infty)=\\mathbb{Z}$.\nThis classification permits us to\ndeform Hamiltonians into each\nother so long as there is no topological obstruction,\nwith no regard to how energetically costly\nthe deformation may be. Thus, the $2N=4$ classification\nwhich we discussed above can perhaps be viewed as\na `hybrid' classification which looks for topological\nobstructions in a class of models with a fixed set of\nbands close to the Fermi energy.\n\nBut even this point of view is not really\nnatural. The discussion above took as\nits starting point an expansion about\na $p_x$ superconductor; the $p_x$ superconducting\norder parameter was assumed to be large and\nfixed while the $ip_y$ and CDW order parameters\nwere assumed to be small. In other words, we\nassumed that the system was at a point in parameter\nspace at which the gap, though non-zero, was small\nat two points in the Brillouin zone (the intersection\npoints of the nodal line in the $p_x$ superconducting\norder parameter with the Fermi surface). This allowed\nus to expand the Hamiltonian about these points\nin the Brillouin zone and write it in Dirac form.\nAnd this may, indeed, be reasonable in a system\nin which $p_x$ superconducting order is strong\n(i.e. highly energetically-favored) and other orders\nare weak. However, a topological classification should\nallow us to take the system into regimes in which\n$p_x$ superconductivity is small and other orders\nare large. Suppose, for instance, that we took\nour model of spin-polarized electrons (which\nwe assume, for simplicity, to be at half-filling\non the square lattice) and went into a regime in\nwhich there was a large\n$d_{{x^2}-{y^2}}$-density-wave (or `staggered\nflux') order parameter \\cite{Nayak00b}\n$\\langle c^\\dagger_{{\\bf k}+{\\bf Q}} c_{\\bf k} \\rangle\n= i\\Phi(\\cos{k_x}a-\\cos{k_y}a)$, where $a$ is the lattice\nconstant and $\\Phi$ is the magnitude of the order parameter.\nWith nearest-neighbor hopping only,\nthe energy spectrum is $E^2_{\\bf k}=\n(2t)^2 (\\cos{k_x}a+\\cos{k_y}a)^2\n+ \\Phi^2 (\\cos{k_x}a-\\cos{k_y}a)^2$.\nThus, the gap vanishes at 4 points,\n$(\\pm\\pi\/2,\\pm\\pi\/2)$ and $(\\mp\\pi\/2,\\pm\\pi\/2)$.\nThe Hamiltonian can be linearized in the\nvicinity of these points:\n\\begin{multline}\nH = {\\psi^\\dagger_1}\\left( i{v_\\Delta}{\\partial_x}{\\tau_x}\n+ i{v_F}{\\partial_y}{\\tau_z}\\right){\\psi_1}\\\\\n+ {\\psi^\\dagger_2}\\left( i{v_\\Delta}{\\partial_y}{\\tau_x}\n+ i{v_F}{\\partial_x}{\\tau_z}\\right){\\psi_2}\n\\end{multline}\nwhere $v_F$, ${v_\\Delta}$ are, respectively, the\nFermi velocity and slope of the gap near the nodes;\nthe subscripts 1,2 refer to the two sets of nodes\n$(\\pm\\pi\/2,\\pm\\pi\/2)$ and $(\\mp\\pi\/2,\\pm\\pi\/2)$;\nand $\\psi_{A}$, $A=1,2$ are defined by:\n\\begin{eqnarray}\n\\psi_{1,2}(k)\n\\equiv \\left(\n\\begin{array}{c}\nc_{(\\pi\/2,\\pm\\pi\/2)+\\vec{k}}\\\\\nc_{(-\\pi\/2,\\mp\\pi\/2)+\\vec{k}}\n\\end{array}\n\\right)\n\\end{eqnarray}\nIf we introduce Majorana fermions\n$\\psi_{A}=\\chi^{}_{A1}+i\\chi^{}_{A2}$, then\nwe can write this Hamiltonian with possible\nmass terms as:\n\\begin{multline}\nH = i\\chi^{}_{1a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right)\\chi^{}_{1a}\\\\\n+ i\\chi^{}_{2a}\\left( {v_\\Delta}{\\partial_y}{\\tau_x}\n+ {v_F}{\\partial_x}{\\tau_z}\\right)\\chi^{}_{2a}\\\\\n+ i\\chi^{}_{Aa}\\, M_{Aa,Bb} \\,  \\chi^{}_{Bb}\n\\end{multline}\nWe have suppressed the spinor indices\n(e.g. $\\chi_{11}$ is a two-component spinor);\nwith these indices included, $M_{Aa,Bb}$\nis an $8\\times 8$ matrix. However, in order for\nthe gap to be the same for all flavors, the\nmass matrix must anticommute with $\\tau_{x,z}$.\nThus, $M_{Aa,Bb}={\\tau_y} {\\tilde M}_{Aa,Bb}$.\nThe matrix ${\\tilde M}_{Aa,Bb}$ can have\n$0,1,2,3$, or $4$ eigenvalues equal to $+1$\n(with the rest being $-1$). The spaces\nof such mass terms are, respectively,\n$0$, $\\text{O}(4)\/(\\text{O}(1)\\times\\text{O}(3))$,\n$\\text{O}(4)\/(\\text{O}(2)\\times\\text{O}(2))$,\n$\\text{O}(4)\/(\\text{O}(3)\\times\\text{O}(1))$,\nand $0$. Mass terms with $0$ or $4$ eigenvalues\nequal to $+1$ correspond physically to\n$\\pm id_{xy}$-density wave order,\n$\\langle c^\\dagger_{{\\bf k}+{\\bf Q}} c_{\\bf k} \\rangle\n= \\pm\\sin{k_x}a\\,\\sin{k_y}a$.\nMass terms with $2$ eigenvalues\nequal to $+1$ correspond physically to\nsuperconductivity, to $Q'=(\\pi,0)$ CDW order,\nand to linear combinations of the two.\nMass terms with $1$ or $3$ eigenvalues\nequal to $+1$ correspond to\n(physically unlikely) hybrid orders with, for instance,\nsuperconductivity at $(\\pm\\pi\/2,\\pm\\pi\/2)$ and\n$\\pm id_{xy}$-density wave order at $(\\pm\\pi\/2,\\mp\\pi\/2)$.\nClearly, this is the $2N=8$ case of the general\nclassification discussed above. Thus, the same\nunderlying physical degrees of freedom -- a single\nband of spin-polarized electrons on a\nsquare lattice -- can correspond to either\n$2N=4$ or $2N=8$, depending on where\nthe system is in parameter space. One\ncan imagine regions of parameter space where\nthe gap is small at an arbitrary number $N$\nof points. Thus, if we restrict ourselves to systems\nwith a single band, then different regions of the parameter\nspace (with different numbers of points at which the gap is\nsmall) will have very different topologies.\nAlthough such a classification may be a necessary evil\nin some contexts, it is far preferable, given the choice,\nto allow topology to work unfettered by energetics.\nThen, we can consider a large number $n$ of bands.\nSuppose that the gap becomes small at $r$ points\nin the Brillouin zone in each band. Then, the low-energy\nHamiltonian takes the Dirac form for $2N=2rn$\nMajorana fermion fields.\nAs we will see below, if $N$ is sufficiently large,\nthe topology of the space of possible mass terms\nwill be independent of $N$. Consequently,\nfor $n$ sufficiently large, the topology of the space\nof possible mass terms will be independent of $r$.\nIn other words, we are in the happy situation\nin which the topology of the space of Hamiltonians\nwill be the same in the vicinity of any gap closing.\nBut any gapped Hamiltonian can be continuously\ndeformed so that the gap becomes small at some\npoints in the Brillouin zone. Thus, the problem of classifying\ngapped free fermion Hamiltonians in $d$-dimensions\nis equivalent to the problem of classifying possible mass terms\nin a generic $d$-dimensional Dirac Hamiltonian\nso long as the number of bands is sufficiently large \\cite{Kitaev09}.\nThis statement can be made more precise and\nput on more solid mathematical footing using ideas\nwhich we discuss in Appendix \\ref{sec:dimension}.\n\n\n\\subsection{Classification of Topological Defects}\n\nThe topological classification described above\nholds not only for classes of translationally-invariant\nHamiltonians such as (\\ref{eqn:Dirac-eqn-generic}),\nbut also for topological defects within a class.\nSuppose, for instance, that we consider\n(\\ref{eqn:Dirac-eqn-generic}) with a mass\nwhich varies slowly as the origin is encircled\nat a great distance. We can ask whether such a\nHamiltonian can be continuously deformed into\na uniform one. In a system in which the mass term\nis understood as arising as a result of some\nkind of underlying ordering such as superconductivity or\nCDW order, we are simply talking about topological\ndefects in an ordered media, but with the caveat that\nthe order parameter is allowed to explore a very large\nspace which may include many physically\ndistinct or unnatural orders, subject only to the condition that\nthe gap not close.\n\nLet us, for the sake of concreteness,\nassume that we have a mass term with $N\/2$\npositive eigenvalues when restricted to the $+1$ eigenspace of\n$\\gamma^{}_2$. (For $N$\nlarge, the answer obviously cannot depend on the\nnumber of positive eigenvalues $k$ so long\nas $k$ scales with $N$. Thus, we will denote the\nspace ${\\cal M}_{2N}$ defined in Eq. \\ref{eqn:BO-def}\nby $\\mathbb{Z}\\times\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwhere the integers in $\\mathbb{Z}$ correspond to the number\nof positive eigenvalues of the mass term when restricted\nto the $+1$ eigenspace of $\\gamma^{}_2$.)\nThen $M(r=\\infty,\\theta)$ defines a loop in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwhich cannot be continuously unwound if it\ncorresponds to a nontrivial element of\n${\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2)))$.\n\n\nTo compute ${\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\n\\text{O}(N\/2)))$, we parametrize\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nby symmetric matrices $K$ which satisfy ${K^2}=1$\nand $\\text{tr}(K)=0$. (Such matrices decompose\n$\\mathbb{R}^{N}$ into their\n$+1$ and $-1$ eigenspaces:\n$\\mathbb{R}^{N}={V_+}\\oplus{V_-}$.\n$K$ can be written in the form:\n$K={O^T} {K_0} O$,\nwhere $K_0$ has $N\/2$ diagonal entries\nequal to $+1$ and $N\/2$ equal to $-1$, i.e\n${K_0}=\\text{diag}(1,\\ldots,1,-1,\\ldots,-1)$.)\nNote that any such $K$ is itself an orthogonal matrix,\ni.e. an element of O$(N)$;\nthus $\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\ncan be viewed as a submanifold of O$(N)$\nin a canonical way. Consider a curve $L(\\lambda)$\nin $\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwith $L(0)=K$ and $L(\\pi)=-K$.\nWe will parametrize it as $L(\\lambda)=K\\, e^{\\lambda A}$,\nwhere $A$ is in the Lie algebra of $\\text{O}(N)$.\nIn order for this loop to remain in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$,\nwe need ${(K\\, e^{\\lambda A})^2}=1$.\nSince ${(K\\, e^{\\lambda A})^2}=K\\, e^{\\lambda A} K\\, e^{i\\lambda A}\n= e^{i\\lambda K A K} \\, e^{\\lambda A}$, this condition\nimplies that $KA=-AK$.\nIn order to have $L(\\pi)=-K$, we need ${A^2}=-1$.\nSuch a curve is, in fact, a minimal\ngeodesic from $K$ to $-K$.\nEach such geodesic can be represented by its\nmidpoint $L(\\pi\/2)=KA$, so the space of such geodesics is\nequivalent to the space of matrices $A$ satisfying\n${A^2}=-1$ and $KA=-AK$. As discussed in\nRef. \\onlinecite{Milnor63}, the space of minimal geodesics\nis a good enough approximation to the entire space of loops\n(essentially because an arbitrary loop can\nbe deformed to a geodesic) that we can compute\n$\\pi_1$ from the space of minimal geodesics\njust as well as from the space of loops. Thus,\nthe loop space of $\\text{O}(N\/2)\/(\\text{O}(k)\\times\\text{O}(N\/2-k))$\nis homotopically equivalent to the space of matrices $A$ satisfying\n${A^2}=-1$ and $KA=-AK$.\nSince it anticommutes with $K$, $KA$\nmaps the $+1$ eigenspace of $K$ to\nthe $-1$ eigenspace. It is clearly a length-preserving map\nsince ${(KA)^2}=1$\nand, since the $\\pm 1$ eigenspaces of $K$ are isomorphic to\n$\\mathbb{R}^{N\/2}$, $KA$ defines an element of\nO$(N\/2)$. Thus a loop in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\ncorresponds to an element of O$(N\/2)$\nor, in other words:\n\\begin{equation}\n\\label{eqn:Bott-step-1}\n{\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2)))=\n{\\pi_0}(\\text{O}(N\/2)).\n\\end{equation}\nThe latter group is simply $\\mathbb{Z}_2$\nsince $\\text{O}(N\/2)$ has two connected components:\n(1) pure rotations and (2) rotations combined with a reflection.\n\nIt might come as a surprise that we find a $\\mathbb{Z}_2$\nclassification for point-like defects in two dimensions.\nIndeed, if we require that the superconducting order parameter\nhas fixed amplitude at infinity, then vortices\nof arbitrary winding number are stable and we have a\n$\\mathbb{Z}$ classification. However,\nin the classification discussed here, we require a weaker\ncondition be satisfied: that the fermionic gap remain constant.\nConsequently, a vortex configuration\nof even winding number can be unwound without\nclosing the free fermion gap by, for instance, `rotating'\nsuperconductivity into CDW order.\n\n\n\\subsection{3D Systems with No Symmetry}\n\nWith these examples under our belts, we now turn to\nthe case which is of primary interest in this paper:\nfree fermion systems in three dimensions\nwithout time-reversal or charge-conservation symmetry.\nWe consider the Dirac Hamiltonian in $3D$\nfor an $2N$-component Majorana fermion field\n$\\chi$:\n\\begin{equation}\n\\label{eqn:3D-Dirac+mass}\nH = i\\chi\\left( \\partial_{1}{\\gamma^{}_1}\n+ \\partial_{2}{\\gamma^{}_2} +\n\\partial_{3}{\\gamma^{}_3}\\right)\\chi + i{\\chi}M{\\chi}\n\\end{equation}\nIn the previous section, we discussed\na lattice model which realizes (\\ref{eqn:3D-Dirac+mass})\nin its continuum limit with\n$2N=8$. Different mass terms\ncorrespond to different quadratic perturbations\nof this model which open a gap (which can be viewed\nas order parameters which we are turning on at the mean-field level).\nWe could classify such terms by considering,\nfrom a physical perspective, all such ways of opening a gap.\nHowever, we will instead determine the topology\nof the space of mass terms (and, thereby, the space\nof gapped free fermion Hamiltonians) by\nthe same mathematical methods by which we analyzed the\n$2D$ case.\n\nSince ${\\gamma_1^2}=1$ and has vanishing\ntrace, this matrix decomposes $\\mathbb{R}^{2N}$\ninto its $\\pm 1$ eigenspaces:\n$\\mathbb{R}^{2N}={X_+} \\oplus {X_-}$.\nNow ${({\\gamma^{}_2}{\\gamma^{}_3})^2}=-1$\nand $[{\\gamma^{}_1},{\\gamma^{}_2}{\\gamma^{}_3}]=0$.\nTherefore, ${\\gamma^{}_2}{\\gamma^{}_3}$ is a\ncomplex structure on ${X_+}$ (and also on ${X_-}$),\ni.e. we can define multiplication of vectors\n$\\vec{v}\\in{X_+}$ by complex scalars\naccording to $(a+bi)\\vec{v}\\equiv\na\\vec{v} + {\\gamma^{}_2}{\\gamma^{}_3}\\vec{v}$.\n(Consequently, we can view ${X_+}$ as\n$\\mathbb{C}^{N\/2}$.)\nNow, consider a possible mass term $M$,\nwith ${M^2}=-1$. ${({\\gamma^{}_2}M)^2}=1$\nand $[{\\gamma^{}_1},{\\gamma^{}_2}M]=0$.\nLet $Y$ be the subspace of ${X_+}$\nwith eigenvalue $+1$ under ${\\gamma^{}_2}M$.\nSince $\\{{\\gamma^{}_2}M,{\\gamma^{}_2}{\\gamma^{}_3}\\}=0$,\n${\\gamma^{}_2}{\\gamma^{}_3}Y$ is the subspace of ${X_+}$\nwith eigenvalue $-1$ under ${\\gamma^{}_2}M$.\nIn other words, ${X_+}=Y\\oplus{\\gamma^{}_2}{\\gamma^{}_3}Y$,\ni.e. $Y$ is a real subspace of ${X_+}$.\nHence, the space of choices of $M$ is the space\nof real subspaces $Y\\subset{X_+}$ (or,\nequivalently, of real subspaces\n$\\mathbb{R}^{N\/2}\\subset\\mathbb{C}^{N\/2}$).\nGiven any fixed real subspace $Y\\subset{X_+}$,\nwe can obtain all others by performing $\\text{U}(N\/2)$\nrotations of ${X_+}$, but two such rotations\ngive the same real subspace if they differ only\nby an $\\text{O}(N\/2)$ rotation of $Y$. Thus,\nthe space of gapped  Hamiltonians\nfor $2N$ free Majorana fermion fields\nin $3D$ with no symmetry is topologically-equivalent\nto $\\text{U}(N\/2)\/\\text{O}(N\/2)$. In the remaining\nsections of this paper, we will be discussing\ntopological defects in such systems and their motions.\n\n\\subsection{General Classification and Bott Periodicity}\n\\label{sec:bott}\n\nBefore doing so, we pause for a minute\nto consider the classification in other dimensions\nand in the presence of symmetries such as\ntime-reversal and charge conservation.\nWe have seen that systems with no symmetry\nin $d=0,2,3$ are classified by the spaces\n$\\text{O}(2N)\/\\text{U}(N)$,\n$\\mathbb{Z}\\times\n\\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$,\nand $\\text{U}(N\/2)\/\\text{O}(N\/2)$. By similar methods,\nit can be shown that the $d=1$ case is\nclassified by $\\text{O}(N)$. As we have seen,\nincreasing the spatial dimension increases the\nnumber of $\\gamma$ matrices by one.\nThe problem of choosing $\\gamma_{1},\\ldots,\\gamma_{d}$\nsatisfying $\\{{\\gamma_i},{\\gamma_j} \\}=2\\delta_{ij}$\nand $M$ which anti-commutes with the $\\gamma_i$s\nand squares to $-1$ leads us to subspaces of\n$\\mathbb{R}^{2N}$ of smaller and smaller\ndimension, isometries between these spaces,\nor complex of quaternionic structures on these spaces.\nThis leads the progression of spaces\nin the top row of Table \\ref{tbl:classifying}.\n\nAt the same time, we have seen\nthat a time-reversal-invariant system in $d=0$\nis classified by $\\text{U}(N)\/\\text{Sp}(N\/2)$.\nSuppose that we add a discrete anti-unitary symmetry\n$S i S^{-1} =-i$ defined by\n\\begin{equation}\n\\label{eqn:symmetries}\nS {a_i} S^{-1} = ({J})_{ij}{a_j}\n\\end{equation}\nwhich squares to ${J^2}=-1$.\nIt must anti-commute with the mass term\n\\begin{equation}\n\\label{eqn:anti-comm}\n\\{J,M\\}=0\n\\end{equation}\nin order to ensure invariance under the symmetry,\nso choosing a $J$\namounts to adding a complex structure,\nwhich leads to the {\\it opposite} progression of\nclassifying spaces.\nConsider, as an example of the preceding statements,\na time-reversal invariant system in $d=3$.\nThen time-reversal symmetry $T$ is an\nexample of a symmetry generator $J$\ndiscussed in the previous paragraph.\nWe define a real subspace $Y\\subset{X_+}$,\nin a similar manner as above, but now\nas the subspace of ${X_+}$\nwith eigenvalue $+1$ under ${\\gamma^{}_2}T$,\nrather than under ${\\gamma^{}_2}M$. Once again,\n${X_+}=Y\\oplus{\\gamma^{}_2}{\\gamma^{}_3}Y$.\nNow, $\\{{\\gamma^{}_3}M,{\\gamma^{}_2}T\\}=0$,\nand ${({\\gamma^{}_3}M)^2}=1$, so the\n$+1$ eigenspace of ${\\gamma^{}_3}M$\nis a linear subspace of $Y$.\nThe set of all such linear subspaces is\n$\\mathbb{Z}\\times\n\\frac{\\text{O}(N\/2)}{\\text{O}(N\/4)\\times\\text{O}(N\/4)}$.\nBut this is the same classifying space as for a system\nwith no symmetry in $d=2$ (apart from a reduction\nof $N$ by a factor of $2$). Thus, we are led\nto the list of classifying spaces for gapped\nfree fermion Hamiltonians\nin Table \\ref{tbl:classifying}.\n\n\\begin{table*}\n\\begin{tabular}{c | c c c c c c c}\ndim.: &  0    &   1 & 2 & 3 & 4 & \\ldots\\\\\n\\hline\\hline\nSU($2$), $T$, $Q$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$\n & $\\text{U}(N\/2)\/\\text{O}(N\/2)$ & $\\text{Sp}(N\/4)\/\\text{U}(N\/4)$ &\n $\\text{Sp}(N\/8)$ & $\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/8)}\n {\\text{Sp}(N\/16)\\times\\text{Sp}(N\/16)}$ \\ldots\\\\\nSU($2$), $T$, $Q$, $\\chi$&  $\\text{O}(N\/4)$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N\/4)}{\\text{O}(N\/8)\\times\\text{O}(N\/8)}$\n & $\\text{U}(N\/8)\/\\text{O}(N\/8)$ & $\\text{Sp}(N\/16)\/\\text{U}(N\/16)$ &\n $\\text{Sp}(N\/32)$ &\\ldots\\\\\nno symm.& $\\text{O}(2N)\/\\text{U}(N)$ &  $\\text{O}(N)$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$\n & $\\text{U}(N\/2)\/\\text{O}(N\/2)$ & $\\text{Sp}(N\/4)\/\\text{U}(N\/4)$ &\n \\ldots\\\\\n$T$ only & $\\text{U}(N)\/\\text{Sp}(N\/2)$  & $\\text{O}(N)\/\\text{U}(N\/2)$\n&$\\text{O}(N\/2)$ & $\\mathbb{Z}\\times\n \\frac{\\text{O}(N\/2)}{\\text{O}(N\/4)\\times\\text{O}(N\/4)}$ & \\ldots\\\\\n$T$ and $Q$ &\n$\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/2)}{\\text{Sp}(N\/4)\\times\\text{Sp}(N\/4)}$\n& $\\text{U}(N\/2)\/\\text{Sp}(N\/4)$ & $\\text{O}(N\/2)\/\\text{U}(N\/4)$\n& $\\text{O}(N\/4)$\\\\\n$T$, $Q$, $\\chi$ & $\\text{Sp}(N\/4)$ &\n$\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/4)}{\\text{Sp}(N\/8)\\times\n\\text{Sp}(N\/8)}$\n& $\\text{U}(N\/4)\/\\text{Sp}(N\/8)$ & $\\text{O}(N\/4)\/\\text{U}(N\/8)$\\\\\n&\\vdots & & & &$\\ddots$\n\\end{tabular}\n\\caption{The period-$8$ (in both dimension and number of\nsymmetries) table of classifying spaces for free fermion Hamiltonians\nfor $N$ complex $=2N$ real (Majorana) fermion fields in dimensions\n$d=0,1,2,3,\\ldots$ with no symmetries; time-reversal symmetry ($T$) only; time-reversal and charge conservation\nsymmetries ($T$ and $Q$); time-reversal, charge conservation,\nand sublattice symmetries ($T$, $Q$, and $\\chi$);\nand the latter two cases with SU($2$) symmetry.\nAs a result of the period-$8$ nature of the table,\nthe top two rows could equally well be the\nbottom two rows of the table.\nMoving $p$ steps to the right and $p$ steps down leads to\nthe same classifying space (but for $1\/2^p$ as many fermion\nfields), which is a reflection of Bott periodicity, as explained\nin the text. The number of disconnected components of any such\nclassifying space -- i.e. the number of different phases in that\nsymmetry class and dimension -- is given by the corresponding\n$\\pi_0$, which may be found in Eq. \\ref{eqn:stable-pi-0}. Higher homotopy groups, which classify defects, can be computed using Eq. \\ref{eqn:Bott periodicity}. Table \\ref{tbl:unitary-classifying}, given in Appendix\n\\ref{sec:QnotT}, is the analogous table\nfor charge-conserving Hamiltonians without time-reversal symmetry.}\n\\label{tbl:classifying}\n\\end{table*}\n\nIn this table, $Q$ refers to charge-conservation symmetry.\nCharge conservation is due to the invariance of\nthe Hamiltonian of a system under the U(1) symmetry\n${c_i}\\rightarrow e^{i\\theta}{c_i}$. In terms of\nMajorana fermions $a_i$ defined according to\n${c_j}=(a_{2j-1}+ia_{2j})\/2$, the symmetry takes the\nform $a_{2j-1}~\\rightarrow~\\cos\\theta a_{2j-1}\n+ \\sin\\theta a_{2j}$, $a_{2j}~\\rightarrow~-\\sin\\theta a_{2j-1}\n+ \\cos\\theta a_{2j}$. However, if a free fermion Hamiltonian\nis invariant under the discrete\nsymmetry ${c_i}\\rightarrow i{c_i}$ or, equivalently,\n$a_{2j-1} \\rightarrow a_{2j}$, $a_{2j} \\rightarrow - a_{2j-1}$,\nthen it is automatically invariant under the full U(1) as well,\nand conserves charge \\cite{Kitaev09}. Thus, we can treat charge\nconservation as a discrete symmetry $Q$\nwhich is unitary, squares to $-1$, and commutes\nwith the Hamiltonian (i.e. with the $\\gamma$ matrices and $M$).\nSince $Q$ transforms ${c_i}\\rightarrow i{c_i}$, it\nanti-commutes with $T$.\nNote further that if a system has time-reversal\nsymmetry, then the product of time-reversal $T$ and\ncharge conservation $Q$ is a discrete anti-unitary symmetry,\n$QT$ which anti-commutes with the Hamiltonian and with $T$\nand squares to $-1$. Then $QT$ is defined by a choice of matrix $J$,\nanalogous to $T$, as in Eq. \\ref{eqn:symmetries}.\nIf the system is not time-reversal-invariant,\nthen charge conservation is a unitary symmetry.\nIt is easier then to work with complex fermions,\nand the classification of such systems falls into an entirely\ndifferent sequence, as discussed in Appendix \\ref{sec:QnotT}.)\n\nIf a system is both time-reversal symmetric\nand charge-conserving, i.e. if it is a time-reversal\ninvariant insulator, then it may have an additional\nsymmetry which guarantees that the eigenvalues\nof the Hamiltonian come in $\\pm E$ pairs, just as in a superconductor.\nAn example of such a symmetry is the sublattice\nsymmetry of Hamiltonians on a bipartite lattice in\nwhich fermions can hop directly from the $A$ sublattice\nto the $B$ sublattice but cannot hop directly between sites on the\nsame sublattice. In such a case, the system is\ninvariant under a unitary symmetry\n$\\chi$ defined as follows. If we block diagonalize\n$\\chi$ so that one block acts on sites in the $A$ sublattice\nand the other on sites in the $B$ sublattice, then\nwe can write $\\chi=\\text{diag}(k,-k)$, i.e.\n${a_i}({\\bf x})\\rightarrow -k_{ij}{a_j}({\\bf x})$ for\n${\\bf x}\\in A$ and\n${a_i}({\\bf x})\\rightarrow k_{ij}{a_j}({\\bf x})$ for ${\\bf x}\\in B$.\nThis symmetry transforms the Hamiltonian to minus itself\nif ${k^2}=1$ or, in other words, if ${\\chi^2}=1$. Then\n$\\chi Q$ is a unitary symmetry which squares to $-1$\nand anti-commutes with the Hamiltonian, $T$,\nand $QT$. Hence $\\chi Q$, too, is defined by a choice of matrix $J$,\nas in Eq. \\ref{eqn:symmetries}. We will call such a\nsymmetry a sublattice symmetry $\\chi$ and a system satisfying\nthis symmetry a `bipartite' system, but the symmetry\nmay have a different microscopic origin.\n\nIn an electron system, time-reversal ordinarily squares\nto $-1$, because the transformation law is \n${c_\\uparrow}\\rightarrow {c_\\downarrow}$,\n${c_\\downarrow}\\rightarrow -{c_\\uparrow}$,\nas we have thus far assumed in taking ${J^2}=-1$.\nHowever, it is possible to have a system of\nfully spin-polarized electrons which has an\nanti-unitary symmetry $T$ which squares to $+1$.\n(One might object to calling this symmetry time-reversal\nbecause it doesn't reverse the electron spins,\nbut $T$ is a natural label because it is a symmetry\nwhich is just as good for the present purposes.)\nThen, since ${J^2}=1$, a choice of $J$ is similar to\na choice of a $\\gamma$ matrix. In general,\nsymmetries (\\ref{eqn:symmetries})\nwhich square to $+1$ have the same effect\non the topology of the space of free fermion Hamiltonians\nas adding dimensions since each such $J$\ndefines a subspace of half the dimension within the\neigenspaces of the $\\gamma$ matrices.\nThis is true for systems with ${T^2}=1$.\n\nSU(2) spin-rotation-invariant and time-reversal-invariant\ninsulators (systems with $T$ and $Q$) effectively\nfall in this category. The Hamiltonian for such\na system can be written in the form $H=h\\otimes {I_2}$\nwhere the second factor is the $2\\times 2$ identity\nmatrix acting on the spin index. Then time-reversal\ncan be written in the form $T= t \\otimes i{\\sigma_y}$,\nwhere ${t^2}=1$, and $Q$ can be written in\nthe form $Q= q \\otimes {I_2}$,\nso that $QT =qt \\otimes i{\\sigma_y}$, where ${(qt)^2}=1$.\nThus, since the matrix $i\\sigma_y$ squares to $-1$,\nthe symmetries $T$ and $QT$ have effectively become\nsymmetries which square to $+1$. They now move the\nsystem through the progression of classifying spaces\nin the same direction as increasing the dimension,\ni.e. in the opposite direction to symmetries which\nsquare to $-1$. Thus, SU(2) spin-rotation-invariant and\ntime-reversal-invariant insulators in $d$ dimensions\nare classified by the same space as\nsystems with no symmetry in $d+2$ dimensions.\nHowever, in a system which, in addition, has\nsublattice symmetry $\\chi=x\\otimes {I_2}$,\nwe have $(qx)^{-1}$. Thus, sublattice symmetry is\nstill a symmetry which squares to $-1$.\nSince the two symmetries which square to $+1$\n($T$ and $QT$) have the same effect as increasing\nthe dimension while the symmetry which squares\nto $-1$ has the same effect as decreasing the dimension,\nSU(2) spin-rotation-invariant and time-reversal-invariant insulators\nwith sublattice symmetry in $d$ dimensions\nare classified by the same space as\nsystems with no symmetry in $d+1$ dimensions\n(but with $N$ replaced by $N\/4$).\nSimilar considerations apply to superconductors with\nSU(2) spin-rotational symmetry.\n\n\\begin{table*}\n \\begin{tabular}[t]{|c|c|c|c|c|}\n \\hline\nSymmetry classes&Physical realizations&$d=1$&$d=2$&$d=3$\n \\\\\\hline\n\\hline D&SC&{\\color{blue}$p$-wave SC}&{\\color{blue}$(p+ip)$-SC}&0\n\\\\\\hline\nDIII&TRI SC&{\\color{red} ${\\rm Z_2}$}&{\\color{blue} $(p+ip)(p-ip)$-SC}&He$^3$-B\n\\\\\\hline AII&TRI\nins.&0&HgTe Quantum well&${\\rm Bi_{1-x}Sb_x}$, ${\\rm Bi_2Se_3}$, etc.\n\\\\\\hline\nCII&Bipartite TRI ins.&Carbon nanotube&0&{\\color{red} ${\\rm Z_2}$}\n\\\\\\hline\nC&Singlet SC&0&{\\color{blue} $(d+id)$-SC}&0\n\\\\\\hline\nCI&Singlet TRI\nSC&0&0&{\\color{red} Z}\n\\\\\\hline\nAI& TRI ins. w\/o SOC&0&0&0\n\\\\\\hline\nBDI&Bipartite TRI\nins. w\/o SOC&Carbon nanotube&0&0\n\\\\\\hline\n \\end{tabular}\n  \\caption{Topological periodic table in physical dimensions $1,2,3$. The first column contains 8 of the 10 symmetry classes in the Cartan notation\n adopted by Schnyder {\\it et al.}\\cite{Ryu08}, following Zirnbauer\n \\cite{Zirnbauer96,Altland97}.\n The second column contains the requirements for physical systems which can\n realize the corresponding symmetry classes. ``w\/o\" stands for ``without\". SC stands for superconductivity, TRI for time-reversal invariant, and SOC for spin-orbit coupling. The three columns $d=1,2,3$ list topological states in the spatial dimensions $1,2,3$ respectively. $0$ means the topological classification is trivial. The red labels {\\color{red} Z} and {\\color{red} ${\\rm Z_2}$} stand for topological states\nclassified by these groups but for which states corresponding\nto non-trivial elements of $\\mathbb{Z}$ or $\\mathbb{Z}_2$\nhave not been realized in realistic materials.\nThe blue text stands for topological states for which\na well-defined physical model has been proposed but convincing experimental candidate has not been found yet. (See text for more discussions on the realistic materials.) }\n  \\label{tbl:periodic}\n\\end{table*}\n\nIn order to discuss topological defects in\nthe systems discussed here,\nit is useful to return to the arguments which led to\n(\\ref{eqn:Bott-step-1}).\nBy showing that the space of loops in\n$\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4))$ is\nwell-approximated by $\\text{O}(N\/4)$,\nwe not only showed that\n${\\pi_1}(\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4)))=\n{\\pi_0}(\\text{O}(N\/4))$ but, in fact, that\n${\\pi_k}(\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4)))=\n\\pi_{k-1}(\\text{O}(N\/4))$ (see Ref. \\onlinecite{Milnor63}).\nContinuing in the same way, we can approximate\nthe loop space of $\\text{O}(N\/4)$ (i.e. the space of loops in\n$\\text{O}(N\/4)$) by minimal\ngeodesics from $\\mathbb{I}$ to $-\\mathbb{I}$:\n$L'(\\lambda)=e^{\\lambda A_1}$ where ${A_1^2}=-1$.\nThe mid-point of such a geodesic, $L'(\\pi\/2)=A_1$\nagain defines a complex structure ${A_1}={O^T}JO$,\nwhere $J$ is given by (\\ref{eqn:canonical-J}) so that\n$\\pi_{k}(\\text{O}(N\/4))=\\pi_{k-1}(\\text{O}(N\/4)\/\\text{U}(N\/8))$.\nIn a similar way, minimal geodesics in\n$\\text{O}(N\/4)\/\\text{U}(N\/8)$ from $A_1$ to $-A_1$\ncan be parametrized by their mid-points $A_2$,\nwhich square to $-1$ and anti-commute\nwith $A_1$, thereby defining a quaternionic\nstructure, so that the loop space of\n$\\text{O}(N\/4)\/\\text{U}(N\/8)$ is equivalent to\n$\\text{U}(N\/8)\/\\text{Sp}(N\/8)$ and, hence\n$\\pi_{k}(\\text{O}(N\/4)\/\\text{U}(N\/8))=\n\\pi_{k-1}(\\text{U}(N\/8)\/\\text{Sp}(N\/8))$.\nThus, we see that {\\it the passage from\none of the classifying spaces to its loop space\nis the same as the imposition of a symmetry\nsuch as time-reversal to a system classified by that space}:\nboth involve the choice of successive anticommuting\ncomplex structures.\nContinuing in this fashion (see Ref. \\onlinecite{Milnor63}),\nwe recover {\\it Bott periodicity}:\n\\begin{multline}\n\\label{eqn:Bott periodicity}\n{\\hskip -0.5 cm} {\\pi_k}(\\text{O}(16N))=\\\\\n\\pi_{k-1}(\\text{O}(16N)\/\\text{U}(8N))\n= \\pi_{k-2}(\\text{U}(8N)\/\\text{Sp}(4N)) \\\\\n= \\pi_{k-3}(\\mathbb{Z}\\times\n\\text{Sp}(4N)\/(\\text{Sp}(2N)\\times\\text{Sp}(2N)))\n= \\pi_{k-4}(\\text{Sp}(2N))\\\\\n= \\pi_{k-5}(\\text{Sp}(2N)\/\\text{U}(2N))\n= \\pi_{k-6}(\\text{U}(2N)\/\\text{O}(2N))\\\\\n = \\pi_{k-7}(\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N)))\\\\\n = \\pi_{k-8}(\\text{O}(N))\n\\end{multline}\nThe approximations made at each step require\nthat $N$ be in the {\\it stable\nlimit}, in which the desired homotopy groups are\nindependent of $N$. For instance,\n${\\pi_k}(\\text{O}(N))$ is\nindependent of $N$ for $N>k\/2$.\n\nIt is straightforward to compute $\\pi_0$ for each of these\ngroups:\n\\begin{eqnarray}\n\\label{eqn:stable-pi-0}\n{\\pi_0}(\\text{O}(N))&=&\\mathbb{Z}_2\\cr\n\\pi_{0}(\\text{O}(2N)\/\\text{U}(N))&=&\\mathbb{Z}_2\\cr\n\\pi_{0}(\\text{U}(2N)\/\\text{Sp}(N))&=&0\\cr\n\\pi_{0}(\\mathbb{Z}\\times\n\\text{Sp}(2N)\/\\text{Sp}(N)\\times\\text{Sp}(N))&=&\\mathbb{Z}\\cr\n\\pi_{0}(\\text{Sp}(N))&=&0\\cr\n\\pi_{0}(\\text{Sp}(N)\/\\text{U}(N))&=&0\\cr\n\\pi_{0}(\\text{U}(N)\/\\text{O}(N))&=&0\\cr\n\\pi_{0}(\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N)))\n &=&\\mathbb{Z}\n\\end{eqnarray}\nCombining (\\ref{eqn:stable-pi-0}) with (\\ref{eqn:Bott periodicity}),\nwe can compute any of the stable homotopy groups\nof the above $8$ classifying spaces. As discussed above,\nthe space of gapped free fermion Hamiltonians in\n$d$-dimensions in a given {\\it symmetry class}\n(determined by the number modulo $8$ of symmetries squaring to $-1$\nminus the number of those squaring to $+1$) is\nhomotopically-equivalent to one of these classifying spaces.\nThus, using (\\ref{eqn:stable-pi-0}) with (\\ref{eqn:Bott periodicity})\nto compute the stable homotopy groups of these classifying spaces\nleads to a complete classification of topological states and\ntopological defects in all dimensions and\nsymmetry classes, as we now discuss.\n\nGapped Hamiltonians with a given symmetry and dimension are classified by\n$\\pi_0$ of the corresponding classifying space in Table \\ref{tbl:classifying}. Due to Bott periodicity, the table is periodic along both directions of dimension and symmetry, so that there are 8 distinct symmetry classes.\nRyu {\\it et. al}\\cite{Ryu08} denoted these classes using the\nCartan classification of symmetric spaces, following\nthe corresponding classification of disordered systems and\nrandom matrix theory \\cite{Zirnbauer96,Altland97} which\nwas applied to the (potentially-gapless) surface states of\nthese systems. In this notation, systems with no symmetry\nare in class D, those with $T$ only are in DIII, and those\nwith $T$ and $Q$ are in AII. The other 5 symmetry classes,\nC, CI, CII, AI, and BDI arise, arise in systems which\nhave spin-rotational symmetry or a sublattice symmetry.\nThere are actually 2 more symmetry classes (denoted by A and AIII in the random matrix theory) which lie on a separate $2\\times 2$ periodic table, which is less relevant to the present work and will be discussed in the Appendix \\ref{sec:QnotT}. In Table \\ref{tbl:periodic} we have listed examples of\ntopologically-nontrivial states in physical dimensions 1,2,3\nin all 8 symmetry classes. To help with\nthe physical understanding of these symmetry classes, we have also listed the physical requirements for the realization of each symmetry class.\nIn each dimension, there are two symmetry classes\nin which the topological states are classified by integer invariants and\ntwo symmetry classes in which the different states are\ndistinguished by $\\mathbb{Z}_2$ invariants. In all the\ncases in which a real material or a well-defined physical model system is known \nwith non-trivial $\\mathbb{Z}$ or $\\mathbb{Z}_2$ invariant,\nwe have listed a typical example in the table.\nIn some of the symmetry classes, non-trivial examples\nhave not been realized yet, in which case we leave the topological classification\n$\\mathbb{Z}$ or $\\mathbb{Z}_2$ in the corresponding position in the table.\n\nIn one dimension, generic superconductors (class D) are classified by $Z_2$, of which the nontrivial example is a $p$-wave superconductor with a single Majorana zero mode on the edge. The time-reversal invariant superconductors (class DIII) are also classified by $Z_2$. The nontrivial example is a superconductor in which spin up electrons pair into a $p$-wave superconductor and spin down electron form another $p$-wave superconductor which is exactly the time-reversal of the spin-up one. Such a superconductor has two Majorana zero modes on the edge which form a Kramers pair and are topologically protected. The two integer classes are bipartite time-reversal invariant insulators with (CII) and without (BDI) spin-orbit coupling. An example of the\nBDI class is a graphene ribbon, or equivalently a carbon nanotube with a zigzag edge. \\cite{Fujita96,Nakada96}. The low-energy band structure of graphene and carbon nanotubes is well-described by a tight-binding model with nearest-neighbor hopping on a honeycomb lattice, which is bipartite. The integer-valued topological quantum number corresponds to the number of zero modes on the edge, which depends on the orientation of the nanotube. Because carbon has negligible spin-orbit coupling, to a good approximation it can be viewed as a system in the BDI class, but it can also be considered as a system in class CII when spin-orbit coupling is taken into account. In two dimensions, generic superconductors (class D) are classified by an integer, corresponding to the number of chiral Majorana edge states on the edge. The first nontrivial example was the $p+ip$ wave superconductor, shown by Read and Green\\cite{Read00} to have one chiral Majorana edge state.\nNon-trivial superconductors in symmetry class D\nare examples of {\\it topological superconductors}.\nSome topological superconductors can be consistent with spin rotation symmetry; singlet superconductors (class C) are also classified by integer, with the simplest physical example a $d+id$ wave superconductor. Similar to the 1D case, the time-reversal invariant superconductors (class DIII) are classified by $Z_2$, of which the nontrivial example is a superconductor with $p+ip$ pairing of spin-up electrons and $p-ip$ pairing of spin-down electrons.\\cite{Roy06,Qi09,Ryu08} The other symmetry class in\n2D with a $Z_2$ classification is composed of time-reversal invariant insulators (class AII), also known as quantum spin Hall insulators\\cite{Kane05A,Kane05B,Bernevig06a}. The quantum spin Hall insulator phase has been theoretically predicted\\cite{Bernevig06b} and experimentally realized\\cite{Koenig07} in HgTe quantum wells. In three dimensions, time-reversal invariant insulators (class AII) are also classified by $Z_2$. \\cite{Fu07,Moore07,Roy09} The $Z_2$ topological invariant corresponds to a topological magneto-electric response with quantized coefficient $\\theta=0,\\pi$\\cite{Qi08}. Several nontrivial topological insulators in this class have been theoretically predicted and experimentally realized, including ${\\rm Bi_{1-x}Sb_x}$ alloy\\cite{Fu07b,Hsieh08} and the family of ${\\rm Bi_2Se_3}$, ${\\rm Bi_2Te_3}$, ${\\rm Sb_2Te_3}$\\cite{Zhang09,Xia09,Chen09}. In 3D, time-reversal invariant superconductors (class DIII) are classified by an integer, corresponding to the number of massless Majorana cones on the surface.\\cite{Ryu08} A nontrivial example with topological quantum number $N=1$ turns out to be the B phase of He$^3$.\\cite{Qi09,Roy08b,Ryu08} The other classes with nontrivial topological classification in 3D are singlet time-reversal invariant superconductors (CI), classified by an integer; and bipartite time-reversal invariant insulators (CII), classified by $Z_2$. Some models have been proposed\\cite{Schnyder09} but no realistic material proposal or experimental realization has been found in these two classes. We would like to note that different physical systems can correspond to the same symmetry class. For example, bipartite superconductors are also classified by the BDI class.\n\nThe two remaining symmetry classes (unitary (A) and chiral unitary (AIII)) corresponds to systems with charge conservation symmetry but without\ntime-reversal symmetry, which forms a separate $2\\times 2$ periodic table. For the sake of completeness, we carry out\nthe preceding analysis for these two classes in Appendix \\ref{sec:QnotT}.\n\nTopological defects in these states are classified by higher homotopy groups of the classifying spaces. Following the convention of Ref. \\cite{Kitaev09}, we name the classifying spaces by $R_q,q=0,1,2,...,7$, with $R_1=O(N),~R_2=O(2N)\/U(N),~...~R_7=U(N)\/O(N),~R_0=\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N))$ in the order of Eq. (\\ref{eqn:stable-pi-0}). The symmetries in Table \\ref{tbl:periodic} can be labeled by $p=0,1,2,...,7$, so that in $d$ dimensions and $p$-th symmetry class, the classifying space is $R_{2+p-d}$. A topological defect with dimension $D$ ($D<d$) is classified by\n\\begin{eqnarray}\n\\pi_{d-D-1}\\left(R_{2+p-d}\\right)=\\pi_0\\left(R_{p-D+1}\\right)\\label{eq:defectclassification}\n\\end{eqnarray}\nwhich is determined by the zero-th homotopy groups in Eq. (\\ref{eqn:stable-pi-0}). The important conclusion we obtain from this formula is that the classification of topological defect is determined by the dimension of the defect $D$ and the symmetry class $p$, and is {\\em independent} of the spatial dimension $d$. For $p$ and $D$, we obtain the same $8\\times 8$ periodic table for the classification of topological defects. \\cite{noteaboutKane}\n\nIn the following\nwe will focus on the 3D system with no symmetry, and discuss the generic cases in Sec. \\ref{sec:discussion}.\nSince a 3D system with no symmetry is classified\nby $R_7=\\text{U}(N)\/\\text{O}(N)$, point-like defects in\nsuch a system are classified by\n${\\pi_2}(\\text{U}(N)\/\\text{O}(N))$.\nIn the stable limit,\n${\\pi_2}(\\text{U}(N)\/\\text{O}(N))=\\mathbb{Z}_2$.\nHowever, we note that, for smaller values of\n$N$ below the stable limit, the classification is a little different,\ne.g. ${\\pi_2}(\\text{U}(2)\/\\text{O}(2))=\n{\\pi_2}(\\text{U}(1)\\times{S^2})=\\mathbb{Z}$.\nThe `8-band model' in Ref. \\onlinecite{Teo10},\nwhich we discussed in the strong-coupling\nlimit in the previous section,\nis an example of this particular `small $N$'\ncase, which is why the hedgehogs\nin that model are classified by a\nwinding number $\\in\\mathbb{Z}$.\n\n\n\\section{Exchanging Particles Connected\nby Ribbons in 3D}\n\\label{sec:tethered}\n\nIn this section, we will take an O(3) non-linear\n$\\sigma$ model, i.e. one with target space\n$S^2$, as a toy model for our problem.\nIt is essentially the `8-band' model\ndiscussed in Ref. \\onlinecite{Teo10}\nand in the strong-coupling limit in Section\n\\ref{sec:strong-coupling}. As noted above,\nit is the $2N=8$ limit of the classification\nreviewed above. It will be more familiar to\nmost readers and easier to visualize\nthan the full problem which we discuss in\nthe next section.\nWe will give a heuristic explanation\nof $\\pi_1$ of the configurations of\ndefects and will make a few comments about\nwhere our toy model goes wrong, compared\nto the full problem. In the next section,\nwe will undertake a full and careful calculation\nof $\\pi_1$ of the configurations of\ndefects of a model with classifying space\n$\\text{U}(N)\/\\text{O}(N)$.\n\nA free fermion Hamitonian with no\nsymmetry, but limited to 8 bands,\ncan be expanded about its minimum\nenergy point in the Brillouin zone as\n\\begin{equation}\nH = i\\chi ({\\partial_i}{\\gamma_i} +\n{n_i}{\\Gamma_i})\\chi\n\\end{equation}\nFrom the considerations in the previous\nsection, we learned that the space of mass terms\nis ${S^2}$, which is why we have written the mass\nterm in the above form with a unit vector $\\vec{n}\\in{S^2}$.\n(More precisely, the space of mass terms is\n$\\text{U}(2)\/\\text{O}(2)=\\text{U}(1)\\times{S^2}$,\nand the mass term can be written\n$e^{\\theta{\\gamma_1}{\\gamma_2}{\\gamma_3}}{n_i}{\\Gamma_i}\ne^{-\\theta{\\gamma_1}{\\gamma_2}{\\gamma_3}}$.\nHowever, the extra U(1) plays no role here and can\nbe ignored.)\nIn the model of Section \\ref{sec:strong-coupling},\nthe three components of $\\vec{n}$ correspond to dimerization\nin each of the three directions on the cubic\nlattice. In Teo and Kane's\nmodel they correspond to the real and imaginary\nparts of the superconducting order parameter\nand the sign of the Dirac mass at a band inversion.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{north-pole.pdf}\n\\caption{The pre-image of the North pole in the $S^2$\ntarget space of $\\vec{n}$ is a collection of arcs\nconnecting hedgehogs. A fixed tangent vector at the North\npole defines a vector field along the arcs, thereby making\nthem framed arcs, which may be viewed as ribbons.}\n\\label{fig:north-pole}\n\\end{figure}\n\nNow consider defects in the $\\vec{n}$ field,\nwhich are classified by ${\\pi_2}({S^2})=\\mathbb{Z}$.\nDefects with winding number $\\pm 1$ are positive and\nnegative hedgehogs. For simplicity, we will focus on\nthese; higher winding number hedgehods can be\nbuilt up from these. (In the real model, as opposed to the\ntoy model, the `hedgehogs' have a $\\mathbb{Z}_2$\nclassification so there are no higher winding number\nhedgehogs and, in fact, they do not even have a sign.)\nAs noted by Teo and Kane \\cite{Teo10}, the\n$\\vec{n}$ field around a hedgehog can be\nvisualized in a simplified way, following Wilczek\nand Zee's discussion of the Hopf term in\na $2+1$-D O(3) non-linear $\\sigma$ model\\cite{Wilczek83}.\nThe field $\\vec{n}$ can be viewed as a map from\nthe physical space where the electrons live,\n$\\mathbb{R}^3$. If we assume\nthat the total winding number is zero\n(equal numbers of + and - hedgehogs)\nand that $\\vec{n}$ approaches a constant\nat $\\infty$, we can compactify the physical space $\\mathbb{R}^3$\nso that it is $S^3$.\nThe target space of the map is $S^2$. The pre-image\nof the north pole $N\\in S^2$ is a set of arcs\nand loops. The choice of the north pole $N$\nis arbitrary, and any other point on the sphere\nwould be just as good for the following discussion.\nLet's ignore the loops for the moment\nand focus on the arcs.\nSince $\\vec{n}$ points in every direction at\na hedgehog, the arcs terminate at hedgehogs.\nIn fact, each arc connects a $+1$ hedgehog to\na $-1$ hedgehog.\nWe now pick an arbitrary unit vector in the tangent\nspace of the sphere at $N$. This vector can be\npulled back to $S^3$ to define a vector field\nalong the arcs which is clearly normal\nto the arcs. This is a {\\it framing}, which allows us to define,\nfor instance, a self-winding number for an arc. Intuitively,\nwe can think of a framing as a thickening of\nan arc into a ribbon.\nThus, the field $\\vec{n}$ allows\nus to to define a set of framed arcs connecting\nthe hedgehogs -- in other words, a set\nof ribbons connecting the hedgehogs. As the\nnormal vector twists around an arc, the ribbon twists,\nas shown in Fig. \\ref{fig:north-pole} (Although we will draw the ribbons\nas bands in the physical space, their width should\nnot be taken seriously; they should really be viewed as\narcs with a normal vector field.)\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2in]{ribbons-_arcs.pdf}\n\\caption{We will depict framed arcs or ribbons\nas arcs with twists accounted for by drawing\nkinks in the arcs, as shown above.}\n\\label{fig:ribbons->arcs}\n\\end{figure}\n\nAlthough these ribbons are strongly reminiscent\nof particle trajectories, it is important to keep in\nmind that they are not. A collection of ribbons\nconnecting hedgehogs defines a state of the system\nat an instant of time. Ribbons, unlike particle trajectories,\ncan cross. They can break and reconnect as the system\nevolves in time. As hedgehogs are moved, the\nribbons move with them.\n\nA configuration of particles connected pairwise\nby ribbons is a seemingly crude approximation\nto the full texture defined by $\\vec{n}$.\nHowever, according to the Pontryagin-Thom\nconstruction, as we describe in the next Section\n(and explain in Appendix \\ref{sec:appendix_pontryagin_thom_construction}),\nit is just as good as the full texture for topological\npurposes. Thus, we focus on the space of\nparticles connected pairwise by ribbons.\n\n\nWe now consider a collection of such particles\nand ribbons. For a topological discussion,\nall that we are interested in about\nthe ribbons is how many times they twist, so we will\nnot draw the framing vector but will, instead, be careful to\nput kinks into the arcs in order to keep track\nof twists in the ribbon, as depicted in Fig. \\ref{fig:ribbons->arcs}.\nThe fundamental group of their configuration space is\nthe set of transformation which return the particles\nand ribbons to their initial configurations,\nwith two such transformations identified if they\ncan be continuously deformed into each other.\nConsider an exchange of two $+1$ hedgehogs, as depicted\nin Fig. \\ref{fig:exchange1}.\nAlthough this brings the particles back to their\ninitial positions (up to a permutation, which\nis equivalent to their initial configuration\nsince the particles are identical), it does not\nbring the ribbons back to their initial configuration.\nTherefore, we need to do a further motion\nof the ribbons. By cutting and rejoining them\nas shown in Fig. \\ref{fig:exchange2}a, a procedure\nwhich we call `recoupling', we now have the ribbons\nconnecting the same particles as in the initial configuration.\nBut the ribbon on the left has a twist in it. So we rotate\nthat particle by $-2\\pi$ in order to undo the twist,\nas in Fig. \\ref{fig:exchange2}b.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.5in]{exchange1.pdf}\n\\caption{When two defects are exchanged, the\n$\\vec{n}$-field around them is modified. This is encapsulated\nby the dragging of the framed arcs as the defects are\nmoved.}\n\\label{fig:exchange1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in]{exchange2.pdf}\n\\includegraphics[width=1.75in]{exchange3.pdf}\n\\caption{(a) In order to restore the framed arcs so that\nthey are connecting the same defects, it is necessary\nto perform a recoupling by which they are reconnected.\nIn order to keep track of the induced twist, it is easiest\nto perform the recoupling away from the overcrossing.\n(b) The particle on the left must be rotated by\n$-2\\pi$ in order to undo a twist in the framed arc\nto which it is attached.}\n\\label{fig:exchange2}\n\\end{figure}\n\nLet us use $t_i$ to denote such a transformation, defined by\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b.\nThe $t_i$s do not satisfy the multiplication rules of the\npermutation group. In particular, ${t_i}\\neq t_i^{-1}$.\nThe two transformations ${t_i}$ and $t_i^{-1}$\nare not distinguished by whether the exchange\nis clockwise or counter-clockwise -- this is immaterial\nsince a clockwise exchange can be\ndeformed into counter-clockwise one -- but rather\nby which ribbon is left with a twist which must be\nundone by rotating one of the particles.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{exchange4.pdf}\n\\caption{The sequence of moves which defines\n$t_i^{-1}$. (Here, the $i^{\\rm th}$ particle is\nat the top left and the $(i+1)^{\\rm th}$ is at the\ntop right.) This may be contrasted with\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b,\nwhich defines ${\\sigma_i^1}$.}\n\\label{fig:exchange4}\n\\end{figure}\n\nTo see that the operations $t_i$, defined by\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b,\nand $t_i^{-1}$, defined by the sequence in \\ref{fig:exchange4},\nare, in fact, inverses, it is useful to note that when they are performed\nsequentially, they involve two $2\\pi$ twists of the same hedgehog.\nIn \\ref{fig:exchange2}b, it is the hedgehog on the left\nwhich is twisted; this hedgehog moves to the right in the first step of\n\\ref{fig:exchange4} and is twisted again in the fourth step.\nOne should then note that a double twist\nin a ribbon can be undone continuously by using\nthe ribbon to ``lasso'' the defect, a famous fact\nrelated to the existence of spin-$1\/2$ and the\nfact that ${\\pi_1}(SO(3))=\\mathbb{Z}_2$.\nThis is depicted in Fig. \\ref{fig:lassomove} in Appendix\n\\ref{sec:Postnikov}. It will be helpful for our late\ndiscussion to keep in mind that $t_i$ not only\npermutes a pair of particles but also rotates one\nof them; any transformation built up by multiplying\n$t_i$s will enact as many $2\\pi$ twists as pairwise\npermutations modulo two.\n\nThus far we have only discussed the $+1$ hedgehogs.\nWe can perform the similar transformations which\nexchange $-1$ hedgehogs. We will not repeat the above\ndiscussion for $-1$ hedgehogs since the discussion\nwould be so similar; furthermore, in the $N\\rightarrow\\infty$\nmodel which is our main interest, defects do not carry\na sign, so they can all be permuted with each other.\n\nWe have concluded that ${t_i}\\neq t_i^{-1}$\nand, therefore, the group of transformations which bring\nthe hedgehogs and ribbons back to their initial configuration\nis not the permutation group. This leaves open the question:\nwhat is ${t_i}^2$? The answer is that $t_i^2$ can be continuously\ndeformed into a transformation which doesn't involve\nmoving any of the particles -- Teo and Kane's\n`braidless operations'. Consider the transformation\n$x_i$ depicted in Fig. \\ref{fig:twist-transfer}. Defect $i$\nis rotated by $2\\pi$, the twist is transferred\nfrom one ribbon to the other,\nand defect $i+1$ is rotated by $-2\\pi$.\nSince a $4\\pi$ rotation can be unwound, as depicted\nin Fig. \\ref{fig:lassomove}, ${x_i^2=1}$.\n\nIntuitively, one expects that ${x_i}={t_i}^2$\nsince neither ${x_i}$ nor ${t_i}^2$\npermutes the particles and both of them involve\n$2\\pi$ rotations of both particles $i$ and $i+1$.\nTo show that this is, in fact, the correct, we need\nto show that the history in Fig. \\ref{fig:twist-transfer}\ncan be deformed into the sequence of Figs.\n\\ref{fig:exchange1}, \\ref{fig:exchange2}a,\n\\ref{fig:exchange2}b repeated twice.\nIf the history in Fig. \\ref{fig:twist-transfer} is\nviewed as a `movie' and the sequence of Figs.\n\\ref{fig:exchange1}, \\ref{fig:exchange2}a,\n\\ref{fig:exchange2}b repeated twice is\nviewed as another `movie', then we need a\none-parameter family of movies -- or a `movie\nof movies' -- which connects the two movies.\nWe will give an example of such a `movie\nof movies' shortly. With this example in hand, the\nreader can verify that  ${x_i}={t_i}^2$ by drawing\nthe corresponding pictures, but we will not do so here\nsince this discussion\nis superseded, in any case, by the\nthe next section, where a similar result is shown for\nthe $N\\rightarrow \\infty$ problem by more general\nmethods. We simply accept this identity for now.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{twist-transfer.pdf}\n\\caption{The sequence of moves which defines\n$x_i$: the defect on the left is rotated by $2\\pi$,\nthe twist is transfered to the ribbon on the right by two\nrecouplings, and then the defect on the right is\nrotated by $-2\\pi$. (Here, the $i^{\\rm th}$ defect is\nat the top left and the $(i+1)^{\\rm th}$ is at the\ntop right.) The defects themselves are not moved\nin such a process.}\n\\label{fig:twist-transfer}\n\\end{figure}\n\nWe now consider the commutation\nrelation for the $x_i$s. Clearly, for $|i-j|\\geq 2$,\n${x_i} {x_j} = {x_j} {x_i}$. It is also\nintuitive to conclude that\n\\begin{equation}\n{x_i} x_{i+1} = x_{i+1} {x_i}\n\\end{equation}\nsince the order in which twists are transferred is\nseemingly unimportant.\nHowever, since this is a crucial point, we verify it\nby showing in Figure \\ref{fig:movie-of-movies}\nthat the sequence of moves which\ndefines ${x_i} x_{i+1}$ (a `movie') can be continuously\ndeformed into the sequence of moves which\ndefines $x_{i+1}{x_i}$ (another `movie'). Such a deformation\nis a `movie of movies'; going from left-to-right in\nFig. \\ref{fig:movie-of-movies} corresponds to going\nforward in time while going from up to down corresponds\nto deforming from one movie to another.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6in]{grid.pdf}\n\\caption{The sequence of moves which defines\n${x_i}x_{i+1}$ is shown in the top row.\nThe sequence of moves which defines\n$x_{i+1}{x_i}$ is shown in the bottom row.\nThe rows in between show how the\ntop row can be continuously deformed\ninto the bottom one. Such a deformation\nof two different sequences is a `movie of movies'\nor a two-parameter family of configurations.\nMoving to the right increases the time parameter while\nmoving down increases the deformation parameter\nwhich interpolates between ${x_i}x_{i+1}$ and\n$x_{i+1}{x_i}$.}\n\\label{fig:movie-of-movies}\n\\end{figure*}\n\nThus, we see that the equivalence\nclass of motions of the defects (i.e. $\\pi_1$\nof their configuration space) has an Abelian\nsubgroup generated by the $x_i$s. Since ${x_i^2}=1$\nand they all commute with each other, this is simply\n$n-1$ copies of $\\mathbb{Z}_2$,\nor, simply, $(\\mathbb{Z}_2)^{n-1}$.\n\nIn order to fully determine\nthe group of transformations which bring\nthe hedgehogs and ribbons back to their initial configuration,\nwe need to check that the $t_i$s generate the full\nset of such transformations -- i.e. that the transformations\ndescribed above and those obtained by combining them\nexhaust the full set. In order to do this, we need\nthe commutation relations of the $t_i$s with\neach other. Clearly, ${t_i}{t_j}={t_j}{t_i}$ for\n$|i-j|\\geq 2$ since distant operations which\ndo not involve the same hedgehogs nor the same\nribbons must commute. On the other hand\noperations involving the same hedgehogs or\nribbons might not commute. For instance,\n\\begin{equation}\n\\label{eqn:t-x-comm}\n{t_i} x_{i+1} = x_{i}x_{i+1}t_i\n\\end{equation}\nTo see why this is true, note that\nif we perform $x_{i+1}$ first, then defects $i+1$ and\n$i+2$ are twisted by $2\\pi$. However, $t_i$ then permutes\n$i$ and $i+1$ and twists $i$ by $2\\pi$. Thus, the left-hand-side\npermutes $i$ and $i+1$ and only twists $i+2$. The\n$(i+1)^{\\rm th}$ hedgehog was twisted by $x_{i+1}$\nand then permuted by $t_i$ so that it ended up in the\n$i^{\\rm th}$ position, where it was twisted again in\nthe last step in $t_i$; two twists can be continuously deformed\nto zero, so this hedgehog is not twisted at all.\nThe right-hand-side similarly permutes $i$ and $i+1$ and\nonly twists $i+2$ by $2\\pi$. The reader may find it instructive\nto flesh out the above reasoning by constructing\na movie of movies.\n\nThe multiplication rule which we have just described\n(but not fully justified) is that of a semi-direct product,\nwhich is completely natural in this context:\nwhen followed by a permutation, a transfer of twists\nends up acting on the permuted defects.\nThe twists $x_i$ form the group $(\\mathbb{Z}_2)^{n-1}$\nwhich we can represent by $n$-component vectors\nall of whose entries are $0$ or $1$ which satisfy the\nconstraint that the sum of the entries is even.\nThe entries tell us whether a given hedgehog is\ntwisted by $2\\pi$ or not. In any product of $x_i$s,\nan even number of hedgehogs is twisted by $2\\pi$.\nNow consider, for $n$ odd, the group elements given by\n\\begin{equation}\n\\label{sigma-i-odd}\n{\\sigma_i} = x_{n-1} x_{n-3} \\ldots x_{i+3}x_{i+1} x_{i-2}x_{i-4}\n\\ldots x_{1}\\, t_{i}\n\\end{equation}\nfor $i$ odd and\n\\begin{equation}\n\\label{sigma-i-even}\n{\\sigma_i} = x_{n-1} x_{n-3} \\ldots x_{i+2}x_{i} x_{i-1}x_{i-3}\n\\ldots x_{1}\\, t_{i}\n\\end{equation}\nfor $i$ even.\nFrom (\\ref{eqn:t-x-comm}), we see that ${{\\sigma_i}^2}=1$.\nThe group element $\\sigma_i$ permutes the\n$i^{\\rm th}$ and $(i+1)^{\\rm th}$\nhedgehogs and twists all of the hedgehogs.\nThus, the $\\sigma_i$s generate\na copy of the permutation group $S_n$. The $\\sigma_i$s do not\ncommute with the $x_i$s, however; instead they act\naccording to the semi-direct product structure noted above.\nOn the other hand, the situation is a bit different for\n$n$ even. This may be a surprise since one might expect\nthat $n$ even is the same as $n$ odd but with the last\nhedgehog held fixed far away. While this is true, exchanging the\nlast hedgehog with the others brings in an additional layer of\ncomplexity which is not present for $n$ odd. The construction\nabove, Eqs. \\ref{sigma-i-odd}, \\ref{sigma-i-even}, does not\nwork. One of the hedgehogs will be left untwisted by such\na construction; since subsequent $\\sigma_i$s will permute this\nuntwisted hedgehog with others, we must keep track of the\nuntwisted hedgehog and, therefore, the $\\sigma_i$s will\nnot generate the permutation group. \nIn the even hedgehog number case, the group of\ntranformations has a $(\\mathbb{Z}_2)^{n-1}$ subgroup,\nas in the odd case, but there isn't an $S_n$ subgroup,\nunlike in the odd case. To understand the even case,\nit is useful to note that in both cases, every transformation\neither (a) twists an even number of ribbons,\nwhich is the subgroup $(\\mathbb{Z}_2)^{n-1}$;\n(b) performs an even permutation, which is the\nsubgroup $A_n$ of $S_n$; or (c)\ntwists an odd number of ribbons and performs\nan odd permutation. Another way of saying this is\nthat the group of transformations is the `even part' of\n$(\\mathbb{Z}_2)^{n}\\rtimes S_n$: the subgroup of\n$(\\mathbb{Z}_2)^{n}\\rtimes S_n$ consisting of\nthose elements whose $(\\mathbb{Z}_2)^{n}$ parity\nadded to their $S_{n}$ parity is even. In the\nodd hedgehog number case, this is the\nsemidirect product $(\\mathbb{Z}_2)^{n-1}\\rtimes S_n$;\nin the even hedgehog number case, it is not.\nAs we will see in Section \\ref{sec:projective}, the\ndifference between the even and odd hedgehog number\ncases is related to the fact that, for an even number\nof hedgehogs, the Hilbert space decomposes into\neven and odd total fermion number parity sectors.\nBy contrast, the situation is simpler for an\nodd number of hedgehogs, where the parity\nof the total fermion number is not well-defined and\nthe representation is irreducible.\n\n\nTo summarize, we have given some plausible\nheuristic arguments that the `statistics' of $+1$ hedgehogs\nin a model of $2N=8$ Majorana fermions\nis governed by a group\n$E((\\mathbb{Z}_2)^{n}\\rtimes S_{n})$,\nthe `even part' of $(\\mathbb{Z}_2)^{n}\\rtimes S_{n}$:\nthose elements of $(\\mathbb{Z}_2)^{n}\\rtimes S_{n}$\nin which the parity of the sum of the entries of the element\nin $(\\mathbb{Z}_2)^{n}$ added to the parity of the\npermutation in $S_{n}$ is even.\n(The same group governs the $-1$\nhedgehogs). Rather than devoting more time\nhere to precisely determining the group for\nthe toy model, we will move on to the problem\nwhich is our main concern here,\na system of $2N\\rightarrow\\infty$ Majorana fermions.\nThis problem is similar, with some important differences.\n(1) The target space is no longer $S^2$ but is, instead,\n$U(N)\/O(N)$. (2) Consequently, the defects do not carry a sign. There\nis no preferred pairing into $\\pm$ pairs; the defects\nare all on equal footing. All $2n$ of them can be exchanged.\n(3) The group obtained by computing $\\pi_1$\nof the space of configurations of $2n$\ndefects then becomes the direct product\nof the `ribbon permutation group'\n${\\cal T}^r_{2n}$ with a trivial ${\\mathbb Z}$,\n${\\cal T}_{2n}={\\mathbb Z} \\times {\\cal T}^r_{2n}$.\nThe ribbon permutation group ${\\cal T}^r_{2n}$ is given by\n${\\cal T}^r_{2n} \\equiv {\\mathbb Z}_2 \\times E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$, where $E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$\nis the `even part' of $(\\mathbb{Z}_2)^{2n}\\rtimes S_{2n}$.\n\n\n\n\\section{Fundamental Group of the Multi-Defect Configuration Space}\n\\label{sec:Kane_space}\n\nIn Section \\ref{sec:free-fermion} we concluded that\nthe effective target space for the order parameter\nof a system of fermions in 3D with no symmetries is\n$U(N)\/O(N)$ -- which, as is conventional, we will simply call $U\/O$,\ndropping the $N$ in the large-$N$ limit. This enables\nus to rigorously define the space of topological configurations,\n$K_{2n}$, of $2n$ hedgehogs in a ball, and calculate\nits fundamental group $\\pi_1(K_{2n})$, thereby elucidating\nTeo and Kane's \\cite{Teo10} hedgehog motions and\nunitary transformations.\n\nWe now outline the steps involved in this calculation:\n\n\\begin{itemize}\n\n\\item We approximate the space $U\/O$ by a {\\it cell complex}\n(or CW complex), ${\\cal C}$, a topological space constructed by\ntaking the union of disks of different dimensions and\nspecifying how the boundary of each higher-dimensional\ndisk is identified with a subset of the lower-dimensional disks.\nThis is a rather crude approximation in some respects,\nbut it is sufficient for a homotopy computation.\n\n\\item We divide the problem into (a) the motion of the hedgehogs\nand (b) the resulting deformation of the field configuration\nbetween the hedgehogs. This is accomplished by expressing\nthe configuration space in the following way. Let us call the\nconfiguration space of $2n$ distinct points\nin three dimensions $X_{2n}$. (For the sake of mathematical convenience,\nwe will take our physical system\nto be a ball $B^3$ and stipulate that the points must lie\ninside a ball $B^3$. Let's denote the space of field configurations\nby ${\\cal M}_{2n}$. This space is the space of maps to $U\/O$\nfrom $B^3$ with $2n$ points (at some standard locations) excised.\nThe latter space is denoted by\n${B^3}\\,\\backslash\\, 2n\\mbox{ standard points}$.\nSince we will be approximating $U\/O$ by ${\\cal C}$,\nwe can take ${\\cal M}_{2n}$ to be the space of\nmaps from ${B^3}\\,\\backslash\\, 2n\\mbox{ standard points}$\nto ${\\cal C}$ with boundary conditions at the $2n$ points\nspecified below. Then, there is a {\\it fibration} of spaces:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{M}_{2n}$}\n    \\obj(2,2)[b]{${K}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\item We introduce another two fibrations which further\ndivide the problem into more manageable pieces:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{${R}_{2n}$}\n    \\obj(2,2)[b]{${K}_{2n}$}\n    \\obj(2,1)[c]{$Y_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{N}_{2n}$}\n    \\obj(2,2)[b]{${Y}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\nThe original fibration is kind of a \"fiber-product\" of the two new fibrations.\nHere, $R_{2n}$ is essentially the space of order parameter textures\ninterpolating between the hedgehogs, and $Y_{2n}$ is the space of\nconfigurations of $2n$ points with infinitesimal spheres\nsurrounding each point and maps from each of these\nspheres to ${\\cal C}$. ${\\cal N}_{2n}$ is the space\nof maps from $2n$ infinitesimal spheres to ${\\cal C}$,\nwith each one of the spheres surrounding a different\none of the $2n$ points (at some standard locations)\nexcised from $B^3$. We call these order parameter\nmaps from infinitesimal spheres to ${\\cal C}$\n``germs''.\n\n\\item Having broken the problem down into smaller\npieces by introducing these fibrations, we use the\nfact that a fibration\n$F \\rightarrow E\\rightarrow B$\ninduces a long exact sequence for homotopy groups\n\\begin{equation*}\n\\ldots \\rightarrow \\pi_{i}(E)\\rightarrow\n{\\pi_i}(B)\\rightarrow\\pi_{i-1}(F)\\rightarrow\\pi_{i-1}(E)\n\\rightarrow ...\n\\end{equation*}\nFor instance, applying this to the\nfibration ${\\cal M}_{2n}\n\\rightarrow K_{2n} \\rightarrow\nX_{2n}$ leads to the exact sequence\n$\\ldots \\rightarrow \\pi_1({\\cal M}_{2n})\n\\rightarrow \\pi_1(K_{2n}) \\rightarrow\n\\pi_1(X_{2n})\\rightarrow 1$.  It follows that $\\pi_1(K_{2n})$ is an extension of the permutation\ngroup $S_{2n}=\\pi_1(X_{2n})$.\nBy itself, the above long exact sequence is not very helpful\nfor computing any of the homotopy groups involved\nunless we can show by independent means that\ntwo of the homotopy groups are trivial. Then the homotopy\ngroups which lie between the trivial ones in the sequence\nare tightly constrained.\n\n\\item We directly compute that\n${\\pi_1}({\\cal N}_{2n})=(\\mathbb{Z}_2)^{2n}$\nand ${\\pi_1}(X_{2n})=S_{2n}$. We show that\nthe homotopy exact sequence then implies that\n${\\pi_1}(Y_{2n})=(\\mathbb{Z}_2)^{2n}\\rtimes S_{2n}$.\n\n\\item We compute the homotopy groups of $R_{2n}$,\ndefined by the fibration $R_{2n}\\rightarrow\nK_{2n} \\rightarrow Y_{2n}$.\nThis computation involves a different\nway from the cell structure of thinking about the topology\nof a space, called the ``Postnikov tower'', explained in\ndetail Appendix \\ref{sec:Postnikov}. The basic idea is to approximate\na space with spaces with only a few non-trivial homotopy groups.\n(This is analogous to the cell structure, which has only a few\nnon-trivial homology groups.) The simplest examples of such spaces\nare Eilenberg-Mac Lane spaces,\nwhich only have a single non-trivial homotopy group.\nThe Eilenberg-Mac Lane space $K(A,m)$ is defined for a\ngroup $A$ and integer $m$ as the space with homotopy group\n${\\pi_m}(K(A,m))=A$ and ${\\pi_k}(K(A,m))=0$ for all $k\\neq m$.\n(The group $A$ must be Abelian for $m>1$.)\nSuch a space exists and is unique up to homotopy.\nA space $T$ with only two non-trivial homotopy groups can\nbe constructed through the fibration\n$K(B,n) \\rightarrow T \\rightarrow K(A,m)$. The space $T$ has ${\\pi_m}(T)=A$\nand ${\\pi_n}(T)=B$, as may be seen from the\ncorresponding long exact sequence for homotopy groups.\nContinuing in this fashion, one\ncan construct a sequence of such approximations\n$M_n$ to a space $M$. They are defined by\n${\\pi_k}({M_n})={\\pi_k}(M)$ for $k\\leq n$ and\n${\\pi_k}({M_n})=0$ for $k>n$. They can be constructed\niteratively from the fibration\n$K(A,n) \\rightarrow {M_n} \\rightarrow M_{n-1}$, where ${\\pi_n}(M)=A$.\n\n\\item With ${\\pi_1}(Y_{2n})$, ${\\pi_2}(Y_{2n})$,\n${\\pi_0}(R_{2n})$ and  ${\\pi_1}(R_{2n})$ in hand,\nwe compute the desired group ${\\pi_1}(K_{2n})$\nfrom the homotopy exact sequence.\n\n\\end{itemize}\n\n\nWe now go through these steps in detail.\n\\vskip 0.25 cm\n\n{\\bf Approximating U\/O by a cell complex}.\nDepending on microscopic details,\ngradients in the overall phase of the\nfermions may be so costly that we wish\nto consider only configurations in which this\noverall phase is fixed. We will refer to this\nas the scenario in which `phase symmetry is broken'.\nIn this case, the effective target space is $SU\/SO$,\nthe non-phase factor of $U\/O \\cong U(1)\/O(1) \\times SU\/SO$.\nIn this case, we simplify matters by replacing $SU\/SO$\nby $\\widetilde{U\/O}$, the universal cover of $U\/O$.\n$\\widetilde{U\/O}$ is homotopy equivalent to $SU\/SO$,\nso this substitution is harmless.  This substitution\nresults in a reduced configuration space $\\widetilde{K}_{2n}$\nand we will concentrate first on calculating $\\pi_1(\\widetilde{K}_{2n})$.\nIn an appendix, we show that this\nreduction essentially makes no difference:\n$\\pi_1(K_{2n}) = \\pi_1(\\widetilde{K}_{2n})\\times {\\mathbb Z}$.\n\nWe now define a cell complex ${\\cal C}$\napproximating $\\widetilde{U\/O}$.\nIn constructing this cell structure, we are not interested\nin the beautiful homogeneous nature of $\\widetilde{U\/O}$\nbut rather only its homotopy type.  The homotopy type\nof a space tells you everything you will need to know\nto study {\\em deformation classes} of maps either\ninto or out of that space.  An important feature of any\nhomotopy type is the list of homotopy groups\n(but these are by no means a complete characterization\nin general). For $\\widetilde{U\/O}$, the\nhomotopy groups are $\\pi_i(\\widetilde{U\/O}) =\n0, {\\mathbb Z}_2, {\\mathbb Z}_2, 0, {\\mathbb Z}, 0, 0, 0, {\\mathbb Z}$ for $i = 1, \\dots, 9$ and\nthereafter $\\pi_i(\\widetilde{U\/O})$ cycles through the last eight groups.\n(For $U\/O$, the first group would be ${\\mathbb Z}$.)\n\nBecause $\\widetilde{U\/O}$ is simply-connected,\nbut has nontrivial $\\pi_2$, it natural in building a\ncellular model for its homotopy type to begin with $S^2$.\nSince $\\pi_2 (S^2) = {\\mathbb Z}$ and we only need\na ${\\mathbb Z}_2$ for $\\pi_2(\\til{U\/O})$, we should kill off the even\nelements by attaching a 3-cell $D^3$ using a\ndegree-2 map of its boundary 2-sphere to the original $S^2$.\nFor future reference, take this map to be\n$(\\theta, \\phi) \\to (2\\theta, \\phi)$ in a polar coordinate\nsystem where the north pole $N = (\\pi,0)$.\nSimilarly, a 4-cell is attached to achieve\n$\\pi_3(\\til{U\/O}) \\cong {\\mathbb Z}_2$.  The necessity of the\n4-cell is proved (Fact 1) below.\n\nThe preceding logic leads us to the cell structure:\n\\begin{equation}\n\\label{eq:U\/O_cell_structure}\n{\\cal C} = S^2 \\bigcup_{\\text{degree}=2} D^3 \\bigcup_{\\text{2Hopf}} D^4 \\bigcup \\text{cells of dimension} \\geq 5\n\\end{equation}\n\nSince we are only trying to compute the fundamental group $\\pi_1(\\widetilde{K}_{2n})$ from our various homotopy long exact sequences, we do not have to figure out the higher cells (dimension $\\geq 5$) of $\\til{U\/O}$.  We will, however, verify that $\\pi_3(\\til{U\/O})$ is generated by the Hopf map into the base $S^2 \\subset \\til{U\/O}$.\n\nTo summarize, we will henceforth assume that the order\nparameter takes values in the cell complex ${\\cal C}$.\nAlthough ${\\cal C}$ is a crude approximation for\nU\/O, it is good enough for the topological calculations\nwhich follow.\n\n{\\bf Dividing the problem into the motion of the hedgehog\ncenters and the deformation of the field configuration.}\nLet us assume that our physical system is a ball\nof material $B^3$. Let $n \\geq 0$ be the number of\nhedgehog pairs in the system.\nA configuration in $\\widetilde{K}_{2n}$ is a texture in the\norder parameter, $\\Phi(x): {B^3}\\rightarrow {\\cal C}$,\nwhich satisfies the following\nboundary conditions at the boundary of $B^3$\nand at the $2n$ hedgehog locations (which\nare singularities in the order parameter).\nThe order parameter has winding number $0$\nat the boundary of the ball, $\\partial B^3$\nand winding number $1$ around each of the\nhedgehog centers. (Recall that ${\\pi_2}({\\cal C})=\\mathbb{Z}_2$,\nso the winding number can only be $0$ or $1$).\n\nFrom its definition, $\\widetilde{K}_{2n}$ is the total space of a fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{M}_{2n}$}\n    \\obj(2,2)[b]{$\\widetilde{K}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent The above diagram suggests that we\nshould think of the fibration\n$\\mathcal{M}_{2n}\n\\rightarrow \\widetilde{K}_{2n} \\rightarrow X_{2n}$\nin the following way: above each point in\n$X_{2n}$ there is a fiber $\\mathcal{M}_{2n}$;\nthe total space formed thereby is $\\widetilde{K}_{2n}$.\n(This is not quite a fiber bundle, since we\ndo not require that there be local coordinate\ncharts in which $\\widetilde{K}_{2n}$ is simply the\ndirect product.) Here, $X_{2n}$ is the simply the\nconfiguration space of $2n$ distinct points in $B^3$.\nWe write this formally as\n$X_{2n} = \\prod_{i=1}^{2n} B^3 \\setminus \\text{big diagonal}$.\n(The big diagonal consists of $2n$-tuples of points in $B^3$\nwhere at least two entries are identical.)\nThe space $\\mathcal{M}_{2n}$ consists of maps\nfrom $B^3 \\setminus 2n \\text{ points in a fixed standard position}$\nto ${\\cal C}$ with the prescribed winding numbers given\nin the preceding paragraph.\n\n\n\\begin{figure}[htpb]\n\\centering\n\\includegraphics[scale=0.5]{M2n.pdf}\n\\caption{$B^3 \\setminus 2n$ points in standard position.\nThe space $\\mathcal{M}_{2n}$ consists of maps from\nthis manifold to ${\\cal C}$.}\n\\label{fig:M2n}\n\\end{figure}\n\n{\\bf Germs of order parameter textures.}\nIt is helpful to introduce an intermediate step in the fibration.\nDefine a point in $Y_{2n}$ as a configuration in\n$X_{2n}$ together with a ``germ''\nof $\\Phi(x)$, which we call $\\widetilde{\\Phi}(x)$,\ndefined only near $\\partial B^3$\nand the $2n$ points. The idea behind the germ\n$\\widetilde{\\Phi}(x)$ is to forget about the order parameter\n$\\Phi(x)$ except for its behavior in an infinitesimal\nneighborhood around each hedgehog center\nand at the boundary of the system.\n$\\widetilde{\\Phi}(x)$ must satisfy the same boundary conditions as\n$\\Phi(x)$ itself. We take $\\widetilde{\\Phi}(x)$ to be\nconstant on $\\partial B^3$ and to have\nwinding number $1$ around each of the\nhedgehog centers. With this definition,\nwe now have the fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{N}_{2n}$}\n    \\obj(2,2)[b]{$Y_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent where $\\mathcal{N}_{2n}$ is the space of\n(germs of) order parameter textures $\\widetilde{\\Phi}$ from the\nneighborhoods of the $2n$ fixed standard points\nand $\\partial B^3$ to ${\\cal C}$.\nWe will henceforth replace discussion of germs\nwith the equivalent and simpler concept of maps\non $\\partial B^3 \\cup \\left( \\bigcup_{i=1}^{2n} S_i^2 \\right)$\nwhere $S_i^2$ is a small sphere surrounding\nthe $i$th standard point.  Thus,\n\\begin{equation}\n\\mathcal{N}_{2n} \\subset\n\\text{Maps}\\biggl(\\Bigl(\\partial B^3 \\cup \\bigcup_{i=1}^{2n} S_i^2\n\\Bigr) \\to {\\cal C} \\biggr).\n \\end{equation}\n\nWe now define $Q_{2n}$ as the ball $B^3$\nwith small balls (denoted below by interior$(S_i^2)$)\ncentered about the hedgehogs\ndeleted:\n\\begin{equation}\nQ_{2n}= \\Bigl(B^3 \\setminus \\bigcup_{i=1}^{2n} \\text{interior}(S_i^2)\n\\Bigr)\n\\end{equation}\nfor fixed standard positions $i = 1, \\dots , 2n$.\nThen $R_{2n}$ is the space of order parameter textures on\n$Q_{2n}$ which satisfy the boundary condition\nthat the winding number is $0$ on $\\partial B^3$\nand $1$ on each of the small spheres.\nWith this definition, we have the fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$R_{2n}$}\n    \\obj(2,2)[b]{$\\widetilde{K}_{2n}$}\n    \\obj(2,1)[c]{$Y_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n    \\label{eqn:R_2n-def}\n\\enddc\\]\n\n\\noindent Given the cell structure ${\\cal C}$, we can\nspecify the boundary conditions for the order\nparameter precisely. On $\\partial B^3$,\nthe order parameter is equal to the North Pole\nin $S^2$. (Recall that ${S^2}\\subset{\\cal C}$ is\nthe bottom cell of the structure ${\\cal C}$ which\nwe are using to approximate U\/O.)  On each of\nthe spheres $S_i^2$, the order parameter\n$\\Phi(x)$ defines a map from ${S_i^2}\\rightarrow S^2$\nwhich is the identity map\n(where, again $S^2$ is understood as a subset\nof the order parameter space ${S^2}\\subset{\\cal C}$).\nThis ensures that the order parameter has the\ncorrect winding numbers at the boundaries of $Q_{2n}$.\nIn essence, what we have done in writing\nEq. \\ref{eqn:R_2n-def} is to break up an\norder parameter texture containing hedgehogs\ninto (a) the hedgehogs together with the order parameter\non infinitesimal neighborhoods around them (i.e. `germs')\nand (b) order parameter textures in the intervening regions\nbetween the hedgehogs. The space of configurations (a)\nis $Y_{2n}$; the space of configurations (b) is $R_{2n}$.\n\nThe name $R_{2n}$ is for ``ribbons.''\nAs we saw in Section \\ref{sec:tethered},\nif the order parameter manifold were $S^2$,\nwe could summarize an order parameter texture\nby looking at the inverse image of the North Pole\n$N \\subset S^2$ and a fixed tangent vector at the North Pole.\nThe inverse images form a collection of ribbons.\nNow, the order parameter manifold is actually\n(approximated by) ${\\cal C}$, but the bottom cell\nin ${\\cal C}$ is $S^2$. The effect of the 3-cell\nis that hedgehogs lose their sign, so  there is no\nwell-defined ``arrow'' running lengthwise along the ribbons.\nThe 4-cell allows the ``twist'' or framings of ribbons to be\naltered at will by $\\pm 2$.\n\n\n{\\bf Long exact sequence for homotopy groups.}\nIt is very convenient to use fibrations to calculate homotopy groups.\n(For those interested in $K$-theory, the last two chapters of\nMilnor's {\\em Morse Theory} \\cite{Milnor63}\nare a must read and exhibit these methods with clarity.) As noted above,\nfibrations have all the homotopy properties of fiber bundles\nbut are (often) found arising between function spaces\nwhere it would be a lot of work -- and probably a distraction from important business -- to attempt to verify the existence of\nlocally trivial coordinate charts. Operationally, fibrations share with fiber bundles the all-important ``homotopy long exact sequence'':\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$F$}\n    \\obj(2,2)[b]{$E$}\n    \\obj(2,1)[c]{$B$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent we have:\n$$\\cdots \\to \\pi_{i+1}(B) \\to \\pi_i(F) \\to \\pi_i(E) \\to \\pi_i(B) \\to \\pi_{i-1}(F) \\to \\cdots$$\n\nWe now compute $\\pi_1(Y_{2n})$ from the exact sequence:\n  \n  \n  \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n\\begin{equation}\n\\label{eq:pi_1(Y_2n)_computation}\n\\begindc{0}[3]\n    \\obj(10,70)[a]{$\\pi_2(X_{2n})$}\n    \\obj(28,70)[b]{$\\pi_1(\\mathcal{N}_{2n})$}\n    \\obj(45,70)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(62,70)[d]{$\\pi_1(X_{2n})$}\n    \\obj(80,70)[e]{$\\pi_0(\\mathcal{N}_{2n})$}\n    \\mor{a}{b}{$\\partial$}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{}\n    \\mor{d}{e}{$\\partial$}\n\\enddc\n\\end{equation}\n\n\\noindent\nWe can compute two of the homotopy\ngroups in this equation by inspection.\n$\\pi_1(X_{2n})$ is clearly the symmetric group of point exchange:\n\\begin{equation}\n\\label{eqn:pi_1_X}\n\\pi_1(X_{2n}) = S_{2n} .\n\\end{equation}\nMeanwhile, $\\pi_1(\\mathcal{N}_{2n})$ amounts to (products of)\nloops of maps from the $S_i^2$ to ${\\cal C}$\nand reduces to $2n$ copies of the third homotopy group\nof ${\\cal C}$ (and, therefore, to ${\\pi_3}(\\til{U\/O})$).\nThus, $\\pi_1(\\mathcal{N}_{2n})=(\\mathbb{Z}_2)^{2n}$:\n\\begin{eqnarray}\n\\label{eqn:pi_1_N}\n\\pi_1(\\mathcal{N}_{2n}) = \\prod_{i=1}^{2n} \\pi_1(\\text{Maps}(S_i^2,\\til{U\/O})) &=& \\prod_{2n\\text{ copies}}\\pi_3(\\til{U\/O})\\cr &=&\n({\\mathbb Z}_2)^{2n} .\n\\end{eqnarray}\n\nTo proceed further, we need to evaluate {\\em boundary maps} in the homotopy exact sequence. In Appendix \\ref{sec:hopf-map},\nwe explain boundary maps through the example of the Hopf map.\nConsider Eq. \\ref{eq:pi_1(Y_2n)_computation}.\n$\\pi_2(X_{2n})$ is generated by the $2n\\choose{2}$\ndifferent 2-parameter motions in which a pair of hedgehogs\ncome close together and explore the 2-sphere of possible relative positions around their center of mass.  This 2-parameter family of motions involves no ``rotation'' of the maps $\\widetilde\\Phi$\nwhich describe $\\pi_1(\\mathcal{N}_{2n})$\n(i.e the order parameter configuration in the neighborhood\nof each hedgehog does not rotate as the hedgehogs are moved),\nso the left most $\\partial$ map in Eq. \\ref{eq:pi_1(Y_2n)_computation}\nis zero.  Similarly, a simple exchange of hedgehogs produces no twist\nof the order parameter configuration in the neighborhood\nof either hedgehog, so the second $\\partial$\nmap of Eq. \\ref{eq:pi_1(Y_2n)_computation} is also zero.\nThus, we have a short exact sequence:\n\\[\\begindc{0}[3]\n    \\obj(10,30)[a]{$1$}\n    \\obj(23,30)[b]{${\\mathbb Z}_2^{2n}$}\n    \\obj(41,30)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(60,30)[d]{$S_{2n}$}\n    \\obj(75,30)[e]{$1$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{$\\alpha$}\n    \\mor{c}{d}{$\\beta$}\n    \\mor{d}{e}{}\n\\enddc\\]\nTo derive this short exact sequence, we used\nthe triviality of the boundary maps noted above\nand Eqs. \\ref{eqn:pi_1_X}, \\ref{eqn:pi_1_N}\nto simplify Eq. \\ref{eq:pi_1(Y_2n)_computation}.\n\nThere is a natural group homomorphism $s$:\n$$s:S_{2n}\\rightarrow\\pi_1(Y_{2n})$$\nwhich associates to each permutation a motion\nof hedgehogs which permutes the hedgehogs in $Y_{2n}$\nbut does not rotate the order parameter configurations\n$\\widetilde\\Phi$ near the hedgehogs. Then\n$\\beta \\circ s = id_{S_{2n}}$. In other words,\nthe sequence is {\\em split}:\n\\[\\begindc{0}[3]\n    \\obj(10,30)[a]{$1$}\n    \\obj(25,30)[b]{${\\mathbb Z}_2^{2n}$}\n    \\obj(40,30)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(60,30)[d]{$S_{2n}$}\n    \\obj(75,30)[e]{$1$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{$\\alpha$}\n    \\mor{d}{e}{}\n    \\cmor((46,31)(51,33)(57,31)) \\pright(51,35){$\\beta$}\n    \\cmor((57,29)(51,27)(46,29)) \\pleft(51,25){$s$}[\\atleft,\\dasharrow]\n\\enddc\\]\nThus, $\\pi_1(Y_{2n})$ is a semi-direct product.\nTo determine $\\pi_1(Y_{2n})$ completely, it only remains to\nidentify how $s(S_{2n})$ acts on the twist factors ${\\mathbb Z}_2^{2n}$\nunder conjugation. It is quite clear that this action is the only\nnatural one available: $s(p)$ acts on ${\\mathbb Z}_2^{2n}$ by applying\nthe permutation $p$ to the $2n$ coordinates of $Z_2^{2n}$.\nSo, $\\pi_1(Y_{2n}) \\cong {\\mathbb Z}_2^{2n} \\rtimes S_{2n}$ with group law:\n\n\\begin{equation}\\label{eq:pi_1(Y2n)_group_multiplication}\n(v,p) \\circ (v',p') = (v + p(v'), p \\circ p')\n\\end{equation}\n\n\\noindent where $v \\in {\\mathbb Z}_2^{2n}$ is a ${\\mathbb Z}_2$-vector, $p \\in S_{2n}$ a permutation, and $p(v')$ the natural action of $S_{2n}$ on ${\\mathbb Z}_2^{2n}$ applied to $v'$. Note that this is precisely the multiplication\nrule which we obtained pictorially in Section \\ref{sec:tethered}.\n\n{\\bf Computing the homotopy groups of $R_{2n}$,\nthe space of order parameter textures interpolating\nbetween the hedgehogs}.\nOf course, computing $\\pi_1(Y_{2n})$ only gets us part\nof the way home. Our ultimate goal is to compute\n$\\pi_1({\\widetilde K}_{2n})$. Thus, we now turn to\nthe homotopy long exact sequence:\n$$\\pi_2(Y_{2n}) \\overset{\\partial_2}{\\longrightarrow} \\pi_1(R_{2n}) \\rightarrow \\pi_1(\\widetilde{K}_{2n}) \\rightarrow \\pi_1(Y_{2n}) \\overset{\\partial_1}{\\longrightarrow} \\pi_0(R_{2n})$$\nFirst consider $\\partial_1$.  The kernel of $\\partial_1$ is represented by loops in $Y_{2n}$ which extend to loops in $R_{2n}$.\nA loop $\\gamma$ in $Y_{2n}$ is a motion of the hedgehogs\ntogether with rotations (about the spheres $S_i^2$)\nof $\\widetilde\\Phi$ which brings the system back to\nits initial configuration. If a loop $\\gamma$ is in\nkernel of $\\partial_1$, then\nthere is a corresponding loop in the configuration space of\nribbons in $B^3$ (obtained by lifting $\\gamma$ to\n$\\widetilde{K}_{2n}$).\n\nThe next steps are to compute ${\\pi_0}(R_{2n})$\nand ${\\pi_1}(R_{2n})$. These computations are\ndetailed in Appendix \\ref{sec:Postnikov}, where we see\nthat using the cell structure\n${\\cal C}$ which we introduced for $\\til{U\/O}$ is\ntricky as a result of the higher cells.\nThus, we instead introduce the\n``Postnikov tower'' for $\\til{U\/O}$ which allows\nus to make all the calculations we need.\nWe find that $\\pi_0(R_{2n})=\\mathbb{Z}_2$\nand ${\\pi_1}(R_{2n})=(\\mathbb{Z}_2)^{2n}$.\n\nThus, $\\pi_1(\\widetilde{K}_{2n})$ sits in the following exact sequence:\n\\[\\begindc{0}[3]\n    \\obj(10,20)[a]{$\\pi_2(Y_{2n})$}\n    \\obj(28,20)[b]{$\\pi_1(R_{2n})$}\n    \\obj(45,20)[c]{$\\pi_1(\\widetilde{K}_{2n})$}\n    \\obj(62,20)[d]{$\\pi_1(Y_{2n})$}\n    \\obj(81,20)[e]{$\\pi_0(R_{2n})$}\n    \\obj(28,15)[b1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(62,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(81,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(28,10)[b2]{$(\\mathbb{Z}_2)^{2n}$}\n    \\obj(62,10)[d2]{$({\\mathbb Z}_2)^{2n} \\rtimes S_{2n}$}\n    \\obj(81,10)[e2]{${\\mathbb Z}_2$}\n    \\mor{a}{b}{$\\partial_2$}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{}\n    \\mor{d}{e}{$\\partial_1$}\n\n\\enddc\\]\nRecall that when we studied $\\pi_2(Y_{2n})$, we found\n(exactly as in the case of $\\pi_2(X_{2n})$) that there are\n$2n\\choose{2}$ generators corresponding to relative\n2-parameter motions of any pair of hedgehogs\naround their center of mass.  This can be\nused to understand the map\n$\\partial_2: \\pi_2(Y_{2n}) \\to \\pi_2(R_{2n})$.\nThe image of any center of mass 2-motion is the ``bag''\ncontaining the corresponding pair of hedgehogs.\nThus, $\\text{coker}(\\partial_2) \\cong {\\mathbb Z}_2$; it is\n${\\mathbb Z}_2^{2n}$ modulo the even sublattice\n(vectors whose coordinate sum is zero in ${\\mathbb Z}_2$).\nThus, we have the short exact sequence:\n\\begin{equation}\\label{eq:extension_problem}\n\\begindc{0}[3]\n    \\obj(10,20)[a]{$1$}\n    \\obj(27,20)[b]{$\\text{coker}(\\partial_2)$}\n    \\obj(44,20)[c]{$\\pi_1(\\widetilde{K}_{2n})$}\n    \\obj(61,20)[d]{$\\text{ker}(\\partial_1)$}\n    \\obj(78,20)[e]{$1$}\n    \\obj(27,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(27,10)[b2]{${\\mathbb Z}_2$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{$\\pi$}\n    \\mor{d}{e}{}\n\\enddc\n\\end{equation}\n\nThe kernel $\\text{ker}(\\partial_1)$ consists of the part of $\\pi_1(Y)$ associated with even ($2\\pi$) twisting.  As shown in Figure \\ref{fig:exchange2}, a simple exchange is associated to a total twisting of ribbons by $\\pm 2\\pi$.\n\n\nThus, $\\partial_1(v,p) = \\sum_{i=1}^{2n} v_i + \\text{parity}(p) \\in {\\mathbb Z}_2$.  We use the notation $E(\\mathbb{Z}_2^m \\rtimes {S_m})$ for $\\text{ker}(\\partial_1)$.\n\n{\\em Note:} If $m=2$, ${\\mathbb Z}_2^{m} \\rtimes S_{m}$ is the dihedral group $D_4$ and its ``even'' subgroup $\\text{ker}(\\partial_1) \\cong {\\mathbb Z}_4$.  This shows that for $m$ even, the induced short exact sequence does not split, and the extension is more complicated:\n\n\\begin{equation*}\n1 \\longrightarrow {\\mathbb Z}_2^{2n-1} \\longrightarrow E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n}) \\longrightarrow S_{2n}\\rightarrow 1\n\\end{equation*}\n\nThere is a final step required to solve the extension problem \\ref{eq:extension_problem} and finish the calculation of $\\pi_1(\\widetilde{K}_{2n})$.  We geometrically construct a homomorphism $s:\\text{ker}(\\partial_1) = E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n}) \\to \\pi_1(\\widetilde{K}_{2n})$ which is a left inverse to the projection.\n\nThis will show that $\\pi_1(\\widetilde{K}_{2n})$ is a semidirect product ${\\mathbb Z}_2 \\rtimes \\text{ker}(\\partial_1)$, but since ${\\mathbb Z}_2$ has no nontrivial automorphism, the semidirect product is actually direct:\n\n\\begin{equation}\\label{eq:pi_1(K_2n)_direct_product}\n\\pi_1(\\widetilde{K}_{2n}) \\cong {\\mathbb Z}_2 \\times E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n})\\end{equation}\n\nTo construct $s$, note that all elements of $\\text{ker}(\\partial_1)$ can be realized by a loop $\\gamma$ of maps into the bottom 2-cell of Eq. \\ref{eq:U\/O_cell_structure} $S^2$.  Still confining the order parameter (map) to lie in $S^2$, such a loop lifts to an arc $\\widetilde{\\gamma}$ of ribbons representing an arc in $\\widetilde{K}_{2n}$.  We may choose the lift so that as the ribbons move, they never ``pass behind'' the $2n$ hedgehogs.  (For example, we may place the hedgehogs on the sphere of radius $=\\frac{1}{2}$ inside the 3-ball $B^3$ (assumed to have radius $=1$) and then keep all ribbons inside $B_{\\frac{1}{2}}^3$.  These arcs may be surgered (still as preimages of $N \\subset S^2$) so that they return to their original position except for a possible accumulation of normal twisting $t2\\pi$.  Since $\\gamma \\in \\text{ker}(\\partial_1)$, $t$ must be even.  Now, allowing the order parameter (map) to leave $S^2$ and pass over the 4-cell (of Eq. \\ref{eq:U\/O_cell_structure}), attached by $2\\text{Hopf}:S^3 \\to S^2$, we may remove these even twists.  (The 4-cell can introduce small closed ribbons with self-linking $=2$ in a small ball.  These small ribbons can be surgered into other ribbons.)  This lifts a generating set of $\\text{ker}(\\partial_1)$ into $\\pi_1(\\widetilde{K}_{2n})$ as a set theoretic cross section (left inverse to $\\pi$). But what about relations?  Because the entire loop is constant outside $B_{\\frac{1}{2}}^3$, the corresponding homology class in $H_2(Q \\times I;{\\mathbb Z}_2) \\cong \\pi_1(R)$ is trivial, so $s$ is actually a group homomorphism.\n\n\n\n\\section{Representation Theory of the Ribbon Permutation Group}\n\\label{sec:projective}\n\nIn this section, we discuss the mathematics of the group\n${\\cal T}_{m}$ and its representation. The\npurpose of this section is to show that\na direct factor of ${\\cal T}_{m}$,\ncalled the even ribbon permutation group,\nis a ghostly recollection of the braid\ngroup and the Teo-Kane representation of the even ribbon permutation group is a projectivized version of\nthe Jones representation of the braid group\nat a $4^{th}$ root of unity, i.e. the representation\nrelevant to Ising anyons.\n\n\\subsection{Teo-Kane fundamental groups}\n\nIn Section V, we consider only even number of hedgehogs for\nphysical reasons.  In this section, we will include\nthe odd case for mathematical completeness.\n\\iffalse For each integer $m \\geq 0$, let $B^3_m$ be the closed complement\nof $m$ disjoint $3$-balls $D^3_i, i=1,2,\\cdots, m$ in the interior\nof $B^3$.  A Teo-Kane configuration is a continuous map $\\Phi:\nB^3_m \\longrightarrow U\/O$ such that $[\\Phi|_{\\partial B^3}]=0\\in\n\\pi_2(U\/O)$, and $[\\Phi|_{\\partial D^3_i}]=1\\in\n\\pi_2(U\/O), i=1,2,\\cdots, m$,  i.e. the map $\\Phi|_{\\partial B^3}: S^2\\rightarrow U\/O$ represents $0$ in\n$\\pi_2(U\/O)\\cong \\mathbb{Z}_2$, and $\\Phi|_{\\partial D_i^3}, i=1,2,\\cdots, m$ representing $1$.\n Then the Teo-Kane configuration space $K_m$ is\nthe topological space of all such maps with the compact-open topology, where the $3$-balls $D^3_i, i=1,2,\\cdots, m$ are not\nfixed.\\fi\n\nThe Teo-Kane fundamental\ngroup is the fundamental group of the Teo-Kane configuration\nspace $K_m$.  As computed in Section V, ${\\cal T}_{m}=\\pi_1(K_m)\\cong \\mathbb{Z}\\times \\mathbb{Z}_2 \\times\nE(\\mathbb{Z}_2^m \\rtimes {S_m})$, where\nthe subgroup ${\\cal T}^r_{m}=\\mathbb{Z}_2 \\times\nE({\\mathbb{Z}_2^m \\rtimes S_m})$ is called the ribbon permutation group. Here, $E({\\mathbb{Z}_2^m \\rtimes S_m})$ is\nthe subgroup of ${\\mathbb{Z}_2^m \\rtimes S_m}$ comprised of\nelements whose total parity in $\\mathbb{Z}_2^m$\nadded to their parity in $S_m$ is even.\nIn the following, we will call\nthe group $G_m=E(\\mathbb{Z}_2^m \\rtimes {S_m})$ the {\\it even} ribbon permutation group because\nit consists of the part of $\\pi_1(Y_m)$ associated with even ($2\\pi$) twisting.\nFor the representations of the Teo-Kane fundamental\ngroups, we will focus on the even ribbon permutation groups $G_m$.\nNo generality is lost if we consider only irreducible representations\nprojectively because irreducibles of $\\mathbb{Z}$ and $\\mathbb{Z}_2$ contribute only overall phases.\nBut for reducible representations, the relative phases from representations of $\\mathbb{Z}$\nand $\\mathbb{Z}_2$ might have physical consequences in interferometer\nexperiments.\n\nThe even ribbon permutation group $G_m$ is an index$=2$ subgroup of\n$\\mathbb{Z}_2^m \\rtimes S_m$.  To have a better understanding of $G_m$, we\nrecall some facts about the important group $\\mathbb{Z}_2^m \\rtimes S_m$.\nThe group $\\mathbb{Z}_2^m \\rtimes S_m$ is the symmetry group of the\nhypercube $\\mathbb{Z}_2^m$, therefore it is called the hyperoctahedral group, denoted as\n$C_m$.  $C_m$ is also a Coxeter group of type\n$B_m$ or $C_m$, so in the mathematical literature it is also\ndenoted as $B_m$ or $BC_m$.  To avoid confusion with the braid\ngroup ${\\cal B}_m$, we choose to use the hyperoctahedral group notation $C_m$.\nThe group $C_m$ has a faithful representation as signed\npermutation matrices in the orthogonal group $O(m)$: matrices with\nexactly one non-zero entry $\\pm 1$ in each row and column.\nTherefore, it can also be realized as a subgroup of the\npermutation group $S_{2m}$, called signed permutations: $\\sigma:\n\\{\\pm 1, \\pm 2, \\cdots, \\pm m\\} \\rightarrow \\{\\pm 1, \\pm 2, \\cdots, \\pm\nm\\}$ such that $\\sigma(-i)=-\\sigma(i)$.\n\nWe will denote elements in $C_m$ by a pair $(b,g)$, where\n$b=(b_i)\\in \\mathbb{Z}_2^m$ and $g\\in S_m$.  Recall the multiplication of\ntwo elements $(b,g)$ and $(c,h)$ is given by $(b,g)\\cdot (c,h)=(b+g.c, gh)$, where $g.c$\nis the action of $g$ on $c$ by permuting its coordinates.  Let $\\{e_i\\}$ be the standard\nbasis elements of $\\mathbb{R}^m$.  To save notation, we will also use it\nfor the basis element of $\\mathbb{Z}_2^m$.  As a signed permutation matrix $e_i$ introduces\na $-1$ into the $i^{th}$ coordinate $x_i$.  Let $s_i$ be the transposition\nof $S_m$ that interchanges $i$ and $i+1$.  As a signed permutation matrix, it\ninterchanges the coordinates $x_i, x_{i+1}$.\nThere is a total parity map $det: C_m\\rightarrow \\mathbb{Z}_2$ defined as\n$det(b,g)=\\sum_{i=1}^m b_i +parity(g)\\; mod \\; 2$.  We denote\nthe total parity map as $det$ because in the realization of $C_m$\nas signed permutation matrices in $O(m)$, the total parity is just the\ndeterminant.  Hence $G_m$, as the kernel of $det$,  can be identified as a subgroup of\n$SO(m)$.  The set of elements\n$t_i=(e_i,s_i), i=1,\\cdots, m-1$ generates $G_m$.  As a signed permutation matrix,\n$t_i(x_1,\\cdots,x_i,x_{i+1},\\cdots, x_m)=(x_1,\\cdots,-x_{i+1},x_i,\\cdots,\nx_m)$.\n\nGiven an element $(b,g)\\in C_m$, let $\\mathbb{Z}_2^{m-1}$ be\nidentified as the subgroup of $\\mathbb{Z}_2^m$ such that $\\sum_{i=1}^{m} b_i$ is\neven.  Then we have:\n\n\\begin{proposition}\\label{presentation}\n\n\\begin{enumerate}\n\n\\item For $m\\geq 2$, the even ribbon permutation group $G_m$ has a presentation as an abstract group\n\\begin{equation*}\n\\begin{split}\n<t_1, \\cdots, t_{m-1}| t_i^4=1, i=1,\\cdots, m-1,\\\\\n(t_it_{i+1}^{-1})^3=(t_i^{-1}t_{i+1})^3=1, i=1, \\cdots, m-2>.\n\\end{split}\n\\end{equation*}\n\n\\item The exact sequence $$1\\rightarrow \\mathbb{Z}_2^{m-1} \\rightarrow G_m \\rightarrow S_m\n\\rightarrow 1$$ splits if and only if $m$ is odd.\n\n\\item When $m$ is even, a normalized $2$-cocycle\n$f(g,h):  S_m\\times S_m \\rightarrow \\mathbb{Z}_2^{m-1}$ for the extension of $S_m$ by $\\mathbb{Z}_2^{m-1}$\nabove can be\nchosen as $f(g,h)=0$ if $g$ or $h$ is even and $f(g,h)=e_{g(1)}$\nif $g$ and $h$ are both odd.\n\n\\end{enumerate}\n\\end{proposition}\n\nWe briefly give the idea of the proof of Prop. \\ref{presentation}.  For $(1)$, first\nwe use a presentation of $C_m$ as a Coxeter group of type\n$B_m$: $<r_1,\\cdots, r_m|r_i^2=1, (r_1r_2)^4=1, (r_ir_{i+1})^3=1,\ni=2,\\cdots, m-1>$.  Then the Reidemeister-Schreier method \\cite{Magnus76} allows\nus to deduce the presentation for $G_m$ above.  For $(2)$, when\n$m$ is odd, a section $s$ for the splitting can be defined as\n$s(g)=(0,g)$ if $g$ is even and $s(g)=((11\\cdots 1),g)$ if $g$ is\nodd.  When $m$ is even, that the sequence does not split follows from\nthe argument in \\cite{Jones83}.  For $(3)$, we choose a set map\n$s(g)=(0,g)$ if $g$ is even and $s(g)=(e_1,g)$ if $g$ is odd.\nThen a direct computation of the associated factor set as on page $91$ of \\cite{Brown82} gives rise\nto our $2$-cocycle.\n\nAs a remark, we note that there are another two obvious maps from the\nhyperoctahedral group  $C_m$ to $\\mathbb{Z}_2$.  One of them is the\nsum of bits in $b$ of $(b,g)$.  The kernel of this map is the\nCoxeter group of type $D_m$, which is a semi-direct product of $\\mathbb{Z}_2^{m-1}$\nwith $S_m$.  The two groups $D_m$ and $G_m$ have\nthe same order, and are isomorphic when $m$ is odd (the two splittings induce the same\naction of $S_m$ on $\\mathbb{Z}_2^{m-1}$), but different\nwhen $m$ is even.  To see the difference, consider the order $2$ automorphism $\\phi: C_m\\rightarrow C_m$  given by $\\phi(x)=det(x) x$.\nIts restriction is the identity on $G_m$, but non-trivial on $D_m$.\n\n\\subsection{Teo-Kane unitary transformations}\n\nSuppose there are $m$ hedgehogs in $B^3$.  A unitary transformation\n$T_{ij}=e^{\\frac{\\pi}{4}\\gamma_i \\gamma_j}$ of the\nMajorana fermions is associated with the interchange of the ($i$,$j$)-pair\nof the\nhedgehogs. Interchanging the same pair twice results in the\n\\lq\\lq braidless\" operation $T_{ij}^2=\\gamma_i \\gamma_j$.  The\nMajorana fermions $\\gamma_i, i=1,2,\\cdots, m$ form the Clifford\nalgebra $Cl_m(\\mathbb{C})$.  Therefore, the Teo-Kane unitary\ntransformations act as automorphisms of the Clifford algebra\n$$T_{ij}: \\gamma \\rightarrow T_{ij} \\gamma T_{ij}^{\\dagger}.$$\nThe projective nature of the Teo-Kane representation rears its\nhead here already as an overall phase cannot be constrained by\nsuch actions.\nDo these unitary transformations afford a linear representation\nof the Teo-Kane fundamental group $K_m$? If so, what are their images?\nThe surprising fact is that the Teo-Kane\nunitary transformations cannot give rise to a linear representation of the\nTeo-Kane fundamental group.  The resulting representation is\nintrinsically projective.   We consider only the even ribbon permutation group\n$G_m$ from now on.\n\nTo define the Teo-Kane representation of $G_m$, we use the presentation of $G_m$\nby $t_i, i=1,\\cdots, m-1$ in Prop. \\ref{presentation}.  The associated unitary matrix for $t_i$ comes from the\nTeo-Kane unitary transformation $T_{i,i+1}$.  As was alluded above, there is a phase\nambiguity of the Teo-Kane unitary matrix.  We will discuss this ambiguity more in the\nnext subsection.  For the discussion below, we choose any matrix realization of the\nTeo-Kane unitary transformation with respect to a basis of the Clifford algebra $Cl_m(\\mathbb{C})$.\nA simple computation using the presentation of $G_m$ in Prop. \\ref{presentation}\nverifies that the assignment of $T_{i,i+1}$ to $t_i$ indeed\nleads to a representation of $G_m$.  Another verification follows from a relation to the Jones\nrepresentation in the next subsection.  We can also check directly that\nthis is indeed the right assignment for\n$T^2_{i,i+1}: \\gamma_i \\rightarrow -\\gamma_i,\\gamma_{i+1} \\rightarrow -\\gamma_{i+1}$.\nIn the Clifford algebra $Cl_m(\\mathbb{C})$, $\\gamma_i, \\gamma_{i+1}$\ncorrespond to the standard basis element $e_i, e_{i+1}$.  The\nelement $t_i^2$ of $G_m$ is $(e_i+e_{i+1},1)$.  As a signed\npermutation matrix, $t_i^2$ sends $(x_1,\\cdots, x_i,\nx_{i+1},\\cdots, x_m)$ to $(x_1,\\cdots, -x_{i},-x_{i+1},\\cdots, x_m)$,\nwhich agrees with the action of $T^2_{i,i+1}$ on $\\gamma_i,\n\\gamma_{i+1}$.\n\nTo see the projectivity, the interchange of the ($i$,$i+1$)-pair hedgehogs corresponds\nto the element $t_i=(e_i, s_i) \\in G_m$.  Performing the\ninterchange twice gives rise to the element $t_i^2=(e_i+e_{i+1},1)$, denoted as $x_i$.  Since\n$x_i$'s are elements of a subgroup of $G_m$ isomorphic $\\mathbb{Z}_2^{m-1}$,\nobviously we have\n$x_ix_{i+1}=x_{i+1}x_i$.  On the other hand,\n$T^2_{i,i+1}T^2_{i+1,i+2}=\\gamma_i \\gamma_{i+1} \\cdot \\gamma_{i+1}\n\\gamma_{i+2}=\\gamma_i \\gamma_{i+2}$, and $T^2_{i+1,i+2}T^2_{i,i+1}=\\gamma_{i+1} \\gamma_{i+2} \\cdot \\gamma_{i}\n\\gamma_{i+1}=-\\gamma_i \\gamma_{i+2}$.  Since it is impossible\nto encode the $-1$ in the $x_i$'s of $G_m$, the representation has to\nbe projective.  Note that an overall phase in\n$T_{ij}$ will not affect the conclusion.\nIn the next subsection, we will see this projective representation\ncomes from a linear representation of the braid group---the Jones\nrepresentation at a $4^{th}$ root of unity and the $-1$ is\nencoded in the Jones-Wenzl projector $p_3=0$.\n\nTo understand the images of the Teo-Kane representation of $G_m$,\nwe observe that the Teo-Kane unitary transformations $T_{ij}=e^{\\frac{\\pi}{4}\\gamma_i \\gamma_j}$ lie inside\nthe even part $Cl_m^0(\\mathbb{C})$ of $Cl_m(\\mathbb{C})$.\nTherefore, the Teo-Kane representation of $G_m$ is just the spinor\nrepresentation projectivized to $G_m \\subset SO(m)$.  It follows\nthat the projective image group of Teo-Kane representation of the even ribbon permutation group is $G_m$ as an\nabstract group.  Recall $Cl_m^0(\\mathbb{C})$ is reducible into two irreducible sectors if $m$\nis even, and irreducible if $m$ is odd.  When $m$ is even, it is important to\nknow how the relative action of the center $Z(G_m)\\cong \\mathbb{Z}_2$ of $G_m$ on the two\nirreducible sectors. The center of $G_m$ is generated by the element $z=((1\\cdots 1), 1)$, whose\ncorresponding Teo-Kane unitary transformation is $U=\\gamma_1 \\gamma_2 \\cdots\n\\gamma_m$ up to an overall phase.  As we show in\nAppendix \\ref{sec:braid-group},\nthe relative phase of $z$ on the two sectors is always $-1$.\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe now review and discuss the main results derived in this\npaper. Using the topological classification of\nfree fermion Hamiltonians \\cite{Ryu08,Kitaev09},\nwe considered a system of fermions in 3D which is allowed to\nhave arbitrary superconducting order parameter\nand arbitrary band structure; and is also allowed to\ndevelop any other possible symmetry-breaking\norder such as charge density-wave, etc. -- i.e. we do\nnot require that any symmetries are preserved.\nWe argued in Section \\ref{sec:free-fermion} that the space\nof possible gapped ground states\nof such a system is topologically equivalent to U($N$)\/O($N$)\nfor $N$ large (for large $N$, the topology of U($N$)\/O($N$)\nbecomes independent of $N$).\nBy extension, if we can spatially vary the superconducting\norder parameter and band structure at will with no\nregard to the energy cost, then there will be topologically\nstable point-like defects classified by\n${\\pi_2}(U($N$)\/O($N$))=\\mathbb{Z}_2$.\n\nThis statement begs the question of whether\none actually can vary the order parameter and\nband structure in order to create such defects.\nIn a given system, the energy cost may simply\nbe too high for the system to wind around\nU($N$)\/O($N$) in going around such a defect.\n(This energy cost, which would include the condensation\nenergy of various order parameters, is not taken into\naccount in the free fermion problem.) If we\ncreate such defects, they may be so costly\nthat it is energetically favorable for them\nto simply unwind by closing the gap over large\nregions. (The energy cost associated with such an unwinding\ndepends on the condensation energy of the order parameters\ninvolved, which is not included in the topological classification.)\nThus, U($N$)\/O($N$) is not the target space\nof an order parameter in the usual sense because\nthe different points in U($N$)\/O($N$) may not correspond to\ndifferent ground states with the same energy.\nHowever, in Section \\ref{sec:strong-coupling},\nwe have given at least one concrete model of free fermions with\nno symmetries in 3D in which the topological defects\npredicted by the general classification are present\nand stable. Furthermore, Teo and Kane \\cite{Teo10}\nhave proposed several devices in which\nthese defects are simply superconducting vortices\nat the boundary of a topological insulator.\n\nIn order to understand the quantum mechanics of\nthese defects, it is important to first understand their\nquantum statistics. To do this,\nwe analyzed the multi-defect configuration\nspace; its fundamental group governs defect\nstatistics. The configuration space of $2n$ point-like\ndefects of a system with `order parameter' taking values\nin U($N$)\/O($N$) is the space which we call $K_{2n}$.\nIt can be understood as a fibration.\nThe base space is $X_{2n}$, the configuration\nspace of $2n$ points (which we know has fundamental\ngroup $S_{2n}$ in dimension three and greater).\nAbove each point in this base space there is\na fiber ${\\cal M}_{2n}$ which is the space of maps from\nthe ball $B^3$ minu $2n$ fixed points to\nU($N$)\/O($N$) with winding number $1$ about each\nof the $2n$ points. $K_{2n}$ is the total space of\nthe fibration. In Section \\ref{sec:Kane_space},\nwe found that its fundamental group is\n${\\pi_1}(K_{2n})=\\mathbb{Z}\\times\\mathbb{Z}_2\n\\times E({\\mathbb{Z}_2^{2n} \\rtimes S_{2n}})$, where\n$E({\\mathbb{Z}_2^{2n} \\rtimes S_m})$ is\nthe subgroup of ${\\mathbb{Z}_2^m \\rtimes S_{2n}}$ comprised of\nelements whose total parity in $\\mathbb{Z}_2^{2n}$\nadded to their parity in $S_{2n}$ is even.\n\nThe fundamental group of the configuration\nspace is the same as the group of equivalence\nclasses of spacetime histories of a system\nwith $2n$ point-like defects. Since these different\nequivalence classes cannot be continuously\ndeformed into each other, quantum mechanics\nallows us to assign them different unitary\nmatrices. These different unitaries form\na representation of the fundamental group of the configuration\nspace of the system. However, we found in Section \\ref{sec:projective}\nthat Teo and Kane's unitary transformations are not\na linear representation of ${\\pi_1}(K_{2n})$,\nbut a {\\it projective representation}, which is to say that\nthey represent ${\\pi_1}(K_{2n})$ only up to a phase.\nEquivalently, Teo and Kane's unitary transformations are\nan ordinary linear representation of a {\\it central\nextension} of ${\\pi_1}(K_{2n})$, as discussed in\nSection \\ref{sec:projective}.\n\nThis surprise lurks in a seemingly innocuous set of\ndefect motions: those in which defects $i$ and $j$ are rotated\nby $2\\pi$ and the order parameter field surrounding\nthem relaxes back to its initial configuration. This has the\nfollowing effect on the Majorana fermion zero mode operators\nassociated with the two defects:\n\\begin{equation}\n{\\gamma_i}\\rightarrow -{\\gamma_i}\\, , \\hskip 0.5 cm\n{\\gamma_j}\\rightarrow -{\\gamma_j}\n\\label{eqn:gamma-trans}\n\\end{equation}\nOne might initially expect that two such motions, one affecting\ndefects $i$ and $j$ and the other affecting $i$ and $k$,\nwould commute since they simply multiply the operators\ninvolved by $-$ signs. However, the unitary operator which\ngenerates (\\ref{eqn:gamma-trans}) is\\cite{footnote1}:\n\\begin{equation}\nU^{ij} = e^{i\\theta} \\,{\\gamma_i} {\\gamma_j}\\, .\n\\end{equation}\nThus, the unitary operators $U^{ij}$ and $U^{ik}$ do not commute;\nthey anti-commute:\n\\begin{equation}\nU^{ij} \\, U^{ik}= -U^{ik}\\,U^{ij} \\,  .\n\\end{equation}\nHowever, as shown in Figure \\ref{fig:movie-of-movies}, the\ncorresponding classical motions can be continuously deformed\ninto each other. Thus, a linear representation of the fundamental group\nof the classical configuration space would have these two\noperators commuting. Instead, the quantum mechanics of this\nsystem involves a projective representation.\n\nProjective quasi-particle statistics were first proposed by\nWilzcek\\cite{Wilczek98}, who suggested a projective representation of the\npermutation group in which generators $\\sigma_j$ and $\\sigma_k$ anti-commute\nfor $|j-k|\\geq 2$, rather than commuting.  Read\\cite{Read03} criticized\nthis suggestion as being in conflict with locality.  We can sharpen Read's\ncriticism as follows.\nSuppose that one can perform the operation $\\sigma_1$ by acting\non a region of space, called $A$, containing particles $1$ and $2$,\nand one can perform $\\sigma_3$ by acting on a region called $B$, containing\nparticles $3$ and $4$, and suppose that regions $A$ and $B$ are disjoint.\nConsider the following thought experiment: let Bob perform operation\n$\\sigma_3$ at time $t=0$ and let Bob then repeat operation $\\sigma_3$ at\ntime $t=1$.  Let Alice prepare a spin in the state $(1\/\\sqrt{2})(|\\uparrow\\rangle+|\\downarrow\\rangle)$ at time $t=-1$, and then let Alice perform the following\nsequence of operations.\nAt time $t=-\\epsilon$, for some small $\\epsilon$, she\nperforms the unitary operation $|\\uparrow\\rangle\\langle \\uparrow| \\otimes \\sigma_1+|\\downarrow\\rangle\\langle\\downarrow | \\otimes I$, where $I$ is the\nidentity operation, leaving the particles alone.\nAt time $t=+\\epsilon$, she\nperforms the unitary operation $|\\uparrow\\rangle\\langle \\uparrow| \\otimes I\n+|\\downarrow\\rangle\\langle\\downarrow | \\otimes \\sigma_1$.   Thus, if\nthe spin is up, she performs $\\sigma_1$ at time $t=-\\epsilon$, while\nif the spin is down, she does it at time $t=+\\epsilon$.\nFinally, at\ntime $t=2$, Alice performs the operation $\\sigma_1$ again.  One may then\nshow that, due to the anti-commutation of $\\sigma_1$ and $\\sigma_3$, that\nthe spin ends in the state\n$(1\/\\sqrt{2})(|\\uparrow\\rangle-|\\downarrow\\rangle)$.  However, if Bob had\nnot performed any operations, the spin would have ended in the original state\n$(1\/\\sqrt{2})(|\\uparrow\\rangle+|\\downarrow\\rangle)$.  Thus, by performing these\ninterchange operations, Bob succeeds in transmitting information to a space-like\nseparated region (if $A$ and $B$ are disjoint, and the time scale in the\nabove thought experiment is sufficiently fast, then Alice and Bob are space-like\nseparated throughout).\n\nHaving seen this criticism, we can also see how Teo and Kane's construction\nevades it.  The fundamental objects for Teo and Kane are not particles, but\nparticles with ribbons attached.  One may verify that, in every case where\noperations in Teo and Kane's construction anti-commute, the two operations\ndo not act on spatially disjoint regions due to the attached ribbons.  That is,\nthe interchange of particles also requires a rearrangement of the order parameter field.\n\nWhile this argument explains why a projective representation\ndoes not violate causality, it does not really explain why\na projective representation actually occurs in this system.\nPerhaps one clue is the fact that the hedgehogs have long-ranged\ninteractions in any concrete model. Even in the `best-case scenario',\nin which the underlying Hamiltonian of the system is U($N$)\/O($N$)-invariant, there will be a linearly-diverging\ngradient energy for an isolated hedgehog configuration.\nThus, there will be a linear long-ranged force between hedgehogs.\nConsequently, one might adopt the point of view that, as a result\nof these long-ranged interactions, the overall phase associated with\na motion of the hedgehogs is not a purely topological quantity\n(but will, instead depend on details of the motion) and, therefore,\nneed not faithfully represent the underlying fundamental group.\nAs one motion is continuously deformed into another\nin Figure \\ref{fig:movie-of-movies}, the phase of the wavefunction\nvaries continuously from $+1$ to $-1$ as the order parameter\nevolution is deformed. It is helpful to compare this to another\nexample of a projective representation: a charged particle in\na magnetic field $B$. Although the system is invariant under the\nAbelian group of translations, the quantum mechanics of the\nsystem is governed by a non-Abelian projective representation\nof this group (which may be viewed as a linear representation of\nthe `magnetic translation group'). A translation by $a$ in the $x$-direction,\nfollowed by a translation by $b$ in the $y$-direction differs in its\naction on the wavefunction from the same translations in\nthe opposite order by a phase $abB\/\\Phi_0$ equal to the magnetic flux through\nthe area $ab$ in units of the flux quantum $\\Phi_0$.\nIf we continuously deform these two sequences of translations\ninto each other, the phase of the wavefunction varies continuously.\nFor any trajectory along this one-parameter family of trajectories\n(or `movie of movies'), the resulting phase of the wavefunction\nis given by the magnetic flux enclosed by the composition of\nthis trajectory and the inverse of the initial one. In our\nmodel of non-Abelian projective statistics, the phase changes\ncontinuously in the same way, but as a result of the evolution\nof the order parameter away from the defects, rather than as a result of a magnetic field.\n\nAn obvious question presents itself: is there a related theory in which the hedgehogs are no longer confined? Equivalently, could 3D objects with non-Abelian ribbon permutation statistics ever be the weakly-coupled low-energy quasiparticles of a system? The most straightforward route will not work: if we had a U($N$)-invariant system and tried to gauge it to eliminate the linear confining force between hedgehogs, we would find that the theory is sick due to the chiral anomaly. If we doubled the number of fermions in order to eliminate the anomaly, there would be two Majorana modes in the core of each\nhedgehog, and their energies could be split away from zero by a local interaction. This is not surprising since ribbon permutation statistics would violate locality if the hedgehogs were truly decoupled (or had exponentially-decaying interactions). On the other hand, if the hedgehogs were to interact through a Coulomb interaction (or, perhaps, some other power-law),\nthey would be neither decoupled nor confined, thereby satisfying the requirements that they satisfy locality and are low-energy particle-like excitations of the system. Elsewhere, we will describe a model which realizes this scenario \\cite{Freedman10}\n\nThe non-Abelian projective statistics studied in the 3D class with no\nsymmetry can be generalized to arbitrary dimension.\nAs shown in Eq. (\\ref{eq:defectclassification}), the classification of topological defects is independent of the spatial dimension.\nThus, in any dimension $d$ with no symmetry ($p=0$), point-like ($D=0$) topological defects are classified by\n$\\pi_0(R_{p-D+1})=\\pi_0(R_1)=\\mathbb{Z}_2$.\nMoreover, it can be proved that analogous topological defects in different dimensions not only carry the same topological quantum number, but also have the same statistics. In Sec.\n\\ref{sec:Kane_space}, we have defined the configuration space\n$\\mathcal{M}_{2n}$ which is the space of maps from\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3$ to $R_7=U\/O$,\nwith specific boundary conditions. Now if we consider point defect\nin the class with no symmetry in 4D, the configuration space\n$\\mathcal{M}_{2n}^{d=4}$ is defined by maps from\n$B^4\\setminus\\bigcup_{i=1}^{2n} B_i^4$ to the classifying space\n$R_6=Sp\/U$. Noticing that $B^4\\setminus\\bigcup_{i=1}^{2n} B_i^4$\nis homotopy equivalent to the suspension of\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3$, we obtain that\n$\\mathcal{M}_{2n}^{d=4}$ is equivalent to the space of maps from\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3\\rightarrow \\Omega\\left(Sp\/U\\right)$, where $\\Omega\\left(Sp\/U\\right)$ is the loop space of\n$Sp\/U$. Since  $\\Omega\\left(Sp\/U\\right)\\simeq U\/O$,\nwe obtain $\\mathcal{M}_{2n}^{d=4}\\simeq \\mathcal{M}_{2n}$. Thus we have proved that the configuration space $\\mathcal{M}_{2n}$ is independent of the spatial dimension $d$. On the other hand, the fundamental\ngroup of the configuration space $X_{2n}$ of $2n$ distinct points in $B^d$ is independent of $d$ as long as $d>2$. Consequently, the space $K_{2n}$ defined by the fibration $\\mathcal{M}_{2n}\\rightarrow K_{2n}\\rightarrow X_{2n}$ is also topologically independent of spatial dimension $d$ for $d>2$. Thus, the proof we did for $d=3$ applies to generic dimension, and non-Abelian projective statistics exist in any spatial dimension $d>3$ for point defects\nin the no-symmetry class. A similar analysis applies to extended\ndefects with dimension $D>0$. When the spatial dimension is increased by $1$ and the symmetry class remains the same, the classifying space is always changed from $R_{2+p-d}$ to $R_{2+p-d-1}\\simeq \\Omega^{-1}(R_{2+p-d})$. Consequently, at least for\nsimple defects with the topology of $S^D$, the statistics is independent of spatial dimension $d$ as long as $d$ is large enough. For point defects, the ``lower critical dimension\" is $d=3$, while for line defects, i.e. $D=1$, the ``lower critical dimension\" is at least $d=4$ since in $d=3$ we can have braiding between loops.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is well known that the canonical quantization procedure is consistent\nonly in Cartesian coordinates \\cite{1}. For most physically\nrelevant systems, it turns out to be possible to find a Cartesian system\nof axes and, hence, successfully\napply canonical quantization. Nevertheless, the\nHamiltonian dynamics of a classical system apparently exhibits,\nat first sight, a\nlarger symmetry than the associated canonically quantized system. Indeed,\nHamiltonian equations of motion are covariant under canonical\ntransformations, while the Heisenberg equations of motion are covariant\n under unitary transformations. Unitary transformations preserve the\nspectrum of the canonical quantum operators, while in the classical case\n canonical\ntransformations do not generally preserve the range of the canonical\nvariables.\n\nIt is worth mentioning in this regard\nthat the old Bohr-Som\\-mer\\-feld quantization postulate\n\\begin{equation}\n\\oint\\limits_{}^{}pdq=2\\pi \\hbar (n+1\/2),\\ \\ n=1,2,\\ldots\n\\label{1.1}\n\\end{equation}\nis invariant with respect to canonical transformations\n\\begin{equation}\np\\rightarrow {\\bar p}(p,q),\\ \\ q\\rightarrow {\\bar p}(q,p)\n\\label{1.2}\n\\end{equation}\nbecause\n\\begin{equation}\n\\oint\\limits_{}^{}pdq =\\oint\\limits_{}^{}{\\bar p}d{\\bar q}\\ .\n\\label{1.3}\n\\end{equation}\n As a consequence, since the result is identical in all canonical\ncoordinate systems, the Bohr-Som\\-mer\\-feld\nquantization is in fact ``coordinate-free\".\nThe characteristic properties of the quantum theory, like the energy\nspectrum, will be independent of the choice\nof canonical coordinates. In this respect, the old quantum dynamics enjoys\nthe same symmetry as classical dynamics.\n\nIn contrast to the Bohr-Sommerfeld procedure, canonical quantization leads\nto a result that is not covariant with respect to the initial\nchoice of canonical coordinates. For example, for a single degree of freedom,\nthe coherent-state phase space path integral representation of the\nevolution operator\n\\begin{eqnarray}\n&\\ &\\hskip-1cm\\<  p'',q'',t|p',q'\\> =\\< p'',q''|\ne^{-it{\\cal H}\/\\hbar}|p',q'\\>  \\nonumber \\\\\n&= & \\int\\limits_{}^{} \\prod\\limits_{\\tau =0}^{t}\\left(\n\\frac{dp(\\tau )dq(\\tau )}{2\\pi \\hbar}\\right)\\exp \\frac{i}{\\hbar}\n\\int\\limits_{0}^{t}d\\tau \\left[p\\dot{q}-h(p,q)\\right]\\ ,\n\\label{1.4} \\\\\n{\\cal H}&=&{\\textstyle\\int} h(p,q)\\,|p,q\\>\\<p,q|\\,dp\\,dq\/(2\\pi\\hbar)\\ ,\n\\label{4.2}\n\\end{eqnarray}\nis not covariant with respect to canonical transformations, although\nthe measure, being the product of local Liouville measures at each moment\nof time, is invariant under canonical transformations. The contradiction\nfollows from the observation that all classical dynamical systems with\npositive energy and one\ndegree of freedom are equivalent to, say, a free particle ($h={\\tilde p}^2\/2$ after\na suitable canonical transformation). Making such a canonical\ntransformation in (\\ref{1.4}), we seem to arrive at the same conclusion\nfor quantum systems because the integral (\\ref{1.4}) is formally\ninvariant. Such a conclusion is certainly incorrect.\n\n\\subsubsection*{Coordinate-free quantization}\n\nTo resolve this paradox, it has been proposed \\cite{2} to\ninterpret the ill-defined path integral (\\ref{1.4}) by means of\nthe regularized expression\n\\begin{equation}\n\\int\\limits_{}^{}{\\cal D}p{\\cal D}q\\,(\\,\\cdot\\,)\n\\rightarrow\\lim _{\\nu \\rightarrow\n\\infty}\n{\\cal M}_\\nu \\int\\limits_{}^{} {\\cal D}p{\\cal D}q\\,(\\,\\cdot\\,)\\, e^{-\n\\frac{1}{2\\nu}\\int _{0}^{t}d\\tau (\\dot{p}^2+ \\dot{q}^2)}\\ ,\n\\label{1.5}\n\\end{equation}\nwhere $\\ {\\cal M}_\\nu $ is a suitable normalization\n and the limit $\\nu\n\\rightarrow \\infty$ must be taken {\\em after} evaluation of the\npath integral. Various factors in (\\ref{1.5})\ncombine to give a Wiener measure\non the two-dimensional\n phase space. In contrast to the integral (\\ref{1.4}), the\ncoherent-state path integral with the regularized measure can be regarded as\na sum over trajectories of a  Brownian particle whose\nflat, two-dimensional configuration space\nis the original phase space of the system.\n\nThe spectrum $E$ of the system can be obtained from\nthe pole structure of\n the Fourier transform\nof the trace of the transition amplitude\n\\begin{equation} Z_t={\\rm\ntr}e^{-it{\\cal H}\/\\hbar}=\\sum\\limits_{E}^{}e^{-itE\/\\hbar}=\n\\int\\limits_{}^{}(dp'dq'\/2\\pi\\hbar)\\< p',q',t|p',q'\\>\\,,\n\\label{1.6}\n\\end{equation}\nwhere $\\< p',q',t|p',q'\\>$ is given by the corresponding path integral.\nUnder canonical transformations (\\ref{1.2}) the\nBrownian motion on a flat two-dimensional phase space\nremains such a Brownian motion, and if one interprets the stochastic\nintegral $\\int pdq$ in the Stratonovich sense,\n then the spectrum of the system is invariant under canonical\ncoordinate transformations.\n\nIn other words, the coherent-state path integral regularized with the\nhelp of the Wiener measure (\\ref{1.5}) provides a ``coordinate-free\"\ndescription of quantum theory \\cite{2}. Such a regularization\nprocedure applies to general theories without constraints.\n\n\\subsubsection*{Gauge theories}\n\nHamiltonian path integrals are often used to quantize\n gauge theories \\cite{3}. We now have in mind a system of $J$ degrees of\nfreedom $p=\\{p_j\\}$, $q=\\{q^j\\}$, $1\\leq j\\leq J$. A\nmain feature of gauge systems is the existence of nonphysical canonical\nvariables. In the standard formulation,\nthe formal path integral (\\ref{1.4}) is divergent because the\nHamiltonian action for gauge systems is invariant with respect to\ntransformations\n\\begin{equation}\nq\\rightarrow q^\\omega ,\\ \\ p\\rightarrow p^\\omega\n\\label{1.7}\n\\end{equation}\nwhose parameters $\\omega$ depend on the time, that is, there are\n orbits traversed by the gauge transformations (\\ref{1.7}) in the phase\nspace along which the action is constant and\ntraditionally have an infinite volume. The nonphysical variables can be\nassociated with these ``gauge\" directions in phase space.\n\nTo factor out such divergencies of the path integral, one should integrate out\nthe nonphysical variables and obtain a measure on the physical phase\nspace\n\\begin{equation}\n[PS]_{ph}=[PS]\/{\\cal G}\\ ;\n\\label{1.8}\n\\end{equation}\nhere ${\\cal G}$ consists of all transformations (\\ref{1.7}).\nTechnically, the\nprocedure amounts to a canonical transformation such that the generators of\n(\\ref{1.7}) become some elements of a new\nset of canonical momenta \\cite{3}. This canonical\ntransformation introduces explicit symplectic coordinates $p^*$ and $q^*$\non the physical phase space (\\ref{1.8}). However, it is important to realize\nthat the canonical coordinates on $[PS]_{ph}$\nare themselves defined only up to a\ncanonical transformation, i.e., the parametrization of the physical phase\nspace is not unique. As we have argued above, the formal integral in the\nHamiltonian path integral cannot provide a\ngenuine invariance with respect to\ncanonical transformations. In the framework of gauge theories, this\ninvariance implies\ngauge invariance because the spectrum of a gauge theory\ncannot depend on one or another particular parametrization of the physical\nphase space.\n\nThus, the regularization of the path integral measure with the help of a\nWiener measure and the invariance\nunder canonical coordinate transformations\nit offers should be extended to gauge theories. The aim of this\nletter is to address this problem. Hereafter, we use units where $\\hbar=1$.\n\n\\section{The projection method}\n\\setcounter{equation}0\n\n\\subsubsection*{Special constraint class}\n\nLet $\\varphi _a=\\varphi _a(p,q)$ be a set of  independent closed first-class\nconstraints, i.e.\n\\begin{equation}\n\\{\\varphi _a,\\varphi _b\\}=f_{abc}\\varphi _c\\ ,\n\\label{2.1}\n\\end{equation}\nand for convenience we also suppose that\nthe Poisson bracket of $\\varphi _a$ with the system Hamiltonian\nvanishes. The constraints\ngenerate gauge transformations on phase space which in their infinitesimal\nform are given by\n\\begin{eqnarray}\n&p &\\rightarrow p+\\delta p=p+\\delta \\omega ^a\\{p,\\varphi_a\\}\\equiv\np^{\\delta \\omega } \\label{2.2} \\\\\n&q & \\rightarrow q+\\delta q=q+\\delta \\omega ^a\\{q,\\varphi_a\\}\\equiv\nq^{\\delta \\omega} \\  ,\\label{2.3}\n\\end{eqnarray}\nfor general $\\{\\omega^a\\}$.\nFrom (\\ref{2.2}) and (\\ref{2.3}) it follows that the infinitesimal\ngauge transformations\ngenerated by the constraints are also infinitesimal canonical transformations\n\\begin{equation}\n\\{p^{\\delta \\omega},q^{\\delta\\omega}\\}=\\{p,q\\}+O(\\delta\\omega ^2)\\ .\n\\label{2.4}\n\\end{equation}\nA finite gauge transformation can be obtained\nby applying the operator $\\exp[ -\n(\\omega ^a ad\\, \\varphi _a)]$, $ ad\\,\\varphi _a\n=\\{\\varphi _a,\\ \\cdot \\}$, to phase\nspace variables.\n\nAs noted at the outset, canonical quantization singles out Cartesian\ncoordinates for special attention. We formulate a special class of\nclosed first-class constraint systems---which we shall refer to\nas constraints of ``Yang-Mills type\"---in such a favored set of\ncoordinates. Specifically, we choose\n\\begin{equation}\n\\varphi_a(p,q)=f^j_a(q)p_j\\equiv(f_a(q),p)\\,,\n\\label{c.1}\n\\end{equation}\nwhere $(\\,,\\,)$ denotes a scalar product in a Euclidean space, and\n$f_a(q)$ {\\it are linear functions of $q$ chosen so that the\nconstraints (\\ref{c.1}) are of the first class}, i.e. they\nsatisfy (\\ref{2.1}). With this choice, the\ngauge transformations (\\ref{1.7}) are linear canonical transformations.\nIt follows for such constraints that\n\\begin{equation}\np_j\\{\\varphi_a,q^j\\}=\\varphi_a(p,q)\n\\label{c.2}\n\\end{equation}\nholds as an identity, which we shall find useful.\nWe also assume that there is no operator ordering ambiguity in\nthe constraints after quantization. This situation is in fact\n entirely realized for a gauge theory based on a compact semi-simple\ngauge group $\\footnote{The formalism applies also to gauge groups\nbeing the direct product a semi-simple and some number of Abelian\ngroups.}$.\n\nSuch constraints enjoy an additional useful property. If\n\\begin{equation}\n|p,q\\>\\equiv e^{-iq^jP_j}e^{ip_jQ^j}|0\\>\\,,\n\\label{c.3}\n\\end{equation}\n where $|0\\>$ is the ground state\nof an harmonic oscillator, i.e, $(Q^j+iP_j)\\,|0\\>=0$ for all $j$,\ndenotes the coherent states in the same Cartesian coordinates,\nthen it follows that\n\\begin{equation}\ne^{-i\\Omega^a{\\hat\\varphi}_a(P,Q)}\\,|p,q\\>=|p^\\Omega,q^\\Omega\\>\\,,\n\\label{c.4}\n\\end{equation}\nnamely the action of any finite gauge transformation is to map one\ncoherent state into another.  Here $\\{{\\hat\\varphi}_a\\}$ denote\nthe constraint operators that generate the gauge transformations.\n\n\\subsubsection*{Coherent state propagator}\n\nThe total Hilbert space of a gauge system can always be split\ninto an orthogonal sum of a subspace formed by gauge invariant states\nand a subspace that consists of gauge variant states.\nTherefore an averaging over the gauge group automatically leads to\n  a projection\noperator onto the physical subspace of gauge invariant\nstates. The physical transition amplitude\nis obtained from the unconstrained propagator by averaging\nthe latter over the gauge group,\n\\begin{eqnarray} \\hskip-.5cm\n\\< p'',q'',t|p',q'\\>^{ph}&\\equiv &\n\\int\\limits_{G}^{} \\frac{d\\mu(\\omega)}{Vol\\ G} \\< p'',q'',t|e^{-i\\omega^a\n\\hat{\\varphi}_a}|p',q'\\>\n \\label{2.6a} \\\\\n&\\equiv &\n\\< p'',q'',t|\\hat{P}_G|p',q'\\>\\\\\n&=& \\int (d^J\\!pd^J\\!q\/(2\\pi)^J)\n\\< p'',q'',t|p,q\\>\\< p,q|\\hat{P}_G |p',q'\\>\\ ,\n\\label{2.6b}\n\\end{eqnarray}\nwhich is a quantum implementation of the classical initial value equation\nfor first-class constraints.\nHere $d\\mu(\\omega)$ is the invariant measure on the space of gauge group\nparameters, and $Vol\\ G = \\int_G d\\mu(\\omega)<\\infty$ is the gauge group\nvolume.\nIn what follows we also adopt a shorthand notation for the\nnormalized Haar measure\n\\begin{equation}\n\\delta \\omega \\equiv \\frac{d\\mu(\\omega)}{Vol\\ G} \\ ,\n\\ \\ \\ \\ \\ \\int\\limits_{G}^{}\\delta\\omega =1\\ .\n\\label{haar}\n\\end{equation}\nThe operator $\\hat{P}_G$\nis a projection operator onto the gauge invariant subspace.\n Its kernel is determined as the\ngauge group average of the unit operator kernel\n\\begin{equation}\n\\<p'',q''|p',q'\\>^{ph}\\equiv\\< p'',q''|\\hat{P}_G|p',q'\\> =\n\\int\\limits_{G}^{} \\delta\\omega\\, \\< p'',q''|e^{-i\\omega_a\n\\hat{\\varphi}_a}|p',q'\\>\\ .\n\\label{2.7}\n\\end{equation}\nFor some gauge systems, it can be calculated explicitly as well as\nthe kernel (\\ref{2.6a}) \\cite{pr}.\n\n\\subsubsection*{The path integral based on the projective method}\n\nApplying the projective formula (\\ref{2.6a}) to an infinitesimal\ntransition amplitude $t\\rightarrow \\epsilon =t\/N$ and making\na convolution of $N$ physical infinitesimal evolution operator\nkernels, we arrive at the following representation of the\namplitude (\\ref{2.6a})\n\\begin{eqnarray}\n\\< p'',q'',t|p',q'\\>^{ph} =&\\ &\n\\int \\prod\\limits_{l=1}^{N-1}(dp^J_ldq^J_l\/(2\\pi)^J)\n\\< p'',q'',\\epsilon |p_{N-1},q_{N-1}\\>^{ph}\n \\nonumber\\\\\n&\\times &\n\\< p_{N-1},q_{N-1},\\epsilon |p_{N-2},q_{N-2}\\>^{ph}\n\\cdots \\< p_1,q_1,\\epsilon |p',q'\\>^{ph}\\  .\n\\label{3.1}\n\\end{eqnarray}\nIn the continuum limit,\nwhere $N\\rightarrow \\infty,\\ \\epsilon\\rightarrow 0$, while\nthe product $t=N\\epsilon$ is kept fixed,\nthe convolution (\\ref{3.1}) of the kernels (\\ref{2.6a})\n$(t=\\epsilon)$ results\nin the coherent state path integral \\cite{kl2}\n\\begin{eqnarray}\n\\< p'',q'',t|p',q'\\>^{ph}&= &{\\cal M}\n\\int {\\cal D}C(\\omega) {\\cal D}p{\\cal D}q\\, e^{iS_H}\\ ,\n\\label{3.6} \\\\\nS_H\n&=&\\int\\limits_{0}^{t}dt'[\\left\n(p,\\dot{q}) - \\omega^a \\varphi_a(p,q) - h(p,q)\n\\right]\\ ,\n\\label{3.7}\n\\end{eqnarray}\nwhere ${\\cal D}C(\\omega)= \\prod_t \\delta\\omega(t)$\nis a formal  (normalized) measure for the gauge group\naverage parameters (cf (\\ref{2.6a})),\nand the symbol $h(p,q)$ is defined in (\\ref{4.2}).\nThus, the gauge group averaging\nparameters $\\omega^a$ become the Lagrange multipliers of the\nclassical theory in the continuum\nlimit.\n\nA relation between the path integral (\\ref{3.6}) and the projective\nformula (\\ref{2.6a}) is found in the boundary condition for the\npath integral. Recall that the integral (\\ref{3.6}) is taken over\nphase space trajectories that obey the boundary conditions\n\\begin{eqnarray}\np(0)&= &p'\\ ,\\ \\ \\ \\ q(0) = q'\\ ; \\label{3.9} \\\\\np(t)&= &p''\\ ,\\ \\ \\ \\ q(t)=q''\\ . \\label{3.10}\n\\end{eqnarray}\nIt is not hard to  find a gauge transformation such that\n\\begin{equation} (p^\\omega,{\\dot q^\\omega})-\n\\omega^a\\varphi_a(p^\\omega,q^\\omega)\n= (p,\\dot{q})\\ .\n\\label{3.11}\n\\end{equation}\nIt is equivalent to solving a linear equation\n\\begin{equation}\n\\dot{q}^\\omega + \\omega^a f_a(q^\\omega) = \\dot{q}\\ .\n\\label{3.12}\n\\end{equation}\nHaving found $q^\\omega$ one easily determines $p^\\omega$ as its\ncanonical momenta.\n\nThe path integral measure is formally\ninvariant under canonical transformations\nand, hence, the explicit dependence on the Lagrange multipliers of the\naction $S_H$ disappears after the canonical transformation constructed\nabove. The residual coherent state path integral represents a\ntransition amplitude in the unconstrained Hilbert space. However the integral\n$\\int {\\cal D}C(\\omega)$ cannot be factored out because a nontrivial\ndependence on the Lagrange multipliers survives at the boundaries.\nTo maintain the boundary conditions (\\ref{3.9}) and (\\ref{3.10}),\none can, say, require\n\\begin{equation}\np^\\omega(t) = p''\\ ,\\ \\ \\ \\ q^\\omega(t) = q''\\ .\n\\label{3.13}\n\\end{equation}\nThen it is impossible to satisfy the boundary condition (\\ref{3.9})\nbecause equation (\\ref{3.12}) admits only one boundary condition,\nsay, at the final time point. Thus, after the canonical transformation\nthe path integral\nmust be taken with boundary conditions that depends on $\\omega_a$\n\\begin{equation}\np^\\omega(0) = p'^\\Omega\\ ,\\ \\ \\ \\ q^\\omega(0) =q'^\\Omega\\ ,\\ \\\n\\ \\ \\Omega = \\Omega[\\omega] \\ ,\n\\label{3.14}\n\\end{equation}\nthat is, one gauge group average ``survives\" the canonical transformation\nthat removes the Lagrange multipliers from the action and provides\nthe equivalence of the path integral (\\ref{3.6})  to the projective\nrepresentation (\\ref{2.6a}).\n\n\\section{Gauge fixing and the path integral over physical phase space}\n\\setcounter{equation}0\n\nIn practice, it often turns out to be useful to integrate out the nonphysical\nphase-space variables associated with pure gauge degrees of freedom\nand work with the path integral over the physical phase space\n(\\ref{1.8}). For this purpose one usually fixes a gauge \\cite{3}\n\\begin{equation}\n\\chi_a(q) = 0\\ .\n\\label{g.1}\n\\end{equation}\nBy a necessary assumption, each gauge orbit $q^\\omega$ must\nintersect the gauge condition surface (\\ref{g.1}) (at least)\nonce. Under this assumption a generic configuration $q$ can be\nparametrized via lifting it onto the gauge condition surface\nalong a gauge orbit passing through $q$\n\\begin{equation}\nq = q_\\chi^\\theta(q^*)\\ ,\n\\label{g.2}\n\\end{equation}\nwhere $\\theta_a$ parametrizes the lift along a gauge orbit, and\npoints $q=q_\\chi(q^*)$ form the surface (\\ref{g.1}), i.e., $q^*$\nparametrizes the surface (\\ref{g.1}).\n\nIn the curvilinear coordinates (\\ref{g.2}) associated with\nthe chosen gauge condition, the constraints are linear combinations\nof canonical momenta for $\\theta_a$, and the Poisson bracket of\nthe canonical variables $p^*$ and $q^*$ with the constraints vanishes,\nthat is, $p^*$ and $q^*$ are gauge invariant according to (\\ref{2.2})\nand (\\ref{2.3}).\nThe $\\theta$-dependence\nof the action can be absorbed by a shift of the Lagrange\nmultipliers $\\omega^a$\non a suitable linear combination of the velocities\n$\\dot{\\theta}_a$ because the canonical one-form assumes the form\n\\begin{equation}\np\\dot{q} +\\omega^a\\varphi_a = p^*\\dot{q}^* + p_\\theta^a\\dot{\\theta}_a\n+\\omega^a\\varphi_a\n\\label{g.3}\n\\end{equation}\nand the Hamiltonian is gauge invariant (the $\\theta_a$'s are\ncyclic variables).\n\nThe integral over $\\theta_a$ yields the gauge group volume that\ncancels the one sitting in the measure ${\\cal D}C(\\omega)$. Finally,\nthe integrals over $\\omega^a$ and $p_\\theta^a$ can also be done, and\none ends up with the integral over physical phase space spanned\nby local symplectic coordinates $p^*, q^*$.\n\nThis result is usually achieved by a formal restriction of the\npath integral measure support in (\\ref{3.6}) to a subspace of\nthe constraint surface $\\varphi_a(p,q) =0$ selected by the gauge\n(supplementary) condition (\\ref{g.1}) \\cite{3}:\n\\begin{equation}\n{\\cal D} p{\\cal D} q{\\cal D} C(\\omega)e^{-i{\\textstyle\\int} dt \\omega^a\\varphi_a}\n\\rightarrow {\\cal D} p{\\cal D} q \\prod_t \\left(\\Delta_{FP}\\prod_a\n\\delta(\\chi_a)\\delta(\\varphi_a)\\right)\\;,\n\\label{f.1}\n\\end{equation}\nwhere $\\Delta_{FP} = \\det \\{\\varphi_a,\\chi_b\\}$ is the Faddeev-Popov\ndeterminant. After the canonical transformation associated with (\\ref{g.2})\nthe Faddeev-Popov measure assumes the form \\cite{3}\n\\begin{equation}\n{\\cal D} p^*{\\cal D} q^*{\\cal D} p^\\theta{\\cal D} \\theta \\prod_t \\delta(p^\\theta)\\delta(\\theta)\\;,\n\\label{f.3}\n\\end{equation}\nand the integration over the nonphysical variables $p^\\theta$ and\n$\\theta$ becomes trivial.\n\nTwo important observations are in order.  First, the procedure\n(\\ref{f.1}) corresponds to a canonical quantization {\\it after}\nthe elimination of all nonphysical degrees of freedom (the so called\nreduced phase-space quantization). As shown above, the physical\nvariables are associated with curvilinear coordinates, while\ncanonical quantization is consistent only in Cartesian coordinates.\nAs a result canonical quantization and the elimination of\nnonphysical degrees of freedom generally do {\\em not} commute \\cite{christ}.\nIn other words, the procedure (\\ref{f.1}) is not, in general, equivalent to\nthe Dirac quantization scheme \\cite{dir} where nonphysical degrees\nof freedom are removed after quantization.\n\nSecond, the geometry and topology of gauge orbits may happen to be\nsuch that there exists no unique gauge condition \\cite{grib}, meaning\nthat for any given $\\chi_a$ the system\n\\begin{equation}\n\\chi_a(q) = \\chi_a(q^{\\omega_s}) =0\n\\label{f.2}\n\\end{equation}\nalways admits nontrivial solutions with respect to $\\omega_s^a$. From\nthe geometrical point of view, the latter implies that the gauge orbit\n$q^\\omega$ intersects the gauge fixing surface more than once, namely, at\npoints $q^{\\omega_s}$. Discrete gauge transformations associated with\nthe gauge variables $\\omega_s^a$ do not reduce the number of physical\ndegrees of freedom, but they do reduce the ``volume\" of the physical\nconfiguration and phase spaces. Therefore the formal measure\n${\\cal D} p^*{\\cal D} q^*$ can no longer be Euclidean and the corresponding path\nintegral should be modified. If the residual discrete gauge transformations\nare explicitly known, then in such cases it appears to be possible to find\na modified path integral formalism that is equivalent to the\nDirac method \\cite{sha2}.\n\nFinally we remark that\nthe Liouville measure ${\\cal D}p^*{\\cal D}q^*= \\prod_tdp^*(t)dq^*(t)$\nis invariant with respect to canonical transformations. This freedom\nin the path integral over physical phase space can be interpreted\nas  gauge invariance. Indeed, another choice of a gauge condition\n(\\ref{g.1}) would induce another parametrization of the physical phase\nspace that is equivalent to the former via a canonical transformation.\nOn the other hand, we have argued in Section 1 that the formal invariance\nof the Liouville measure in the path integral is not sufficient to ensure\nthe invariance of the quantum theory\nwith respect to canonical transformations.\nIn the framework of gauge systems, it implies that, to achieve  gauge\ninvariance of the path integral over physical phase space, the measure\n should be regularized {\\em before} integrating out pure gauge degrees of\nfreedom with the help of a canonical transformation associated\nwith a chosen parametrization of the physical phase space by local\nsymplectic coordinates.\n\nIn the next section we propose a generalization of the path integral\nmeasure regularization with a Wiener measure to gauge theories.\n\n\\section{The Wiener measure for gauge theories}\n\\setcounter{equation}0\n\nThe Wiener measure regularized phase space path integral for a\ngeneral phase function $G(p,q)$ is  given by\n\\bn\n&&\\hskip-.3cm\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\n \\int\\exp\\{i{\\textstyle\\int}_0^T[p_j{\\dot q}^j+{\\dot G}(p,q)-h(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip1.5cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[{\\dot p}^2\n+{\\dot q}^2]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\nonumber\\\\\n &&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J e^{J\\nu T\/2}\n \\int\\exp\\{i{\\textstyle\\int}_0^T[p_jdq^j+dG(p,q)-h(p,q)dt]\\}\\,d\\mu^\\nu_W(p,q)\\nonumber\\\\\n&&\\hskip.3cm=\\<p'',q''|e^{-i{\\cal H} T}|p',q'\\>\\;\\ ,\n\\label{4.1}\n\\en\nwhere the last relation involves a coherent state matrix element.\nIn this expression we note that ${\\textstyle\\int} p_j\\,dq^j$ is a\n{\\it stochastic integral}, and as such we need to give it a definition.\nAs it stands both the It\\^o (nonanticipating) rule and the Stratonovich\n(midpoint) rule of definition for stochastic integrals yield the same\nresult (since $dp_j(t)dq^k(t)=0$ is a valid It\\^o rule in these\ncoordinates). Under any change of canonical coordinates,\nwe consistently will interpret this stochastic integral\nin the Stratonovich sense because it will then obey the ordinary\nrules of calculus.\n\nWhy does the representation of the propagator as well as the Hamiltonian\noperator involve coherent states\n\\bn |p,q\\>\\equiv e^{-iG(p,q)}e^{-iq^jP_j}\ne^{ip_jQ^j}|0\\>\\;,\\hskip1cm(Q^j+iP_j)|0\\>=0\\;?\n\\label{4.3}\n\\en\nOne simple argument is as follows. The Wiener measure\nis on a flat {\\it phase space}, and is pinned at both ends thus resulting\nin the boundary conditions $p(T),q(T)=p'',q''$ and $p(0),q(0)=p',q'$.\nHolding this many end points fixed is incompatible with a Schr\\\"odinger\nrepresentation, which holds just $q(T)$ and $q(0)$ fixed, or with a\nmomentum space representation, which holds just $p(T)$ and $p(0)$ fixed.\nIt turns out, as a consequence of the Wiener measure regularization,\nthat the propagator is {\\it forced} to be in a coherent state\nrepresentation. We also emphasize the covariance of this expression\nunder canonical coordinate transformations. In particular,\nif ${\\overline p}d{\\overline q}=pdq+dF({\\overline q},q)$ characterizes a canonical\ntransformation from the variables $p,q$ to ${\\overline p},{\\overline q}$,\nthen with the Stratonovich rule the path integral becomes\n\\bn\n&&\\<{\\overline p}'',{\\overline q}''|e^{-i{\\cal H} T}|{\\overline p}',{\\overline q}'\\>\\nonumber\\\\\n&&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J e^{J\\nu T\/2}\\int\n\\exp\\{i{\\textstyle\\int}_0^T[{\\overline p}_jd{\\overline q}^j+d{\\overline G}({\\overline p},{\\overline q})-{\\overline h}({\\overline p},\n{\\overline q})dt]\\}\\,d\\mu^\\nu_W({\\overline p},{\\overline q})\\nonumber\\\\\n&&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}_0^T[{\\overline p}_j\n{\\dot{\\overline q}}^j+{\\dot{\\overline G}}({\\overline p},{\\overline q})-{\\overline h}({\\overline p},{\\overline q})dt]\\}\\nonumber\\\\\n&&\\hskip3.2cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[d\\sigma({\\overline p},{\\overline q})^2\/dt^2]\\,\ndt\\}\\,{\\cal D}{\\overline p}\\,{\\cal D}{\\overline q}\\,,\n\\label{4.4}\n\\en\nwhere $\\overline G$ incorporates both $F$ and $G$.\nIn this expression we have set $d\\sigma({\\overline p},{\\overline q})^2=dp^2+dq^2$,\nnamely, the new form of the flat metric in curvilinear phase space\ncoordinates. We emphasize that this path integral regularization\ninvolves Brownian motion on a flat space whatever\nchoice of coordinates is made. Our transformation has also made\nuse of the formal -- and in this case valid -- invariance of the\nLiouville measure.\n\nIf we have auxiliary terms in the classical action representing\nconstraints, then the expression of interest would seem to be\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}_0^T[p_j\n{\\dot q}^j-h(p,q)-\\omega^a\\varphi_a(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[{\\dot p}^2+{\\dot q}^2]\\,\ndt\\}\\,{\\cal D} p\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;,\n\\label{4.5}\n\\en\nwhere the formal measure ${\\cal D} C(\\omega)=\\prod_t\\delta\\omega(t)$\nmay be proposed.\nWe expect some expression of this sort to hold; however,\nthe explicit proposal in (\\ref{4.5}) is incorrect\n as we now proceed to demonstrate.\n\nAccording to the discussion of the previous sections it is clear\nthat the physical propagator may also be given by\n\\begin{equation}\n\\lim_{\\nu\\rightarrow\\infty}\n{\\cal M_\\nu}\\int\\limits_{G}^{}\\delta\\Omega\n\\int\\exp\\{i{\\textstyle\\int}[p_j\n{\\dot q}^j-h(p,q)]\\,dt\\}\\exp\\{-(1\/2\\nu){\\textstyle\\int}{[\\dot p}^2+{\\dot q}^2]\n\\,dt\\}\\,{\\cal D} p{\\cal D} q\\ ;\n\\label{4.6}\n\\end{equation}\nhere all the paths satisfy $p(T),q(T)=p'',q''$ and $p(0),q(0)=p'^{\\,\n\\Omega},q'^{\\,\\Omega}$, where following the notation introduced in\nSection 2, we define\n\\bn\np^\\Omega=e^{-\\Omega^aad\\,\\varphi_a}p\\;,\\hskip1.5cm\nq^\\Omega=e^{-\\Omega^aad\\,\\varphi_a}q\\;.\n\\label{4.7}\n\\en\nIn short, we have used the fact that the unitary operators\nrepresenting the finite gauge group transformations satisfy\nthe condition (\\ref{c.4})\nmapping any coherent state into another coherent state.\n\nBased on the mapping property (\\ref{4.7}), we can give another formulation\nto the path integral (\\ref{4.6}). With the Wiener measure regularization\npresent, the path integral for any finite $\\nu$ is well defined,\nand as such we are free to change variables of integration.\nIn particular, let us make a canonical change of variables so that\n\\bn\n&&p(t)\\rightarrow e^{{\\textstyle\\int}_t^T ds\\omega^a(s)ad\\,\\varphi_a}p(t)\\;,\\nonumber\\\\\n&&q(t)\\rightarrow e^{{\\textstyle\\int}_t^T ds\\omega^a(s)ad\\,\\varphi_a}q(t)\\;,\n\\label{4.9}\n\\en\nwhere $\\omega^a$ are functions of time subject only to the\nrequirement that\n\\bn\n{\\textstyle\\int}_0^T\\omega^a(s)\\,ds\\equiv\\Omega^a\\;.\n\\label{4.10}\n\\en\nClearly, there are infinitely many functions $\\omega^a$ that will\nsatisfy such a criterion, and in a certain sense we will be led to\naverage over ``all'' of them. Note what this change of variables\naccomplishes. In the new variables, whatever the choice of\n$\\omega^a$ may be, the final values remain unchanged, $p(T),q(T)=p'',\nq''$, while the initial values have become $p(0),q(0)=p',q'$ since\n$(p'^{\\,\\Omega})^{-\\Omega}\\equiv p'$ and $(q'^{\\,\\Omega})^{-\\Omega}\n\\equiv q'$. Thus we have transformed all the gauge dependence\nfrom the initial points $p'^{\\,\\Omega},q'^{\\,\\Omega}$ and have\ndistributed it throughout the time interval $T$. This discussion is\nreminiscent of that in Sections 2 and 3.\n\nIt should be remarked that the condition (\\ref{4.10})\n may also be avoided if so desired.\nSuppose we drop the condition (\\ref{4.10}). Let $\\bar{\\Omega}^a$\nbe the value of the integral in the right-hand side of (\\ref{4.10}).\nSince the integral (\\ref{4.6}) involves the average over the gauge\norbit that goes through the initial point $p',q'$, the explicit\ndependence of the boundary condition on $\\bar{\\Omega}^a$ at the initial\ntime can be removed by an appropriate shift of the average parameters\n$\\Omega^a$. The initial boundary condition remains intact\n$p(0),q(0) = p'^\\Omega, q'^\\Omega$ in contrast to the case when\nthe condition (\\ref{4.10}) is imposed. Nevertheless, we proceed on the\nbasis of (\\ref{4.10}).\n\nLet us next see\nwhat are the consequences for the path integral of such a change\nof integration variables. We first observe that\n\\bn\n&&{\\dot p}(t)\\rightarrow{\\dot p}(t)-\\omega^aad\\,\\varphi_ap(t)={\\dot p}\n(t)-\\omega^a\\{\\varphi_a,p\\}(t)\\ ,\\nonumber\\\\\n&&{\\dot q}(t)\\rightarrow{\\dot q}(t)-\\omega^aad\\,\\varphi_aq(t)={\\dot q}(t)-\n\\omega^a\\{\\varphi_a,q\\}(t)\\;.\n\\label{4.11}\n\\en\nSuch a change leads to a new form for the path integral given by\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\n\\int\\limits_{G}^{}\\delta\\Omega\n\\int\\exp\\{i{\\textstyle\\int}[p_j({\\dot q}^j\n-\\omega^a\\{\\varphi_a,q^j\\})-h(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}[({\\dot p}-\\omega^a\\{\\varphi_a,\np\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\;.\n\\label{4.12}\n\\en\nThis relation holds because the formal measure remains invariant\nunder this canonical transformation of coordinates. We recall\nthat in this form the fixed end points are $p(T),q(T)=p'',q''$\nand $p(0),q(0)=p',q'$.\nThis equation is true for any choice of $\\omega^a$ which fulfills\nthe required integral condition (\\ref{4.10}),\nand {\\it a fortiori} it is\nstill true if we average (\\ref{4.12}) over ``all'' functions which satisfy\nthe required integral condition. In so doing let us at the same\ntime incorporate the integral over $\\Omega$ and simply average\nover ``all'' functions $\\omega^a$ directly without any\ncondition on the overall integral value. For now let us continue\nto treat such an average in a formal manner; we will return to\nthe question of a proper average at a later stage. Thus we may\nreplace (\\ref{4.12}) by\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}[p_j({\\dot q}^j\n-\\omega^a\\{\\varphi_a,q^j\\})-h(p,q)]\\,dt\\}\n\\label{4.13}\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}[({\\dot p}\n-\\omega^a\\{\\varphi_a,p\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,\n{\\cal D} p\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;, \\nonumber\n\\en\nwhere $C(\\omega)$ denotes a measure which averages over all\nfunctions $\\omega^a$ as required.\nSince the object under discussion is manifestly gauge invariant,\nit is noteworthy that we can explicitly display such invariance\nunder the gauge transformations\n\\bn\n\\delta p=\\{\\varphi_a,p\\}\\delta\\lambda^a\\;,\\hskip1cm\n\\delta q=\\{\\varphi_a,q\\}\\delta\\lambda^a\\;,\\hskip1cm\n\\delta\\omega^a=\n\\delta\\dot{\\lambda}^a-f_{abc}\\omega^b\\delta\\lambda_c\\;,\n\\label{4.14}\n\\en\nfor general infinitesimal functions $\\delta\\lambda^a(t)$\nwhich vanish at the end points, and\nfor which the indicated path integral is invariant for\nall values of $\\nu$, hence in the limit. Although the\npath integral is invariant under the gauge transformations\nindicated, the reader should not jump to the conclusion that\nthe path integral diverges. In fact, the integral over the\ngauge functions $\\omega^a$ is an {\\it average},\nthat is, ${\\textstyle\\int} {\\cal D}C(\\omega)$ is finite, as we have\nstressed, and for a bounded integrand no divergences are possible.\n\nEquation (\\ref{4.13}) represents a manifestly gauge invariant\nexpression that is covariant under a general canonical\nchange of variables. For the class of constraints under\ndiscussion, we can also present another useful expression.\n Using the identity (\\ref{c.2}) leads to the equivalent relation\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}[p_j\n {\\dot q}^j -\\omega^a\\varphi_a(p,q)-h(p,q)]\\,dt\\}\n \\label{4.15}\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[({\\dot p}-\\omega^a\n\\{\\varphi_a,p\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,{\\cal D}\np\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;, \\nonumber\n\\en\n and once again we recognize the parameters $\\{\\omega^a\\}$ as the\nLagrange mutipliers of the classical theory.\n\nAdditionally, we observe that the drift terms in the\nWiener measure cannot be neglected. For the Brownian motion\nwe have the It\\^o rule $dp(t)^2=\\nu dt$, and the connected expectation value\n$E(p(t)p(s))_{\\rm conn}=\\nu s(1-t\/T)$ for $s<t$. Thus the (Stratonovich)\nstochastic integral $(1\/\\nu){\\textstyle\\int} \\omega^a\\{\\varphi_a,p\\}\\,dp$\nand the term\n$(1\/2\\nu){\\textstyle\\int}[\\omega^a\\{\\varphi_a,p\\}]^2\\,dt$ are both of order\nunity for all values of $\\nu$ since in the general case\n$\\omega^a\\{\\varphi_a,p\\}\\simeq p$. A similar discussion holds\nfor $q$ as well. It is for this reason that our initial naive\nproposal (\\ref{4.5}) is not acceptable.\n\n\\subsubsection*{Choice of measure for the gauge variables}\n\nFinally we take up the question of the choice of the measure\n$C(\\omega)$ and its associated integral. Although we have\nloosely stated that ${\\cal D} C(\\omega)=\\prod_t\\delta\\omega(t)$ and that\nwe should integrate over all functions\n$\\omega^a$, this is still an imprecise concept. Despite appearances, there\nis actually a great deal of choice in this measure.\nThis freedom arises\nbecause the only real requirement on this measure is that it\nsimulates a {\\it single group invariant integral over the initial\nparameters} $p'^{\\,\\Omega},q'^{\\,\\Omega}$ as discussed in (\\ref{4.6}).\nWe shall consider two possible choices. The first one will provide\nus with a manifestly gauge invariant measure, while the second choice\ngives an example of a gauge noninvariant measure which nonetheless leads\nto the gauge invariant transition amplitude. The latter amounts to\nsome specific gauge fixing that is manifestly\n{\\it free} of any Gribov problem.\n\nTo define an appropriate measure\nthat is invariant under general gauge transformations,\nwe appeal to the classical theory of Kolmogorov \\cite{kol}\non stochastic processes, which will ensure that we obtain\na well-defined probability measure on the gauge path space.\nKolmogorov's theorem asserts that an\nunderlying probability measure on paths exists\nprovided the set of multi-time joint probability\ndensities satisfies certain basic consistency conditions.\nTo show the needed consistency let us again use $\\delta\\omega\n$ as the normalized Haar measure (\\ref{haar})\nfor the compact semi-simple gauge group under consideration.\nThen let us introduce a stochastic process defined by the following\nset of multi-time joint probability densities\n\\bn\n{\\cal P}_n(\\omega_n,t_n;\\ldots;\\omega_2,t_2;\\omega_1,t_1)\\equiv 1\n\\label{k.1}\n\\en\nfor all $n\\geq 1$. Here $T\\geq t_n>\\cdots t_2>t_1\\geq 0$.\nThe left-hand side of this equation is\nthe joint probability density for the gauge field\nto have value $\\omega_1$ at time $t_1$, value $\\omega_2$\nat time $t_2$, etc.\nIn this terminology $\\omega=\\{\\omega^a\\}$.\nThe right-hand side of this joint probability density\nrelation is simply unity, meaning that {\\it any} set of values\nat {\\it any} set of distinct times is equally likely.\nThis is the proper mathematical statement of a uniform\naverage over all gauge paths.\nConsistency of the given joint probability\ndensities is simply the trivial\nobservation that\n\\bn\n&&\\int {\\cal P}_n(\\omega_n,t_n;\\ldots;\\omega_r,t_r;\\dots;\\omega_1,t_1)\\,\n\\delta\\omega_r\\nonumber\\\\\n&&\\hskip1cm=1\\nonumber\\\\\n&&\\hskip1cm={\\cal P}_{n-1}(\\omega_n,t_n;\\ldots;\n\\omega_{r+1},t_{r+1};\\omega_{r-1},t_{r-1};\\dots;\\omega_1,t_1)\\;,\n\\label{k.2}\n\\en\nfor any choice or $r$, $n\\geq r\\geq 1$, and all $n$,\n$n\\geq2$; for $n=1$ the last line should be ignored.\nThe evident consistency of this set of joint\nprobability densities is then sufficient to guarantee for\nus a (countably additive) probability measure\non gauge fields, which we denote by $\\rho(\\omega)$,\nthat exhibits these joint probability distributions.\n\nAccepting this choice for the integration over gauge fields\nleads to the fact that the physical propagator\nmay be given the mathematically well-defined formulation\n\\bn\n\\hskip-1cm&&\\<p'',q''|e^{-i{\\cal H} T}|p',q'\\>^{ph}\\nonumber\\\\\n&&=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J\ne^{J\\nu T\/2}\\int\\exp\\{i{\\textstyle\\int}[p_jdq^j+dG(p,q)\n-\\omega^a\\varphi_a(p,q)dt-h(p,q)dt]\\}\\nonumber\\\\\n&&\\hskip3.5cm\\times\\,d\\mu^\\nu_W(p,q,\\omega)\\,d\\rho\n(\\omega)\\;;\n\\label{k.3}\n\\en\nhere we have added $\\omega$ to the argument of $\\mu^\\nu_W$\nto acknowledge the presence of the drift terms.\nThe result only depends on the initial and final values of\n$p$ and $q$ since we have integrated over the set of gauge paths\nwithout any boundary conditions;\nthis result is still invariant under\ncontinuous and differential gauge transformations (\\ref{4.14}).\n\nFinally we note that the relation between\nthe physical Hamiltonian operator and\nthe classical expression $h(p,q)$ is given by\n\\bn\n{\\cal H}_{ph}\\equiv\n\\int h(p,q)\\,|p,q\\>^{ph}\\,^{ph}\n\\<p,q|\\,d^J\\!p\\,d^J\\!q\/(2\\pi)^J\\;,\n\\label{k.4}\n\\en\nwhere the physical coherent state $|p,q\\>^{ph}$ is obtained\nby the average of the coherent state (\\ref{c.4}) over the group\n$G$ with the normalized measure $\\delta\\Omega$.\n\nFormally, the measure $d\\rho(\\omega)$ constructed above comes naturally\nfrom the convolution formula (\\ref{3.1}) where each infinitesimal\ntransition amplitude is to be replaced by the corresponding infinitesimal\namplitude (\\ref{4.6}) with the Wiener measure. In this construction\nthe projection operator is inserted at each moment of time, that is, formally,\n$d\\rho(\\omega) = \\prod_t \\delta \\omega(t)$. Clearly, this formal measure\nsatisfies the conditions (\\ref{k.1}) and (\\ref{k.2}), and in addition it is\nmanifestly gauge invariant and normalized ${\\textstyle\\int} d\\rho(\\omega) =1$.\n\n  However, from the calculational point of view the measure $\\rho(\\omega)$\nis not always convenient. Sometimes it is also useful to have\na measure for the gauge variables that is not explicitly gauge\ninvariant (gauge fixing). A conventional gauge fixing discussed\nin Section 3 may suffer from Gribov ambiguities. Next we\nshow an example of a Gaussian probability measure free of such a disease.\n\nSince we want the measure to have at least one average over\nthe group manifold $G$, it is natural to assume\nthat for any time slice the measure must be the  group\ninvariant measure, but what is at our disposal is the\nrelationship of the functions at neighboring points of time.\nAs one set of examples, it would suffice to restrict our\nintegration to the set, or even a subset, of {\\it continuous\nfunctions}. A natural way to achieve it is to choose ${\\cal D} C(\\omega)$\nto be a Wiener measure on the manifold $G$\n\\bn\n{\\cal D} C(\\omega)= d\\rho_W(\\omega)=\n{\\cal N}\\exp[-\\textstyle{\\frac{1}{2}}{\\textstyle\\int} g_{ab}(\\omega)\n{\\dot\\omega}^a{\\dot\\omega}^b\\,dt]\\,\\Pi_t\\,\\delta\\omega(t)\\;.\n\\label{b.1}\n\\en\nHere the metric $g_{ab}(\\omega)$ is the positive-definite\nmetric associated with a homogeneous space determined by the\ncompact semi-simple gauge group.\nThe measure can also be regarded as the imaginary time quantum dynamics\n of a free particle propagating on the compact\nhomogeneous manifold $G$.\n\nLet us now establish a relation between the projection formula\n(\\ref{4.6}) and (\\ref{4.15}) with the choice (\\ref{b.1})\nfor the measure. Let $g_\\omega$ be an element of the gauge group in a\nmatrix representation. Then the action in the exponential\nin (\\ref{b.1}) can also be rewritten as\n\\begin{equation}\nS_W =-c\\,{\\rm tr}\\int\\limits_{0}^{T}(\\dot{g}_\\omega\ng_\\omega^{-1})^2\/2dt\\ ,\n\\label{b.2}\n\\end{equation}\nwhere $c=1\/{\\rm tr}(1)$ is a normalization factor.\nConsider a transition amplitude of\na free particle on the manifold $G$\n\\begin{equation}\nK_T(g_\\Omega, g_{\\Omega'}) ={\\cal N}\n\\int\\limits_{g_\\omega(0)=g_{\\Omega'}}^{g_\\omega(T)= g_\\Omega}\n\\prod\\limits_{t=0}^T \\delta\\omega(t) e^{-S_W}\\ ,\n\\label{b.3}\n\\end{equation}\nnormalized so as to satisfy\n\\begin{equation}\nK_T(g_{\\Omega''},g_{\\Omega'})=\\int K_{T-t}\n (g_{\\Omega''},g_\\Omega)K_t(g_\\Omega,g_{\\Omega'})\\,\\delta\\Omega\\,.\n\\label{b.3a}\n\\end{equation}\nDue to the global invariance of the action with respect to the left and\nright shifts, $g_\\omega\\rightarrow g_0g_\\omega$ and\n$g_\\omega\\rightarrow g_\\omega g_0$,  the amplitude (\\ref{b.3}) is\nalso invariant under these transformations\n\\begin{equation}\nK_T(g_\\Omega, g_{\\Omega'}) =\nK_T(g_0g_\\Omega,g_0 g_{\\Omega'}) =\nK_T(g_\\Omega g_0, g_{\\Omega'}g_0) \\ .\n\\label{b.4}\n\\end{equation}\nFrom (\\ref{b.4}) we deduce the identity\n\\begin{equation}\n\\int\\limits_{G}^{}\\delta\\Omega''\nK_T(g_{\\Omega''}, g_{\\Omega'}) =\n\\int\\limits_{G}^{}\\delta\\Omega'\nK_T(g_{\\Omega''}, g_{\\Omega'}) =  1\\ ,\n\\label{b.5}\n\\end{equation}\n which can be easily seen from the Feynman-Kac representation\nof the transition amplitude (\\ref{b.3}) as a spectral sum.\nThe integral (\\ref{b.5}) determines an action of the evolution\noperator on the ground state of the system. So, only the ground\nstate will contribute to the integral. We naturally assume that\nthe Casimir energy (the ground state energy) can always be\nsubtracted and included into the path integral normalization.\n\nNow we insert the identity (\\ref{b.5}) into the measure of the\npath integral (\\ref{4.6}) and then proceed with the change\nof variables (\\ref{4.9}). Since in the identity (\\ref{b.5})\neither $\\Omega''$ or $\\Omega'$ is a free parameter, we can\nalways choose it to coincide with the parameter $\\Omega$\nof the $G$-average in (\\ref{4.6}). Substituting\nthe path integral representation of $K_T$ (\\ref{b.3})\nin the appropriately transformed integral (\\ref{4.6}), we arrive\nat the expression (\\ref{4.15}) with the Wiener measure\n(\\ref{b.1}) for the gauge variables.\n\nTypically we\nencounter Wiener measures that are pinned at either the\ninitial time or at both end points; in the present case,\nthe measure for gauge variables\nis neither pinned at the initial nor the final\ntime as seen from the derivation of (\\ref{4.15}).\nSince the group is compact, the group volume is finite\nand we may therefore normalize such a Wiener measure that is\nnot pinned; our normalization is such that\n\\begin{equation}\n\\int{\\cal D} C(\\omega)=\n\\int\\limits_{G}^{}\\delta\\Omega''\\delta\\Omega'\nK_T(g_{\\Omega''}, g_{\\Omega'}) =  1\\ .\n\\label{b.6}\n\\end{equation}\nIn that case the formal measure ${\\cal D} C(\\omega)$ is actually a\nwell-defined (countably additive) probability measure which\nwe denote by $d\\rho_W(\\omega)$.\nWith this choice we note that\nthe physical propagator may also be given the well-defined\ndefinition (\\ref{k.3}) where $d\\rho(\\omega)\\rightarrow\nd\\rho_W(\\omega)$.\nThe result only depends on the\ninitial and final values of $p,q$ since we have integrated\nover the set of continuous $\\omega^a$ paths without any\nboundary conditions.\n\nThe measure is not invariant under the gauge transformations\n(\\ref{4.14}), nonetheless the transition amplitude is gauge\ninvariant because the measure provides the necessary projection\nonto gauge invariant states. In contrast to the conventional\nprocedure of section 3, there is no explicit gauge condition\nimposed on the system of phase space variables, and hence the\nGribov problem is avoided.\n\nOne should add that two such propagators, one from $t=0$ to\n$t=T$ and the second from $t=T$ to $t=2T$, for example,\nseems to\nnot compose to a propagator of the same form as (\\ref{k.3})\ndue to the discontinuity of paths at the interface.\nHowever, the resultant propagator  is nonetheless correct;\nit simply involves another acceptable form for the measure\n${\\cal D}C(\\omega)$.\n\n\n\\subsubsection*{Conclusion}\n\n\nWith (\\ref{4.15}) and two  choices\nof the measure for the gauge variables, we have arrived at\nour coordinate-free and mathematically\nwell-defined formulation for the path integral representation of\nthe special class of first-class constraints that was our goal.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}