{"text":"\\section{Introduction}\nThe emergence of hierarchies is a common phenomenon in societies and animal\ngroups. In a pioneering work, Bonabeau {\\it et al.}\\cite{bonabeau} have\nshown that a hierarchical society can emerge spontaneously from an\nequal society by a simple algorithm of fighting between individuals\ndiffusing on a square lattice.\nOn the basis of results of Monte Carlo simulation and an analysis by a mean\nfield theory, they concluded that subcritical or supercritical bifurcations\nexist in the formation of the hierarchical structure as the density of\nindividuals is varied. \nIn their model, each individual is assumed to have some wealth or power which\nincreases or decreases by winninng or losing in a fight.\nThe essential processes of the model are diffusion, fighting\nand spontaneous relaxation of the wealth.\nVarious societies can be modelled by specifying each process and\nthe emergence of the hierarchy depends strongly on the\nspecifications.\\cite{sousa,stauffer}\n\nIn this paper, we investigate a variation of the model introduced\nby Bonabeau {\\it et al.}\\cite{bonabeau},\nwhere the diffusion algorithm is modified to\ninclude the effect of the trend of society. Namely, we study the emergence of\nhierarchies in a timid society, in which an individual always tries\nto avoid fighting and to fight with the weakest among the neighbors\nif he\/she cannot avoid fighting.\nBy Monte Carlo simulation, we show that the emergence of the hierarchy\nis retarded in the timid society compared to the no-preference society\ninvestigated by Bonabeau {\\it et al.} and that the transition\nto the hierarchical state occurs in two successive transitions of\na continuous and a discontinuous ones.\nConsequently, there exist\nthree different states in the society, one equal and two hierarchical states.\nIn the first hierarchical states, we see no winners but losers and\npeople in the middle class. In the second hierarchical states, \nmany winners emerge from the middle class.\nWe also show that the distribution of wealth in the second hierarchical\nstate is wider compared to the hierarchical state of the no-preference society.\n\nIn Sec. 2, our model is explained in detail.\nResults of Monte Carlo simulation are presented in Sec. 3.\nIn Sec. 4 the characteristics of the hierarchical\nstates is analyzed in detail. Section 5 is devoted to discussion.\n\n\\section{A timid society}\nWe consider $N$ individuals diffusing on an $L \\times L$ square lattice,\nwhere every lattice site accomodates at most one individual.\nAn individual is to move to one of nearest neighbor sites\naccording to the following protocol.\nWhen individual $i$ tries to move to a site occupied by $j$,\n$i$ and $j$ fight each other. If $i$ wins, $i$ and $j$ exchange their\npositions, and if $i$ loses, they keep their original positions.\nWe associate each individual a quantity which we call power or wealth.\nThe power increases by unity for every victory and decreases\nby unity for every loss.\nThe probability $Q_{ij}$ that $i$ wins the fight against $j$ is determined\nby the difference of their powers $F_i$ and $F_j$ as\n\\begin{equation}\nQ_{ij} = \\frac{1}{1 + \\exp[\\eta(F_j - F_i)]} ,\n\\label{probability}\n\\end{equation}\nwhere $\\eta$ is introduced as a controlling parameter.\nWhen $\\eta = \\infty$, the stronger one always wins the fight\nand when $\\eta = 0$, the winning probability of both ones are equal.\nWe also assume that the power of individuals relaxes to zero when \nthey do not fight, namely power $F_i(t+1)$ at time $t+1$ is given by\n$F_i(t)$  through\\cite{bonabeau}\n\\begin{equation}\nF_i (t+1) = F_i(t) - \\mu \\tanh[F_i(t)] .\n\\end{equation}\nHere, the unit of time is defined by one Monte Carlo step\nduring which every individual is accessed once for move and\n$\\mu$ represents an additional controlling parameter.\nThis relaxation rule indicates that people lose their wealth\nof a constant amount when their power is large, and\nwhen their power is small, they lose it at a constant fraction,\nnamely they behave rather miserly.\nIt also indicates that the negative wealth (debt) can relax to zero\nin the similar manner.\nNote that this rerlaxation rule is critical to the emergence of\nhierarchical society.\\cite{sousa}\n\nWe characterize the timid society by the preference of\nindividuals in diffusion. In the timid society, every individual favors\nnot to fight and thus it moves to a vacant site if it exists. If no vacant\nsites exist in the nearest neighbors, then it moves to a site occupied by\nan individual whose power is the smallest among the neighbors.\nWhen more than two neighbors have the equal power, then an opponent\nis chosen randomly from them.\n\nWe characterize the static status of the society by an order parameter $\\sigma$\nwhich is defined by\\cite{bonabeau, sousa}\n\\begin{equation}\n\\sigma^2 = \\frac{1}{N}\\sum_{i}\\left\\{\n\\frac{D_i}{D_i+S_i} -\\frac{1}{2}\\right\\}^2 .\n\\label{order}\n\\end{equation}\nHere, $N$ is the number of individuals, and\n$D_i$ and $S_i$ are the number of fights won and lost, respectively,\nby individual $i$. Note that $\\sigma = 0$ corresponds to an egalitarian\nstatus and $\\sigma = 1\/\\sqrt{12} \\simeq 0.2887$ when the chance for victory\n$\\frac{\\displaystyle D_i}{\\displaystyle D_i+S_i}$ is distributed\nuniformly in $[0, 1]$. After sufficiently long Monte Carlo simulation,\nvariation of $\\sigma$ is stabilized and one can use it as an order parametrer.\n\nWe also monitor the population profile by focusing on the winning probability.\nWe classify individuals into three groups by the number of fights which\nan individual won; winners are individuals who won more than 2\/3 of fights\nand losers are individuals who won less than 1\/3 of fights.\nIndividuals between these groups are called middle class.\n\n\\section{Monte Carlo simulation}\nMonte Carlo simulation was performed for $N = 3500$ individuals\non the square lattice\nwith periodic boundary conditions from $L = 60$ to $L = 180$.\nWe obtained the order parameter $\\sigma^2$ and other quantities for\n$10^6$ Monte Carlo steps. \n \n\\begin{figure}[thb] \n\\begin{center} \n\\includegraphics[height=8cm]{Fig1.eps} \n\\end{center} \n\\caption{Order parameter $\\sigma ^2$ as a function of $\\rho=N\/L^2$ with\n $\\mu=0.1$, for four different values of $\\eta$:\n$\\eta =$ $50(\\Box)$, $0.5(\\times)$, $0.05(\\bigcirc)$, $0.005(\\triangle)$.\nError bars are much smaller than the size of symbols.}\n\\end{figure} \n\nFigure 1 shows the dependence of the order parameter $\\sigma^2$\non the density $\\rho = N\/L^2$ for several values of $\\eta$, where\n$\\mu$ is fixed to $\\mu = 0.1$.\nWe can see two clear transitions; one at a lower critical density\n$\\rho_{C1}$ and the other at a higher critical density $\\rho_{C2}$.\nThe transition at $\\rho_{C1}$ is continuous and the transition\nat $\\rho_{C2}$ is discontinuous.\nThe dependence of the critical densities $\\rho_{C1}$ and $\\rho_{C2}$\non parameter $\\eta$ is shown in Fig. 2.\n\n\\begin{figure}[bht] \n\\begin{center} \n\\includegraphics[height=8cm]{Fig2.eps} \n\\end{center} \n\\caption{The dependence of the critical densities $\\rho_{C1}$\n (the circles) and $\\rho_{C2}$ (the crosses) on parameter $\\eta$,\nwhere $\\mu=0.1$. The curves are the guide for eyes.}\n\\end{figure} \n\nWe can identify three states for a given value of $\\eta$;\nan egalitarian state for $\\rho \\underline{<} \\rho_{C1}$, a hierarchical\nsociety of type I for $\\rho_{C1} \\underline{<} \\rho \\underline{<} \\rho_{C2}$\nand a hierachical society  of type II for $\\rho_{C2} \\underline{<} \\rho \\underline{<} 1$.\nIn the egalitarian society, winners and losers lose their\nmemory of previous fight before they engage in the next fight, and\nthus they changes their status in time.\nIn the hierarchical state, a winner keeps winning and a loser keeps losing.\nWe discuss the difference between type I and type II hierarchical society\nin the next section.\n\nThe results strongly depend on $\\mu$. We show the phase\nboundary on the $\\rho$-$\\mu$ plane for $\\eta = 0.05$ in Fig. 3.\n\\begin{figure}[bht] \n\\begin{center} \n\\includegraphics[height=8cm]{Fig3.eps} \n\\end{center} \n\\caption{The dependence of the critical densities $\\rho_{C1}$\n (the circles) and $\\rho_{C2}$ (the crosses) on parameter $\\mu$\nfor $\\eta = 0.05$. The curves are the guide for eyes.}\n\\end{figure} \n\n\\section{Two hierarchical societies}\nIn order to investigate the structure of the hierarchical states,\nwe analyze profile of population.\nThe dependence of the population of each class is plotted\nagainst the density in Fig. 4.\nRapid changes of the populations signify emergence of different\nstate of the hierarchical societies.\nIn the egalitarian state $\\rho \\underline{<} \\rho_{C1}$, \nall individuals belong to the middle class as expected.\nIn the hierachical society I $\\rho_{C1} \\underline{<} \\rho \\underline{<} \\rho_{C2}$,\nsome individuals become losers whose number increases as\nthe density is increased, but no winners are seen.\nIn the hierachical state II $\\rho \\geq \\rho_{C2}$,\nmany winners appear and the population in the middle\nclass is reduced significantly.\n\\begin{figure}[thb] \n\\begin{center} \n\\includegraphics[height=8cm]{Fig4.eps} \n\\end{center} \n\\caption{Dependence of the population in each class on the density\nwhen $\\mu = 0.1 $ and $\\eta = 0.1$.\n}\n\\end{figure} \n\nFigures 5 (a), (b) and (c) show the spacial distribution of individuals after\n $10^6$ Monte Carlo steps in the egalitarian, the first hierarchical and the\nsecond hierarchical states, respectively. No specific spatial\ninhomogeneity is observed in the timid society.\n\n\\begin{figure}[htb] \n\\begin{center}\n(a) \\includegraphics[height=6.3cm]{Fig5-a.eps} \\\\\n(b) \\includegraphics[height=6.3cm]{Fig5-b.eps} \\\\\n(c) \\includegraphics[height=6.3cm]{Fig5-c.eps}\n\\end{center}\n\\caption{Structure in the quilibrium state. (a) the eagalitarian state\n$\\rho = 0.3$,\n(b) the hierarchical socity I $\\rho = 0.5$\nand (c) the hierarchical socity II $\\rho = 0.7$.\nthe circles are the winner, the traiangles are individuals in the\nmiddle class and the crosses are the loser.}\n\\end{figure} \n\n\nIn order to see details of the hierarchical structure, we plot the\npopulation as a function of the density and the winning probablity in Fig. 6.\nFrom this plot, we conclude that (1) in the hierarichical state I,\npeople in the middle class with slightly higher winning probability\nincrease, but no winners are seen and (2) in the hierarchical state II,\nthe most of winners have very high winning probability, while people in the\nlosers and the middle class are distributed in a wide region of the winning\nprobability, \n\n\\begin{figure}[bht] \n\\begin{center} \n\\includegraphics[height=8cm]{Fig6.eps}\n\\end{center} \n\\caption{Population as a function of the density and\nthe winning rate. $X = D_i\/(D_i + S_i)$.}\n\\end{figure}\n\\section{Discussion}\nWe have investigated the emergence of self-organized hierarchies\nin the timid society.\nOur results show that the emergence of the hierarchical state\nin the timid society is retarded compared to the no-preference society.\nThis delay is natural since individuals in the timid society\ntend to avoid fighting and thus the wealth is distriruted\nmore or less evenly among individuals when the population is low.\nFurthermore, the emergence of the hierarchical society in the timid\nsociety occurs in two steps, and the first transition is continuous\nand the second one is discontinuous.\nThe strength of the hierarchy in the high density region is stronger\nin the timid society compared to the no-preference society.\nFor the same choice of $\\eta = 0.05$ and $\\mu = 0.1$, $\\sigma^2$\nfor the former case is twice as large as the latter\\cite{bonabeau}.\n\nTo understand these behaviors, we first remind the fact that the\nthe hierarchical society emerges when the power cannot relax before the\nsubsequet fight. In the timid society, an idividual can avoid fighting\nwhen the density is low, and  thus the ealitarian state is favored\nfor low densities. In the timid society, weaker individuals have more chance\nto be challenged and thus to lose their power, and stronger ones has less\nchance to fight and their power stay near zero. This situation corresponds\nto the hierarchical state I.\nWhen the density is increased above the upper critical density,\nall individuals have more chance to fight and thus stronger individuals become\nmuch stronger.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\\label{section1}\n\nA vast number of astronomical observations suggests that magnetic\nfields play a crucial role in the dynamics of many phenonema of\nrelativistic astrophyics, either on stellar scales, such as for\npulsars, magnetars, compact X-ray binaries, short and long\/gamma-ray\nbursts (GRBs) and possibly for the collapse of massive stellar cores,\nbut also on much larger scales, as it is the case for radio galaxies,\nquasars and active galactic nuclei (AGNs). A shared aspect in all\nthese phenomena is that the plasma is essentially electrically neutral\nand the frequency of collisions is much larger than the inverse of the\ntypical timescale of the system. The MHD approximation is then an\nexcellent description of the global properties of these plasmas and\nhas been employed with success over the several decades to describe\nthe dynamics of such systems well in their nonlinear regimes. Another\nimportant common aspect in these systems is that their flows are\ncharacterized by large magnetic Reynolds numbers ${\\cal R}_{\\rm M} = L\nV\/ \\lambda = 4 \\pi \\sigma L V\/c^2$, where $L$ and $V$ are the typical\nsizes and velocities, respectively, while $\\lambda$ is the magnetic\ndiffusivity and $\\sigma$ is the electrical conductivity.\nFor a typical relativistic compact object, ${\\cal R}_{\\rm M} \\gg 1$ and,\nunder these conditions, the magnetic field is\nessentially advected with the flow, being continuosly distorted and\npossibly amplified, but also essentially not decaying. We note that\nthese conditions are very different from those traditionally produced\nin the Earth's laboratories, where ${\\cal R}_{\\rm M} \\ll 1$, and the\nresistive diffusion represents an important feature of the\nmagnetic-field evolution.\n\nA particularly simple and yet useful limit of the MHD approximation is\nthat of the \\textit{``ideal-MHD''} limit. This is mathematically\ndefined as the limit in which the electrical resistivity $\\eta \\equiv\n1\/\\sigma$ vanishes or, equivalently, by an infinite electrical\nconductivity. It is within this framework that many multi-dimensional\nnumerical codes have been developed over the last decade to study a\nnumber of phenomena in relativistic astrophysics and in fully\nnonlinear regimes \\citep{Kom:1999b,KoiShiKud:1999,Kom:2001,\n  KolRomUstLov:2002,GamMckTot:2003,DelBucLon:2003,AnnFraSal:2005,\n  DueLiuShaSte:2005,\n  ShiSek:2005,NeiHirMil:2006,AZMMIFP:2006,McKinney:2006, MigBod:2006,\n  Noble:2007zx, GiaRez:2007, DelZanBucLon:2007, FarLiLiuShap:2008}.\nThe ideal-MHD approximation is not only a convenient way of writing\nand solving the equations of relativistic MHD, but it is also an\nexcellent approximation for any process that takes place over a\ndynamical timescale. In the case of an old and ``cold'' neutron star,\nfor example, the electrical and thermal transport properties of the\nmatter are mainly determined by the transport properties of the\nelectrons, which are the most important carriers of charge and\nheat. At temperatures above the crystallization temperature of the\nions, the electrical (and thermal) conductivities are governed by\nelectron scattering off ions and an approximate expression for the\nelectrical conductivity is given by~\\citep{L:91} $\\sigma \\approx\n10^{24} \\left({10^9\\ \\rm K}\/{T}\\right)^2 \\left({\\rho}\/{10^{14}\\ {\\rm\n    g\\ cm}^{-3}}\\right)^{3\/4} \\ {\\rm s}^{-1}$, where $T$ and $\\rho$\nare the stellar temperature and mass density\\footnote{Note that this\n  expression for the electrical conductivity is roughly correct for\n  densities in the range $10^{10}-10^{14}$ g cm$^{-3}$ and\n  temperatures in the range $10^{6}-10^{8}$ K, but provides a\n  reasonable estimate also at larger temperatures of $\\sim\n  10^{9}-10^{10}$ K [\\textit{cf.}~.~\\citet{PBHY:99}].}. Even for a magnetic\nfield that varies on a length-scale as small as $L \\simeq 0.1 R$,\n(where $R$ is the stellar radius) the magnetic diffusion timescale is\n$\\tau_{\\rm diff} = {4\\pi L^2 \\sigma}\/{c^2} \\approx 3 \\times 10^{6}\n{\\rm yr}$.\n\nClearly, at these temperatures and densities, Ohmic diffusion will be\nneglible for any process taking place on a dynamical timescale for the\nstar, \\textit{i.e.},~ $\\lesssim {\\rm few\\ s}$, and thus the conductivity can be\nconsidered as essentially infinite. However, \ncatastrophic events, such as\nthe merger of two neutron stars, or of a neutron star\nwith a black hole, can produce plasmas with regions\nat much larger temperatures (\\textit{e.g.},~ $T\\sim 10^{11-13}\\ {\\rm K}$) and much\nlower densities (\\textit{e.g.},~ $\\rho \\sim 10^{8-10}\\ {\\rm g cm}^{-3})$. In such\nregimes, all the transport properties of the matter will be\nconsiderably modified and non-ideal effects, absent in perfect-fluid\nhydrodynamics (such as bulk viscosity) and ideal MHD (such as Ohmic\ndiffusion on a much shorter timescale $\\tau_{\\rm diff} \\sim 10^3 {\\rm\n  s}$) will need to be taken into account. Similar conditions are\nlikely not limited to binary mergers but, for instance, be present also behind\nprocesses leading to long GRBs, thus extending the range of\nphenomena for which resistive effects could be important. Note also\nthat these non-ideal effects in hydrodynamics (MHD) are proportional\nnot only to the viscosity (resistivity) of the plasma, but also to the\nsecond derivatives of the velocity (magnetic) fields. Hence, even in\nthe presence of a small viscosity (resistivity), their contribution to\nthe overall conservation of energy and momentum can be considerable if\nthe velocity (magnetic) fields undergo very rapid spatial variations\nin the flow. A classical example of the importance of resistive MHD\neffects in plasmas with high but finite conductivities is offered by\n{\\it current sheets}. These phenomena are often observed in the solar\nactivity and are responsible for the reconnection of magnetic field\nlines and changes in the magnetic field topology. While these\nphenomena are behind the emission of large amounts of energy, they are\nstrictly forbidden within the ideal-MHD limit due to magnetic flux\nconservation and so can not be studied employing this limit.\n\nBesides having considerably smaller conductivities, low-density higly\nmagnetized plasmas are present rather generically around magnetized\nobjects, constituting what is referred to as the ``magnetosphere''. \nIn such regions magnetic stresses are much larger than magnetic\npressure gradients and cannot be properly balanced; as a result, the magnetic\nfields have to adjust themselves so that the magnetic stresses\nvanish identically. This scenario is known as the \\textit{force free}\nregime (because the Lorentz force vanishes in this case) and while the\nequations governing it can be seen as the low-inertia limit of the\nideal-MHD equations~\\citep{Kom:2002,McKinney:2006tf}, the force-free\nlimit is really distinct from the ideal-MHD one. This represents a\nconsiderable complication since it implies that it is usually not\npossible to decribe, within the same set of equations, both the\ninterior of compact objects and their magnetospheres.\n\nTheoretical work to derive a fully relativistic theory of non-ideal\nhydrodynamics and non-ideal MHD has been carried out by several\nauthors in the past \\citep{Israel:1976,Stewart:1977,Carter:1991,Lichnerowizc:1967,Anile:1989}\nand is particularly simple in the case of the resistive MHD description. The purpose of\nthis work is indeed that of proposing the solution of the relativistic\nresistive MHD equations as an important step towards a more realistic\nmodelling of astrophysical plasmas. There are a number of advantages\nbehind such a choice. First, it allows one to use a single\nmathematical framework to describe both regions where the conductivity is\nlarge (as in the interior of compact objects) and small (as\nin magnetospheres), and even the vacuum regions outside the\ncompact objects where the MHD equations trivially reduce to the\nMaxwell equations. Second, it makes it possible to account\nself-consistently for those resistive effects, such as current sheets,\nwhich are energetically important and could provide a substantial\nmodification of the whole dynamics. Last but not least, the numerical\nsolution of the resistive MHD equations provides the only way to\ncontrol and distinguish the physical resistivity from the numerical\none. The latter, which is inevitably present and proportional to\ntruncation error, is also completely dependent on the specific details\nof the numerical algorithm employed and on the resolution used for the\nsolution.\n\nAs noted already by several authors, the numerical solution of the\nideal-MHD equations is considerably less challenging than that of the\nresistive MHD equations. In this latter case, in fact, the equations\nbecome mixed hyperbolic-parabolic in Newtonian physics or hyperbolic\nwith stiff relaxation terms in special relativity. The presence of\nstiff terms is the natural consequence of the fact that the diffusive\neffects take place on timescales that are intrinsically larger than\nthe dynamical one. Stated differently, in such equations the\nrelaxation terms can dominate over the purely hyperbolic ones,\nposing severe constraints on the timestep for the evolution. While\nconsiderable work has already been made to introduce numerical\ntechniques to achieve efficient implementations in either\nregime~\\citep{Kom:2004, KomBarLyu:2007, Kom:2007, ReySamWoo:2006, GraTreMilCol:2008},\nthe use of these techniques in fully three-dimensional simulations is still\ndifficult and expensive.\n\nIn order to benefit from the many advantages discussed above in the use\nof the resistive MHD equations, we here present a novel approach for\nthe solution of the relativistic resistive MHD equations exploiting\nthe properties of implicit-explicit (IMEX) Runge Kutta methods. This\napproach represents a simple but effective solution to the problem of\nthe vastly different timescales without sacrificing either\nsimplicity in the implementation or the numerical efficiency. By\nexamining a number of tests we illustrate the accuracy of our approach\nunder a variety of conditions and demonstrate its robustness. In\naddition, we also compare it with the alternative method proposed\nby~\\citet{Kom:2007} for the solution of the same set of relativistic\nresistive MHD equations. This latter approach employs\nStrang-splitting techniques and the analytical integration of a reduced\nform of Ampere's law. While it works well\nin a number of cases, it has revealed to be unstable when\napplied to discontinuous flows with large conductivities; such\ndifficulties were not encountered when solving the same problem within the\nIMEX implementation.\n\nBecause our approach effectively treats within a unified framework\nboth those regions of the flow which are fluid-pressure dominated and\nthose which are instead magnetic-pressure dominated, it could find a\nnumber of applications and serve as a first step towards a more\nrealistic modeling of relativistic astrophysical plasmas.\n\nOur work is organized as follows. In Sect.~\\ref{section2} we present\nthe system of equations describing a resistive magnetized fluid, while\nin Section \\ref{section3} we discuss the problems related to the\nnumerical evolution of this system of equations and the numerical\napproaches developed to solve them. In particular, we introduce\nthe basic features of the IMEX Runge-Kutta schemes and recall their\nstability properties. In Sect~\\ref{section4} we instead explain in\ndetail the implementation of the IMEX scheme to the resistive MHD\nequations. Finally, in Sect.~\\ref{section5} we present the numerical\ntests carried out either in one or two dimensions and that span\nseveral prescriptions for the conductivity. Section \\ref{section5} is\nalso dedicated to the comparison with the Strang-splitting technique.\nThe conclusions and the perspectives for future improvements are\npresented in Sect.~\\ref{section6}, while Appendix~\\ref{appendixB}\nreviews our space discretization of the equations.\n\nHereafter we will adopt Gaussian units such that $c=1$ and employ the\nsummation convention on repeated indices. Roman indices $a,b,c,...$\nare used to denote spacetime components (\\textit{i.e.},~ from $0$ to $3$), while\n$i,j,k,...$ are used to denote spatial ones; lastly, bold italics\nletters represent vectors, while bold letters represent tensors.\n\n\n\n\n\\section{The resistive MHD description}\\label{section2}\n\nAn effective description of a fluid in the presence of electromagnetic\nfields can be made by considering three different sets of equations\ngoverning respectively the electromagnetic fields, the fluid variables\nand the coupling between the two. In particular, the electromagnetic\npart can be described via the Maxwell equations, while the\nconservation of energy and momentum can be used to express the\nevolution of the fluid variables. Finally, Ohm's law, whose exact form\ndepends on the microscopic properties of the fluid, expresses the\ncoupling between the electromagnetic fields and the fluid\nvariables. In what follows we review these three sets of equations\nseparately, discuss how they then lead to the resistive MHD\ndescription, and how the latter reduces to the well-known limits of\nideal-MHD and of the Maxwell equations in vacuum. Our presentation\nwill be focussed on the special-relativistic regime, but the extension\nto general relativity is rather straightforward and will be presented\nelsewhere.\n\n\n\\subsection*{The Maxwell equations}\n\nThe special relativistic Maxwell equations can be written\nas~\\citep{LanLif:1980}\n\\begin{eqnarray}\\label{maxwell_covariant1}\n   \\partial_b F^{ab} &=& I^a\\,, \\\\\n\\label{maxwell_covariant2}\n   \\partial_b ^{~*}\\!F^{ab} &=& 0\\,, \n\\end{eqnarray}\nwhere $F^{ab}$ and $^{~*}\\!F^{ab}$ are the Maxwell and the Faraday\ntensor respectively and $I^a$ is the electric current 4-vector. A\nhighly-ionized plasma has essentially zero electric and magnetic\nsusceptibilities and the Faraday tensor is then simply the dual of the\nMaxwell tensor. This tensor provides information about the electric\nand magnetic fields measured by an observer moving along any timelike\nvector $n^a$, namely\n\\begin{eqnarray}\\label{maxwell_tensor} \n   F^{ab} = n^a E^b - n^b E^a + \\epsilon^{abc} B_c\\,.\n\\end{eqnarray}\nWe are considering  $n^a$ to be the time-like traslational\nkilling vector field in a flat (Minkowski) spacetime, so $n_a=(-1,0,0,0)$\nand the Levi-Civita symbol $\\epsilon^{abc}$ is non-zero only for spatial indices.\nNote that the electromagnetic fields have no components\nparallel to $n^a$ (\\textit{i.e.},~ $E^a~n_a=0=B^a~n_a$).\n\n\nBy using the decomposition of the Maxwell tensor\n(\\ref{maxwell_tensor}), the equations\n(\\ref{maxwell_covariant1})--(\\ref{maxwell_covariant2}) can be split\ninto directions which are parallel and orthogonal to $n^a$ to yield\nthe familiar Maxwell equations\n\\begin{eqnarray}\n   \\nabla \\cdot {\\boldsymbol E} &=& q\\,,\n\\label{maxwell_clasic1} \\\\    \n   \\nabla \\cdot {\\boldsymbol B} &=& 0\\,,\n\\label{maxwell_clasic2} \\\\\n   \\partial_t {\\boldsymbol E} - \\nabla \\times {\\boldsymbol B}  &=& - {\\boldsymbol J} \\, ,\n\\label{maxwell_clasic3} \\\\\n   \\partial_t {\\boldsymbol B} + \\nabla \\times {\\boldsymbol E} &=& 0 \\,,\n\\label{maxwell_clasic4} \n\\end{eqnarray}\nwhere we have decomposed also the current vector $I^a = q n^a + J^a$, with\n$q$ being the charge density, $qn^a$ the convective current and $J^a$\nthe conduction current satisfying $J^a~n_a=0$.\n\nThe current conservation equation $\\partial_a I^a = 0$ follows from\nthe antisymmetry of the Maxwell tensor and provides the evolution of\nthe charge density $q$\n\\begin{eqnarray}\\label{current_conservation_clasic}\n   \\partial_t q + \\nabla \\cdot {\\boldsymbol J} = 0\\,,\n\\end{eqnarray}\nwhich can be obtained also directly by taking the divergence of\n(\\ref{maxwell_clasic3}) when the constraints\n(\\ref{maxwell_clasic1})--(\\ref{maxwell_clasic2}) are satisfied.\n\n\n\\subsection*{The hydrodynamic equations}\n\nThe evolution of the matter follows from the conservation of the stress-energy\ntensor\n\\begin{eqnarray}\\label{conservation_Tmunu} \n   \\partial_b  T^{ab} = 0 \\, ,\n\\end{eqnarray}\nand the conservation of baryon number\n\\begin{eqnarray}\\label{conservation_baryons} \n   \\partial_a  (\\rho u^a) = 0 \\, ,\n\\end{eqnarray}\nwhere $\\rho$ is the rest-mass density (as measured in the rest frame\nof the fluid) and $u^a$ is the fluid 4-velocity. The stress-energy\ntensor $T^{ab}$ describing a perfect fluid minimally coupled to an\nelectromagnetic field is given by the superposition\n\\begin{eqnarray}\\label{full_Tmunu} \n   T_{ab} &=& T_{ab}^{\\rm fluid} + T_{ab}^{\\rm em} \\, , \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{Tmunu_em} \n    T^{ab}_{\\rm em} &\\equiv& F^{ac} F^b_c - \\frac{1}{4} (F^{cd} F_{cd}) g^{ab} \\, , \\\\\n\\label{Tmunu_fluid} \n    T^{ab}_{\\rm fluid} &\\equiv& h u^a u^b + p~g^{ab} \\, .\n\\end{eqnarray}\nHere $h \\equiv \\rho (1 + \\epsilon) + p$ is the enthalpy, with $p$ the\npressure and $\\epsilon$ the specific internal energy.\n\nThe conservation law (\\ref{conservation_Tmunu}) can be split into\ndirections parallel and orthogonal to $n^a$ to yield the familiar\nenergy and momentum conservation laws\n\\begin{eqnarray}\n    \\partial_t \\tau + \\nabla \\cdot \\boldsymbol{F}_{\\tau} = 0 \\, , \n\\label{fluid_tau} \\\\\n    \\partial_t {\\boldsymbol S} + \\nabla \\cdot {\\bf F}_{\\boldsymbol S} = 0 \\, ,\n\\label{fluid_S} \n\\end{eqnarray}\nwhere we have introduced the conserved quantities $\\{ \\tau,\n{\\boldsymbol S} \\}$, which are essentially the energy density $ \\tau\n\\equiv T_{ab} n^a n^b $and the energy flux density $S_i \\equiv T_{ai}\nn^a$, and whose expressions are given by\n\\begin{eqnarray}\n    \\tau &\\equiv& \\frac{1}{2} (E^2 + B^2) + h~W^2 - p \\, ,\n\\label{def_e} \\\\\n    {\\boldsymbol S} &\\equiv& {\\boldsymbol E} \\times {\\boldsymbol B} + h~W^2~{\\boldsymbol v} ~.\n\\label{def_S} \n\\end{eqnarray}\nHere ${\\boldsymbol v}$ is the velocity measured by the inertial\nobserver and $W \\equiv -n_a u^a = 1\/ \\sqrt{1-v^2}$ is the Lorentz\nfactor. The fluxes can then be written as\n\\begin{eqnarray}\n    && \\hskip -0.5cm\n\\boldsymbol{F}_{\\tau} \\equiv {\\boldsymbol E} \\times {\\boldsymbol B} + h~W^2~{\\boldsymbol v} \\, ,\n\\label{def_Fe} \\\\\n    && \\hskip -0.5cm\n{\\bf F}_{\\boldsymbol S} \\equiv -\\boldsymbol{E E} - \\boldsymbol{B B} + h W^2 \\boldsymbol{v v}  \n+ \\left[ \\frac{1}{2} (E^2 + B^2) + p \\right] {\\bf g} \\, .\n\\label{def_FS} \n\\end{eqnarray}\nFinally, the conservation of the baryon number\n(\\ref{conservation_baryons}) reduces to the continuity equation\nwritten as\n\\begin{eqnarray}\\label{fluid_baryons} \n    \\partial_t D + \\nabla \\cdot \\boldsymbol{F}_{D} = 0 \\, ,\n\\end{eqnarray}\nwhere we have introduced another conserved quantity $D \\equiv  \\rho W$ and its\nflux $\\boldsymbol{F}_{D} \\equiv  \\rho W {\\boldsymbol v}$.\n\n\\subsection*{Ohm's law}\n\nAs mentioned above, Maxwell equations are coupled to the fluid ones by\nmeans of the current 4-vector ${I}^a$, whose explicit form will depend\nin general on the electromagnetic fields and on the local fluid\nproperties. A standard prescription is to consider the current to be\nproportional to the Lorentz force acting on a charged particle and the\nelectrical resistivity $\\eta$ to be a scalar function. Ohm's law,\nwritten in a Lorentz invariant way, then reads\n\\begin{equation}\\label{ohm_relativistic_covariant}\n I_a + (I^b~u_b) u_a=  \\sigma~F_{ab}~u^b \\, ,\n\\end{equation}\nwith $\\sigma \\equiv 1\/\\eta$ being the electrical conductivity of the medium.\nExpressing (\\ref{ohm_relativistic_covariant}) in terms of the electric\nand magnetic fields one obtains the familiar form of Ohm's law in a general inertial frame  \n\\begin{equation}\\label{ohm_relativistic}\n {\\boldsymbol J} = \\sigma~W [{\\boldsymbol E} + {\\boldsymbol v} \\times {\\boldsymbol B}\n         - ({\\boldsymbol E} \\cdot {\\boldsymbol v}) {\\boldsymbol v}] + q~{\\boldsymbol v} \\, .\n\\end{equation}\nNote that the conservation of the electric charge\n(\\ref{current_conservation_clasic}) provides the evolution equation\nfor the charge density $q$ (\\textit{i.e.},~ the projection of the 4-current\n$\\boldsymbol{I}$ along the direction ${\\boldsymbol n}$), while Ohm's\nlaw provides a prescription for the (spatial) conduction current\n${\\boldsymbol J}$ (\\textit{i.e.},~ the components of ${\\boldsymbol I}$ orthogonal\nto ${\\boldsymbol n}$).\n\nIt is important to recall that in deriving\nexpression~(\\ref{ohm_relativistic}) for Ohm's law we are implicitly\nassuming that the collision frequency of the constituent particles of\nour fluid is much larger that the typical oscillation frequency of the\nplasma. Stated differently, the timescale for the electrons and ions\nto come into equilibrium is much shorter than any other timescale in\nthe problem, so that no charge separation is possible and the fluid is\nglobally neutral. This assumption is a key aspect of the MHD\napproximation.\n\nThe well-known ideal-MHD limit of Ohm's law can be obtained by\nrequiring the current to be finite even in the limit of infinite\nconductivity ($\\sigma \\rightarrow \\infty $).  In this limit Ohm's law\n(\\ref{ohm_relativistic}) then reduces to\n\\begin{equation}\n  {\\boldsymbol E} + {\\boldsymbol v} \\times {\\boldsymbol B} - ({\\boldsymbol E} \\cdot {\\boldsymbol v}) {\\boldsymbol v} = 0 \\, .\n\\end{equation}\nProjecting this equation along ${\\boldsymbol v}$ one finds that the\nelectric field does not have a component along that direction and then from\nthe rest of the equation one recovers the well-known ideal-MHD\ncondition\n\\begin{equation}\n\\label{ef_imhd}\n {\\boldsymbol E} = - {\\boldsymbol v} \\times {\\boldsymbol B} \\,,\n\\end{equation}\nstating that in this limit the electric field is orthogonal to both\n${\\boldsymbol B}$ and ${\\boldsymbol v}$. Such a condition also\nexpresses the fact that in ideal MHD the electric field is not an\nindependent variable since it can be be computed via a simple\nalgebraic relation from the velocity and magnetic vector fields.\n\nSummarizing: the system of equations of the relativistic resistive MHD\napproximation is given by the constraint equations\n(\\ref{maxwell_clasic1})--(\\ref{maxwell_clasic2}),\n evolution equations\n(\\ref{maxwell_clasic3})--(\\ref{current_conservation_clasic}),\n(\\ref{fluid_tau})--(\\ref{fluid_S}) and (\\ref{fluid_baryons}), where\nthe fluxes are given by Eqs.~(\\ref{def_Fe})--(\\ref{def_FS}) and the\n3-current is given by Ohm's law (\\ref{ohm_relativistic}). These\nequations, together with a equation of state (EOS) for the fluid and\na reasonable model for the conductivity, completely describe the system\nunder consideration provided consistent initial and boundary data are defined.\n\n\n\\subsection*{Different limits of the resistive MHD description}\n\nAt this point it is useful to point out some properties of the\nrelativistic resistive MHD equations discussed so far, to underline\ntheir purely hyperbolic character and to contrast them with those of\nother forms of the resistive MHD equations which contain a parabolic\npart instead. To do this within a simple example, we adopt the\nNewtonian limit of Ohm's law (\\ref{ohm_relativistic}),\n\\begin{equation}\\label{ohm_classic}\n {\\boldsymbol J} = \\sigma [{\\boldsymbol E} + {\\boldsymbol v} \\times {\\boldsymbol B}] \\, ,\n\\end{equation}\nwhere we have neglected terms of order ${\\cal O}(v^2\/c^2)$, obtaining \nthe following potentially stiff equation for the electric field\n\\begin{equation}\n   \\partial_t {\\boldsymbol E} - \\nabla \\times {\\boldsymbol B}  =\n    - \\sigma [{\\boldsymbol E} + {\\boldsymbol v} \\times {\\boldsymbol B}] \\, .\n\\label{maxwell_stiff} \n\\end{equation}\nAssuming now a uniform conductivity and taking a time derivative of\nEq.~(\\ref{maxwell_clasic4}), we obtain the following hyperbolic\nequation with relaxation terms (henceforth referred simply \nas hyperbolic-relaxation equation) for the magnetic field\n\\begin{equation}\\label{relativistic_rMHD} \\\\\n -\\frac{1}{\\sigma} [\\partial_{tt} {\\boldsymbol B} - \\nabla^2 {\\boldsymbol B}]\n  = [\\partial_t {\\boldsymbol B}  - \\nabla \\times ({\\boldsymbol v} \\times {\\boldsymbol B})]\\,.\n\\end{equation}\n\nIf the displacement current can be neglected, \\textit{i.e.},~ $\\partial_t {\\boldsymbol E}\n\\simeq \\partial_{tt} {\\boldsymbol B} \\simeq 0$, equation\n(\\ref{relativistic_rMHD}) reduces to the familiar\nparabolic equation for the magnetic field \n\\begin{equation}\\label{newtonian_rMHD} \\\\\n   \\partial_t {\\boldsymbol B} \n    - \\nabla \\times ({\\boldsymbol v} \\times {\\boldsymbol B})\n    - \\frac{1}{\\sigma} \\nabla^2 {\\boldsymbol B} = 0  \\,,\n\\end{equation}\nwhere the last term is responsible for the diffusion of the magnetic\nfield. It is important to stress the significant difference in the\ncharacteristic structure between equations (\\ref{relativistic_rMHD})\nand (\\ref{newtonian_rMHD}). Both equations reduce to the same\nadvection equation in the ideal-MHD limit of infinite conductivity\n($\\sigma \\rightarrow \\infty$) indicating the flux-freezing\ncondition. However, in the opposite limit of infinite resistivity\n($\\sigma \\rightarrow 0$) Eq.~(\\ref{newtonian_rMHD}) tends to the\n(physically incorrect) elliptic Laplace equation $\\nabla^2\n{\\boldsymbol B} = 0$ while Eq.~(\\ref{relativistic_rMHD}) reduces to\nthe (physically correct) hyperbolic wave equation for the magnetic\nfield.\n\n\n\\subsection{The augmented MHD system}\n\nThe set of Maxwell equations described above can also be cast in an\nextended fashion which includes two additional fields, $\\psi$ and\n$\\phi$, introduced to control dynamically the constraints of the\nsystem, \\textit{i.e.},~ Eqs~ (\\ref{maxwell_clasic1}) and (\\ref{maxwell_clasic2}). This\n\\textit{``augmented''} system reads\n\\begin{eqnarray}\\label{maxwell_augmented1}\n   \\partial_b (F^{ab} + \\psi g^{ab}) &=& I^a - \\kappa \\psi n^a\\,, \\\\\n\\label{maxwell_augmented2}\n   \\partial_b (^{~*}\\!F^{ab} + \\phi g^{ab}) &=& - \\kappa \\phi n^a\\,.\n\\end{eqnarray}\nClearly, the standard Maxwell equations\n(\\ref{maxwell_covariant1})--(\\ref{maxwell_covariant2}) are recovered\nwhen $\\psi = \\phi = 0$ and we are in this way extending the space of\nsolutions of the original Maxwell equations to include those with\nnon-vanishing $\\{\\psi,\\phi\\}$.\n\nThe evolution of these extra scalar fields can be obtained by taking a\npartial derivative $\\partial_a$ of the augmented Maxwell equations\n(\\ref{maxwell_augmented1})--(\\ref{maxwell_augmented2}) and using the\nantisymmetry of the Maxwell and Faraday tensors together with the conservation\nof charge to obtain\n\\begin{eqnarray}\n   \\partial_a \\partial^a \\psi &=& - \\kappa \\partial_a (\\psi n^a)\\,, \\\\\n   \\partial_a \\partial^a \\phi &=& - \\kappa \\partial_a (\\phi n^a)\\,.\n\\end{eqnarray}\nIt is evident that these represent wave equations with sources for the\nscalar fields $\\{\\psi,\\phi\\}$, which propagate at the speed of light\nwhile being damped if $\\kappa > 0$. In particular, for any positive\n$\\kappa$, they decay exponentially over a timescale $\\sim 1\/\\kappa$ to\nthe trivial solution $\\psi = \\phi = 0$ and the augmented system then\nreduces to the standard Maxwell equations, including the constraints\n(\\ref{maxwell_clasic1}) and (\\ref{maxwell_clasic2}). This approach,\nnamed hyperbolic divergence cleaning in the context of ideal\nMHD~\\citep{DKKMSW:2002}, was proposed as a simple way of solving\nthe Maxwell equations and enforcing the conservation of the\ndivergence-free condition for the magnetic field.\n\nAdopting this approach and following the formulation proposed\nby~\\citet{Kom:2007}, the evolution equations of the augmented Maxwell\nequations (\\ref{maxwell_augmented1})--(\\ref{maxwell_augmented2}) can\nthen be written as\n\\begin{eqnarray}\n   \\partial_t \\psi + \\nabla \\cdot {\\boldsymbol E}  &=& q - \\kappa~\\psi \\, ,\n\\label{maxwell_aug1} \\\\    \n   \\partial_t \\phi + \\nabla \\cdot {\\boldsymbol B} &=& -\\kappa~\\phi \\, ,\n\\label{maxwell_aug2} \\\\\n   \\partial_t {\\boldsymbol E} - \\nabla \\times {\\boldsymbol B} + \\nabla \\psi &=& - {\\boldsymbol J} \\, ,\n\\label{maxwell_aug3} \\\\\n   \\partial_t {\\boldsymbol B} + \\nabla \\times {\\boldsymbol E} + \\nabla \\phi &=& 0 \\, .\n\\label{maxwell_aug4} \n\\end{eqnarray}\nThe system of equations (\\ref{maxwell_aug1})--(\\ref{maxwell_aug4}), together\nwith the current conservation (\\ref{current_conservation_clasic}), is\nthe one we will use for the numerical evolution of the electromagnetic\nfields within the set of relativistic resistive MHD equations.\n\n\n\n\\section{Evolution of hyperbolic-relaxation equations}\\label{section3}\n\nWhile the ideal-MHD equations are well suited to an efficient\nnumerical implementation, the general system of relativistic resistive\nMHD equations brings about a delicate issue when the conductivity in\nthe plasma undergoes very large spatial variations. In the regions\nwith high conductivity, in fact, the system will evolve on timescales\nwhich are very different from those in the low-conductivity\nregion. Mathematically, therefore, the problem can be regarded as a\nhyperbolic one with stiff relaxation terms which requires special\ncare to capture the dynamics in a stable and accurate manner. In the\nnext Section we discuss a simple example of a hyperbolic equation with\nrelaxation which exhibits the problems discussed above and then\nintroduce implicit-explicit (IMEX) Runge Kutta methods to deal with\nthese kind of equations. In essence, these methods treat the advection\ncharacter of the system with strong-stability preserving (SSP)\nexplicit schemes, while the relaxation character with an L-stable\ndiagonally implicit Runge Kutta (DIRK)\nscheme. After presenting the scheme, its properties and some examples,\nwe discuss in detail its application to the resistive MHD equations.\n\n\n\n\\subsection{Hyperbolic systems with relaxation terms}\n\nA prototypical hyperbolic equation with relaxation is given by \n\\begin{eqnarray}\\label{stiff_equation}\n    \\partial_t {\\boldsymbol U} = F({\\boldsymbol U}) + \\frac{1}{\\epsilon} R({\\boldsymbol U})\\,,\n\\end{eqnarray}\nwhere $\\epsilon >0$ is the \\textit{relaxation time} (not necessarily\nconstant either in space or in time), $F({\\boldsymbol U})$ gives\nrise to a quasilinear system of equations (\\textit{i.e.},~ $F({\\boldsymbol U})$\ndepends linearly on first derivatives of ${\\boldsymbol U}$), and $R$\ndoes not contain derivatives of ${\\boldsymbol U}$.\n\nIn the limit $\\epsilon \\rightarrow \\infty$ (corresponding for the\nresistive MHD equations to the case of vanishing conductivity) the\nsystem is hyperbolic with propagation speeds bounded by $c_h$. This\nmaximum bound, together with the length scale $L$ of the system,\ndefine a characteristic timescale $\\tau_h \\equiv L \/ c_h$ of the\nhyperbolic part. In the opposite limit $\\epsilon \\rightarrow 0$\n(corresponding to the case of infinite conductivity), the system is\ninstead said to be \\textit{stiff}, since the timescale $\\epsilon$ of\nthe relaxation (or stiff) term $R({\\boldsymbol U})$ is in general much\nlarger than the timescale $\\tau_h$ of the hyperbolic part\n$F({\\boldsymbol U})$. In such a limit, the stability of an explicit\nscheme is only achieved\n\\footnote{Implicit schemes could avoid this issue at an increased\ncomputational cost; however, an explicit second order accurate \nmethod approaching iteratively the Crank-Nicholson scheme\nhas been shown, in a simple model with hyperbolic-relaxation terms, \nto work well when dealing with smooth profiles without being too costly \n(M. Choptuik, private communication)}\nwith a timestep size $\\Delta t \\leq \\epsilon$.\nThis requirement is certainly more restrictive than the\nCourant-Lewy-Friedrichs (CFL) stability condition $\\Delta t \\leq\n\\Delta x \/ c_h $ for the hyperbolic part and makes an explicit\nintegration impractical.  The development of efficient numerical\nschemes for such systems is challenging, since in many applications\nthe relaxation time can vary by several orders of magnitude across the\ncomputational domain and, more importantly, to much beyond the one \ndetermined by the speed $c_h$.\n\nWhen faced with this issue several strategies can be adopted.  The\nmost straightforward one is to consider only the stiff limit $\\epsilon\n\\rightarrow 0$, where the system is well approximated by a suitable\nreduced set of conservation laws called \\textit{``equilibrium system''}\n\\citep{CheLevLiu:1994} such that\n\\begin{eqnarray}\\label{equilibrium_system}\n     R(\\boldsymbol{\\bar{U}}) &=& 0 \\,,\\\\\n    \\partial_t \\boldsymbol{\\bar{U}} &=& G(\\boldsymbol{\\bar{U}}) \\,.\n\\end{eqnarray}\nwhere $\\boldsymbol{\\bar{U}}$ is a reduced set of variables. This\napproach can be followed if the resulting system is also hyperbolic.\nThis is precisely the case in the resistive MHD equations for\nvanishing resistivity $\\eta \\rightarrow 0$ (or $\\sigma \\rightarrow\n\\infty$). In this case, the equations reduce to those of\nideal MHD and describe indeed an ``equilibrium system'' in which the magnetic\nfield is simply advected with the flow. As discussed earlier, this\nlimit is often adequate to describe the behaviour of dense\nastrophysical plasmas, but it may also stray away in the\nmagnetospheres. A more general approach could consist of dividing the\ncomputational domain in regions in each of which a simplified set of\nequations can be adopted. As an example, the ideal-MHD equations could\nbe solved in the interior of compact objects, the force-free MHD\nequations could be solved in the magnetosphere, and finally the\nMaxwell equations for the vacuum regions outside the compact\nobject. However, this approach requires the overall scheme to suitably\nmatch the different regions so as to obtain a global solution. This\ntask, unfortunately, is far from being straightforward and, to date,\nit lacks a rigorous definition.\n\nAn alternative approach consists of considering the original \nhyperbolic-relaxation\nsystem in the whole computational domain and then employ suitable\nnumerical schemes that work for all regions. Among such schemes is the\nStrang-splitting technique~\\citep{Strang:1968}, which has been\nrecently applied by~\\citet{Kom:2007} for the solution of the (special)\nrelativistic resistive MHD equations. The Strang-splitting scheme\nprovides second-order accuracy if each step is at least second-order\naccurate, and this property is maintained under suitable assumptions\neven for stiff problems~\\citep{JahLub:2000}. In practice, however,\nhigher-order accuracy is difficult to obtain even in non-stiff regimes\nwith this kind of splitting. Moreover, when applied to hyperbolic\nsystems with relaxation, Strang-splitting schemes reduce to\nfirst-order accuracy since the kernel of the relaxation operator is\nnon-trivial and corresponds to a singular matrix in the linear case,\ntherefore invalidating the assumptions made by~\\citet{JahLub:2000} to\nensure high-order accuracy.~\\citet{Kom:2007} avoided this problem by\nsolving analytically the stiff part in a reduced form of Ampere's law.\nAlthough this procedure works well for smooth solutions, our\nimplementation of the method has revealed problems when evolving\ndiscontinuous flows (shocks) for large-conductivities\nplasmas. Moreover, it is unclear whether the same procedure can be\nadopted in more general configurations, where an analytical solution\nmay not be available.\n\nAs an alternative approach to the methods solving the relativistic\nresistive MHD equations on a single computational domain, we here\nintroduce an IMEX Runge-Kutta method\n\\citep{AshRuuWet:1995,AshRuuSpi:1997,Par:2001,ParRus:2005} to cope\nwith the stiffness problems discussed above. These methods, which are\neasily implemented, are still under development and have few\n(relatively minor) drawbacks. The most serious one is a degradation to\nfirst or second-order accuracy for a range of values of the relaxation\ntime $\\epsilon$. However, since High-Resolution Shock-Capturing (HRSC) schemes\nusually employed for the solution of the hydrodynamic equations already suffer from similar\neffects at discontinuities, the possible degradation of the IMEX\nschemes does not spoil the overall quality numerical solution when\nemployed in conjunction with HRSC schemes. The next sections\nreview in some detail the IMEX schemes and our specific implementation\nfor the relativistic resistive MHD equations.\n\n\\subsection{The IMEX Runge-Kutta methods}\n\nThe IMEX Runge-Kutta schemes rely on the application of an implicit\ndiscretization scheme to the stiff terms and of an explicit one to the\nnon-stiff ones. When applied to system (\\ref{stiff_equation}) it takes\nthe form \\citep{ParRus:2005}\n\\begin{eqnarray}\\label{IMEX}\n&& \\hskip -0.5cm\n   {\\boldsymbol U}^{(i)} = {\\boldsymbol U}^n + \\Delta t \\sum_{j=1}^{i-1} {\\tilde{a}}_{ij} F({\\boldsymbol U}^{(j)}) \n     + \\Delta t  \\sum_{j=1}^{\\nu} a_{ij} \\frac{1}{\\epsilon}\n     R({\\boldsymbol U}^{(j)})\\,, \\nonumber \\\\\n && \\hskip -0.5cm\n{\\boldsymbol U}^{n+1} = {\\boldsymbol U}^n + \\Delta t \\sum_{i=1}^{\\nu} {\\tilde{\\omega}}_{i} F({\\boldsymbol U}^{(i)})\n     + \\Delta t  \\sum_{i=1}^{\\nu} \\omega_{i} \\frac{1}{\\epsilon} R({\\boldsymbol U}^{(i)})\\,, \n\\nonumber\\\\\n\\end{eqnarray}\nwhere ${\\boldsymbol U}^{(i)}$ are the auxiliary intermediate values of\nthe Runge-Kutta scheme.  The matrices $\\tilde{A}= (\\tilde{a}_{ij})$\nand $A= (a_{ij})$ are $\\nu \\times \\nu$ matrices such that the\nresulting scheme is explicit in $F$ (\\textit{i.e.},~ $\\tilde{a}_{ij} = 0$ for $j\n\\geq i$) and implicit in $R$. An IMEX Runge-Kutta scheme is\ncharacterized by these two matrices and the coefficient vectors\n$\\tilde{\\omega}_i$ and $\\omega_i$.  Since simplicity and efficiency in\nsolving the implicit part at each step is important, it is\nnatural to consider diagonally implicit Runge-Kutta (DIRK) schemes\n(\\textit{i.e.},~ $a_{ij}=0$ for $j > i$) for the stiff terms. \n\nA particularly convenient way of describing an IMEX Runge-Kutta scheme\nis offered by the Butcher notation, in which the scheme is by a double\ntableau of the type~\\citep{But:1987,But:2003}\n\\begin{equation}\n\\begin{minipage}{1.2in}\n\\begin{tabular} {c c c}\n${\\tilde c}$  & \\vline & ${\\tilde A}$  \\\\\n\\hline \n              & \\vline & ${\\tilde \\omega}^T$  \n\\end{tabular}\n\\end{minipage} \n\\hskip 1.0cm\n\\begin{minipage}{1.2in}\n\\begin{tabular} {c c c}\n${c}$  &  \\vline & ${A}$  \\\\\n\\hline \n       &  \\vline & ${\\omega}^T$  \n\\label{butcher_tableau}\n\\end{tabular}\n\\end{minipage}\n\\end{equation}\nwhere the index $T$ indicates a transpose and where the coefficients\n$\\tilde{c}$ and $c$ used for the treatment of non-autonomous systems\nare given by \n\\begin{equation}\n\\label{definition_cs}\n   {\\tilde c}_{i} = \\sum_{j=1}^{i-1}~ {\\tilde{a}}_{ij}\\,, \\hskip 2.0cm\n   {c}_{i} = \\sum_{j=1}^{i}~ {a}_{ij} ~~~.\n\\end{equation}\nThe accuracy of each of the Runge-Kutta is achieved by imposing \nrestrictions in some of the coefficients of their respective\nButcher tableaus. Although each of them separately can have an arbitrary\naccuracy, this does not ensure that the combination of the two schemes\nwill preserve the same accuracy. In addition to the above conditions for\neach Runge-Kutta scheme, there are also some additional conditions combining terms\nin the two tableaus which must be fulfilled in order to achieve a global\naccuracy order for the complete IMEX scheme.\n\nSince the details of these methods are not widely known, we first\nconsider a simple example to fix ideas. A second-order IMEX scheme can\nbe written in the tableau form given in Table \\ref{SSP2-222}. The\nintermediate and final steps of this IMEX Runge-Kutta scheme would\nthen be written explicitly as\n\\begin{eqnarray}\\label{IMEX-example}\n&&\n{\\boldsymbol U}^{(1)} = {\\boldsymbol U}^n + \\frac{\\Delta t}{\\epsilon} \\gamma R({\\boldsymbol U}^{(1)}) \\,,\n\\nonumber \\\\\n&&\n{\\boldsymbol U}^{(2)} = {\\boldsymbol U}^n + \\Delta t F({\\boldsymbol U}^{(1)}) \n\\nonumber \\\\\n && \\hskip 1.6cm  + \\frac{\\Delta t}{\\epsilon} [(1 - 2 \\gamma) R({\\boldsymbol U}^{(1)}) + \\gamma R({\\boldsymbol U}^{(2)})] \\,,\n\\nonumber \\\\ \n  &&{\\boldsymbol U}^{n+1} = {\\boldsymbol U}^n + \\frac{\\Delta t}{2} [ F({\\boldsymbol U}^{(1)}) + F({\\boldsymbol U}^{(2)}) ]\n\\nonumber \\\\\n  && \\hskip 1.8cm   + \\frac{\\Delta t}{2 \\epsilon} [R({\\boldsymbol U}^{(1)}) + R({\\boldsymbol U}^{(2)})] \\,.\n\\nonumber\n\\end{eqnarray}\nNote that at each sub-step an implicit equation for the auxiliary\nintermediate values ${\\boldsymbol U}^{(i)}$ must be solved.  The\ncomplexity of inverting this equation will clearly depend on the\nparticular form of the operator $R({\\boldsymbol U})$.\n\n\\subsubsection{Stability properties of the IMEX schemes}\n\nStable solutions of conservation-type equations are usually analyzed\nin terms of a suitable norm being bounded in time.  With ${\\boldsymbol\n  U}^n$ representing the solution vector at the time $t= n~\\Delta t$,\nthen a sequence $\\{{\\boldsymbol U}^n\\}$ is said to be\n\\textit{``strongly stable''} in a given norm $\\| \\cdot \\|$ provided\nthat $\\| {\\boldsymbol U}^{n+1} \\| \\leq \\| {\\boldsymbol U}^n \\|$ for\nall $n \\geq 0$.\n\n\\begin{table}\n\\caption{Tableau for the explicit (left) implicit (right) IMEX-SSP2 $(2,2,2)$\nL-stable scheme}\n\\begin{minipage}{1.2in}\n\\begin{tabular} {c c c c}\n $0$ & \\vline & $0$ & $0$  \\\\\n $1$ & \\vline & $1$ & $0$  \\\\\n\\hline \n   & \\vline & $1\/2$ & $1\/2$  \\\\\n\\end{tabular}\n\\end{minipage}\n\\begin{minipage}{1.2in}\n\\begin{tabular} {c c c c}\n $\\gamma$ & \\vline & $\\gamma$ & $0$  \\\\\n $1 - \\gamma$ & \\vline & $1 - 2 \\gamma$ & $\\gamma$  \\\\\n\\hline \n   & \\vline & $1\/2$ & $1\/2$  \\\\\n\\end{tabular}\n\\end{minipage}\n\\begin{eqnarray}\n\\gamma \\equiv 1 - \\frac{1}{\\sqrt{2}}\\,. \\nonumber\n\\end{eqnarray}\n\\label{SSP2-222}\n\\end{table}\n\nThe most commonly used norms for analyzing schemes for nonlinear\nsystems are the Total-Variation (TV) norm and the infinity norm.  A\nnumerical scheme that maintains strong stability at the discrete level is\ncalled Strong Stability Preserving (SSP) (see \\citet{SpiRuu:2002} for\na detailed description of optimal SSP schemes and their properties).\nBecause of the stability properties of the IMEX\nschemes~\\citep{ParRus:2005}, it follows that if the explicit part of\nthe IMEX scheme is SSP, then the method is SSP for the equilibrium\nsystem in the stiff limit. This property is essential to avoid\nspurious oscillations during the evolution of non-smooth data.\n\nThe stability of the implicit part of the IMEX scheme is ensured\nby requiring that the Runge-Kutta is ``L-stable'' and  this represents an\nessential condition\nfor stiff problems. In practice, this amounts to requiring that the numerical approximation\nis bounded in cases when the exact solution is bounded.\nA more strict definition can be derived starting from a linear scalar\nordinary differential equation,\nnamely\n\\begin{equation}\n\\label{ode}\n    {\\rm d}_t \\Psi = q \\Psi \\,.\n\\end{equation}\nIn this case it is easy to define the stability (or\namplification) function $C(z)$ as the ratio of the solutions \nat subsequent timesteps \n$C(z)\\equiv\\Psi^{n+1}\/\\Psi^{n}$, where $z \\equiv \\Delta t\\,q$.\nA Runge-Kutta scheme is then said to be \\textit{L-stable} if\n$|C(z)|<1$ (\\textit{i.e.},~ it is bounded) and $C(\\infty)=0$\n\\citep{But:1987,But:2003}.\n\nThere are a number of IMEX Runge-Kutta schemes available in the\nliterature and we report here only some of the second and third-order\nschemes which satisfy the condition that in the limit $\\epsilon\n\\rightarrow 0$, the solution corresponds to that of the equilibrium\nsystem (\\ref{equilibrium_system})~\\citep{ParRus:2005}. These are\ngiven in their Butcher tableau form in Table~\\ref{SSP2-322} and are\ntaken from~\\citet{ParRus:2005}.  In all these schemes the implicit\ntableau corresponds to an L-stable scheme.\nThe tableaus are reported in the notation SSP$k(s,\\sigma,p)$, where $k$\ndenotes the order of the SSP scheme and the triplet $(s,\\sigma,p)$ characterizes\nrespectively the number of stages of the implicit scheme ($s$), the number of stages of\nthe explicit scheme ($\\sigma$), and the order of the IMEX scheme\n($p$).\n\n\\begin{table}\n\\caption{Tableaux for the explicit (first row) and implicit (second\n  row) IMEX SSP-schemes. We use the standard notation\n  SSP$k(s,\\sigma,p)$, where $k$ denotes the order of the SSP scheme\n  and the triplet $(s,\\sigma,p)$ characterizes respectively the number\n  of stages of the implicit scheme ($s$), the number of stages of the\n  explicit scheme ($\\sigma$), and the order of the IMEX scheme ($p$).}\n\\begin{minipage}{1.6in}\n{SSP2 $(3,3,2)$} \n\\vskip 0.125cm\n\\begin{tabular} {c c c c c}\n$ 0 $ & \\vline & $0  $ & $0  $ & $0$ \\\\\n$1\/2$ & \\vline & $1\/2$ & $0  $ & $0$ \\\\\n$ 1 $ & \\vline & $1\/2$ & $1\/2$ & $0$ \\\\\n\\hline \n   & \\vline & $1\/3$ & $1\/3$ & $1\/3$ \n\\end{tabular}\n\\end{minipage}\n\\vskip 0.2cm\n\\begin{minipage}{1.6in}\n\\begin{tabular} {c c c c c}\n$1\/4$ & \\vline & $1\/4$ & $0  $ & $0  $ \\\\\n$1\/4$ & \\vline & $ 0 $ & $1\/4$ & $0  $ \\\\\n$ 1 $ & \\vline & $1\/3$ & $1\/3$ & $1\/3$ \\\\\n\\hline \n   & \\vline & $1\/3$ & $1\/3$ & $1\/3$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\label{SSP2-322}\n\\vskip 0.5cm\n\\begin{minipage}{1.6in}\n{SSP3 $(3,3,2)$}\n\\vskip .125cm\n\\begin{tabular} {c c c c c}\n $0  $ & \\vline & $ 0 $ & $ 0 $ & $0$ \\\\\n $1  $ & \\vline & $ 1 $ & $ 0 $ & $0$ \\\\\n $1\/2$ & \\vline & $1\/4$ & $1\/4$ & $0$ \\\\\n\\hline \n   & \\vline & $1\/6$ & $1\/6$ & $2\/3$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\vskip 0.2cm\n\\begin{minipage}{1.6in}\n\\begin{tabular} {c c c c c}\n$\\gamma$   & \\vline & $\\gamma$       & $0$      & $0$   \\\\\n$1-\\gamma$ & \\vline & $1-2 \\gamma$   & $\\gamma$ & $0$   \\\\\n$ 1\/2$     & \\vline & $1\/2 - \\gamma$ & $0$      & $\\gamma$ \\\\\n\\hline \n   & \\vline & $1\/6$ & $1\/6$ & $2\/3$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\vskip 0.5cm\n\\begin{minipage}{1.6in}\nSSP3 $(4,3,3)$\n\\vskip 0.125cm\n\\begin{tabular} {c c c c c c}\n $0  $ & \\vline & $0$  & $ 0 $ & $ 0 $ & $0$  \\\\\n $0  $ & \\vline & $0$  & $ 0 $ & $ 0 $ & $0$  \\\\\n $1  $ & \\vline & $0$  & $ 1 $ & $ 0 $ & $0$  \\\\\n $1\/2$ & \\vline & $0$  & $1\/4$ & $1\/4$ & $0$  \\\\\n\\hline \n   & \\vline &  $0$ & $1\/6$ & $1\/6$ & $2\/3$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\vskip .2cm\n\\begin{minipage}{1.6in}\n\\centering\n\\begin{tabular} {c c c c c c}\n $\\alpha$   & \\vline & $\\alpha$  &  $0$  &  $0$  & $0$  \\\\\n $0$        & \\vline & $-\\alpha$  &  $\\alpha$  &  $0$  & $0$  \\\\\n $1$        & \\vline & $0$  &  $1-\\alpha$  &  $\\alpha$  & $0$ \\\\\n $1\/2$      & \\vline & $\\beta$  & $\\eta$ & $1\/2-\\beta-\\eta-\\alpha$ & $\\alpha$ \\\\\n\\hline \n   & \\vline &  $0$ & $1\/6$ & $1\/6$ & $2\/3$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\begin{eqnarray}\n &\\alpha \\equiv 0.24169426078821\\,, &\\beta \\equiv 0.06042356519705\\,, \\nonumber \\\\\n &\\gamma \\equiv 1 - {1}\/{\\sqrt{2}}\\,, &\\eta \\equiv 0.12915286960590\\,. \\nonumber\n\\end{eqnarray}\n\\end{table}\n\n\n\n\\section{IMEX Runge-Kutta scheme for the augmented resistive MHD equations}\\label{section4}\n\nHaving reviewed the main properties of the IMEX schemes, we now apply\nthem to the particular case of the special relativistic resistive MHD\nequations. Our goal is to consider a numerical implementation of the\ngeneral system that can deal with standard hydrodynamic issues (like\nshocks and discontinuities) as well as those brought up by the stiff\nterms discussed in the previous Section. Hence, we adopt high-resolution\nshock-capturing algorithms (see Appendix~\\ref{appendixB}) together\nwith IMEX schemes. Because the first ones involve the introduction of\nconserved variables in order to cast the equations in a conservative\nform, we first discuss how to implement the IMEX scheme within our\ntarget system and subsequently how to perform the transformation \nfrom the conserved variables to the primitive ones.\n\n\\subsection{IMEX schemes for the Maxwell-Hydrodynamic equations and \ntreatment of the implicit stiff part}\n\nFor our target system of equations it is possible to introduce a\nnatural decomposition of variables in terms of those whose evolution\ndo not involve stiff terms and those which do. More specifically, with\nthe electrical resistivity $\\eta$ playing the role of the relaxation\nparameter $\\epsilon$, the vector of fields $\\boldsymbol{U}$ can be\nsplit in two subsets $\\{\\boldsymbol{X},\\boldsymbol{Y}\\}$, with\n${\\boldsymbol X}= \\{ {\\boldsymbol E} \\}$ containing the stiff terms,\nand ${\\boldsymbol Y} = \\{ {\\boldsymbol B}, \\psi, \\phi, q, \\tau,\n{\\boldsymbol S}, D \\}$ the non-stiff ones.\n\nFollowing the prototypical Eq.~(\\ref{stiff_equation}), the evolution\nequations for the relativistic resistive MHD equations can then be\nschematically written as\n\\begin{eqnarray}\\label{split}\n    \\partial_t {\\boldsymbol Y} &=& F_{\\!_{Y}}({\\boldsymbol X},{\\boldsymbol Y})\\,,\\\\\n    \\partial_t {\\boldsymbol X} &=& F_{\\!_{X}}({\\boldsymbol X},{\\boldsymbol Y}) + \\frac{1}{\\epsilon({\\boldsymbol Y})} R_{\\!_{X}}({\\boldsymbol X},{\\boldsymbol Y})\\,, \n\\end{eqnarray}\nwhere the relaxation parameter $\\epsilon$ is allowed to depend also on\nthe ${\\boldsymbol Y}$ non-stiff fields. The vector ${\\boldsymbol Y}$\ncan be evolved straightforwardly as it involves no stiff term. We\nfurther note that for our particular set of equations, it is\nconvenient to write the stiff part as\n\\begin{eqnarray}\\label{stiff_part}\n  R_{\\!_{X}}({\\boldsymbol X},{\\boldsymbol Y}) = A({\\boldsymbol Y})\n  {\\boldsymbol X} + S_{\\!_{X}}({\\boldsymbol Y})\\,.\n\\end{eqnarray}  \nAs a result, the procedure to compute each stage\n${\\boldsymbol U}^{(i)}$\nof the IMEX scheme can be performed in two steps:\n\\begin{enumerate}\n  \\item Compute the explicit intermediate values $\\{ {\\boldsymbol\n    X}^{*}, {\\boldsymbol Y}^{*} \\}$ from all the\n   previously known levels, that is\n\\begin{eqnarray}\\label{first_stepa}\n&& \\hskip -.7cm\n  {\\boldsymbol Y}^{*} = {\\boldsymbol Y}^n + \\Delta t~\n  \\sum_{j=1}^{i-1}~ {\\tilde{a}}_{ij} F_{\\!_{Y}}({\\boldsymbol U}^{(j)}) \\,, \n \\\\\n\\label{first_stepb}\n&& \\hskip -.7cm\n  {\\boldsymbol X}^{*} = {\\boldsymbol X}^n + \\Delta t~ \\sum_{j=1}^{i-1}~ {\\tilde{a}}_{ij} F_{\\!_{X}}({\\boldsymbol U}^{(j)}) \n     + \\Delta t~ \\sum_{j=1}^{i-1}~ \\frac{a_{ij} }{\\epsilon^{(j)}} R_{\\!_{X}}({\\boldsymbol U}^{(j)})  \\,,\n\\nonumber \\\\\n   \\end{eqnarray} \nwhere we have defined $ \\epsilon^{(j)} \\equiv \\epsilon({\\boldsymbol\n  Y}^{(j)})$ and $a_{ij}\/\\epsilon^{(j)}$ in Eq.~(\\ref{first_stepb})\nis a simple division and not a contraction on dummy indices.\n\n   \\item Compute the implicit part, which involves only ${\\boldsymbol X}$, by solving\n  \\begin{eqnarray}\n    \\label{second_step_a}\n   {\\boldsymbol Y}^{(i)} &=& {\\boldsymbol Y}^{*} \\, , \\\\\n    \\label{second_step_b}\n   {\\boldsymbol X}^{(i)} &=& {\\boldsymbol X}^{*} + \\Delta t~ \\frac{a_{ii} }{\\epsilon^{(i)}} R_{\\!_{X}}({\\boldsymbol U}^{(i)}) \\, .  \n   \\end{eqnarray} \n   Note that the implicit equation, with the previous assumption (\\ref{stiff_part}),\n   can be inverted  explicitly\n   \\begin{eqnarray}\\label{invert_matrix}\n       {\\boldsymbol X}^{(i)} &=& M ({\\boldsymbol Y}^*) ~\n                    ( \\boldsymbol{X}^* + a_{ii}~\\frac{\\Delta t}{\\epsilon^{(i)}}~{\\boldsymbol S}_{\\!_{X}}({\\boldsymbol Y}^*) ) \\, , \\\\\n       M({\\boldsymbol Y}^*) &=& [I - a_{ii}~\\frac{\\Delta t} {\\epsilon^{(i)}} A({\\boldsymbol Y}^*)]^{-1} \\,,\n   \\end{eqnarray}\nsince the form of the matrix $[I - a_{ii}~{\\Delta t} A({\\boldsymbol\n    Y}^*)\/{\\epsilon^{(i)}} ]$ is known explicitly in terms of the\nevolved fields.\n\\end{enumerate}\n\nThe explicit expressions for stiff part are then given simply by\n\\begin{eqnarray}\\label{matrix1}\n  {\\boldsymbol R}_{\\!_{E}} &=& -W {\\boldsymbol E} + W~({\\boldsymbol E} \\cdot {\\boldsymbol v}) {\\boldsymbol v}\n         - W {\\boldsymbol v} \\times {\\boldsymbol B} \\, , \\\\\n  {\\boldsymbol S}_{\\!_{E}}  &=& - W {\\boldsymbol v} \\times {\\boldsymbol B} \\, ,\n\\end{eqnarray} \nwith the matrix $A$ defined as\n\\begin{equation}\nA \\equiv W \\left( \\begin{array}{ccc}\n -1+v_x^2 & v_x~v_y & v_x~v_z \\\\\n  v_x~v_y &-1+v_y^2 & v_y~v_z \\\\\n  v_z~v_x & v_z~v_y &-1+v_z^2 \\end{array} \\right)\\,.\n\\end{equation}\nHence, the matrix $M$ can be computed explicitly to obtain\n\\[ \\frac{1}{m} \\left( \\begin{array}{ccc}\n a+W+a W^2 v_x^2 &\\hskip-3mm a W^2 v_x v_y   & \\hskip-3mm a W^2 v_x v_z \\\\\n  a W^2 v_x v_y  &\\hskip-3mm a+W+a W^2 v_y^2 & \\hskip-3mm a W^2 v_y v_z \\\\\n  a W^2 v_z v_x  &\\hskip-3mm a W^2 v_z v_y   & \\hskip-3mm a+W+a W^2 v_z^2  \\end{array} \\right)\\]\nwhere $m \\equiv W^2 a+ W a^2 + W + a$ and $a \\equiv a_{ii}~\\sigma^{(i)}~\\Delta t$.\\\\\n\nSummarizing: First, an intermediate state $\\{ \\boldsymbol{E}^*\\}$ is\nfound through the evolution of the non-stiff part for the electric\nfield. Second, if the velocity $ {\\boldsymbol v}$ is known, the\nevolution of the stiff part can be performed by acting with $M$ to obtain\n\\begin{eqnarray}\\label{invert_matrix_E}\n  {\\boldsymbol E} = M ({\\boldsymbol v}) ~\n[ \\boldsymbol{E^*} + a_{ii}~\\Delta t~\\sigma^{(i)}~{\\boldsymbol S}_{\\!_{E}}({\\boldsymbol v},{\\boldsymbol B}) ]\\,.\n\\end{eqnarray}\nAt this point the approach proceeds with the conversion from the\nconserved variables to the primitive ones. Because of the coupling\nbetween the electric and the velocity fields, such a procedure is\nrather involved and more complex than in the ideal-MHD case; a\ndetailed discussion of how to do this in practice will be presented in\nSect.~\\ref{inversion_con2prim}.\n\nIt is interesting to highlight the consistency\nat two known limits of the implicit solution of the stiff part. In the\nideal-MHD limit (\\textit{i.e.},~ $\\sigma \\rightarrow \\infty$) the first term of\nEq.~(\\ref{invert_matrix_E}) vanishes, while the contribution of the\nsecond term leads to the ideal-MHD condition (\\ref{ef_imhd}). On the\nother hand, in the vanishing conductivity limit (\\textit{i.e.},~ $\\sigma\n\\rightarrow 0$) the second term in Eq.~(\\ref{invert_matrix_E})\nvanishes, and the matrix reduces to the identity one $M({\\boldsymbol\n  v}) = I$. In this case, the electric field is obtained only by\nevolving the explicit part, \\textit{i.e.},~ ${\\boldsymbol E} = \\boldsymbol{E^*}$.\n\nFinally, it is important to stress that one could, in principle, have considered\nthe alternative route of adopting instead \n${\\boldsymbol X}= \\{ {\\boldsymbol E}, q\\}$, so that the\nright-hand-side of $q$ would be considered stiff with $R_q = 0$ and\n$S_q = \\nabla \\cdot {\\boldsymbol R}_{\\!_{E}}$. However, this choice could lead\nto spurious numerical oscillations in the solution since the fluxes of\n$q$ can be discontinuous, while they would be evolved with an implicit\nRunge-Kutta. As it has been shown under fairly general conditions,\nhigh-order SSP schemes are necessarily explicit~\\citep{GotShuTad:2001}, so\nit follows that this part of the equations cannot be evolved with the implicit Runge-Kutta\nunless a low-order scheme is implemented.\n\n\n\\subsection{Transformation of conserved variables to primitive ones}\\label{inversion_con2prim}\n\nAs mentioned in the previous Section, in order to evolve our system of\nequations, the fluxes $\\{ \\boldsymbol{F}_{\\tau}, {\\bf F}_{\\boldsymbol\n  S} ,\\boldsymbol{F}_{D} \\}$ must be computed at each timestep. These\nfluxes depend on the primitive fields $\\{ \\rho, ~ p, ~ {\\boldsymbol v}\n, ~ {\\boldsymbol E} , ~{\\boldsymbol B}\\}$, which must be recovered\nfrom the evolved conserved fields $\\{ D, ~ \\tau , ~ {\\boldsymbol S}, ~\n{\\boldsymbol E}, ~{\\boldsymbol B} \\}$. These quantities are related by\ncomplicated equations which become transcendental except for\nparticularly simple equations of state (EOS). As a result, the\nconversion must be in general pursued numerically and the primitive\nvariables are then given by the roots of the function\n\\begin{equation}\\label{trascendental}\n   f ( {\\bar p}) = p(\\rho,\\epsilon) - {\\bar p}\\,,\n\\end{equation}\nwhere $p(\\rho,\\epsilon)$ is given by the chosen EOS and ${\\bar p}$ is\nthe trial value for the pressure eventually leading to the primitive\nvariables. \n\nNote that since ${\\boldsymbol Y}^{(i)} = {\\boldsymbol Y}^{*}$ [\\textit{cf.}~\n  Eq.~(\\ref{second_step_a})], the values of the conserved quantities\n$\\{ D, ~ \\tau , ~ {\\boldsymbol S},~{\\boldsymbol B} \\}$ at time\n$(n+1)\\Delta t$ are obtained by evolving their non-stiff evolution\nequations which, however, provide only an approximate solution for the\nelectric field $\\{ \\boldsymbol{E^*} \\}$. As discussed in the previous\nSection, the final solution for the electric field ${\\boldsymbol E}$\nrequires the inversion of an implicit equation and, hence, is a\nfunction of the velocity ${\\boldsymbol v}$ and of the fields $\\{\n{\\boldsymbol B}, \\boldsymbol{E^*} \\} $ [\\textit{cf.}~\n  Eq.~(\\ref{invert_matrix_E})].  However, the velocity is a primitive\nquantity and thus not known at the time $(n+1)\\Delta t$. It is clear,\ntherefore, that it is necessary to obtain, at the same time, the\nevolution of the stiff part of the equations and the conversion of the\nconserved quantities into to the primitive ones. In what follows we\ndescribe how to do this in practice using an iterative procedure.\n\n\\begin{enumerate}\n  \\item Adopt as initial guess for the velocity its value at the\n    previous time level ${\\boldsymbol v} = {\\boldsymbol v}^n$.  The\n    electric field ${\\boldsymbol E}$ is computed by\n    Eq.~(\\ref{invert_matrix_E}) as a function of $({\\boldsymbol\n      E}^*,{\\boldsymbol v},{\\boldsymbol B})$.\n\n   \\item Adopt as initial guess for the pressure its value at the\n     previous time level $p = p^{n}$. Compute in the\n     following order\n\n     \\begin{eqnarray}\\label{guesses}\n       {\\boldsymbol v} &=& \\frac{{\\boldsymbol S} - {\\boldsymbol E} \\times {\\boldsymbol B}}\n                      {\\tau - (E^2 + B^2)\/2 + p} \\, , \\nonumber \\\\\n       W &=& \\frac{1}{\\sqrt{1 - v^2}}\\,,  \\nonumber \\\\\n       \\rho &=& \\frac{D}{W} \\, , \\nonumber \\\\\n       \\epsilon &=& \\frac{\\tau - (E^2 + B^2)\/2 - D~W + p~(1-W^2)}\n                           {D~W} \\, .\n     \\end{eqnarray}  \n    \\item Solve numerically Eq.~(\\ref{trascendental}) by means of an\n      iterative Newton-Raphson solver, so that the solution at the\n      iteration $m+1$ can be computed as\n     \\begin{equation}\\label{newton-raphson}\n       p_{m+1} = p_m - \\frac{f(p_m)}{f'(p_m)} \\, .\n     \\end{equation}  \n    The derivative of the function $f(p)$ needed for the Newton-Raphson solver\n    can be computed as\n    \\begin{equation}\\label{derivativef}\n       f' (p) = v^2 c_{s}^2 - 1\\,,\n    \\end{equation}\n    with $c_{s}$ being the local speed of the fluid which, for an\n    ideal-fluid EOS $p(\\rho,\\epsilon) = (\\Gamma -1)~\\rho~\\epsilon$ is\n    given by\n    \\begin{equation}\n       \\label{soundspeed}\n       c_{s}^2 = \\frac{\\Gamma (\\Gamma-1)  \\epsilon}{ 1 + \\Gamma~\\epsilon}  \\, .\n    \\end{equation}\n\n    \\item With the newly obtained values for the velocity\n      ${\\boldsymbol v}$ and the pressure $p$, the steps (i)--(iii) can\n      be iterated until the difference between two successive values\n      falls below a specified tolerance.\n\\end{enumerate}\n\nThe approach discussed above is a simple procedure that can be implemented\nstraightforwardly and works well for moderate ratios of\n$|{\\boldsymbol B}|^2\/p$, converging in less than $10$ iterations both for\nsmooth electromagnetic fields and for discontinuous ones. \nFaster and more robust procedures to obtain the\nprimitive variables certainly ca be implemented, but this is beyond the\nscope of this work.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width = 75mm]{alfven.eps}}\n\\caption{Magnetic field component $B_y$ for a large-amplitude CP\n  Alfv\\'en wave and for three different resolutions $\\Delta\n  x=\\{1\/50,1\/100,1\/200 \\}$.  The conductivity is constant with a\n  magnitude of $\\sigma=10^6$. The agreement betweem\n  the exact solution and that corresponding to the high resolution one\n  is excellent.}\n\\label{alfven}\n\\end{figure}\n\n\\section{Numerical tests}\\label{section5}\n\nIn this section we present several one-dimensional (1D) or\ntwo-dimensional (2D) tests which have been used to validate the\nimplementation of the IMEX Runge-Kutta schemes in the different\nregimes of relativistic resistive MHD.  In all these tests we employ\nthe ideal-fluid EOS with $\\Gamma = 2$ for the 1D tests and $\\Gamma =\n4\/3$ in the 2D ones. The different tests span several prescriptions\nfor the conductivity and compare the solutions obtained either with\nthose expected in the ideal-MHD limit or with those obtained with the\nStrang-splitting technique.\n\nMore specifically, in 1D we consider large-amplitude circularly\npolarized (CP) Alfv\\'en waves to test the ability of the code to\nreproduce the ideal-MHD results when adopting a very large\nconductivity.  The intermediate conductivity regime is instead tested\nby simulating a self-similar current sheet. Finally, a large range of\nuniform and non-uniform conductivities are used for a representative\nshock-tube problem.  In 2D, on the other hand, we first consider a\ncommonly employed test for ideal-MHD codes corresponding to a\ncylindrical explosion. Subsequently, we simulate a toy model for a\n``magnetized neutron star'' when modelled as a cylindrically symmetric\ndensity distribution obeying a Gaussian-profile. The behaviour of the\nmagnetic field is studied again for a range of constant and\nnon-uniform conductivities.\n\n\\subsection{One-dimensional tests}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width = 75mm]{sheet.eps}}\n\\caption{Magnetic field component $B_y$ in a self-similar current\n  sheet. The solution is computed with $N=200$ gridpoints ($\\Delta x = 1\/200$) \n  and is shown\n  at the initial time $t=1$ and at $t=10$. The conductivity is uniform\n  with a magnitude of $\\sigma=10^2$ (\\textit{i.e.},~ $\\eta = 1\/\\sigma=0.01$). The\n  numerical solution is in excellent agreement with the\n  exact one.}\n\\label{sheet}\n\\end{figure}\n\n\n\\subsubsection{Large amplitude CP Alfv\\'en waves}\n\nThis test is discussed in detail by \\citet{DelZanBucLon:2007} and we\nreport here only a short summary. The solution describes the\npropagation of a large amplitude circularly-polarized Alfv\\'en waves\nalong a uniform background field $B_0$ in a domain with periodic\nboundary conditions. The exact solution in the ideal-MHD limit and\nassuming $v_x = 0$ for simplicity, is given by\n\\citep{DelZanBucLon:2007}\n\\begin{eqnarray}\n   (B_y,B_z) &=& \\eta_A B_0 ~ (\\cos[k(x-v_A~t)], \\sin[k(x-v_A~t)])\\,, \\nonumber \\\\\n   (v_y,v_z) &=& -\\frac{v_A}{B_0} ~(B_y, B_z) \\,,\n\\end{eqnarray}\nwhere $B_x=B_0$, $k$ is the wave vector, $\\eta_A$ is the amplitude\nof the wave and the special relativistic Alfv\\'en speed $v_A$ is\ngiven by\n\\begin{equation}\n   v_A^2 = \\frac{2 B_0^2}{h + B_0^2 (1 + \\eta_A^2)} \\left(1 + \\sqrt{1 - \\left(\\frac{2 \\eta_A B_0^2}{h + B_0^2 (1 + \\eta_A^2)} \\right)^2 } \\right)^{-1}\\,.\n\\end{equation}\nIn practice, using such ideal-MHD solution it is possible to assess\nthe accuracy of evolution of the resistive equations by requiring that\nfor very large conductivities the numerical solution approaches the\nexact one as the resolution is progressively increased. It is also\nworth remarking that although we do not expect the solution of the\nresistive MHD equations to converge to that of ideal MHD for any\nfinite value of $\\sigma$, we also expect the differences between the\ntwo to be ${\\cal O}(v\/\\sigma)$ and thus negligibly small for\nsufficiently large values. For this reason, we have performed the\nevolution with a high uniform conductivity of $\\sigma = 10^6$ for three\ndifferent resolutions $N=\\{50,100,200\\}$ covering the computational\ndomain $x \\in [-0.5,0.5]$.  In addition, the initial data parameters\nhave been chosen so that $\\rho = p = \\eta_A = 1$ and $B_0 = 1.1547$,\nthus yielding $v_A =1\/2$, with a full period being achieved at $t=2$.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[width = 75mm]{shock_convergence.eps}\n\\hskip 1.0cm\n\\includegraphics[width = 75mm]{shock_condcte.eps}\n}\n\\caption{\\textit{Left panel:} Magnetic field component $B_y$ in the\n  solution of the shock-tube problem. Different lines refer to three\n  different resolutions and to the exact ideal-MHD solution at\n  $t=0.4$.  The conductivity is uniform  with a magnitude of\n  $\\sigma_0=10^6$. \\textit{Right panel:} The same as in the left panel\n  but for different uniform conductivities. Note that for\n  $\\sigma_0=0$ the solution describes a discontinutiy \n  propagating at\n  the speed of light and corresponding to Maxwell equations in\n  vacuum. As the conductivity increases, the solution tends to the\n  ideal-MHD one.}\n\\label{shock_convergence}\n\\end{figure*}\n\nFig.~\\ref{alfven} confirms this expectation by reporting the component\n$B_y$ after one period and thus overlapping with the initial one (at\n$t=0$) for the highest resolution.  This test shows clearly that in\nthe limit of very high conductivity the resistive MHD equations tend\nto a solution which is very close to the same solution obtained in the\nideal-MHD limit. The convergence rate measured for the different\nfields is consistent with the second-order spatial discretization\nbeing used as expected for smooth flows (see\nAppendix~\\ref{appendixB}).\n\n\n\\subsubsection{Self-similar current sheet}\n\n\\begin{figure}\n\\centerline{\\includegraphics[width = 75mm]{error_n.eps}}\n\\caption{Differences in the magnetic field component $B_y$ between the\n  numerical solution computed with either the Strang or the IMEX\n  schemes and the exact solution of the shock-tube in the ideal-MHD\n  limit. The differences are computed for several uniform\n  conductivities, although the Strang-splitting technique does not\n  yield a stable solution for values larger than $\\sigma_0 \\sim\n  7000$ for the reference resolution of $\\Delta x = 1\/400$ (i.e. with\n  $400$ gridpoints). Shown in the\n  inset is the maximum conductivity for which a solution was possible,\n  $\\sigma_{\\rm max}$, as a function of the number of gridpoints, $N$.}\n\\label{error}\n\\end{figure}\n\nThe details of this test are described by~\\citet{Kom:2007}, so again\nwe provide here only a short description for completeness. We assume\nthat the magnetic pressure is much smaller than the fluid pressure\neverywhere, with a magnetic field given by ${\\boldsymbol B} = (0,\nB_y(x,t), 0)$, where $B_y(x,t)$ changes sign within a thin current\nlayer of width $\\Delta l$. Provided the initial solution is in\nequilibrium ($p={\\rm const.}$), the evolution is a slow diffusive\nexpansion of the layer due to the resistivity and described by the\ndiffusion equation [\\textit{cf.}~ Eq.~(\\ref{newtonian_rMHD}) with ${\\boldsymbol\n    v}=0$]\n\\begin{eqnarray}\\label{diffusion_eq}\n    \\partial_t B_y - \\frac{1}{\\sigma} \\partial_x^2 B_y = 0 ~~.\n\\end{eqnarray}\nAs the system expands, the width of the layer becomes much larger\nthan $\\Delta l$ and it evolves in a self-similar fashion. For $t > 0$, \nthe analytical exact solution is given by\n\\begin{eqnarray}\\label{diffusion_sol}\n   B_y(x,t) = B_0 \\, {\\rm erf} \\,\\left(\n   \\frac{1}{2}\\sqrt{\\frac{\\sigma}{\\xi}} \\right )\\,,\n\\end{eqnarray}\nwhere $\\xi = t\/x^2$ and ``${\\rm erf}$'' is the error function. This\nsolution can be used for testing the moderate resistive\nregime. Following~\\citet{Kom:2007}, and in order to avoid the singular\nbehaviour at $t=0$, we have chosen as initial data the solution at\n$t=1$ with $p=50$, $\\rho=1$, $E={\\boldsymbol v}=0$ and\n$\\sigma=100$. The domain covers the region $x \\in [-1.5,1.5]$ with\n$N=200$ points.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[width = 75mm]{shock_condpot.eps}\n\\hskip 1.0cm\n\\includegraphics[width = 75mm]{shock_pot.eps}\n}\n\\caption{\\textit{Left panel:} Evolution of a non-uniform conductivity\n  $\\sigma$ in the shock-tube problem for different values of $\\gamma$\n  and indicated by the different lines ($\\sigma_0=10^6$ for all\n  lines). Notice the large variability on the magnitude of the\n  conductivity. \\textit{Right panel:} The same as in the left panel\n  but for the magnetic field component $B_y$.}\n\\label{shock_condpot}\n\\end{figure*}\n\nThe numerical simulation is evolved up to $t=10$ and then the\nnumerical and the exact solution are compared in Fig.~\\ref{sheet}.\nThe two solutions match so well that they are not distinguishable on\nthe plot, thus, showing that the intermediate-conductivity regime is\nalso well described by our method.\n\n\n\\subsubsection{Shock-tube problem}\n\nAs prototypical shock-tube test we consider a simple MHD\nversion of the Brio and Wu test \\citep{BriWu:1988}, where\nthe initial left and right states are separated at $x=0.5$ and are given by\n\\begin{eqnarray}\n   (\\rho^L,p^L,B_y^L) &=& (1.0, 1.0, 0.5)\\,, \\nonumber \\\\\n   (\\rho^R,p^R,B_y^R) &=& (0.125, 0.1, -0.5)\\,, \\nonumber\n\\end{eqnarray}\nwhile all the other fields set to $0$. We consider both uniform and\nnon-uniform conductivities. In the latter case we adopt the following\nprescription\n\\begin{equation}\\label{def_cond}\n   \\sigma = \\sigma_0 D^{\\gamma} \\, ,\n\\end{equation}\nthus allowing for nonlinearities in the dependence of the conductivity\non the conserved quantity $D$. This is one of the simplest cases, but\nin realistic situations a more general expression for the conductivity\ncan be assumed, where $\\sigma$ is a function of both the rest-mass\ndensity and of the specific internal energy, \\textit{i.e.},~ $\\sigma = \\sigma\n(\\rho, \\epsilon)$.\n\nThe exact solution of the ideal MHD Riemann problem was found\nby~\\citep{GiaRez:2006}, and in our particular case it has been\ncomputed with a publicly available code\n[see~\\citet{GiaRez:2006}]. When $B_x=0$, the structure of the solution\ncontains only two fast waves, a rarefaction moving to the left and a\nshock moving to the right, with a tangential discontinuity between\nthem. More demanding Riemann problems have also been performed but the\nprocedure to convert the conserved variables into the primitive ones\nhas shown in these case a lack of robustness for large ratios of\n$|{\\boldsymbol B}|^2\/p$.\n\nWe have first considered the case of uniform ($\\gamma=0$) and very\nlarge conductivity ($\\sigma_0=10^6$) as in this case we can use the\nsolution in the ideal-MHD limit as a useful guide. The profile of the\nmagnetic field component $B_y$ for three different resolutions $\\Delta\nx=\\{1\/100,1\/200,1\/400\\}$ and the exact solution are shown in the left\npanel of Fig.~\\ref{shock_convergence} at $t=0.4$. Overall, the results\nindicate that even in the presence of shocks our numerical solution of\nthe resistive MHD tends to the ideal-MHD solution as the resolution is\nincreased. It is also interesting to study the behaviour of the\nsolution for different values of the constant $\\sigma_0$ while still\nkeeping a uniform conductivity (\\textit{i.e.},~ $\\gamma=0$). This is shown in the\nright panel of Fig.\\ref{shock_convergence}, which displays the\ndifferent solutions obtained, and where it is possible to see how they\nchange smoothly from a wave-like solution for $\\sigma_0=0$ to the\nideal-MHD one for $\\sigma_0=10^6$.\n\nThis set up is also useful to perform a comparison between the IMEX\nand the Strang-splitting approaches. In Fig.~\\ref{error} we show the\n$L^1$-norm of the difference between the numerical solution obtained\nwith both schemes and the ideal-MHD exact solution, for different\nvalues of the conductivity with $N=400$ points.\n\nSeveral comments are in order. Firstly, the reported difference\nbetween the numerical solution for the resistive MHD equations and the\nideal-MHD equations should not be interpreted as an error given that\nthe latter is not the correct solution of the equations. Hence, the\nfact that the use of a Strang-splitting method yields smaller\ndifferences is simply a measure of its ability of better capture steep\ngradients. Secondly, while the IMEX approach does not show any sign of\ninstability for $\\sigma_0$ ranging between $10^2$ and $10^9$, the\nimplementation adopting the Strang-splitting technique becomes\nunstable for moderately high values of the conductivity and, at least\nfor the shock-tube problem, no numerical solution was possible for\n$\\sigma_0 \\gtrsim 7000$ at the above resolution. Increasing the\nresolution can help increase the maximum value of the resistivity\nwhich can be handled, but since this gain is only linear with the\nnumber of gridpoints aiming for higher conductivities results impractical. \nThis is shown in\nthe inset of Fig.~\\ref{error}, which reports the maximum conductivity\nfor which a solution was possible, $\\sigma_{\\rm max}$, as a function\nof the number of gridpoints, $N$. Finally, we note that the difference\nbetween the IMEX numerical solution and the exact ideal-MHD one\nsaturates between $\\sigma_0 \\sim 10^5-10^6$. This is not surprising\nsince the differences are expected to be ${\\cal O}(1\/\\sigma)$, and\nthus the saturation in the differences essentially provides a measure\nof our truncation error at the resolution used.\n\nA more challenging test is offered by the solution of the shock-tube\nin the presence of a non-uniform conductivity. In particular, we have\nconsidered the same initial states and the same non-uniform\nconductivity discussed above, but used different values for the\nexponent $\\gamma$ in (\\ref{def_cond}) while keeping $\\sigma_0$\nconstant. The results of this test are shown in the left panel of\nFig.~\\ref{shock_condpot}, where the conductivity is plotted at $t=0.4$\nfor several values of $\\gamma$. Note that the conductivity traces the\nevolution of the rest-mass density and that the solution can be found\nalso when $\\sigma$ varies of almost $12$ orders of magnitude across\nthe grid. Similarly, the right panel of Fig.~\\ref{shock_condpot}\ndisplays the component $B_y$ for the different values of $\\gamma$. It\nshould be stressed that because of the relation (\\ref{def_cond})\nbetween $\\sigma$ and $\\rho$, the region on the left has at this time a\nvery high conductivity and the numerical solution tends to the\nideal-MHD one. The opposite happens on the right region, where the\nconductivity is lower for higher values of $\\gamma$. Clearly, the\nresults presented in Fig.~\\ref{shock_condpot} show that our\nimplementation can handle non-uniform (and quite steep) conductivity\nprofiles even in the presence of shocks.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[width = 75mm]{explosion_Bx_2D.eps}\n\\hskip 1.cm\n\\includegraphics[width = 75mm]{explosion_By_2D.eps} \n}\n\\caption{Magnetic field components $B_x$ (left panel) and $B_y$ (right\n  panel) for the cylindrical explosion test at time $t=4$.}\n\\label{explosion_Bs_2D}\n\\end{figure*}\n\n\n\\subsection{Two-dimensional tests}\n\n\\subsubsection{The cylindrical explosion}\n\nWe now consider problems involving shocks in more than one dimension.\nA demanding test for the relativistic codes is the cylindrical blast\nwave expanding in a plasma with an initially uniform magnetic\nfield. Although there is no exact solution for this problem, strong\nsymmetric explosions are useful tests since shocks are present in all\nthe possible directions and the numerical implementation is therefore\ntested in all of its parts.  For this test we set a square domain\n$(x,y) \\in [-6,6]$ with a resolution $\\Delta x=\\Delta y=1\/200$. The\ninitial data is such that inside the radius $r<0.8$ the pressure is\nset to $p=1$ while the density to $\\rho=0.01$. In the intermediate\nregion $0.8 \\leq r \\leq 1.0$ the two quantities decrease exponentially\nup to the exterior region $r>1$, where the ambient fluid has\n$p=\\rho=0.001$. The magnetic field is uniform with only one nontrivial\ncomponent ${\\boldsymbol B}= (0.05, 0 ,0 )$. The other fields are set\nto be zero (\\textit{i.e.},~ ${\\boldsymbol E} = q = 0$), which is consistent within\nthe ideal-MHD approximation.\n\nThe evolution is performed with a high conductivity $\\sigma=10^6$ in\norder to recover the solution from the ideal-MHD approximation. As\nshown in Fig.~\\ref{explosion_Bs_2D}, which reports the magnetic field\ncomponents $B_x$ (left panel) and $B_y$ (right panel) at time $t=4$,\nwe obtain results that are qualitatively similar to those published in\ndifferent\nworks~\\citep{Kom:1999,NeiHirMil:2006,DelZanBucLon:2007,Kom:2007}. While\na strict comparison with an exact solution is not possible in this\ncase, the solution found matches extremely well the one obtained with\nanother 2D code solving the ideal MHD equations.\nMost importantly, however, the figure\nshows that the solution is regular everywhere and that similar results\ncan be obtained also with smaller values of the conductivity (\\textit{e.g.},~ no\nsignificant difference was seen for $\\sigma \\gtrsim 10^4$).\n\n\\subsubsection{The cylindrical star}\n\nWe next consider a toy model for a star, thought as an infinite column\nof fluid aligned with the $z$-axis but with compact support in other\ndirections. Because of the symmetry in the $z$-direction, $\\partial_z\n\\boldsymbol{U} = 0$ for all the fields and the problem is therefore\ntwo-dimensional. More specifically, we consider initial data given by\n\\begin{eqnarray}\n   \\rho &=& \\rho_o e^{-(r\/r_o)^2} \\, ,\n\\label{ID_fluid1} \\\\\n   {\\boldsymbol v} &=& (v^r,v^\\phi,v^z) = \\rho~( 0, \\omega^\\phi, 0) \\, ,\n\\label{ID_fluid2} \\\\\n   {\\boldsymbol B} &=& (B^r,B^\\phi,B^z) = \\rho~\\left( 0, 0, 2~B_o~(1 - \\frac{r^2}{r_o^2})\\right) \\,,\n\\label{ID_fluid3} \n\\end{eqnarray}\nwhere $r \\equiv \\sqrt{x^2+y^2}$ is the cylindrical radial coordinate.\nThe other fields can be computed at the initial time by using the\npolytropic EOS $p = \\rho^\\Gamma$, the ideal-MHD\nexpression~(\\ref{ef_imhd}) for the electric field, and the electric\ncharge from the constraint equation $q = \\nabla \\cdot {\\boldsymbol\n  E}$.  We have chosen $r_o=0.7$, $\\rho=1.0$, $\\omega^\\phi=0.1$ and\n$B_0 = 0.05$. An atmosphere ambient fluid with $\\rho=0.01$ is added\noutside the cylinder. Finally, the resolution is $\\Delta x = 1\/200$\nand the domain is $(x,y)\\in [-3,3]$.\n\nThis simple problem exhibits some of the issues present in a\nmagnetized rotating neutron star: a compactly supported rest-mass\ndensity distribution, an azimuthal velocity field and a poloidal\nmagnetic field.  Suitable source terms describing a gravitational\npotential have been added to the Euler equations in order to get, at\nleast at the initial time, a stationary solution. In the ideal-MHD\nlimit the magnetic lines are frozen in the fluid and thus a static\nprofile is also expected for the magnetic field.\n\nIn the left panel of Fig.~\\ref{star_Bz} we plot the slice $y=0$ of the\nmagnetic field component $B^z$ at $t=14$ as obtained from the\nevolution of the resistive MHD system for different uniform\nconductivities in the range $\\sigma_0~\\epsilon~[10^2, 10^6]$. In the\nlimiting case $\\sigma_0 = 0$ the solution corresponds to a wave\npropagating at the speed of light (\\textit{i.e.},~ the solution of the Maxwell\nequations in vacuum), while for large values of $\\sigma_0$ the\nsolution is stationary (as expected in the ideal-MHD limit). The\nbehaviour observed in the left panel Fig.~\\ref{star_Bz} is also the\nexpected one: the higher the conductivity, the closer the solution is\nto the stationary solution of the ideal-MHD limit. For low\nconductivities, on the other hand, there is a significant diffusion of\nthe solution, which is quite rapid for $\\sigma_0 < 10^2$ and for this\nreason those values are not plotted here. We note that values of the\nconductivity larger than $\\sigma_0 > 10^7$ lead to numerical\ninstabilities that we believe are coming from inaccuracies in the\nevolution of the charge density $q$, and which contains spatial\nderivatives of the current vector.  In addition, the stiff quantity\n$E_x$ is seen to converge only to an order $ \\sim 1.5$. This can be\ndue to the ``final layer'' problem of the IMEX methods, which is known\nto produce a degradation on the accuracy of the stiff\nquantities. Luckily, this does not spoil the convergence of the\nnon-stiff fields, which are instead second-order convergent. It is\npossible that the use of stiffly-accurate schemes can solve this\ndegradation of the convergence and this is an issue we are presently\nexploring.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[width = 75mm]{star_Bz.eps}\n\\hskip 1.0cm\n\\includegraphics[width = 75mm]{star_Bp.eps}\n}\n\\caption{\\textit{Left panel:} Slice, at $y=0$, of the magnetic field\n  component $B^z$ for different conductivities $\\sigma$ and the exact\n  solution in the ideal-MHD limit.  The resolution is $\\Delta x=1\/200$\n  and the solution is plotted at $t=14$. \\textit{Right panel:} the\n  same configuration as in the left panel but with a non-uniform\n  conductivity with $\\sigma_0=10^6$ and $\\gamma=[0,3,6,9]$. The values\n  inside the star are essentially the same for any $\\gamma$, while there\n  are significant differences outside.}\n\\label{star_Bz}\n\\end{figure*}\n\n\nWe finally consider the same test, but now employing the non-uniform\nconductivity given by Eq.~(\\ref{def_cond}) with $\\sigma_0=10^6$ and\ndifferent values for $\\gamma$. The results are presented in the right\npanel of Fig.~\\ref{star_Bz}, which shows that the magnetic fields\ninside the star are basically the same in all the cases, stressing the\nfact that the interior of the star will not be significantly affected\nby the exterior solution, which has much smaller\nconductivity. However, the electromagnetic fields outside the star do\nchange significantly for different values of $\\gamma$, underlining the\nimportance of a proper treatment of the resistive effects in those\nregions of the plasma where the ideal-MHD approximation is not a good\none.\n\n\n\\section{Conclusions}\n\\label{section6}\n\nWe have introduced Implicit-Explicit Runge-Kutta schemes to solve\nnumerically the (special) relativistic resistive MHD equations and\nthus deal, in an effective and robust way, with the problems inherent\nto the evolution of stiff hyperbolic equations with relaxation terms.\nSince for these methods the only limitation on the size of the\ntimestep is set by the standard CFL condition, the approach suggested\nhere allows to solve the full system of resistive MHD equations\nefficiently without resorting to the commonly adopted limit of the\nideal-MHD approximation.\n\nMore specifically, we have shown that it is possible to split the\nsystem of relativistic resistive MHD equations into a set of equations\nthat involves only non-stiff terms, which can be evolved\nstraightforwardly, and a set involving stiff terms, which can also be\nsolved explicitly because of the simple form of the stiff\nterms. Overall, the only major difficulty we have encountered in\nsolving the resistive MHD equations with IMEX methods arises in the\nconversion from the conserved variables to the primitive ones. In this\ncase, in fact, there is an extra difficulty given by the fact that\nthere are four primitive fields which are unknown and have to be\ninverted simultaneously. We have solved this problem by using extra\niterations in our 1D Newton-Raphson solver, but a multidimensional\nsolver is necessary for a more robust and efficient implementation of the\ninversion process.\n\nWith this numerical implementation we have carried out a number of\nnumerical tests aimed at assessing the robustness and accuracy of the\napproach, also when compared to other equivalents ones, such as the\nStrang-splitting method recently proposed by~\\citet{Kom:2007}. All of\nthe tests performed have shown the effectiveness of our approach in\nsolving the relativistic resistive MHD equations in situations\ninvolving both small and large uniform conductivities, as well as\nconductivities that are allowed to vary nonlinearly across the\nplasma. Furthermore, when compared with the Strang-splitting\ntechnique, the IMEX approach has not shown any of the instability\nproblems that affect the Strang-splitting approach for flows with\ndiscontinuities and large conductivities.\n\nWhile the results presented here open promising perspectives for the\nimplementation of IMEX schemes in the modelling of relativistic\ncompact objects, at least two further improvements can be made with\nminor efforts. The first one consists of the generalization of the\n(special) relativistic resistive MHD equations with a scalar isotropic\nOhm's law to the general relativistic case, and its application to\ncompact astrophysical bodies such a magnetized binary neutron\nstars~\\citep{AHLLMNPT:2008,LiuShaZacTan:2008}. The solution of the\nresistive MHD equations can yield different results not only in the\ndynamics of the magnetosphere produced after the merger, but also\nprovide the possibility to predict, at least in some approximation,\nthe electromagnetic radiation produced by the merger of these\nobjects. The second improvement consists of considering a non-scalar\nand anisotropic Ohm's law, so that the behaviour of the currents in\nthe magnetosphere can be described by using a very high conductivity\nalong the magnetic lines and a negligibly small one in the transverse\ndirections~\\citep{Kom:2004}. Such an improvement may serve as a first\nstep towards an alternative modelling of force-free plasmas.\n\n\n\\section*{Acknowledgments}\nWe would like to thank Eric Hirschmann, Serguei Komissarov, Steve\nLiebling, Jonathan McKinney, David Neilsen and Olindo Zanotti for\nuseful comments and Bruno Giacomazzo for comments and for providing\nthe code computing the exact solution of the Riemann problem in ideal\nMHD. LL and CP would like to thank FaMAF (UNC) for hospitality. CP is\nalso grateful to Lorenzo Pareschi for the many clarifications about\nthe IMEX schemes. This work was supported in part by NSF grants\nPHY-0326311, PHY-0653369 and PHY-0653375 to Louisiana State\nUniversity, the DFG grant SFB\/Transregio~7, CONICET and Secyt-UNC.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{}\n\\addcontentsline{toc}{section}{Introduction}\n\\normalsize\n\nAnyons \\cite{A1,A2,A3,A4,A5} are probably the most interesting objects I have encountered in physics.\n\nAnyons captivate because of their mathematical beauty, because studying them means to enter into the heart of topology, of knot and braid theories, of tensor categories.\n\nMeanwhile anyons are fascinating because of their deep physical meaning. They emerge as quasiparticles of an exceptional type of organization of matter, which is purely quantum in nature:  topological order \\cite{TO1, TO2}.\nIn contrast to all other types of orders (crystals, ferromagnets, or even superfluids or superconductors) topological order cannot be described by a classical local field. It is not readable from the individual components (atoms, electrons, photons), it is hidden in the pattern of many-particle entanglement established among them.\n\nIn a topologically ordered state a system of individuals constitute a global entity, which acquires a macroscopic self-identity that transcends the identities of the microscopic constituents. The laws that govern the emergent collectivity are topological laws: they are invariant under local deformations of the system. This is in radical opposition to the nature of the original microscopical laws (electromagnetic or gravitational forces), which are strongly dependent on geometrical details such as distances or angles.\n\nAnyons are profoundly counterintuitive. Their braiding statistics has dramatically shattered our system of beliefs regarding the possible statistics for quantum particles \\cite{A1,A2,A3,A4,A5}. Specially striking is the case of non-Abelian anyons \\cite{NA1,NA2,NA3,NA4,NA5,NA6,NA7,NA8,NA9,NA10}: How is it possible that the result of sequentially exchanging pairs in a set of indistinguishable particles might depend on the order in which the exchanges were performed?\n\nWe do have mathematical languages to describe anyons and represent their bizarre properties. We could say, for example, that an anyon can be identified with an irreducible representation of the group of braids, and that its non-Abelian character is a natural consequence of the composition of braids being non-commutative. \nBut that we are able to name or represent anyons with appropriate mathematical tools does not mean that we thoroughly understand what they are, nor, specially, does it mean that we can explain under which conditions they emerge from a physical system. \n\nThere is still a large gap between the topological mathematical rules governing anyons and the physical laws dictating the behavior of the underlying physical system. \nWhat is the correspondence between a certain type of anyon statistics and the pattern of many-particle entanglement that gave it birth? Which particular combination of microscopic degrees of freedom, interactions among them and external fields, made such pattern emerge? We can still not give precise answers to these questions.\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{1cm}\nIt appears to me that finding physically meaningful languages to describe anyons can help us sharpen our knowledge about them. I believe that if we are able to capture the essence of anyons using an intuitive, comprehensive physical vocabulary, we will be closer to fill the lacunas between the mathematical and the physical, between the global and the local faces of topological orders.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{IntroAnyons.jpg}\n\\end{center}\n\n\n\n\\newpage\n\\thispagestyle{plain}\n\\section*{}\n\\addcontentsline{toc}{section}{Introduction}\n\nThe set of anyon types emerging from a certain topological order satisfies a collection of {\\em fusion and braiding rules} \\cite{Kitaev1,Kitaev2,Preskill,Wang,Bonderson}. These rules determine the way in which anyons fuse with each other to give rise to other anyon types, and the form in which they braid around each other. A set of anyon types together with their fusion and braiding rules define an anyon model. \nDifferent topological orders give rise to distinct anyon models, which perfectly mirror their corresponding underlying physical states.\n\n\nA first natural question we can ask  is: What are the possible anyon models that can exist? The answer to this question can be easily expressed in a formal manner.\nFor an anyon model to exist its fusion and braiding rules cannot be arbitrary. They have to fulfill a collection of consistency conditions which can be written as a set of equations, known as the Pentagon and Hexagon equations \\cite{Preskill, Bonderson}. An anyon model is therefore a solution of these equations. \nWang \\cite{Wang} has tabulated all {\\em modular} anyon models with up to four anyon types. These correspond, for example, to truncated Lie algebras such as $\\mathbf{SU}(2)_2$, or $\\mathbf{SO}(3)_3$, the celebrated Fibonacci model.\nBonderson \\cite{Bonderson} has developed an algorithm to numerically solve the pentagon and hexagon equations under certain conditions, tabulating a series of very interesting anyon models for up to ten topological charges. \n\nFrom a purely mathematical perspective, it is known that the language underlying anyon models is {\\em modular tensor category}. Firstly, the structures of anyon models originated from conformal field theory \\cite{CFT1, CFT2} and Chern-Simons theory \\cite{CHS1}. They were further developed in terms of algebraic quantum field theory \\cite{AT1, AT2} and then made mathematically rigorous in the language of braided tensor categories \\cite{TC1,TC2,TC3}. Within this beautiful (and complex) mathematical formalism the answer to the question above can be also simply phrased:\nany possible anyon model corresponds to a unitary braided modular tensor category. \n\nBut if we want to delve into our physical knowledge of anyons, we should refine the question above.  We should ask ourselves not only what the possible anyon models are, but, more importantly, what the relational architecture of possible anyon models is. We should be able to find answers to questions such as: Is there a hidden organization in the set of anyon models? Can we construct complex anyon models from other simpler ones? Which are the elementary pieces? What is the glue mechanism of these pieces?\nIn clarifying these questions, it seems unclear whether generating solutions to the Pentagon and Hexagon equations or enumerating possible unitary braided theories can be by themselves illuminating enough.\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{1cm}\n\nIn establishing relations between different anyon models, something that we know is how to disintegrate certain complex anyon models into other simpler ones.  This procedure, called {\\em anyon condensation} \\cite{AC1,AC2,AC3,AC4,AC5,AC6,AC7,AC8,AC9,AC10}, works by making two or more different anyon types become the same. \nThough no fully general description is known, for the special case in which the condensing anyons have trivial statistics, it is possible to systematically obtain a condensed anyon model from a more complex (uncondensed) one.\nAnyon condensation does not tell us, however, about the reverse process, that is, about how to build up more complex anyon models by putting simpler ones together. To go in this \"up\" direction, we have only straightforward operations at our disposal, such as, for instance, making the tensor product of two or more given anyon models. \n\nHere, I believe it is crucial to develop pathways to orderly construct anyon models.  This can help us enormously to apprehend the subjacent texture of anyon models and thereby the anatomy of topological orders.\n\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{0cm}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{IntroConstruction.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{plain}\n\\section*{}\n\\addcontentsline{toc}{section}{Introduction}\n\nThis work is an attempt to find out the skeleton of anyon models. \n\nI present a construction to systematically generate anyon models.\nThe construction uses a set of elementary pieces or fundamental anyon models, which constitute the building blocks to construct other, more complex, anyon models.\nA principle of assembly is established that dictates how to articulate the building blocks, setting out the global blueprint for the whole structure. \nRemarkably, the construction generates essentially all tabulated anyon models \\cite{Wang, Bonderson}. Moreover, novel anyon models (non-tabulated, to my knowledge) arise.\n\nThe construction is formulated in a very physical, visual and intuitive manner. An anyon model corresponds to a system of bosons in a lattice.\nBy varying the number of bosons and the number of lattice sites, towers of more and more complex anyon models are built up. It is a Boson-Lattice construction.\nImportantly, the Boson-Lattice systems used in the construction are not real physical systems from which anyon models emerge. Here, Boson-Lattice systems {\\em are} themselves  anyon models.\n\nTo formulate the construction I develop a language for anyon models. In this language an anyon model is represented by a graph or a collection of graphs, which encode the properties of the anyon model. Topological charges (anyon types) are represented by graph vertices. Fusion rules can be read from the connectivity pattern of the graphs.\nBraiding rules are obtained through diagonalization of the graphs.\nThis graph language is the first contribution of this work. It provides an enlightening way to encode anyon models,  allowing to both easily visualize and extract their properties. \n\n\n\nThe elementary pieces of the Boson-Lattice construction are the Abelian $\\mathbb{Z}_n$ anyon models. In the language of graphs these models are represented by periodic one-dimensional lattices, in which each lattice site is connected to its next (to the right) neighbour. \nTriggered by this graph representation, the first key idea to develop the construction arises: I make a conceptual leap by identifying a $\\mathbb{Z}_n$ model with a single particle in a periodic lattice with $n$ sites.\nThis visual image condenses the essence of the anyon model into a particle in a lattice. It inspires the next crucial step: the conception of the principle of assembly. The assembly of the building blocks is defined as a {\\em bosonization procedure}, in which particles corresponding to different building blocks are made indistinguishable. The resulting Boson-Lattice system  characterizes the constructed anyon model.\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{2.3cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{IntroBosonLattice.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{0.5cm}\n\nI give a prescription to assign a graph to the Boson-Lattice system. The graph is defined as the {\\em connectivity graph} of the corresponding Fock states.\nThe central result of this work states that, with this definition, the Boson-Lattice graph always embodies a modular anyon model. A dictionary is established between the elements of the Boson-Lattice graph (Fock states, connectivity pattern, eigenvalues and eigenstates), and the properties of the anyon model (topological charges, fusion rules, braiding rules).\nThe special features of the Boson-Lattice graph assure that the extracted properties are well defined and satisfy the required consistency conditions.\n\nTo illustrate the construction I consider several examples of anyon models arising within the Boson-Lattice formalism. It is pleasing to see how series of well known anyon models are generated by varying the number of bosons and the number of lattice sites. $\\mathbf{SU}(2)_k$ anyon models are constructed as Boson-Lattice systems of $k$ bosons in a lattice of $2$ sites. The Fibonacci anyon model corresponds to a Boson-Lattice system of $2$ bosons in a lattice of $3$ sites. The series of $\\mathbf{SO}(3)_k$ models are built up as systems of $2$ bosons in lattices of $k$ sites. Furthermore, other non-tabulated anyon models emerge, as, for instance, those corresponding to Boson-Lattice systems of $2$ bosons in $4$ lattice sites, or $3$ bosons in $3$ lattice sites.\n\nThe Boson-Lattice construction is a fractal construction. Anyon models created by assembling the building blocks $\\mathbb{Z}_n$ can be used themselves as elementary pieces to generate new models at a second level of the construction. Nicely, the principle of assembly is the same at any level of the construction, giving rise to a self-similar pattern that replicates itself at any scale. Might this fractal architecture be the one behind anyon models? \n\nThe formalism can be generalized by adding internal degrees of freedom to the bosons participating, by using multidimensional lattices, or, additionally, by considering fermions instead of bosons. It is very interesting to see how further series of anyon models, such as, for instance, {\\em quantum double models} \\cite{Kitaev1}, can arise from such generalizations.\n\nThe construction reveals an anatomy for anyon models. \nI have focused here on building up {\\em modular} anyon models, for which corresponding topological field theories and conformal field theories exist. It would be revealing  to\ninvestigate how known structures and concepts in topological quantum field theory and conformal field theory can be interpreted within the language of the Boson-Lattice construction. And, conversely, to see how the construction might shed light onto the skeleton of topological field theories and conformal field theories themselves.\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{3cm}\n\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{IntroGraph.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{plain}\n\\vspace*{2cm}\n\nThe Boson-Lattice approach can guide us to develop a construction for topological states and Hamiltonians at the microscopic level. The Boson-Lattice blueprint can serve as a dual blueprint to design the corresponding many-particle wave functions.\nMoreover, the actual Boson-Lattice system that abstractly represents the anyon model, can help us to design the microscopic degrees of freedom and interactions composing the topological model from which the anyon model emerges. I believe the Boson-Lattice system can be regarded as a dual physical entity, able to encode at the same time the global and the local, the mathematical and the physical ingredients of a topological order.\n\nI find extremely interesting to draw a map of connections among the many-body wave functions and Hamiltonians generated by the Boson-Lattice construction and those arising in seminal topological systems and models, such as {\\em fractional quantum Hall systems} \\cite{NA1,NA2,NA3,NA4,NA5,NA6,NA7}, {\\em quantum loop models} \\cite{Kitaev2, QLM1, QLM2, QLM3} and {\\em string-net models} \\cite{SNC1,SNC2,SNC3,SNC4}.\n\n\nAs an essential outcome, this work reveals that the mathematical language describing anyon models can be identical to the one describing bosonic lattice systems. It states that the fusion rules and braiding rules characterizing anyon models can be represented by simple, intuitive physical objects, such as Fock states or tunneling Hamiltonians.\n\nIt is indeed remarkable that the connectivity graph of Fock states of a bosonic lattice system can encode the non-trivial consistency conditions required for an anyon model to exist.\nWhile developing the Boson-Lattice formalism I was thrilled to see how the construction succeeded in always generating anyon models with the correct properties, for any number of bosons and lattice sites, at any level of the construction.\nI was urged to understand the reason behind this extraordinary coincidence. \nNicely, in trying to find intuitive grounds for it, an unexpected connection emerged: a correspondence between Boson-Lattice graphs and {\\em curved space geometries}. \n\nThis connection anticipates an intriguing duality between anyon models and curved space-time geometries, between anyon models and {\\em gravity}. Understanding and developing this duality is a challenge I feel compelled to achieve.\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{10cm}\n\\hspace*{7cm}\n\\includegraphics[width=0.4\\textwidth]{ConstructionTitleFigure.jpg}\n\n\\newpage\n\\thispagestyle{plain}\n\n\\section*{}\n\\large\n\\includegraphics[width=0.5\\textwidth]{TitleIndex.jpg}\n\n\\hspace*{1cm}\n\\begin{tabular}{m{2.5cm}m{10cm}m{1cm}}\n\\includegraphics[width=0.1\\textwidth]{TitlePreliminaries.jpg}&\\large Preliminaries & \\normalsize17\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleGraphs.jpg}&\\large Topological graphs. The language of the construction & \\normalsize 21\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleBlocks.jpg}&\\large The building blocks &  \\normalsize 41\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleConstruction.jpg}&\\large The Boson-Lattice construction & \\normalsize 49\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleExamples.jpg}&\\large Boson-Lattice examples & \\normalsize 73\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleWhy.jpg}&\\large Why does the construction work? & \\normalsize 123\\\\\n\\includegraphics[width=0.1\\textwidth]{TitleLevels.jpg}&\\large Higher levels of the construction &\\normalsize 129 \\\\\n\\includegraphics[width=0.1\\textwidth]{TitleClosures.jpg}&\\large Closures and openings & \\normalsize 133\\\\\n\\end{tabular}\n\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Preliminaries}\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{AnyonModel.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{An Anyon model}\n\\rfoot{\\thepage}\n\\section*{An anyon model}\n\\addcontentsline{toc}{section}{An anyon model}\n\\normalsize\n\nAn anyon model \\cite{Kitaev1, Kitaev2, Preskill, Wang, Bonderson} is characterized by a finite set of conserved topological charges or anyon types:\n\\begin{eqnarray}\n\\{a,b,\\cdots,c\\}.\n\\end{eqnarray}\nThese charges obey the fusion algebra:\n\\begin{eqnarray}\na \\times b =\\sum_c N_{ab}^c \\, c,\n\\end{eqnarray}\nwhere the multiplicities $N_{ab}^c$ are non-negative integers that indicate the number of ways that charge $c$ can be obtained from fusion of the charges $a$ and $b$.\nThe fusion algebra is commutative and associative. \n\nThere exists a unique trivial charge $0$ that satisfies $N_{a0}^b=\\delta_{ab}$.\n\nEach charge $a$ has a conjugate charge $\\bar a$ such that $N_{a b}^0=\\delta_{b \\bar a}$.\n\nThe fusion multiplicities obey the relations:\n\\begin{eqnarray}\n&N_{ab}^c=N_{ba}^c=N_{b\\bar c}^{\\bar a}=N_{\\bar a\\bar b}^{\\bar c}&\\\\\n&\\sum_e N_{ab}^eN_{ec}^d=\\sum_f N_{af}^dN_{bc}^f.&\n\\end{eqnarray}\n\nMeanwhile, the charges obey a set of braiding rules that determine the way in which they braid around each other. Self-braiding of charge $a$ with itself is encoded in the topological spin $\\theta_a$, which is a root of unity. The diagonal matrix of topological spins is called the topological $T$-matrix of the anyon model:\n\\begin{eqnarray}\nT_{ab}=\\theta_a \\delta_{ab}.\n\\end{eqnarray}\nThe mutual braiding of charges $a$ and $b$ is given by the elements of the topological $S$-matrix, which is a symmetric matrix defined as:\n\\begin{eqnarray}\nS_{ab}=\\sum_c N_{a\\bar b}^c \\frac{\\theta_c}{\\theta_a\\theta_b} d_c,\n\\end{eqnarray}\nwhere $d_c$ is the {\\em quantum dimension} of charge $c$, determined through the fusion multiplicities.\n\n\nWhen the topological $S$-matrix is {\\em unitary}, the anyon model is called {\\em modular}. A modular anyon model corresponds to a {\\em topological quantum field theory}.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Preliminaries}\n\\includegraphics[width=0.7\\textwidth]{TablesAnyonModels.jpg}\n\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{TablesAnyonModels_2.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Tabulating Anyon models}\n\\rfoot{\\thepage}\n\\vspace*{2cm}\n\\section*{Tabulating anyon models}\n\\addcontentsline{toc}{section}{Tabulating anyon models}\n\n\nThe fusion rules and braiding rules of an anyon model fulfill a set of multivariate polynomial equations known as the {\\em Pentagon and Hexagon equations}.\nFinding all possible anyon models seems then easy. We just need to find all possible solutions to these equations.\nHowever, the number of variables and equations involved grows rapidly with the number of charges, which makes difficult to systematically solve them.\n\nBy classifying all topological quantum field theories up to four topological charges, Wang \\cite{Wang} has tabulated all possible modular anyon models with up to four particle types.\n\nBy using a numerical program, Bonderson \\cite{Bonderson} has been able to solve the Pentagon and Hexagon equations for many interesting fusion rules. This has allowed to tabulate a list of anyon models with up to 6 particles restricted to multiplicity-free fusion rules ($N_{ab}^c=0,1$). Several additional models relevant for non-Abelian quantum Hall states \\cite{NA1,NA4} have been listed for $10$ and $12$ particles.\n\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{3,5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.9\\textwidth]{TitleLanguageFigure.jpg}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\section*{}\n\\addcontentsline{toc}{section}{The language of the construction}\n\\vspace*{-3cm}\n\\hspace*{1cm}\n\\includegraphics[width=10cm]{TitleLanguage.jpg}\n\n\\hspace*{2cm}\n\\parbox{13cm}\n{I introduce a language to describe anyon models. \n\\parskip=5pt\nThis language encodes the properties of an anyon model in a collection of graphs, which I call {\\bf topological graphs}.\nGraph encoding provides a visual and enlightening way of representing anyon models.\n\nFirst, I introduce the concept of a {\\bf topological algebra}, an algebra of operators able to encode the fusion rules of an anyon model. I analyze in depth its properties, as well as the conditions for an algebra to be topological. I give special focus to the fact that a topological algebra can also encode valuable information about the braiding properties of an anyon model. Everyone familiar with anyon models has learnt as a mantra the beautiful result by Verlinde \\cite{Verlinde}: {\\em the topological $S$-matrix of a modular anyon model diagonalizes the fusion rules}. Yet I think that the meaning and the consequences or extensions of this idea have been neither realized nor exploited enough.\nThe results and connections I present  are not the review from texts I have read. They have followed from the genuine need to give an orderly structure to the concepts that were naturally emerging in my way to conform (from the pure definition of fusion and braiding rules) an appropriate language to express the Boson-Lattice construction.\n\nFinally, I represent a topological algebra with a collection of graphs. This representation allows to easily visualize the properties of the anyon model. Topological charges are represented by graph vertices. Fusion rules can be read from the connectivity patterns of the graphs. Braiding rules are obtained through diagonalization of the graphs.\n\nGraphs have been extensively used in physics and mathematics to represent matrices. Here, I reveal that topological graphs compose a useful language to embody anyon models. They constitute a befitting language to formulate the Boson-Lattice construction I will develop in the following sections.}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\newgeometry{bottom=0.1cm}\n\\vspace*{0,5cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{Algebra.jpg}\n\n\\hspace*{0.5cm}\n\\includegraphics[width=0.8\\textwidth]{AlgebraDefinition.jpg}\n\\end{center}\n\n\n\\newpage\n\\restoregeometry\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Algebra}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Topological algebras}\n\\addcontentsline{toc}{section}{Topological algebras}\n\\normalsize\n\nI introduce the concept of topological algebra, as a useful way to encode, firstly, the fusion rules of an anyon model.\n\n{\\em \\bf Definition.} An algebra of operators \n\\begin{eqnarray}\n\\mathcal{A}=\\{X_a,X_b,\\ldots\\},\n\\end{eqnarray}\nis a topological algebra if it fulfills:\n\\begin{eqnarray}\nX_aX_b=\\sum_c N_{ab}^c\\,X_c,\n\\label{AlgebraCondition1}\n\\end{eqnarray}\nwith the tensors $N_{ab}^c$ defining a set of well defined fusion rules as described above. \nThe operators in the algebra are in one to one correspondence with the topological charges of the anyon model.\n\n{\\em \\bf The topological algebra of an anyon model: matrix representation.}\n\nGiven an anyon model with topological charges $\\{a,b,\\ldots\\}$ and fusion rules $a \\times b= \\sum_c N_{ab}^c \\,c$, \nwe can always find a topological algebra associated with the anyon model in the following way.\n\nLet me consider a Hilbert space of dimension $n$ equal to the number of topological charges in the anyon model.\nLet me denote the canonical basis in this Hilbert space by\n\\begin{eqnarray}\n\\{\\ket{a},\\ket{b},\\ldots\\},\n\\end{eqnarray}\nwhere each state is associated with a charge in the anyon model.\nI define a set of $n\\times n$ matrices with matrix elements given by:\n\\begin{eqnarray}\n\\braket{c|X_a|b}=\\ N_{ab}^c.\n\\label{AlgebraDef}\n\\end{eqnarray}\n\n\\begin{result}\nWith the definition (\\ref{AlgebraDef}) the set of matrices $\\{X_a\\}$ satisfies the condition (\\ref{AlgebraCondition1}) and is thus a topological algebra associated with the anyon model.\n\\end{result}\n\\begin{proof}\nUsing the associative property of the fusion rules we have:\n\\begin{eqnarray}\n\\braket{i|X_aX_b|j}&=&\\sum_k\\braket{i|X_a|k}\\braket{k|X_b|j}=\\nonumber\\\\\n&=&\\sum_kN_{ak}^iN_{bj}^k=\\sum_cN_{ab}^cN_{cj}^i=\\sum_cN_{ab}^c\\braket{i|X_c|j}.\\qedhere\n\\end{eqnarray}\n\\end{proof}\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\includegraphics[width=\\textwidth]{AlgebraProperties.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Algebra}\n\\rfoot{\\thepage}\n\\subsection*{Properties of a topological algebra}\n\\addcontentsline{toc}{subsection}{Properties of a topological algebra}\n\nA topological algebra exhibits the following properties, which are inherited from the properties of fusion rules:\n\\begin{itemize}\n\\item {\\bf the algebra is Abelian}. From the commutativity property of the fusion rules encoded by the algebra, it follows that:\n\\begin{eqnarray}\nX_aX_b=\\sum_c N_{ab}^c\\,X_c=\\sum_c N_{ba}^c\\,X_c=X_bX_a.\n\\end{eqnarray}\n\n\\item {\\bf the identity operator} is the operator corresponding to the trivial charge $0$:\n\\begin{eqnarray}\n\\braket{a|X_0|b}=N_{0b}^a=\\delta_{ab}.\n\\end{eqnarray}\n\\item {\\bf acting on the trivial state}, the operator $X_a$ gives \n\\begin{eqnarray}\nX_a\\ket{0}=\\ket{a},\n\\end{eqnarray}\nsince we have:\n\\begin{eqnarray}\n\\braket{b|X_a|0}=N_{a0}^b=\\delta_{ab}.\n\\end{eqnarray}\n\\item {\\bf the operator associated with the conjugate charge is the adjoint operator}. From the properties of the fusion rules it follows that:\n\\begin{eqnarray}\n\\braket{c|X_{\\bar a}^{}|b}=N_{\\bar ab}^c=N_{a\\bar b}^{\\bar c}=N_{ac}^b=\\braket{b|X_{a}^{}|c}=\\braket{c|X^\\dagger_a|b},\n\\end{eqnarray}\nand therefore\n\\begin{eqnarray}\nX_{\\bar a}=X_a^\\dagger.\n\\end{eqnarray}\n\\item {\\bf the operators in the algebra are normal}. Combining the properties above we have:\n\\begin{eqnarray}\n[X_a^{},X_a^\\dagger]=[X_a^{},X_{\\bar a}]=0.\n\\end{eqnarray}\n\\end{itemize}\n\nIt is interesting to see how the properties of the fusion rules are translated into a set of illuminating properties of the algebra: the operators are {\\bf normal and mutually commuting}, which allows for their simultaneous diagonalization.\n\n\n\\newpage\n\\thispagestyle{empty}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\subsection*{When is an algebra a topological algebra?}\n\\addcontentsline{toc}{subsection}{When is an algebra a topological algebra?}\n\nWe have seen above how to define an algebra encoding the fusion rules of an anyon model.\n\nBut if we are given a certain algebra, how do we know that this algebra is topological?, that is, how do we know that it defines a set of well defined fusion rules?\n\nIn the following result I give the {\\bf necessary and sufficient conditions} that an algebra needs to fulfill in order to be topological.\nThis result will be very useful when building up topological algebras in the Boson-Lattice construction I develop in the next sections.\n\n\\indent\n\\hangindent=0,6cm\n\\begin{fresult}\n\nLet us consider the following set of $n$ linear operators on a Hilbert space of dimension $n$:\n\\begin{eqnarray}\n\\mathcal{A}=\\{X_0=\\mathds{1},X_1,\\cdots,X_{n-1}\\}. \n\\end{eqnarray}\nThis set defines a topological algebra if and only if the following conditions are satisfied:\n\\begin{enumerate}\n\\item The operators commute with each other: $\\left[X_a,X_b\\right]=0$, $\\forall a,b$.\n\\item For each $X_a\\in \\mathcal{A}$ there exists $X_{\\bar a}\\in \\mathcal{A}$ such that $X_{\\bar a}=X_a^\\dagger$.\n\\item There exists a state $\\ket{0}$ for which the set of states $\\{\\ket{a}=X_a\\ket{0}\\}$ defines an orthonormal basis.\n\\item In such basis the operators have natural entries: $\\braket{c|X_a|b}=0,1,2,\\cdots.$\n\\end{enumerate}\n\nWith these conditions the set of operators $\\mathcal{A}$ is an algebra satisfying:\n\\begin{eqnarray}\n\\label{AlgebraCondition}\nX_aX_b=\\sum_cN_{ab}^cX_c,\n\\end{eqnarray}\nwith $N_{ab}^c=\\braket{c|X_a|b}$ defining a set of well defined {\\em fusion rules}.\n\\end{fresult}\n\n\\begin{proof}\n\\small\nIt is clear that a topological algebra fulfills the set of conditions listed above.\n\nTo see that an algebra fulfilling conditions 1-4 is topological, we proceed as follows.\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Algebra}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\n\nFirst, we prove that $\\mathcal{A}$ fulfills Eq.(\\ref{AlgebraCondition}). From properties 1. and 3. we have:\n\\begin{eqnarray}\nX_a\\ket{b}=X_aX_b\\ket{0}=X_b\\ket{a},\n\\end{eqnarray}\nand therefore\n\\begin{eqnarray}\nX_aX_b\\ket{d}=X_aX_d\\ket{b}=\\sum_cX_d\\ket{c}\\braket{c|X_a|b}=\\sum_c\\braket{c|X_a|b}X_c\\ket{d}=\\sum_c N_{ab}^c X_c\\ket{d}.\n\\end{eqnarray}\n\nSecond, we prove that $N_{ab}^c=\\braket{c|X_a|b}$ are well defined fusion rules, since they fulfill:\n\\begin{itemize}\n\\item $N_{a b}^c=N_{b a}^c$,\\\\\nsince we have $N_{a b}^c=\\braket{c\\,|X_a|\\,b}=\\braket{c\\,|X_b|\\,a}=N_{b a}^c,$.\n\\item $N_{a 0}^b=\\delta_{ab}$,\\\\\nsince we have  $N_{a 0}^b=\\braket{b\\,|X_a|\\,0}=\\delta_{ab}$.\n\\item $N_{ab\\phantom{\\bar b}}^c=N_{a \\bar c\\phantom{\\bar b}}^{\\bar b}=N_{\\bar a \\bar b}^{\\bar c}$,\\\\\nsince we have\n$\\braket{c\\,|X_a|\\,b}=\\braket{\\bar b\\,|X_a|\\,\\bar c}=\\braket{\\bar c\\,|X_a^\\dagger|\\,\\bar b}$\\\\\n$\\leftrightarrow \\braket{0\\,|X^\\dagger_cX_aX_b|\\,0}=\\braket{0\\,|X_bX_aX_c^\\dagger|\\,0}=\\braket{0\\,|X_cX^\\dagger_aX^\\dagger_b|\\,0}$.\\qedhere\n\n\\end{itemize}\n\\end{proof}\n\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{AlgebraConditions.jpg}\n\\end{center}\n\n\n\n\n\n\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\vspace*{1.5cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{AlgebraDiagonalization.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\subsection*{Diagonalization of a topological algebra}\n\\addcontentsline{toc}{subsection}{Diagonalization of a topological algebra}\n\nA topological algebra $\\mathcal{A}$ is a collection of {\\em mutually commuting normal operators}. Therefore there exists an {\\em orthonormal basis of eigenstates common to all operators} in the algebra. \nFor an anyon model, these eigenstates encode valuable information about the {\\bf braiding properties} of the model.\n\nThe following result holds for any topological algebra.\n\n\\begin{fresult}\nThere exists a unitary matrix $U$ that simultaneously diagonalizes the operators $\\{X_a\\}$ of the topological algebra, so that:\n\\begin{eqnarray}\nU^\\dagger X_aU=F_a,\n\\end{eqnarray}\nwith $\\{F_a\\}$ a collection of diagonal matrices.\nThe unitary matrix $S$ satisfies the equation:\n\\begin{eqnarray}\nN_{ab}^{c}=\\sum_\\ell U_{\\ell c}^{\\dagger} \\,\\frac{U_{a\\ell}^{}}{U_{0\\ell}^{}}\\,U_{b\\ell}^{},\n\\label{VerlindeEquation}\n\\end{eqnarray}\nwhere $U_{\\ell \\ell^\\prime}=\\braket{\\ell|U|\\ell^\\prime}$ are the matrix elements of $U$.\n\\end{fresult}\n\nEquation (\\ref{VerlindeEquation}) reminds us of {\\bf Verlinde's equation} \\cite{Verlinde, Witten, VerlindeEquation1, VerlindeEquation2} for a modular anyon model, which relates the fusion rules $N_{ab}^{c}$ with the topological $S$-matrix.\nIndeed, as we will see later, for a modular anyon model, the topological $S$-matrix exactly corresponds to a symmetric choice of the unitary matrix $U$. Here, it is important to note that the result above is valid for any topological algebra, independently of whether it corresponds to a modular anyon model or not. \n\n\\begin{proof}\nLet me denote the orthonormal basis of common eigenstates of the algebra $\\{X_a\\}$ by $\\{\\ket{\\psi_a}\\}$, with\n\\begin{eqnarray}\nX_b\\ket{\\psi_a}=\\lambda_b^{(a)}\\ket{\\psi_a},\n\\end{eqnarray}\nand $\\lambda_b^{(a)}$ the eigenvalue corresponding to the operator $X_b$.\nEncoding the eigenvectors in the unitary matrix $U$ and the set of eigenvalues in the set of diagonal matrices $F_b$:\n\\begin{eqnarray}\n\\braket{a^\\prime|U_{}^{}|a}&=&\\braket{a^\\prime|\\psi_a}\\nonumber\\\\\n\\braket{a^\\prime|F_b^{}|a}&=&\\lambda_b^{(a)}\\delta_{aa^\\prime},\\nonumber\n\\end{eqnarray}\nwe have:\n\\begin{eqnarray}\nU^\\dagger X_aU=F_a.\n\\end{eqnarray}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\nTo prove equation (\\ref{VerlindeEquation}) we solve $X_a$ from the above expression and take matrix elements to obtain:\n\\begin{eqnarray}\n\\braket{c|X_a|b}&=&\\sum_\\ell\\braket{c|U|\\ell}\\braket{\\ell|F_a|\\ell}\\braket{\\ell|U^\\dagger|b}\\nonumber\\\\\nN_{ab}^c&=&\\sum_\\ell U^\\dagger_{\\ell c}\\braket{\\ell |F^*_a|\\ell}U^{}_{b\\ell}\\label{VerlindePrevious}.\n\\end{eqnarray}\n\nComparing (\\ref{VerlindePrevious}) with (\\ref{VerlindeEquation}) it suffices to prove that:\n\\begin{eqnarray}\n\\frac{U_{a\\ell}}{U_{0\\ell}}=\\braket{\\ell |F^*_a|\\ell}.\n\\label{1DRepresentation_1}\n\\end{eqnarray}\n\nTo prove (\\ref{1DRepresentation_1}) let me consider a set of states of the form:\n\\begin{eqnarray}\n\\ket{\\varphi_\\ell}=\\sum_a\\braket{\\ell|F_a^*|\\ell}\\ket{c}.\n\\end{eqnarray}\nSince we have that\n\\begin{eqnarray}\n\\braket{c|X_a|\\varphi_\\ell}&=&\\sum_b\\braket{c|X_a|b}\\braket{b|\\varphi_\\ell}=\\sum_b\\braket{\\ell|N_{ab}^cF_b^*|\\ell}=\\sum_b\\braket{\\ell|N_{a\\bar c}^{\\bar b}F_{\\bar b}|\\ell}\\nonumber\\\\\n&=&\\braket{\\ell|F_aF_{\\bar c}|\\ell}=\\braket{\\ell|F_a|\\ell}\\braket{c|\\varphi_\\ell},\n\\end{eqnarray}\nit follows that \n\\begin{eqnarray}\nX_a\\ket{\\varphi_\\ell}=\\lambda_a^{(\\ell)}\\ket{\\varphi_\\ell},\n\\end{eqnarray}\nand therefore the state $\\ket{\\varphi_\\ell}$ is proportional to the eigenstate $\\ket{\\psi_\\ell}$. \nWe thus have\n\\begin{eqnarray}\n\\frac{U_{a\\ell}}{U_{0\\ell}}=\\frac{\\braket{a|\\psi_\\ell}}{\\braket{0|\\psi_\\ell}}\n=\\frac{\\braket{\\ell|F_a^*|\\ell}}{\\braket{\\ell|F_0^*|\\ell}}=\\braket{\\ell|F_a^*|\\ell}.\\qedhere\n\\end{eqnarray}\n\\end{proof}\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Algebra}\n\\subsubsection*{One dimensional representations of the topological algebra}\n\\addcontentsline{toc}{subsubsection}{One dimensional representations of the topological algebra}\n\nThe result above tells us that the common eigenstates of a topological algebra are in one to one correspondence with the \\textit{one-dimensional representations of the algebra}.\n\nTo see this we note that the set of eigenvalues \n\\begin{eqnarray}\n\\{\\lambda_a^{(\\ell)}=\\braket{\\ell|F_a|\\ell}\\,\\,\\}_{a=0,\\cdots,n-1}\n\\end{eqnarray}\nis (for each $\\ell=0,\\cdots,n-1$) a one dimensional representation of the topological algebra $\\mathcal{A}$, since we have \n\\begin{eqnarray}\nF_aF_b=\\sum_cN_{ab}^cF_c,\n\\end{eqnarray}\nand therefore\n\\begin{eqnarray}\n\\braket{\\ell|F_a|\\ell}\\braket{\\ell|F_b|\\ell}=\\sum_cN_{ab}^c\\braket{\\ell|F_c|\\ell}.\n\\end{eqnarray}\nSince equation (\\ref{1DRepresentation_1}), proven above, states that the common eigenstates of the algebra have components proportional to the eigenvalues:\n\\begin{eqnarray}\n\\braket{\\psi_\\ell|a}\\propto\\braket{\\ell|F_a|\\ell}=\\lambda_a^{(\\ell)},\n\\end{eqnarray}\nit follows that these eigenstates are in one to one correspondence with the one-dimensional representations of the algebra.\n\n\\subsubsection*{A common eigenvector with all positive components}\n\\addcontentsline{toc}{subsubsection}{A common eigenvector with all positive components}\n\nA beautiful property of a topological algebra is the existence of a common eigenvector, whose components are all positive. Without loss of generality, this eigenvector can be written as\n\\begin{eqnarray}\n\\ket{\\psi_0}=\\frac{1}{\\mathcal{D}}\\sum_a d_a \\ket{a},\n\\end{eqnarray}\nwhere $d_a>0$, $d_0=1$, and $\\mathcal{D}=\\sqrt{\\sum_a d_a^2}$.\nThe positive numbers $d_a$ correspond to the largest eigenvalues of the operators $X_a$.\n\nThis property follows from Perron-Frobenius theorem \\cite{Perron, Frobenius}, which applies to non-negative irreducible matrices. \nIn the language of graphs that I introduce later, it becomes transparent that the operators of a topological algebra are direct sums of irreducible operators, so that the theorem applies.\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\subsection*{Diagonalization and the S-matrix}\n\\addcontentsline{toc}{subsection}{Diagonalization and the S-matrix}\n\nIn light of the results above and taking into account the definition of the topological $S$-matrix, we can state the following useful connections between the $S$-matrix of an anyon model and the eigenvectors of the corresponding topological algebra \\footnote{The proof of the results above is straightforward by combining the definition of the topological $S$-matrix with the information given before regarding diagonalization of a topological algebra. I will present the details of this proof elsewhere.}\n\n\\begin{itemize}\n\n\\item\nThe S-matrix of an anyon model is a symmetric matrix of eigenvectors of the topological algebra. \nWe have:\n\\begin{eqnarray}\nS_{ab}=\\braket{a|\\psi_b},\n\\end{eqnarray}\nwhere $\\{\\ket{\\psi_b}\\}$ is a set of eigenvectors fulfilling\n$\\braket{a|\\psi_b}=\\braket{b|\\psi_a}$.\n\nThe {\\em quantum dimensions} of the anyon model correspond to the components of the common eigenvector $\\ket{\\psi_0}$, whose components are all positive:\n\\begin{eqnarray}\nd_c=\\frac{\\braket{c|\\psi_0}}{\\braket{0|\\psi_0}}.\n\\end{eqnarray}\n\nThese results hold for any anyon model, modular or not.\n\n\\item For a modular anyon model, the $S$-matrix is a unitary symmetric matrix. It therefore corresponds to a symmetric orthonormal basis of eigenstates of the topological algebra.\nAs any orthonormal basis of the topological algebra, it fulfills:\n\\begin{eqnarray}\nN_{ab}^{c}=\\sum_\\ell S_{\\ell c}^{\\dagger} \\,\\frac{S_{a\\ell}^{}}{S_{0\\ell}^{}}\\,S_{b\\ell}^{}.\n\\end{eqnarray}\nThis equation is Verlinde's equation \\cite{Verlinde}.\n\n\\item For a non-modular anyon model the $S$-matrix is not unitary. It corresponds to a non-orthonormal set of eigenvectors of the topological algebra. \nIt fulfills the equation:\n\\begin{eqnarray}\nX_aS=SE_a,\n\\end{eqnarray}\nwhere $E_a$ is a diagonal matrix with elements $\\braket{b |E_a| c}=\\delta_{bc} \\braket{\\psi_b | X_a |\\psi_b}$, corresponding to a set of eigenvalues of $X_a$.\nElement by element  we have:\n\\begin{eqnarray}\n\\sum_b N_{ab}^c S_{b\\ell}^{}=S_{c\\ell}\\frac{S_{a\\ell}^*}{S_{0\\ell}}.\n\\end{eqnarray}\n\nThis is a generalization of Verlinde's equation. It is valid for any anyon model, modular or not.\n\\end{itemize}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Algebra}\n\\rfoot{\\thepage}\n\\subsection*{Topological algebras and anyon models}\n\\addcontentsline{toc}{subsection}{Topological algebras and anyon models}\n\nGiven an anyon model there is always a topological algebra associated with it, which encodes its fusion rules.\nHowever, given a topological algebra (a set of well defined fusion rules) there is not necessarily an anyon model satisfying the corresponding fusion rules.\n\nFor example, it is clear that if the topological algebra does not admit a symmetric set of eigenvectors, there will be no anyon model corresponding to it.\nAlso, if the algebra does not admit a symmetric eigenbasis, we can be sure that there will be no modular anyon model with such fusion rules.\n\nRemarkably, even if there is a symmetric set of eigenvectors, it is not guaranteed that an anyon model exists \\footnote{It is illuminating to construct examples of topological algebras which do not correspond to anyon models, even when a symmetric eigenbasis exists. In a forthcoming work I will give explanatory examples of different interesting situations.}.\nThe following result summarizes the conditions under which an anyon model can exist with the fusion rules of a given topological algebra.\n\n\\begin{mdframed}\nGiven a topological algebra, an anyon model associated with it corresponds to a symmetric choice $S_{ab}$ of one dimensional representations of the algebra satisfying the equation:\n\\begin{eqnarray}\nS_{ab}=\\sum_c N_{a\\bar b}^c \\frac{\\theta_c}{\\theta_a\\theta_b} d_c, \\label{S-T-Relation}\n\\end{eqnarray}\nwhere $N_{ab}^c$ are the fusion multiplicities, $\\theta_a$ are roots of unity, with $\\theta_0=1$, and $d_c$ are the components of the algebra eigenvector with all positive components.\n\\end{mdframed}\n\nThis result is indeed an alternative formulation of the Pentagon and Hexagon equations. \nIt can provide us with a guided route to obtain the possible anyon models associated with a given set of fusion rules. First, we search for the possible symmetric sets of eigenvectors of the algebra. This step highly reduces the possible candidates for topological $S$-matrices. Then, we check whether these matrices fulfill equation (\\ref{S-T-Relation}) for a certain choice of the $\\theta_a$.\n\nThe phrasing above can be enlightening. For example, it becomes clear that anyon models with the same fusion rules have $S$-matrices corresponding to different symmetric choices of a set of eigenvectors (for example, they can correspond to reorderings of the same set of eigenvectors). \n\nThis formulation will be very useful to prove the existence of modular anyon models corresponding to topological algebras in the Boson-Lattice construction I describe in the next sections. \n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\vspace*{1.5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{TopologicalGraphs.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Graphs}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\normalsize\n\\section*{Topological graphs}\n\\addcontentsline{toc}{section}{Topological graphs}\n\nI represent a topological algebra with a collection of graphs. \n\nEach operator $X_a$ in the topological algebra is represented by a {\\bf weighted directed graph} $\\mathcal{G}_a$ defined as follows:\n\\hspace*{0,5cm}\n\\begin{mdframed}\n{\\bf Vertices}. The vertices of the graph are in one to one correspondence to the states of the canonical basis, each of them corresponding to a charge of the anyon model.\n\n{\\bf Connectivity}. Two vertices $\\ket{b}$ and $\\ket{c}$ are connected if the matrix element \n$\\braket{c|X_a|b}$ is different from zero. The link is oriented from $\\ket{b}$ to $\\ket{c}$.\n\n{\\bf Links-weight}. The link connecting vertex $\\ket{b}$ to vertex $\\ket{c }$ has weight $\\braket{c|X_a|b}=N_{ab}^c$.\nA link with weight $n=0,1,2,\\cdots$ is represented by a $n$-multiple line.\n\n\\end{mdframed}\n\nThe table below shows an example (for an anyon model of five charges) of graph encoding of a set of fusion rules.\n\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{TopologicalGraphsExample.jpg}\n\\end{center}\n\n\n\n\\newpage\n\\newgeometry{bottom=0.1cm}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{GraphProperties.jpg}\n\n\\includegraphics[width=0.895\\textwidth]{GraphProperties2.jpg}\n\\end{center}\n\\newpage\n\\restoregeometry\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Graphs}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\subsection*{Properties of topological graphs}\n\\addcontentsline{toc}{subsection}{Properties of topological graphs}\n\\normalsize\nTopological graphs exhibit the following properties, which are inherited from the properties of the topological algebra.\n\n\\begin{itemize}\n\\setlength{\\itemindent}{-0.1in}\n\\item {\\bf From the vertex $\\ket{0}$}  there is only one outgoing link (to the vertex $\\ket{a}$ in graph $\\mathcal{G}_a$) and one incoming link (from the vertex $\\ket{\\bar a}$).\n\\item {\\bf Current conservation law}. The sum of the squares of the multiplicities of links entering a vertex is equal to the one of links going out from it. For a graph $\\mathcal{G}_a$ and a vertex $\\ket{b}$ we have:\n\\begin{eqnarray}\n\\sum_cN_{ab}^cN_{ab}^c=\\braket{b|X_a^\\dagger X_a|b}=\\braket{b|X_aX_a^\\dagger|b}=\\sum_dN_{ad}^bN_{ad}^b.\n\\end{eqnarray}\nFor a graph with weights either $0$ or $1$, the number of links is conserved at each vertex:\n\\begin{eqnarray}\n\\sum_cN_{ab}^c=\\sum_d N_{ad}^b.\n\\end{eqnarray}\n\\item {\\bf Loops.} A vertex can be connected to itself, forming a loop. This occurs for non-vanishing diagonal matrix elements \n\\begin{eqnarray}\n\\braket{b|X_a|b}\\ne0.\n\\end{eqnarray}\n\\item {\\bf Conjugate graphs.} Graphs corresponding to conjugate charges have the same links, with arrows reversed:\n\\begin{eqnarray}\n\\braket{c|X_a|b}=\\braket{b|X_{\\bar a}|c}.\n\\end{eqnarray}\nSimilarly, conjugate vertices share the same links with arrows reversed:\n\\begin{eqnarray}\n\\braket{\\bar c|X_a| \\bar b}=\\braket{b|X_{a}|c}.\n\\end{eqnarray}\n\\item {\\bf Connectivity.} A topological graph is always a disjoint union of {\\em connected} graphs.\n\n\\end{itemize}\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The language}\n\\lfoot{\\thepage}\n\\vspace*{2cm}\n\\includegraphics[width=\\textwidth]{GraphGenerating.jpg}\n\n\\newpage\n\\vspace*{3cm}\n\\subsection*{Generating topological graph}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological Graphs}\n\\rfoot{\\thepage}\n\\addcontentsline{toc}{subsection}{Generating topological graph}\n\nA graph is called {\\bf connected} if there is a path that connects any pair of vertices.\n\nThe operator $X$ corresponding to a connected graph is {\\em irreducible}. It fulfills that for every pair $i,j$ there exists a positive integer $m$ such that:\n\\begin{eqnarray}\n\\braket{i|X^m| j}\\ne0.\n\\end{eqnarray}\n\nA connected topological graph defines a very interesting kind of topological graph. All other graphs in the topological algebra can be derived from it. They are indeed {\\em polynomials} of this graph.\nI will call it generating or fundamental graph, since it encodes the complete topological algebra.\n\nIn the Boson-Lattice construction I develop here, an anyon model will be encoded in a generating graph, from which all properties of the model can be read.\n\nA general topological graph is always the disjoint union of connected graphs \n\\footnote{The graph language provides an enlightening way to prove this property. The details of this proof will be presented elsewhere.}. Therefore the corresponding operator is the direct sum of irreducible operators.\nThanks to this property, the Perron-Frobenius theorem applies, and the existence of an eigenvector with all positive components is guaranteed.\n\n\n\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{6cm}\n\\includegraphics[width=0.95\\textwidth]{TitleBuildingBlocksFigure.jpg}\n\n\\newpage\n\\section*{}\n\\thispagestyle{empty}\n\\addcontentsline{toc}{section}{Building blocks}\n\\vspace*{2.5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.75\\textwidth]{TitleBuildingBlocks.jpg}\n\\vspace*{1cm}\n\n\\large\n\\hspace*{2.5cm}\n\\parbox{11cm}{I introduce the building blocks of the Boson-Lattice construction.}\n\n\\hspace*{2.5cm}\n\\parbox{11cm}{These are the Abelian anyon models $\\mathbb{Z}_n$. I describe them using the language of topological graphs introduced in the previous section.}\n\n\\hspace*{2.5cm}\n\\parbox{11cm}{A conceptual leap is made by identifying a $\\mathbb{Z}_n$ model with {\\em a particle in a one-dimensional periodic lattice} of $n$ sites. With this identification, the elementary pieces of the Boson-Lattice construction are defined as {\\em particles in lattices}.}\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The building blocks}\n\\lfoot{\\thepage}\n\\vspace*{2.5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{ZnModels.jpg}\n\\end{center}\n\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$\\mathbb{Z}_n$ models}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{$\\mathbb{Z}_n$ models}\n\\addcontentsline{toc}{section}{$\\mathbb{Z}_n$ models}\nA $\\mathbb{Z}_n$ anyon model \\footnote{The $\\mathbb{Z}_n$ anyon models I define here are Abelian modular anyon models corresponding to $\\mathbf{SU}(n)_1$ conformal field theories.}\nis characterized by a set of $n$ charges:\n\\begin{eqnarray}\n\\{0,1,\\cdots,n-1\\},\n\\end{eqnarray}\nwhich fulfill the fusion rules:\n\\begin{eqnarray}\na\\times b=a+b\\,(\\text{mod}\\,\\,n).\n\\end{eqnarray}\n\nThe mutual braiding statistics of charges $a$ and $b$ is given by the element $S_{ab}$ of the $S$-matrix:\n\\begin{eqnarray}\nS_{ab}=\\frac{1}{\\sqrt{n}}e^{i\\frac{2\\pi}{n}a\\cdot b}.\n\\end{eqnarray}\n\nThe self statistics of charge $a$ is given by the topological spin $\\theta_a$:\n\\begin{eqnarray}\n\\theta_{a}^2=e^{-i\\frac{2\\pi}{n}a^2}.\n\\end{eqnarray}\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n\\vspace*{-0.9cm}\n\\includegraphics[width=0.48\\textwidth]{ZnModelsQuantumHall.jpg}\n\\end{wrapfigure}\n\\vspace*{0.5cm}\nA physical realization of a $\\mathbb{Z}_n$ model can be obtained by representing the charges of the model with fractional electric charges attached to magnetic fluxes \\cite{A1,A2,Preskill}.\nA topological charge $a$ is represented by a fractional charge $q_a=a\\frac{e}{n}$ attached to a flux $\\phi_a=a\\phi_0$, where $e$ is the electron charge and $\\phi_0$ is the quantum of flux.\n\nThe mutual statistics $S_{ab}$ is obtained as the Aharonov-Bohm phase that charge-flux composites $q_a$ and $q_b$ acquire when going around each other.\n\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The building blocks}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{ZnModelsGraph.jpg}\n\\end{center}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{ZnModelsSmatrix.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$\\mathbb{Z}_n$ models}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Graph representation of $\\mathbb{Z}_n$ models}\n\\addcontentsline{toc}{section}{Graph representation of $\\mathbb{Z}_n$ models}\n{\\em Hilbert space}. The Hilbert space corresponding to a $\\mathbb{Z}_n$ model has dimension $n$. I denote the canonical basis by:\n\\begin{eqnarray}\n\\{\\ket{0},\\ket{1},\\cdots,\\ket{n-1}\\}.\n\\end{eqnarray}\n{\\bf Topological algebra}. Following the definition introduced in the previous section, the topological algebra of the model is given by the set of operators:\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},X,X^2,\\cdots,X^{n-1}\\},\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\nX=\\sum_{x=0}^{n-1}\\ket{x+1}\\bra{x},\n\\end{eqnarray}\nwhere $\\ket{x}=\\ket{x \\,(\\text{mod}\\,n)}$. The operator $X$ fulfills $X^n=\\mathds{1}$.\n\n{\\bf Generating graph}. \nThe charge $1$ represented by the operator $X$ is a generating or fundamental charge. \n\nThe graph associated with it, \n$\\mathcal{G}_X$, is the one in which each vertex is linked to its next (to the right) neighbour.\nThat is, it is an {\\bf oriented lattice with periodic boundary conditions.}\nFusion rules and braiding rules of the model are encoded in this graph.\n\n{\\bf S-Matrix}. The $S$-matrix of the model is directly obtained by diagonalizing the operator $X$. The eigenstates of $X$ are Fourier transformed states \nof the form:\n\\begin{eqnarray}\n\\ket{q}=\\frac{1}{\\sqrt{n}}\\sum_{x=0}^{n-1} e^{i\\frac{2\\pi}{n}q\\cdot x}\\ket{x}.\n\\end{eqnarray}\nThe unitary and symmetric matrix diagonalizing the algebra is thus:\n\\begin{eqnarray}\nS_{qx}=\\frac{1}{\\sqrt{n}}e^{i\\frac{2\\pi}{n}q\\cdot x},\n\\end{eqnarray}\nwhich corresponds to the $S$-matrix of the $\\mathbb{Z}_n$ anyon model.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The building blocks}\n\\lfoot{\\thepage}\n\\vspace*{3.5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{TheIdea.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$\\mathbb{Z}_n$ models}\n\\rfoot{\\thepage}\n\\vspace*{1.5cm}\n\\section*{The leap to a particle in a lattice}\n\\vspace*{0.3cm}\n\\addcontentsline{toc}{section}{The leap to a particle in a lattice}\nThe generating graph contains complete information of the anyon model.\nFusion and braiding rules can be read from the graph.\n\nThe operator $X$ is the (chiral) translation operator in a one-dimensional lattice with periodic boundary conditions.\n\n{\\em Abstraction}. I identify a $\\mathbb{Z}_n$ anyon model with a particle in a lattice of $n$ sites with periodic boundary conditions and chiral tunneling operator:\n\n\\begin{eqnarray}\nX=\\sum_{x=0}^{n-1}a^\\dagger_{x+1}a_{x}^{},\n\\end{eqnarray}\n\nwhere $a_{x}$($a^\\dagger_{x}$) is the annihilation (creation) operator of a particle in the lattice site $x$.\n\nThis identification establishes the essence of the elementary pieces of the construction.\n\n{\\bf The building blocks are particles in lattices.}\n\n{\\bf The construction will assemble particles in lattices.}\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{5.5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{ConstructionTitleFigure.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{The construction}\n\\rfoot{\\thepage}\n\\section*{}\n\\addcontentsline{toc}{section}{The Construction}\n\\hspace{1cm}\n\\includegraphics[width=0.8\\textwidth]{ConstructionTitle.jpg}\n\n\\hspace*{2cm}\n\\parbox{12cm}{Using the language and the building blocks described in the previous sections, I present a formalism to systematically construct anyon models.\n\\parskip=8pt\n\nAn anyon model is built up by assembling $k$ identical building blocks of length $n$. The Hilbert space of the model is obtained by {\\bf bosonization} of the Hilbert spaces of the building blocks.\n\nBased on the graphs of the building blocks, I give a prescription to construct a graph in the bosonized Hilbert space. This graph is conceived such that it always corresponds to the generating graph of a modular anyon model. \n\nA one-to-one correspondence is established between the properties of the Boson-Lattice system (Fock states, tunneling connectivity patterns, eigenvalues and eigenstates) and the properties of the anyon model (topological charges, fusion rules, quantum dimensions, $S$ and $T$ matrices).\n\nThis Boson-Lattice construction systematically generates, by varying the number of bosons and the number of lattice sites, a series of well known tabulated anyon models. In particular, it generates anyon models corresponding to truncated Lie algebras such as $\\mathbf{SU}(2)_k$, Fibonnaci, $\\mathbf{SO}(3)_k$, or $\\mathbf{SO}(5)_k$.\nInterestingly, the construction also yields anyon models which are not tabulated.}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{2cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.85\\textwidth]{ConstructionBosonization.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Bosonization}\n\\rfoot{\\thepage}\n\\vspace*{0,5cm}\n\\section*{The bosonization idea}\n\\addcontentsline{toc}{section}{The bosonization idea}\nIn the last section I have shown that a building block anyon model (a $\\mathbb{Z}_n$ model) is completely characterized by:\n\n\\begin{itemize}\n\\item {a Hilbert space $\\mathcal{H}(1,n)$, corresponding to a single particle in a one-dimensional lattice of $n$ sites with periodic boundary conditions.}\n\\item {a generating graph $\\mathcal{G}(1,n)$, corresponding to the chiral tunneling operator of the particle in such a lattice.}\n\\end{itemize}\n\nI give now a prescription to assemble these building blocks in order to sequentially generate new anyon models.\n\nTo construct a new anyon model I consider $k$ identical building blocks of length $n$.\n\nI define the Hilbert space $\\mathcal{H}(k,n)$ associated with the new anyon model as the one resulting from bosonization (symmetrization) of the tensor product of the $k$ identical Hilbert spaces of the building blocks:\n\n\\begin{eqnarray}\n\\mathcal{H}(k,n)=\\mathcal{S}\\,\\,\\underbrace{\\mathcal{H}(1,n)\\otimes\\cdots\\otimes\\mathcal{H}(1,n)}_{k\\,\\, \\text{copies}}.\n\\end{eqnarray}\n\nThe Hilbert space $\\mathcal{H}(k,n)$ is the one of $k$ bosons in a  one-dimensional lattice of $n$ lattice sites with periodic boundary conditions.\n\nThe new Hilbert space is constructed by making the $k$ particles become indistinguishable.\nIt is important to emphasize that in this bosonization strategy the particles that are made indistinguishable are not physical objects, but mathematical constructions used to encode an anyon model.\n\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{0.7cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{ConstructionHilbertSpace.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Bosonization}\n\\rfoot{\\thepage}\n\\section*{The Boson-Lattice Hilbert space}\n\\addcontentsline{toc}{section}{The Boson-Lattice Hilbert space}\n\nI give here some definitions in the Hilbert space $\\mathcal{H}(k,n)$ which will be useful to describe the anyon model associated with it.\n\n\n{\\bf The Fock basis}. I consider the basis of Fock states. Each Fock state is characterized by the corresponding occupation numbers of the lattice sites:\n\\begin{eqnarray}\n\\ket{i}\\equiv\\ket{n_0^{(i)},n_1^{(i)},\\cdots,n_{n-1}^{(i)}},\n\\end{eqnarray}\nwith $n_\\ell^{(i)}$ being the occupation number of site $\\ell$, and $\\ell=0,\\cdots,n-1$. The total number of bosons is equal to $k$, $\\sum_\\ell n_\\ell^{(i)}=k$.\n\n{\\bf The trivial state}. I choose a reference state, $\\ket{0}$, as the Fock state with all bosons occupying the same lattice site (for example, the site $\\ell=0$.)\n\\begin{eqnarray}\n\\ket{0}\\equiv\\ket{k,0,\\cdots,0}.\n\\end{eqnarray}\nI call this state the trivial state.\n\n{\\bf The generating state}. I denote as $\\ket{1}$ the Fock state obtained from the state $\\ket{0}$ by transferring one boson to site $1$:\n\\begin{eqnarray}\n\\ket{1}\\equiv\\ket{k-1,1,\\cdots,0}.\n\\end{eqnarray}\nI call this state the generating state.\n\n{\\bf Conjugation}. I define the unitary operation $C$ as the one mapping each Fock state to its mirror image with respect to the site $0$:\n\\begin{eqnarray}\n\\ket{i}&\\overset{C}{\\longrightarrow} &\\ket{\\bar i}=C\\ket{i} \\nonumber\\\\\nn_\\ell^{(i)}&\\overset{C}{\\longrightarrow} &n_{n-\\ell}^{(i)}.\n\\end{eqnarray}\nAs a reflection, the unitary $C$ fulfills $C^\\dagger=C$ and thus $C^2=\\mathds{1}$.\n\n{\\bf Global translation}. I define the unitary operation $T$ as the one mapping each Fock state to the one in which each boson has been moved one site to the right:\n\\begin{eqnarray}\n\\ket{i}&\\overset{T}{\\longrightarrow} &T\\ket{i}\\nonumber\\\\\nn_\\ell^{(i)}&\\overset{T}{\\longrightarrow} &n_{\\ell-1}^{(i)}.\n\\end{eqnarray}\nThe unitary $T$ fulfills $T^\\dagger=T^{n-1}$.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLAGraphDefinition.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{The Graph}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{The Boson-Lattice graph}\n\\addcontentsline{toc}{section}{The Boson-Lattice graph}\n\nassociated with the Hilbert space $\\mathcal{H}(k,n)$ I define a graph $\\mathcal{G}(k,n)$ which will be the generating graph of the corresponding anyon model. The conception of this graph is an essential step to develop the Boson-Lattice construction. \n\n\\hspace*{0,5cm}\\begin{mdframed}\n\n{\\bf Boson-Lattice Graph: Definition.}\n\nThe Boson-Lattice Graph $\\mathcal{G}(k,n)$ associated with the Hilbert space $\\mathcal{H}(k,n)$ of $k$ bosons in a periodic lattice of $n$ sites is defined as follows:\n\n{\\bf Vertices}. The vertices of the graph are in one to one correspondence with the Fock states in the Hilbert space $\\mathcal{H}(k,n)$.\n\n{\\bf Connectivity and links-weight}. Two vertices are connected if the corresponding Fock states are connected by tunneling of one boson to the next (to the right) lattice site. \nThe link  is given a weight $1$.\n\n\\end{mdframed}\n\nThe connectivity pattern of the Boson-Lattice graph is inspired by the connectivity of the building block graph, the generating graph of the anyon model $\\mathbb{Z}_n$. There, two vertices are connected if the corresponding one-particle states are connected by tunneling of the particle to the next (to the right) lattice site.\n\nIt is illuminating to write down the expression for the operator $X$ corresponding to the graph $\\mathcal{G}$. It can be written as:\n\\begin{eqnarray}\nX=\\sum_{i^{\\vphantom{\\prime}} \\leadsto i^\\prime}\\ket{i^\\prime} \\bra{i^{\\vphantom{\\prime}}}&\\longleftrightarrow &\\mathcal{G},\n\\end{eqnarray}\nwhere $i \\leadsto i^\\prime$ indicates that the sum runs over pairs of Fock states $\\ket{i}$, $\\ket{i^\\prime}$, such that $\\ket{i^\\prime}$ can be obtained from $\\ket{i}$ through tunneling of one particle to the next (to the right) lattice site.\n\n\nUsing creation and annihilation operators we can write $X$ as:\n\\begin{eqnarray}\nX=\\sum_{\\ell=0}^{n-1}A^\\dagger_{\\ell+1}A^{\\vphantom{\\dagger}}_\\ell.\n\\end{eqnarray}\n\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\nHere,\n\\begin{eqnarray}\n&&A^\\dagger_{\\ell}\\ket{\\,\\cdots, n_\\ell, \\cdots\\,}=\\ket{\\,\\cdots, n_\\ell+1, \\cdots\\,}\\nonumber\\\\\n&&A^{}_{\\ell}\\ket{\\,\\cdots, n_\\ell, \\cdots\\,}=\n\\begin{cases}\n\\hspace*{0.1cm} 0 \\hspace*{3cm} \\text{if} \\,\\,n_\\ell=0\\\\\n\\hspace*{0.1cm}\\ket{\\,\\cdots, n_\\ell-1, \\cdots\\,} \\hspace*{1cm} \\text{otherwise}.\n\\end{cases}\n\\label{CreationOperators}\n\\end{eqnarray}\nThe operators $A_\\ell$ satisfy the commutation relations:\n\\begin{eqnarray}\n\\left[A^{\\vphantom{\\dagger}}_{\\ell^{\\vphantom{\\prime}}},A^\\dagger_{\\ell^\\prime}\\right]=\\delta_{\\ell^{\\vphantom{\\prime}}\\ell^\\prime}P_\\ell,\n\\label{CommutationRelationAoperators}\n\\end{eqnarray}\nwhere $P_\\ell$ is the projector onto the subspace of Fock states with $n_\\ell=0$.\nIt is crucial to note that the operators $A_\\ell$ are not one-particle bosonic operators. The operator $X$ is therefore a many-body operator different from the one-particle tunneling operator:\n\\begin{eqnarray}\nX\\ne\\sum_{\\ell=0}^{n-1}a^\\dagger_{\\ell+1}a_{\\ell}^{\\vphantom{\\dagger}}.\n\\end{eqnarray}\nHere, $a^\\dagger_{\\ell}$ $(a^{\\vphantom{\\dagger}}_{\\ell})$ is the creation (annihilation) operator of one boson at site $\\ell$.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLAGraphProperties2.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{The Graph}\n\\rfoot{\\thepage}\n\n\\vspace*{0.5cm}\n\\subsection*{Properties of the operator $X$}\n\\addcontentsline{toc}{section}{Properties of the operator $X$}\n\nThe operator $X$ corresponding to the Boson-Lattice graph fulfils the following properties. \n\n\\begin{itemize}\n\n\\item {\\bf T-invariance}: $T^\\dagger XT=X$\n\nTaking into account that $T^\\dagger A_\\ell T=A_{\\ell-1}$, we have:\n\\begin{eqnarray}\nT^\\dagger XT=\\sum_\\ell A^\\dagger_{\\ell}A^{\\vphantom{\\dagger}}_{\\ell-1}=X.\n\\end{eqnarray}\n\n\\item  {\\bf Conjugation}: $CXC=X^\\dagger$\n\nGiven that $CA_\\ell C=A_{n-\\ell}$, we have:\n\\begin{eqnarray}\nCXC=\\sum_\\ell A^\\dagger_{n-\\ell-1}A^{\\vphantom{\\dagger}}_{n-\\ell}=\\sum_\\ell A^\\dagger_{\\ell-1}A^{\\vphantom{\\dagger}}_{\\ell}=X^\\dagger.\n\\end{eqnarray}\n\n\\item {\\bf $X$ is normal}: $\\left[X,X^\\dagger\\right]=0$\n\nFrom the commutation relations of the operators $A_\\ell$ (\\ref{CommutationRelationAoperators}), we have:\n\\begin{eqnarray}\n\\left[X,X^\\dagger\\right]&=&\\sum_{\\ell^{\\vphantom{\\prime}},\\ell^\\prime}\n\\left[\nA^\\dagger_{\\ell^{\\vphantom{\\prime}}+1}\nA^{\\vphantom{\\dagger}}_{\\ell^{\\vphantom{\\prime}}},\\,\nA^\\dagger_{\\ell^\\prime+1}\nA^{\\vphantom{\\dagger}}_{\\ell^\\prime}\n\\right]=\n\\sum_{\\ell^{\\vphantom{\\prime}}}\n\\left[\nA^\\dagger_{\\ell^{\\vphantom{\\prime}}+1}\nA^{\\vphantom{\\dagger}}_{\\ell^{\\vphantom{\\prime}}},\\,\nA^\\dagger_{\\ell^{\\vphantom{\\prime}}}\nA^{\\vphantom{\\dagger}}_{\\ell^{\\vphantom{\\prime}}+1}\\right]=\\nonumber\\\\\n&=&\n\\sum_{\\ell^{\\vphantom{\\prime}}}\nP_\\ell-P_{\\ell+1}+P_\\ell P_{\\ell+1}-P_\\ell P_{\\ell+1}=0.\n\\end{eqnarray}\n\n\\item $X\\ket{0}=\\ket{1}$\n\\item $\\braket{a|X|b}=0,1$\n\\end{itemize}\n\nThese properties imply properties for the Boson-Lattice graph which will entitle it to be the generating graph of an anyon model.\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{3cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLAGraphProperties.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{The Graph}\n\\rfoot{\\thepage}\n\\subsection*{Boson-Lattice graph properties}\n\\addcontentsline{toc}{section}{Boson-Lattice graph properties}\n\nAs defined above the Boson-Lattice graph exhibits the following properties:\n\n\\begin{itemize}\n\\item {\\bf $T$ invariance.} The graph is invariant under the unitary $T$, the global translation by one site. Given a vertex $\\ket{a}$ the graph looks the same from the translated vertex $\\ket{a^\\prime}=T\\ket{a}$:\n\\small\n\\begin{eqnarray}\n\\braket{c|X|a}=\\braket{c|T^\\dagger X T|a}=\\braket{c^\\prime|X|a^\\prime}.\n\\end{eqnarray}\n\\normalsize\n\\item {\\bf Conjugate graph.} Under the conjugation operation $C$, the arrows of the links are reversed. Given a vertex $\\ket{a}$ the graph looks the same from the conjugate vertex $\\ket{\\bar a}=C\\ket{a}$, but arrows are reversed:\n\\small\n\\begin{eqnarray}\n\\braket{c|X|a}=\\braket{c|CCXCC|a}=\\braket{\\bar c|X^\\dagger| \\bar a}=\\braket{\\bar a|X| \\bar c}.\n\\end{eqnarray}\n\\normalsize\n\\item {\\bf From the vertex $\\ket{0}$}  there is only one outgoing link (to the vertex $\\ket{1}$) and one incoming link (from the vertex $\\ket{\\bar 1}=C\\ket{1}$):\n\\small\n\\begin{eqnarray}\n\\delta_{1 b}=\\braket{b|X|0}=\\braket{0|X|\\bar b}=\\delta_{\\bar1 \\bar b}.\n\\end{eqnarray}\n\\normalsize\n\\item {\\bf Links have weight 1}. There are no links with multiple lines.\n\\item {\\bf Current conservation law}. At each vertex the number of incoming links is equal to the number of outgoing links. This number is equal to the number of occupied sites in the corresponding Fock state. \n\nThis can be easily seen by noticing that a Fock state with $r$ occupied states can lead (by chiral tunneling of one particle) to $r$ different Fock states. Reversely, such Fock state can be obtained (by chiral tunneling of one particle) from $r$ different Fock states. \n\n\\small\nFormally, equality of number of incoming links $\\#_i$ and outcoming links $\\#_o$ can be shown as a consequence of $X$ being normal:\n\\begin{eqnarray}\n&&\\braket{a|XX^\\dagger |a}=\\braket{a|X^\\dagger X |a} \\\\\n&\\Longleftrightarrow& \\sum_b\\braket{a|X|b}\\braket{a|X|b}=\\sum_b\\braket{b|X|a}\\braket{b|X |a} \\label{CurrentSquare}\\\\\n&\\Longleftrightarrow& \\#_i=\\sum_b\\braket{a|X|b}=\\sum_b\\braket{b|X |a}=\\#_o\\label{Current},\n\\end{eqnarray}\nwhere (\\ref{Current}) follows from (\\ref{CurrentSquare}) since $\\braket{a|X|b}=0,1$.\n\\normalsize\n\\item {\\bf The graph is connected.} Given two arbitrary Fock states, there exists a sequence of consecutive tunneling moves of one particle to the next (to the right) lattice site that connects one Fock state with the other.\n\n\\end{itemize}\n\n\n\n\\newpage\n\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLACentralResult.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Central result}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\hspace*{3cm}\n\\section*{Boson-Lattice central result}\n\\addcontentsline{toc}{section}{Boson-Lattice central result}\n\n\nThe central result of the Boson-Lattice construction I present here, states that the Boson-Lattice graph I have defined above is the generating graph of a modular anyon model.\nThe result is formulated as follows:\n\n\\vspace*{1cm}\n\\hspace*{1,5cm}\n\\parbox{12cm}{ {\\bf The Boson-Lattice graph $\\mathcal{G}(k,n)$ associated with the Hilbert space $\\mathcal{H}(k,n)$ of $k$ bosons in a one-dimensional lattice of $n$ sites is the generating graph of a modular anyon model for any number of bosons $k$ and any number of lattice sites $n$.}\n\\parskip=8pt\n\nThe {\\em topological charges} of the Boson-Lattice anyon model are in one to one correspondence with the {\\em Fock states} of the Boson-Lattice system.\n\nThe {\\em fusion rules and braiding rules} of the anyon model are encoded in the graph $\\mathcal{G}$.\n\nThe graph $\\mathcal{G}$ can be completed to a set of graphs which define the {\\em topological algebra} of the anyon model.\n\nThe {\\em $S$-matrix} of the anyon model is obtained from {\\em diagonalization} of the topological algebra.}\n\n\\vspace{1,2cm}\nIt is remarkable that a graph defined through connectivity rules between Fock states of a boson lattice system is able to encode an anyon model. Furthermore,  series of tabulated modular anyon models can all be encoded in Boson-Lattice graphs.\n\nIn the following I describe a blueprint to obtain the properties of a Boson-Lattice anyon model from the Boson-Lattice graph. A correspondence is established between the elements characterizing the graph and the properties of the anyon model. The graph features guarantee that the obtained fusion and braiding rules are well defined and satisfy the required consistency equations.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.93\\textwidth]{BOLACharges.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Topological charges}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Topological charges}\n\\addcontentsline{toc}{section}{Topological charges}\n\n\\normalsize\nThe topological charges of the  anyon model associated with the Boson-Lattice system $\\mathcal{H}(k,n)$ are in one to one correspondence with the Fock states of the system:\n\\begin{eqnarray}\n\\{a,b,\\cdots,c\\} &\\longleftrightarrow &\\{\\ket{a},\\ket{b},\\cdots,\\ket{c}\\}.\n\\end{eqnarray}\n\nThe number of topological charges equals the dimension of $\\mathcal{H}(k,n)$.\n\nThe trivial charge $0$ is represented by the Fock state $\\ket{0}$, with all bosons in the same lattice site.\n\nThe charge $1$, represented by the Fock state $\\ket{1}$, will be the generating charge of the model.\n\n{\\bf Conjugation}. Given a charge $a$, the conjugate charge $\\bar a$ corresponds to the conjugate Fock state:\n\\begin{eqnarray}\n\\bar a \\longleftrightarrow C\\ket{a}=\\ket{\\bar a}.\n\\end{eqnarray}\nSince $C^2=1$, we have that $\\bar {\\bar a}=a$.\n\n{\\bf Translation equivalence relation}. The global translation $T$ defines an {\\em equivalence relation} in the set of charges. Two charges are equivalent if the corresponding Fock states are obtained from each other by applying a power of the operator $T$:\n\\begin{eqnarray}\na \\sim b \\Longleftrightarrow \\ket{b}=T^r \\ket{a} \\hspace*{0.5cm} \\text{for some} \\hspace*{0.2cm} r=0,1,\\cdots,n-1.\n\\end{eqnarray}\nFor each charge $a$ the class of {\\em translated charges} is denoted by\n\\begin{eqnarray}\n\\{a, ta, \\cdots, t^{n-1} a\\},\n\\end{eqnarray}\nin one to one correspondence with the set of translated Fock states\n\\begin{eqnarray}\n\\{\\ket{a}, T\\ket{a},  \\cdots, T^{n-1}\\ket{a}\\}.\n\\end{eqnarray}\nThe class of the trivial charge will be denoted by:\n\\begin{eqnarray}\n\\{0, t, \\cdots, t^{n-1} \\}.\n\\end{eqnarray}\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\subsection*{Fusion rules of charge $1$}\n\\addcontentsline{toc}{section}{Fusion rules of charge $1$}\n\nThe Boson-Lattice graph $\\mathcal{G}(k,n)$ is the topological graph corresponding to charge $1$:\n\\begin{eqnarray}\n1\\leftrightarrow\\mathcal{G}\\leftrightarrow X.\n\\end{eqnarray}\nThe fusion rules of charge $1$ can be therefore read from the connectivity pattern of $\\mathcal{G}$, or, equivalently, from the matrix elements of the operator $X$:\n\\begin{eqnarray}\nN_{1a}^b=\\braket{b\\,|X|\\,a}.\n\\end{eqnarray}\n\nThe conjugate charge $\\bar 1$ is assigned the topological graph $\\mathcal{G}^*$, which corresponds to the operator $X^\\dagger$:\n\\begin{eqnarray}\n\\bar1\\leftrightarrow\\mathcal{G}^*\\leftrightarrow X^\\dagger.\n\\end{eqnarray}\nIts fusion rules are:\n\\begin{eqnarray}\nN_{\\bar 1 a}^b=\\braket{b\\,|X^\\dagger|\\,a}=\\braket{a\\,|X|\\,b}.\n\\end{eqnarray}\n\n\n\nThe special properties of the graph assure that these fusion rules are well defined, since they fulfill:\n\\begin{itemize}\n\\item $N_{1 0}^a=\\delta_{1a}$\\\\\nsince we have $X\\ket{0}=\\ket{1}$ and therefore $N_{1 0}^a=\\braket{a\\,|X|\\,0}=\\delta_{1a}$.\n\\item $N_{1 a}^0=\\delta_{a \\bar 1}$\\\\\nsince we have $N_{1 a}^0=\\braket{0\\,|X|\\,a}=\\braket{\\bar a\\,|X|\\,0}=\\braket{\\bar 1|a}=\\delta_{a \\bar 1}$.\n\\item $N_{1 a}^b=N_{\\bar 1 \\bar a}^{\\bar b}=N_{1 \\bar b}^{\\bar a}$\\\\\nsince we have $\\braket{b\\,|X|\\,a}=\\braket{\\bar b\\,|X^\\dagger|\\,\\bar a}=\\braket{\\bar a\\,|X|\\,\\bar b}$.\n\\end{itemize}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Fusion rules}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\subsection*{Fusion rules of charge $t$}\n\\addcontentsline{toc}{section}{Fusion rules of charge $t$}\n\nThe charge $t^r$ is assigned the operator $T^r$. \nSince we have that \n$N_{t^r a}^b=\\braket{b\\,|T^r|\\,a}=\\delta_{b,t^ra}$,  the charge $t^r$ is thus an Abelian charge with fusion rules:\n\\begin{eqnarray} \nt^r\\times t^s&=&t^{r+s\\,(n)}\\nonumber\\\\\na\\times t^r&=&at^r.\n\\end{eqnarray}\n\n\\vspace*{1cm}\n\\subsection*{The nucleus of the topological algebra}\n\\addcontentsline{toc}{section}{The nucleus of the topological algebra}\n\nThe set of operators \n\\begin{eqnarray} \n\\{X,X^\\dagger,T\\}\n\\end{eqnarray}\nconstitute the nucleus of the topological algebra of the anyon model.\nThey are normal, they commute with each other, and their fusion rules are well defined. As we see below, they can be completed to a topological algebra.\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{X can be completed to a topological algebra}\n\\addcontentsline{toc}{section}{X can be completed to a topological algebra}\n\nThe operator $X$ as defined above can be completed to a topological algebra of operators:\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},X_1,X_2,\\cdots\\},\n\\end{eqnarray}\nwhere the operator $X_a$ is associated with the charge $a$, and $X_1\\equiv X$.\nThe fusion rules of the model are then given by\n\\begin{eqnarray}\nN_{a b}^c=\\braket{c|X_a|b}.\n\\end{eqnarray}\n\n\nThe following result shows how to complete the topological algebra:\n\\hspace*{0,5cm}\\begin{mdframed}\n{\\bf The algebra of polynomials}. \nFor each charge $a$ there exists a unique operator $X_a$ of the form:\n\\begin{eqnarray}\nX_a=\\text{\\large p}_a(X,X^\\dagger,T),\n\\end{eqnarray}\nwhere $\\text{\\large p}_a$ is a polynomial of integer coefficients of the operators $X,X^\\dagger$ and $T$ satisfying:\n\\begin{eqnarray}\n&&X_a\\ket{0}=\\ket{a},\\\\\n&&\\braket{c|X_a|b}=0,1,2,\\cdots.\n\\end{eqnarray}\nThis set of polynomials defines the topological algebra of the anyon model.\n\\end{mdframed}\n\nThe existence of the algebra of polynomials follows from the connectivity properties of the Boson-Lattice graph. Since the graph is connected, every state $\\ket{a}$ can be reached from the state $\\ket{0}$ by consecutive application of the operator $X$. Equivalently, a combination of consecutive applications of the operators $X$, $X^\\dagger$ and $T$ connects any state $\\ket{a}$ with the state $\\ket{0}$. Thus there is always a polynomial $X_a$ of integer coefficients of these operators that fulfills $X_a\\ket{0}=\\ket{a}$.\n\nIn the following section I will explicitly find these polynomials for a series of examples of Boson-Lattice models. \n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Fusion rules}\n\\rfoot{\\thepage}\n\\thispagestyle{empty}\n\n\n{\\em The polynomials are unique.} Once we have found an algebra of polynomials satisfying the conditions above, we can be sure that no other exists. To prove that, we consider two different sets of polynomials $\\{X_a\\}$ and $\\{X_a^\\prime\\}$. Since the  operators $X$,$X^\\dagger$, and $T$ commute with each other, we have that $[X^{\\phantom{\\prime}}_a,X^{\\phantom{\\prime}}_b]=[X^\\prime_a,X_b]=[X^\\prime_a,X^\\prime_b]=0$. Therefore:\n\\begin{eqnarray}\nX_a^{\\phantom{\\prime}}\\ket{b}=X_a^{\\phantom{\\prime}}X_b^{\\phantom{\\prime}}\\ket{0}=X_b^{\\phantom{\\prime}}X_a^{\\phantom{\\prime}}\\ket{0}=X_b^{\\phantom{\\prime}}X_a^\\prime\\ket{0}=X_a^\\prime\\ket{b},\n\\end{eqnarray}\nand thus $X_a=X_a^\\prime$.\n\n{\\em The polynomials define a topological algebra.}  By definition, the polynomials fulfill the necessary and sufficient conditions given in the first section for a topological algebra. The only non-trivial property we need to prove is that: \n\\begin{itemize}\n\\item For each $X_a$  there exists $X_{\\bar a}$ such that $X_{\\bar a}=X_a^\\dagger$.\n\\end{itemize}\nThis is shown by noticing that:\n$X_a^\\dagger=p_a^\\dagger=Cp_a^{\\phantom{\\dagger}}C$,\nand therefore\n\\begin{eqnarray}\nX_a^\\dagger\\ket{0}=CX_a^{\\phantom{\\dagger}}C\\ket{0}=\\ket{\\bar a},\n\\end{eqnarray}\nso that $X_{\\bar a}^{\\phantom{\\dagger}}=X_a^\\dagger$.\n\n\\includegraphics[width=\\textwidth]{BOLAFusionRules.jpg}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{BOLABraiding.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Braiding}\n\\rfoot{\\thepage}\n\\vspace*{0.5cm}\n\\section*{S and T matrices of the Boson-Lattice model}\n\\addcontentsline{toc}{section}{S and T matrices of the Boson-Lattice model}\nThe braiding rules of the Boson-Lattice model are encoded in the Boson-Lattice graph $\\mathcal{G}$. The special properties of this graph  guarantee that the braiding rules are well defined and correspond to those of a modular anyon model.\n\n{\\bf Quantum dimensions}.\nThe Boson-Lattice graph $\\mathcal{G}$ is a strongly connected graph. Therefore the corresponding operator $X$ is a non-negative irreducible matrix. The Perron-Frobenius theorem states that the operator $X$ has a real eigenvalue $\\lambda_0$ (largest in absolute value) with a corresponding eigenvector $\\ket{\\psi_0}$ whose components are all positive. Without loss of generality we can write this state as:\n\\begin{eqnarray}\n\\ket{\\psi_0}=\\frac{1}{\\mathcal{D}}\\sum_ad_a\\ket{a},\n\\end{eqnarray}\nwhere $d_a>0$, $d_0=1$ and $\\mathcal{D}=\\sqrt{\\sum_a d_a^2}$. The components of this vector define the quantum dimensions of the anyon model. The topological charge $a$ has quantum dimension $d_a$ and the anyon model has quantum dimension $\\mathcal{D}$.\n\n{\\bf The S-matrix}. The operators in the Boson-Lattice topological algebra are normal operators. Since the algebra is Abelian, it follows that there exists an orthonormal basis of common eigenvectors $\\{\\ket{\\psi_a}\\}$. The characteristic features of the Boson-Lattice algebra assure that these eigenvectors can be chosen such that the unitary matrix $S$:\n\\begin{eqnarray}\nS_{ab}=\\braket{a|\\psi_b}\n\\end{eqnarray}\nis a {\\em symmetric} matrix. Moreover, a Boson-Lattice algebra is such that there is essentially a unique way of choosing this unitary matrix as a symmetric \none\\footnote{Different symmetric choices of a unitary matrix correspond to anyon models that are trivially related, for example, they can be mirror image models under parity ($S^*=CSC$).}. This matrix defines the S-matrix of the anyon model.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{The construction}\n\\lfoot{\\thepage}\n\n{\\bf The T-matrix}. The special properties of the Boson-Lattice algebra also guarantee that the $S$-matrix defined above satisfies the following property. It can be written as:\n\\begin{eqnarray}\n(S\\mathcal{T})^3=\\Theta S^2,\n\\label{ModularSTEquation}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\mathcal{T}_{ab}=\\theta_{a}\\delta_{ab}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\Theta=\\frac{1}{\\mathcal{D}}\\sum_a d_a^2 \\theta_a=e^{i2\\pi c\/8}.\n\\end{eqnarray}\n\nThe diagonal matrix $\\mathcal{T}$ defines the $T$-matrix of the Boson-Lattice model \n\\footnote{It can be shown that for a modular anyon model equation (\\ref{ModularSTEquation}) is equivalent to equation (\\ref{S-T-Relation}). Details of proof will be given elsewhere.}. \nThe diagonal elements $\\theta_a$ define the topological spins of the charges, and the constant $c$ is the central charge of the modular anyon model.\n\n\\vspace{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{BOLATMatrix.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Braiding}\n\\rfoot{\\thepage}\n\\section*{The special properties of Boson-Lattice graphs}\n\\addcontentsline{toc}{section}{The set of Boson-Lattice models}\n\n\nGiven an arbitrary graph, the set of conditions it has to fulfill to be the generating graph of a modular anyon model is highly demanding.\nFirst, the graph has to be the generating graph of a topological algebra. Second, it has to admit a symmetric eigenbasis of eigenvectors. Finally, such eigenbasis has to fulfill the non-trivial condition given in equation (\\ref{S-T-Relation}).\n\nThis array of conditions is so restrictive that it seems clear that a randomly chosen graph for a Boson-Lattice system has low chances to represent a modular anyon model. Moreover, there is in principle no reason to think that such fortunate graph could even exist for a Boson-Lattice system.\n\nThe Boson-Lattice graph I have defined succeeds in generating well defined modular anyon models for any number of bosons and lattice sites. The special connectivity properties of the graph make it possible to fulfill the non-trivial constellation of conditions that guarantee the existence of a modular anyon model.\n\nNot less surprising is the fact that, as I show in the next section, series of known tabulated anyon models can all be encoded into Boson-Lattice graphs.\n\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{SetBOLAModels.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{ExamplesTitleFigure.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{0.5cm}\n\\section*{}\n\\addcontentsline{toc}{section}{Examples}\n\\hspace*{0.5cm}\n\\includegraphics[width=0.75\\textwidth]{ExamplesTitle.jpg}\n\n\\large\n\\vspace*{0cm}\n\\hspace*{2cm}\n\\parbox{11cm}{To see the Boson-Lattice construction at work I consider several examples of Boson-Lattice anyon models constructed with the formalism introduced above. \n\\parskip=8pt\n\nGiven a system with $k$ bosons in a lattice with $n$ sites, I analyze the corresponding bosonic Hilbert space.\nI identify the Boson-lattice graph and show that the corresponding operator $X$ can be completed to a topological algebra. This algebra encodes the fusion rules of the model.\n{\\em Diagonalization} of the topological algebra allows us to derive the braiding properties of the Boson-Lattice anyon model.\n\nThe set of fusion and brading rules obtained with the Boson-Lattice formalism define a well-defined anyon model. In some cases, the constructed Boson-Lattice models correspond to tabulated models, such as $\\mathbf{SU}(2)_k$ or $\\mathbf{SO}(3)_k$. Interestingly, we will see how the construction also yields other well-defined anyon models that are not tabulated.}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{6.5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.85\\textwidth]{kBosons2SitesTitleFigure.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\section*{}\n\\addcontentsline{toc}{section}{$k$ bosons in $2$ lattice sites}\n\\vspace*{3cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.7\\textwidth]{kBosons2SitesTitle.jpg}\n\n\\vspace{1cm}\n\\hspace*{2cm}\n\\parbox{11cm}{ \n\\parskip=8pt\nI analyze the anyon model corresponding to $k$ bosons in a lattice of $2$ sites.\n\nI identify the Boson-Lattice generating graph, construct the topological algebra and characterize the fusion and braiding rules of the anyon model.\nI show that it corresponds to the modular anyon model $\\mathbf{SU}(2)_k$.\n\nIt is illuminating to see how in the language of the Boson-Lattice construction, the $\\mathbf{SU}(2)_k$ anyon model corresponds to a particle in a lattice of $k+1$ sites with open boundary conditions.\nThe elements of the $S$-matrix of the anyon model acquire a physical interpretation as the eigenfunctions of such a particle.\n}\n\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\vspace*{2.5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{kBosons2SitesBOLAGraph.jpg}\n\\end{center}\n\n\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$k$ bosons in $2$ sites}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\n\nThe Hilbert space of $k$ bosons in $2$ lattice sites has dimension $k+1$. The Fock states can be labelled by:\n\\begin{eqnarray}\n\\ket{x}=\\frac{(a^\\dagger_0)^{k-x}}{\\sqrt{(k-x)!}}\\frac{(a^\\dagger_1)^{x}}{\\sqrt{x!}}\\ket{\\text{vac}}, \\quad x=0,\\cdots,k,\n\\end{eqnarray}\nwhere $a^\\dagger_0$ $(a^\\dagger_1)$ creates a particle in the $0$ ($1$) lattice site, and $\\ket{x}$ denotes the Fock state with $k-x$ bosons in the $0$ site and $x$ bosons in the $1$ site.\n\nThe corresponding anyon model has $k+1$ topological charges, in one to one correspondence with the Fock states. We label them by:\n\\begin{eqnarray}\n\\{0,1,\\cdots, k\\} \\quad \\longleftrightarrow \\quad \\{\\ket{0},\\ket{1},\\cdots, \\ket{k}\\}.\n\\end{eqnarray}\n\nFollowing the prescription of the construction, the Boson-Lattice graph corresponds to a one-dimensional lattice of $k+1$ sites in which each vertex is connected to its two next neighbors. The two ending vertices are not connected to each other.\n\nThe operator $X$ corresponding to the Boson-Lattice generating graph can be written as:\n\\begin{eqnarray}\nX=\\sum_{x=0}^{k-1}\\ket{x+1}\\bra{x} + \\text{h.c.}\n\\end{eqnarray}\nThis is a hermitian operator that corresponds to the (real) {\\bf tunneling operator of one particle in a one-dimensional  lattice of $k+1$ sites with open boundary conditions}.\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{kBosons2SitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$k$ bosons in $2$ sites}\n\\rfoot{\\thepage}\n\n\\vspace*{1.5cm}\n\\section*{Topological algebra}\n\\addcontentsline{toc}{section}{Topological algebra}\n\nThe generating operator $X$ can be completed to a topological algebra. To show this we search for the algebra of polynomials of $X$,\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},\\text{p}_1[X],\\cdots,\\text{p}_k[X]\\},\n\\end{eqnarray} \nthat satisfy \n\\begin{eqnarray}\n\\text{p}_\\ell[X]\\ket{0}=\\ket{\\ell}, \\quad \\ell=0,\\cdots,k.\n\\end{eqnarray}\n\nBy inspection of the generating graph $\\mathcal{G}$ it is straightforward to see that the polynomials are obtained by the recursive relation:\n\\begin{eqnarray}\n\\text{p}_{\\ell+1}[X]=X\\text{p}_\\ell[X]-\\text{p}_{\\ell-1}[X],\n\\end{eqnarray}\nwith $\\text{p}_0[X]=\\mathds{1}$ and $\\text{p}_1[X]=X$. \n\nExplicitly, we obtain:\n\\begin{eqnarray}\n\\text{p}_0=\\mathds{1},\\,\\,\\,\\text{p}_1=X, \\,\\,\\,\\text{p}_2=X^2-1, \\,\\,\\,\\text{p}_3=X^3-2X,\\,\\,\\,\\cdots\n\\end{eqnarray} \n\nAdditionally, the following {\\em boundary} equation is fulfilled:\n\\begin{eqnarray}\nX\\text{p}_k[X] -\\text{p}_{k-1}[X]=0.\n\\end{eqnarray} \n\n\nThe polynomials above define by construction a topological algebra. To explicitly show that they correspond to non-negative matrices, we draw the corresponding topological graphs.\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{kBosons2SitesTopGraphs.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$k$ bosons in $2$ sites}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\section*{Topological graphs and Fusion rules}\n\\addcontentsline{toc}{section}{Topological graphs and Fusion rules}\n\nFrom the Boson-Lattice generating graph we can directly read the fusion rules of the generating charge $1$. These are:\n\n\\begin{center}\n\\begin{tabular}{llll}\n& & Fusion rules &\\\\\n& & &\\\\\n$X\\ket{0}=\\ket{1}$ & &$1 \\times 0=1$ &\\\\\n$X\\ket{x}=\\ket{x-1}+\\ket{x+1}$ &$\\longleftrightarrow$& $1 \\times x=(x+1)+(x-1)$& $1<x<k$\\\\\n$X\\ket{k}=\\ket{k-1}$ & &$1 \\times k=k-1$&\n\\end{tabular}\n\\end{center}\n\nInterestingly, they exactly correspond to those of the topological charge $\\frac{1}{2}$ in the anyon model \n$\\mathbf{SU}(2)_k$ \\footnote{Note that the charges of the anyon model $\\mathbf{SU}(2)_k$ are usually labelled by $\\{0,\\frac{1}{2},1,\\cdots,\\frac{k}{2}\\}$.\nHere, they are labelled by $\\{0,1,2,\\cdots,k\\}$.}.\n\nBy composing the generating graph with itself we can generate the topological graphs depicted in the figure.  \nThe fusion rules of the anyon model are directly obtained from these graphs. They are:\n\n\\begin{eqnarray}\nx_1\\times x_2=\\sum_{x=|x_1-x_2|}^m x,\n\\end{eqnarray} \nwhere $m=\\text{min}\\{x_1+x_2,2k-x_1-x_2\\}$.\n\n\nThey exactly correspond to the fusion rules of the anyon model $\\mathbf{SU}(2)_k$.\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{kBosons2SitesS-matrix.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$k$ bosons in $2$ sites}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\section*{S-matrix}\n\\addcontentsline{toc}{section}{S-matrix}\n\nDiagonalization of the operator $X$ yields eigenstates of the form:\n\\begin{eqnarray}\n\\ket{\\psi_q}=\\sum_x S_{qx}\\ket{x},\n\\end{eqnarray} \nwhere\n\\begin{eqnarray}\nS_{qx}=\\frac{\\sqrt{2}}{\\sqrt{k+2}}\\sin\\frac{\\pi}{k+2}(q+1)(x+1).\n\\end{eqnarray} \nThey are sine functions that vanish at the boundaries of the one-dimensional lattice. They are the eigenmodes of a particle in a lattice of $k$ sites with open boundary conditions. \n\nPleasingly, they define a unitary symmetric matrix that exactly corresponds to the $S$-matrix of the $\\mathbf{SU}(2)_k$ anyon model.\n\n\n{\\bf It is remarkable that something as physical (and simple) as the tunneling Hamiltonian of a particle in a one-dimensional lattice with open boundaries, can encode the apparently complex mathematical properties of the anyon model $\\mathbf{SU}(2)_k$.}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{4.5cm}\n\\begin{center}\n\\includegraphics[width=0.78\\textwidth]{2Bosons3SitesTitleFigure.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\section*{}\n\\addcontentsline{toc}{section}{$2$ bosons in $3$ lattice sites}\n\\hspace*{1cm}\n\\includegraphics[width=0.75\\textwidth]{2Bosons3SitesTitle.jpg}\n\n\\large\n\\hspace*{3cm}\n\\parbox{9.5cm}{\\parskip=8pt\n\nI analyze the anyon model corresponding to $2$ bosons in a lattice of $3$ sites.\n\nI identify the Boson-Lattice generating graph, construct the topological algebra and characterize the fusion and braiding rules of the anyon model.\n\nI show that it corresponds to the modular anyon model \\bf{Fib} $\\times \\,\\mathbb{Z}_3$.}\n\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\n\nThe Hilbert space of $2$ bosons in $3$ lattice sites has dimension $6$. I denote the states in the Fock basis by\n\\begin{equation}\n\\{\\ket{0},\\ket{1},\\ket{2},\\ket{3},\\ket{4},\\ket{5}\\}.\n\\end{equation}\nThe corresponding anyon model has thus $6$ topological charges.\n\nFollowing the prescription given in the previous section, the Boson-Lattice generating graph $\\mathcal{G}$ is the one depicted below.\n\n\\vspace{0.5cm}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{2Bosons3SitesBOLAGraph.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\section*{Topological Algebra}\n\\addcontentsline{toc}{section}{Topological Algebra}\n\nFor this model, the completion of the generating operator $X$ to a topological algebra is straightforward. \nWe have that\n\\begin{equation}\n\\begin{array}{lll}\n\\ket{0}=\\mathds{1}\\ket{0} & \\ket{2}=T\\ket{0} & \\ket{4}=T^2\\ket{0}  \\\\\n\n\\ket{1}=X\\ket{0} & \\ket{3}=XT\\ket{0} & \\ket{5}=XT^2\\ket{0}.\n\\end{array}\n\\end{equation}\n\nTherefore the set of operators\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},X,T,XT,T^2,XT^2\\}\n\\end{eqnarray}\nis the algebra of polynomials (of the operators $X$ and $T$) we are looking for. They commute with each other, they have positive entries (since both $X$ and $T$ have positive entries), and we have $X^\\dagger=XT^2$, $T^\\dagger=T^2$ and $(XT)^\\dagger=XT$.\nThey define the topological algebra.\n\n\\begin{center}\n\\includegraphics[width=1.03\\textwidth]{2Bosons3SitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Topological graphs}\n\\addcontentsline{toc}{section}{Topological graphs}\nIt is illuminating to draw the set of graphs corresponding to the topological algebra. This can be done just by composition of the graphs corresponding to the operators $X$ and $T$. \n\nNicely, the graph corresponding to the operator $XT$ decomposes into three identical copies of the generating graph of the Fibonacci model.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.97\\textwidth]{2Bosons3SitesTopGraphs.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Equivalence to  $\\text{Fib}\\times\\mathbb{Z}_3$}\n\\addcontentsline{toc}{section}{The anyon model $\\text{Fib}\\otimes\\mathbb{Z}_3$}\n\nThe topological algebra\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},X,T,XT,T^2,XT^2\\}\n\\end{eqnarray}\nhas two subalgebras:\n\\begin{eqnarray}\n\\begin{array}{lll}\n\\text{{\\bf Fibonacci} subalgebra} &\\{\\mathds{1},XT\\} &XT \\cdot XT=\\mathds{1}+XT\\\\\n&&\\\\\n\\mathbb{Z}_3 \\,\\,\\text{subalgebra}&\\{\\mathds{1},T,T^2\\}& T \\cdot T=T^2\\\\\n&&T\\cdot T^2=\\mathds{1}.\n\\end{array}\n\\end{eqnarray}\nIt is useful to write the operators $XT$ and $T$ as:\n\\begin{eqnarray}\n\\begin{array}{rlrll}\nXT&=&X_{\\text{Fib}}&\\otimes &\\mathds{1}_3\\\\\nT&= &\\mathds{1}_2 &\\otimes &X_{\\mathbb{Z}_3},\n\\end{array}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nX_{\\text{Fib}}=\\left[\n\\begin{array}{cc}\n0&1\\\\\n1&1\n\\end{array}\n\\right],\n\\quad\n\\quad\nX_{\\mathbb{Z}_3}=\\left[\n\\begin{array}{ccc}\n0&0&1\\\\\n1&0&0\\\\\n0&1&0\n\\end{array}\n\\right],\n\\end{eqnarray}\nand $\\mathds{1}_2$ ($\\mathds{1}_3$) is the $2\\times 2$ ($3\\times 3$) identity matrix.\nDefining \n\\begin{eqnarray}\n&&\\mathcal{A}_{\\text{Fib}}=\\{\\mathds{1}_2,X_{\\text{Fib}}\\}\\\\\n&&\\mathcal{A}_{\\mathbb{Z}_3}=\\{\\mathds{1}_3,X_{\\mathbb{Z}_3}^{\\phantom{2}},X_{\\mathbb{Z}_3}^2\\},\n\\end{eqnarray}\nthe topological algebra $\\mathcal{A}$ can be written as the tensor product of the topological algebra of the Fibonacci model and the topological algebra of the $\\mathbb{Z}_3$ model:\n\\begin{eqnarray}\n\\mathcal{A}=\\mathcal{A}_{\\text{Fib}}\\otimes \\mathcal{A}_{\\mathbb{Z}_3}.\n\\end{eqnarray}\n\nThe fusion rules of the Boson-lattice model are therefore those of \nthe anyon model \n\\begin{eqnarray}\n\\text{Fib}\\times\\mathbb{Z}_3.\n\\end{eqnarray}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\section*{S-Matrix}\n\\addcontentsline{toc}{section}{S-Matrix}\nDiagonalization of the graphs yields a unique (up to conjugation) symmetric and unitary matrix, which defines de $S$-matrix of the anyon model.\nThis matrix can be written as the tensor product:\n\\begin{eqnarray}\nS=S_{\\text{Fib}}\\otimes S_{\\mathbb{Z}_3}, \n\\end{eqnarray}\nwhere $S_{\\text{Fib}}$ and $S_{\\mathbb{Z}_3}$ are, respectively, the $S$-matrix of the Fibonnaci anyon and the $\\mathbb{Z}_3$ anyon models.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{2Bosons3SitesSMatrix.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{T-Matrix}\n\\addcontentsline{toc}{section}{T-Matrix}\nThe topological $T$-matrix is determined by equation (\\ref{S-T-Relation}), relating the $S$-matrix to the $T$-matrix. It can be written as:\n\\begin{eqnarray}\nT=T_{\\text{Fib}}\\otimes T_{\\mathbb{Z}_3}, \n\\end{eqnarray}\nwhere $T_{\\text{Fib}}$ and $T_{\\mathbb{Z}_3}$ are, respectively, the $T$-matrix of the Fibonnaci anyon and the $\\mathbb{Z}_3$ anyon models.\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{2Bosons3SitesTMatrix.jpg}\n\\end{center}\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.8\\textwidth]{2BosonsNSitesTitleFigure.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\section*{}\n\\addcontentsline{toc}{section}{$2$ bosons in $n$ lattice sites}\n\\includegraphics[width=0.7\\textwidth]{2BosonsNSitesTitle.jpg}\n\n\\large\n\\hspace*{3cm}\n\\parbox{9cm}{\\parskip=8pt\n\nI analyze the  anyon model corresponding to $2$ bosons in a lattice of $n$ sites, with $n$ odd.\n\nI identify the Boson-Lattice generating graph, construct the topological algebra, and characterize the fusion and braiding rules of the anyon model.\n\nI show that it corresponds to the modular anyon model  {\\bf SO}$(3)_n\\times \\,\\mathbb{Z}_n$.\n\nNotice that for $n=3$ we have {\\bf SO}$(3)_3 \\equiv $ {\\bf Fib}, recovering the result of the previous section.}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Hilbert space}\n\\addcontentsline{toc}{section}{Fock basis}\nThe Hilbert space of $2$ bosons in $n=2\\ell+1$ lattice sites has dimension $(\\ell+1)\\times n$. The Fock states can be labelled by the relative distance $r$ between the two particles and the position $x$ of one of them (for example, the one with the smallest value of $x$):\n\\begin{eqnarray}\n\\ket{r,x}=T^x\\ket{r,0}, \\quad &&r=0,\\cdots,\\ell \\nonumber\\\\\n &&x=0,\\cdots,n-1.\n\\end{eqnarray}\nSince $n$ is odd, for a fixed relative position $r$ there are always $n$ possible Fock states. \n\n\\vspace*{0.5cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{2BosonsNSitesFock.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $n$ sites}\n\\rfoot{\\thepage}\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\nThe corresponding anyon model has $(\\ell+1)\\times n$ topological charges.\nFollowing the prescription of the construction,  the Boson-Lattice generating graph $\\mathcal{G}$ is depicted below for\n$n=5$, $n=7$ and $n=9$.\n\n\\vspace*{1cm}\n\\includegraphics[width=1.02\\textwidth]{2BosonsNSitesBOLAGraph.jpg}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\n\\section*{Topological Algebra}\n\\addcontentsline{toc}{section}{Topological Algebra}\n\nBy inspection of the generating graph $\\mathcal{G}$ we can directly see that the algebra of polynomials we are looking for is:\n\\begin{eqnarray}\n\\mathcal{A}=\\{X_rT^x \\quad r=0,\\cdots,\\ell;\\,x=0,\\cdots,n-1\\},\n\\end{eqnarray}\nwhere $X_0=\\mathds{1}$, $X_1=X$ and $X_r,$ (for $r=2,\\cdots,\\ell$)  are polynomials of the operators $X$ and $T$, obtained through the recursive relation:\n\\begin{eqnarray}\nX_r=XX_{r-1}-X_{r-2}\\,T.\n\\end{eqnarray}\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{2BosonsNSitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $n$ sites}\n\\rfoot{\\thepage}\n\n\\section*{The $\\mathbf{SO}(3)_k$ subalgebra}\n\\addcontentsline{toc}{section}{The $\\mathbf{SO}(3)_k$ subalgebra}\n\nTo show that $\\mathcal{A}$ is a topological algebra, let me consider the operator $Q=XT^\\ell\\in \\mathcal{A}$. This is the generating operator of the subalgebra\n\\begin{eqnarray}\n\\{\\mathds{1},Q=Q_1,Q_2,\\cdots, Q_\\ell\\}, \\quad Q_r=X_rT^{r\\ell}.\n\\end{eqnarray}\nIt fulfills:\n\\begin{eqnarray}\n\\mathds{1} \\times Q &=& Q \\nonumber\\\\\n\\,Q \\times Q_r&=&Q_{r-1}+Q_{r+1} \\nonumber\\\\\n\\,Q \\times Q_\\ell&=&Q_{\\ell-1}+Q_\\ell.\n\\end{eqnarray}\nThe corresponding graph is depicted below. It is very similar to the generating graph of the anyon model $\\mathbf{SU}(2)_\\ell$, except for the important fact that the graph has now a loop at the last vertex. Is this graph a topological graph?\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{2BosonsNSitesSubalgebra.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\n\\vspace*{0cm}\nLet me consider the operator $Q_2$ in the subalgebra above. Reordering the elements of the subalgebra\n\\begin{eqnarray}\nQ_i \\longrightarrow \\widetilde{Q}_i=Q_{p(i)},\n\\end{eqnarray}\nwith the permutation $p$ such that\n\\begin{eqnarray}\n\\{0,1,\\cdots,\\ell\\} &\\longrightarrow &\\{0,2,4,\\cdots, \\ell, \\ell-1,\\cdots,3,1\\} \\hspace{1cm}\\text{$\\ell$ even}\\nonumber\\\\\n\\{0,1,\\cdots,\\ell\\} &\\longrightarrow &\\{0,2,4,\\cdots, \\ell-1, \\ell,\\cdots,3,1\\} \\hspace{1cm}\\text{$\\ell$ odd},\n\\end{eqnarray}\nwe have that $\\widetilde{Q}\\equiv Q_2$ fulfills:\n\\begin{eqnarray}\n\\mathds{1} \\times \\widetilde{Q} &=& \\widetilde{Q} \\nonumber\\\\\n\\,\\widetilde{Q} \\times \\widetilde{Q}_r&=&\\widetilde{Q}_{r-1}+\\widetilde{Q}_r+\\widetilde{Q}_{r+1} \\nonumber\\\\\n\\,\\widetilde{Q} \\times \\widetilde{Q}_\\ell&=&\\widetilde{Q}_{\\ell-1}+\\widetilde{Q}_\\ell.\n\\end{eqnarray}\nThese fusion rules exactly correspond to the ones of the anyon model $\\mathbf{SO}(3)_n$.\n\n\\vspace*{1cm}\n\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{2BosonsNSitesSubalgebra_2.jpg}\n\\end{center}\n\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $n$ sites}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\nThe generating operator $Q$ corresponds to the tunneling Hamiltonian of one particle in a one-dimensional lattice of $\\ell+1$ sites with open boundaries, and in the presence of an impurity potential at the last site, $\\ell$. \nThe eigenmodes of this Hamiltonian define a unitary symmetric matrix, which exactly coincides with the topological $S$-matrix of the modular anyon model  $\\mathbf{SO}(3)_n$.\n\nIt is again remarkable that the Hamiltonian of a particle in a one-dimensional lattice with open boundaries (and an impurity potential at the last site) can encode the mathematical properties of the anyon model $\\mathbf{SO}(3)_n$.\n\nFrom the results above, we can conclude that the anyon model corresponding to the Boson-Lattice system of $2$ particles in $n$ sites, with $n$ odd, is $\\mathbf{SO}(3)_n \\times \\mathbb{Z}_n$.\n\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{2BosonsNSitesSMatrix.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.8\\textwidth]{2Bosons4SitesTitleFigure.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\section*{}\n\\addcontentsline{toc}{section}{$2$ bosons in $4$ lattice sites}\n\\includegraphics[width=0.7\\textwidth]{2Bosons4SitesTitle.jpg}\n\n\\large\n\\hspace*{3cm}\n\\parbox{10cm}{\\parskip=8pt\n\nI analyze the anyon model corresponding to $2$ bosons in a lattice of $4$ sites, as an illustration of the interesting case of $2$ bosons in a lattice with an {\\em even} number of sites.\n\nI identify the Boson-Lattice  graph, construct the topological algebra, and characterize the fusion and braiding rules of the anyon model.\n\nIn contrast to the case of $2$ bosons in a lattice with an odd number of sites, the topological algebra of this Boson-Lattice model is not decomposable as the tensor product of two topological algebras.\n\nThe Boson-Lattice model of $2$ bosons in $4$ sites is a well defined anyon model, which is to my knowledge {\\em not tabulated}.}\n\n\\normalsize\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\nThe Hilbert space of $2$ bosons in $4$ lattice sites has dimension $10$. The corresponding anyon model has therefore $10$ topological charges.\nFollowing the prescription given in the previous section, we construct the Boson-Lattice generating graph $\\mathcal{G}$ depicted below.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{2Bosons4SitesBOLAGraph_2.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $4$ sites}\n\\rfoot{\\thepage}\n\\section*{Topological Algebra}\n\\addcontentsline{toc}{section}{Topological Algebra}\nBy inspection of the generating graph $\\mathcal{G}$ we can see that the set of polynomials  we are looking for is:\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},T, T^2,T^3,X,XT,XT^2,XT^3,Q,TQ\\},\n\\end{eqnarray}\nwhere the operator $Q$ fulfills:\n\\begin{eqnarray}\n&&Q=XX-T \\\\\n&&QT^2=Q.\n\\end{eqnarray}\n\n\\vspace*{0.3cm}\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{2Bosons4SitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\vspace*{0cm}\n\\section*{Topological graphs}\n\\addcontentsline{toc}{section}{Topological graphs}\nTo show that $\\mathcal{A}$ is a topological algebra, it suffices to show that $Q$ has non-negative entries. By composing the graph $X$ with itself and subtracting the graph corresponding to the operator $T$, we obtain the graph of the operator $Q$.\nThis is a non-negative graph (links have weight either $0$ or $1$) that decomposes into two independent graphs.\n\n\\vspace*{1cm}\n\\includegraphics[width=0.9\\textwidth]{2Bosons4SitesTopGraphs.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $4$ sites}\n\\rfoot{\\thepage}\n\\vspace*{0cm}\n\\normalsize\n\\section*{Fusion rules}\n\\addcontentsline{toc}{section}{Fusion Rules}\n\nThe fusion rules of the model can be directly read from the topological graphs below.\n\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{2Bosons4SitesTopGraphs_2.jpg}\n\\end{center}\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{S-Matrix}\n\\addcontentsline{toc}{section}{S-Matrix}\nDiagonalization of the graphs yields a unique (up to conjugation) symmetric and unitary matrix, which defines the $S$-matrix of the anyon model.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{2Bosons4SitesSMatrix.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$2$ bosons in $4$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{T-Matrix}\n\\addcontentsline{toc}{section}{T-Matrix}\nThe topological $T$-matrix is uniquely determined by equation (\\ref{S-T-Relation}), relating the $S$-matrix to the $T$-matrix. \n\n\\vspace*{1.5cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{2Bosons4SitesTMatrix.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.8\\textwidth]{3Bosons3SitesTitleFigure.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\section*{}\n\\addcontentsline{toc}{section}{$3$ bosons in $3$ lattice sites}\n\\hspace*{1cm}\n\\includegraphics[width=0.7\\textwidth]{3Bosons3SitesTitle.jpg}\n\n\\large\n\\hspace*{3cm}\n\\parbox{9cm}{\\parskip=8pt\n\nI analyze the  anyon model corresponding to $3$ bosons in a lattice of $3$ sites.\n\nI identify the Boson-Lattice  graph, construct the topological algebra, and characterize the fusion and braiding rules of the anyon model.\n\nThe anyon model corresponding to $3$ bosons in a lattice of $3$ sites is a well defined anyon model, which is to my knowledge {\\em not tabulated}.}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\nThe Hilbert space of $3$ bosons in $3$ lattice sites has dimension $10$. The corresponding anyon model has therefore $10$ topological charges.\nFollowing the prescription given in the previous section, we construct the Boson-Lattice generating graph $\\mathcal{G}$ as depicted below.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{3Bosons3SitesBOLAGraph.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$3$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\section*{Topological Algebra}\n\\addcontentsline{toc}{section}{Topological Algebra}\nBy inspection of the generating graph $\\mathcal{G}$ we can see that the set of polynomials  we are looking for is:\n\\begin{eqnarray}\n\\mathcal{A}=\\{\\mathds{1},T, T^2,X,XT,XT^2,X^\\dagger,X^\\dagger T,X^\\dagger T^2,Q\\},\n\\end{eqnarray}\nwhere the operator $Q$ fulfills:\n\\begin{eqnarray}\n&&Q=XX^\\dagger-\\mathds{1}\\\\\n&&QT=Q.\n\\end{eqnarray}\n\n\n\\begin{center}\n\\includegraphics[width=1.05\\textwidth]{3Bosons3SitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Topological graphs}\n\\addcontentsline{toc}{section}{Topological graphs}\nBy composing the graphs of  $X$ and $X^\\dagger$ and subtracting the identity graph, we obtain the graph of the operator $Q$.\nThis is a non-negative graph that decomposes into three independent graphs.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{3Bosons3SitesTopGraphs.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$3$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{Fusion rules}\n\\addcontentsline{toc}{section}{Fusion Rules}\n\nThe fusion rules of the model can be directly read from the topological graphs below.\nInterestingly, this model has multiplicities larger than $1$, as it can be seen from the double loop in the graph corresponding to the charge $Q$. We have:\n\\begin{eqnarray}\nQ\\times Q=1+T+T^2+2Q.\n\\end{eqnarray}\nThis model is not tabulated in the tables by Bonderson \\cite{Bonderson}, which are restricted to multiplicity-free anyon models.\n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{3Bosons3SitesTopGraphs2.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{S-Matrix}\n\\addcontentsline{toc}{section}{S-Matrix}\nDiagonalization of the graphs yields a unique (up to conjugation) symmetric and unitary matrix, which defines the $S$-matrix of the anyon model.\n\n\\vspace*{1cm}\n\\includegraphics[width=0.9\\textwidth]{3Bosons3SitesSMatrix.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$3$ bosons in $3$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{T-Matrix}\n\\addcontentsline{toc}{section}{T-Matrix}\nThe topological $T$-matrix is uniquely determined by equation (\\ref{S-T-Relation}), relating the $S$-matrix to the $T$-matrix. \n\n\\vspace*{1cm}\n\\includegraphics[width=0.9\\textwidth]{3Bosons3SitesTMatrix.jpg}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{5cm}\n\\hspace*{1cm}\n\\includegraphics[width=0.8\\textwidth]{3BosonsNSitesTitleFigure.jpg}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Boson-Lattice examples}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\section*{}\n\\addcontentsline{toc}{section}{$3$ bosons in $N$ lattice sites}\n\\hspace*{1cm}\n\\includegraphics[width=0.65\\textwidth]{3BosonsNSitesTitle.jpg}\n\n\\large\n\\hspace*{3cm}\n\\parbox{9cm}{\\parskip=8pt\n\nI analyze the  anyon model corresponding to $3$ bosons in a lattice of $N$ sites.}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Boson-Lattice graph}\n\\addcontentsline{toc}{section}{Boson-Lattice graph}\nFollowing the prescription given in the previous section, we construct the Boson-Lattice generating graph $\\mathcal{G}$ as depicted below.\nBy considering the graph of equivalence classes (a vertex corresponds to the class of Fock states that can be obtained from each other by a global translation) the graph takes the form of a pyramid with a triangular tiling. \n\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{3BosonsNSitesBOLAGraph.jpg}\n\\end{center}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$3$ bosons in $n$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{Topological Algebra}\n\\addcontentsline{toc}{section}{Topological Algebra}\nThe pyramidal structure of the graph allows to obtain the topological algebra in a recursive way.\nPolynomials corresponding to charges at a certain level of the pyramid are obtained from polynomials at the previous levels as depicted below.\n\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.92\\textwidth]{3BosonsNSitesAlgebra.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Boson-Lattice examples}\n\\lfoot{\\thepage}\n\\normalsize\n\\section*{Topological graphs}\n\\addcontentsline{toc}{section}{Topological graphs}\nThe pyramidal structure assures that the polynomials correspond to non-negative matrices.\nThis can be graphically seen by drawing the corresponding graphs. Graphs at a certain level of the pyramid contain the graphs corresponding to the previous level, so that the operators remain non-negative after subtraction of graphs from the upper level. \n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{3BosonsNSitesAlgebra_2.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{$4$ bosons in $n$ sites}\n\\rfoot{\\thepage}\n\\normalsize\n\\section*{The general Boson-Lattice graph}\n\\addcontentsline{toc}{section}{4 Bosons in N Lattice sites}\nThe Boson-Lattice graphs for $k$ bosons in $n$ lattice sites correspond to multidimensional pyramidal structures. For $4$ bosons in $n$ lattice sites, the graph corresponds to a pyramid with a tetrahedral tiling. \nThe pyramidal structure allows for the existence of a topological algebra of polynomials of the operators $X$, $X^\\dagger$ and $T$. \n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{4BosonsNSites.jpg}\n\\end{center}\n\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2,5cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{GravityFigureTitle.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Why does the construction work?}\n\\cfoot{\\thepage}\n\\section*{}\n\\addcontentsline{toc}{section}{Boson-Lattice construction and gravity}\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{GravityTitle.jpg}\n\\end{center}\n\n\\large\n\\hspace*{2cm}\n\\parbox{11.5cm}{\\parskip=8pt\n\nThe Boson-Lattice construction succeeds in systematically generating well defined anyon models.\nThe Boson-Lattice graph is conceived such that 1) it defines a set of well defined fusion rules, and 2) there is always a symmetric choice of eigenvectors of the graph that can represent the $S$-matrix of a modular anyon model.\n\nBut why is this always like this? Why does the connectivity graph of Fock states of a bosonic lattice system always have a symmetric matrix of eigenvectors? Beyond possible mathematical general proofs that I can give, I am here interested in finding an intuitive argument able to explain this fact.\n\nTo this aim I take a closer look into the generating Boson-Lattice operator. I discover an illuminating connection between Boson-Lattice graphs and curved-space geometries.\nThis connection throws light into the success of the construction. \n\nBut even more, this connection anticipates a beautiful duality between anyon models and curved space-time geometries, between anyon models and gravity.}\n\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Why does the construction work?}\n\\lfoot{\\thepage}\n\n\\section*{Tunneling in a curved space}\n\\addcontentsline{toc}{section}{Tunneling in a curved space}\n\n\\normalsize\nThe generating operator corresponding to the Boson-Lattice graph has the form:\n\\begin{eqnarray}\nX=\\sum_{i^{\\vphantom{\\prime}} \\leadsto i^\\prime}\\ket{i^\\prime} \\bra{i^{\\vphantom{\\prime}}}=\\sum_{x}A^\\dagger_{x+1}A^{\\vphantom{\\dagger}}_x,\n\\end{eqnarray}\nwhere $i \\leadsto i^\\prime$ indicates that Fock state $\\ket{i^\\prime}$ can be obtained from $\\ket{i}$ by tunneling of one particle, and the operators\n$A^{\\vphantom{\\dagger}}_x$ are defined in Eq. \\ref{CreationOperators}.\n\nThough the operator $X$ connects Fock states that are connected by one-particle tunneling, $X$ is a many-body operator different from the one-particle tunneling operator:\n\\begin{eqnarray}\nX\\ne\\sum_{x}a^\\dagger_{x+1}a_{x}^{\\vphantom{\\dagger}}.\n\\end{eqnarray}\nBut how is then $X$ written in terms of bosonic creation and annihilation operators $a_x$? It is not difficult to see that the operator $A_x$ can be written as:\n\\begin{eqnarray}\nA_x=\\frac{1}{\\sqrt{n_x+1}}\\,a_x,\n\\end{eqnarray}\nwhere $n_x=a^\\dagger_{x}a_{x}^{\\vphantom{\\dagger}}$ is the density operator at site $x$.\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Gravity_OneParticle.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Tunneling in a curved space}\n\\rfoot{\\thepage}\nThis means that the generating operator has the form:\n\\begin{eqnarray}\nX=\\sum_{x}a^\\dagger_{x+1}\\,g_x\\,a_{x}^{\\vphantom{\\dagger}},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\ng_x=\\frac{1}{\\sqrt{n_x+1}}\\frac{1}{\\sqrt{n_{x+1}+1}}.\n\\end{eqnarray}\nThis reveals that the Boson-Lattice graph is a {\\bf correlated tunneling} operator: the tunneling amplitude for a particle to hope from site $x$ to site $x+1$ depends on the density of particles at sites $x$ and $x+1$. Effectively, {\\em a particle tunnels in a space that has been curved by the presence (by the mass) of the other particles}. The metric in that curved space is given by $g_x$. \n\n\\vspace*{1cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Gravity_CurvedSpace.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Why does the construction work?}\n\\lfoot{\\thepage}\n\\vspace*{2,5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{GravitySymmetricMatrix.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Flat vs. curved space}\n\\rfoot{\\thepage}\n\\vspace*{1,6cm}\n\\section*{Flat space vs. curved space}\n\\addcontentsline{toc}{section}{Flat space versus curved space}\n\nThe connection above sheds light on the success of the Boson-Lattice construction.\nA Boson-Lattice graph can be obtained from a one-particle tunneling operator by deforming the metric from flat to curved.\n\\begin{eqnarray}\n\\sum_{x}a^\\dagger_{x+1}a_{x}^{\\vphantom{\\dagger}}\\longleftrightarrow \\sum_{x}a^\\dagger_{x+1}g_xa_{x}^{\\vphantom{\\dagger}}.\n\\end{eqnarray}\nIt is clear that for the one-particle tunneling operator there exists a symmetric unitary matrix $U$ of eigenvectors. This corresponds to the Fock basis in momentum space:\n\\begin{eqnarray}\nU_{ab}=\\braket{a|\\widetilde{b}}=\\braket{\\widetilde{a}| b},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\ket{a}&\\equiv&\\ket{n_0,\\cdots,n_{n-1}}\n\\propto[a^\\dagger_0]^{n_0}\\cdots [a^\\dagger_{n-1}]^{n_{n-1}}\\ket{\\text{vac}}\\\\\n\\ket{\\widetilde{b}}&\\equiv&\\ket{\\widetilde{n}_0,\\cdots,\\widetilde{n}_{n-1}}\\propto\n[\\widetilde{a}^\\dagger_0]^{\\widetilde{n}_0}\\cdots [\\widetilde{a}^\\dagger_{n-1}]^{\\widetilde{n}_{n-1}}\\ket{\\text{vac}},\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n\\widetilde{a}_q=\\frac{1}{\\sqrt{n}}\\sum_{x=0}^{n-1} e^{i\\frac{2\\pi}{n}q\\cdot x}a_x.\n\\end{eqnarray}\nThis strongly suggests the existence of a unitary symmetric matrix of eigenvectors of the operator $X$, which is obtained after the metric transformation from flat to curved space:\n\\begin{eqnarray}\nU_{ab}\\longleftrightarrow S_{ab}.\n\\end{eqnarray}\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2,5cm}\n\\begin{center}\n\\includegraphics[width=1.05\\textwidth]{BOLAHigherLevels.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Higher levels of the construction}\n\\rfoot{\\thepage}\n\\section*{}\n\\addcontentsline{toc}{section}{Higher levels of the construction}\n\\hspace*{1cm}\n\\includegraphics[width=0.9\\textwidth]{BOLALevelsTitle.jpg}\n\n\\hspace*{2.5cm}\n\\parbox{11.5cm}{\\parskip=8pt The Boson-Lattice construction is a fractal construction. Anyon models arising from assembling the building blocks $\\mathbb{Z}_n$ can be used themselves as elementary pieces to generate more complex anyon models at a second level of the construction. \n\nRemarkably, the principle of assembly is the same at any level of the construction. An anyon model generated at a certain level is identified with a particle in a lattice given by the corresponding Boson-Lattice graph. Anyon models at the next level are then constructed by assembling bosons in that graph.\nThe bosonization principle of assembly generates well defined anyon models at any level of the construction.}\n\n\\begin{center}\n\\includegraphics[width=0.65\\textwidth]{BOLASecondLevel.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Higher levels of the construction}\n\\lfoot{\\thepage}\n\\vspace*{1,5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLALevel2_SU2_k.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Higher levels of the construction}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\section*{Second level of Boson-Lattice construction}\n\\addcontentsline{toc}{section}{Boson-Lattice construction at the second level}\nTo illustrate how the construction works at higher levels, let me consider the case in which the anyon models $\\mathbf{SU}(2)_k$ are used as building blocks. \nThe Boson-Lattice graph encoding the anyon model $\\mathbf{SU}(2)_k$ is a one-dimensional lattice of $k+1$ sites with open boundaries. \nFollowing the bosonization principle, the Boson-Lattice graph of the model obtained by assembling $k^\\prime$ copies of $\\mathbf{SU}(2)_k$ corresponds to the connectivity graph of Fock states of $k^\\prime$ bosons in a one-dimensional lattice of $k+1$ sites with open boundaries. \n\nFor example, the anyon model resulting from assembling two identical copies of $\\mathbf{SU}(2)_2$ is described by the Hilbert space of $2$ bosons in a lattice of $3$ sites with open boundary conditions.\nThis Hilbert space has dimension $6$. The corresponding anyon model has therefore $6$ topological charges.\nThe Boson-Lattice graph is constructed following the connectivity pattern prescription.\nThe fusion rules and braiding rules can be obtained. They exactly correspond to the anyon model $\\mathbf{SO}(5)_2$.\n\nBy varying the number $k^\\prime$ of $\\mathbf{SU}(2)_k$ building blocks the whole tower of anyon models $\\mathbf{SO}(5)_{k^\\prime}$ can be constructed.\n\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{BOLALevel2_SU2_2.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2,5cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{IntroGraph.jpg}\n\\end{center}\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\section*{}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{ClosuresTitle.jpg}\n\\end{center}\n\\addcontentsline{toc}{section}{Closures and openings}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{ClosuresTopGraphs.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{2cm}\n\\subsection*{The language of topological graphs}\n\\addcontentsline{toc}{section}{The language of graphs}\n\nThe language of topological graphs I have developed provides a visual and enlightening way to encode anyon models.\n\nOne single graph is able to encode complete information about the fusion rules of an anyon model. \nUnlike tables of fusion rules in which the essential information about an anyon model might be sometimes difficult to grasp, hidden inside a huge collection of numbers, a glance at a topological graph can rapidly tell us about the fundamental properties of an anyon model. Is the model Abelian or non-Abelian? Does it decompose into a product of simpler models? Which charges are obtained from which? They are all questions that can be immediately answered after a quick examination of the connectivity pattern of a topological graph. \n\nBraiding properties can also be nicely extracted from topological graphs. I have emphasized and discussed how diagonalization of a topological graph can yield important information, sometimes complete, about the topological $S$ and $T$ matrices of an anyon model. In some cases diagonalization of a topological graph might even turn obvious, as it is the case for anyon models such as $\\mathbb{Z}_n$, $\\mathbf{SU(2)}_k$, or $\\mathbf{SO}(3)_k$, for which the corresponding graphs are familiar lattice patterns, whose eigenstates and eigenvalues are evident to us.\n\nGraph representation of matrices is often used in physics and mathematics. Here, I have revealed the strength of a graph language to represent anyon models.\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{IntroBosonLattice.jpg}\n\\includegraphics[width=0.6\\textwidth]{ClosuresBOLAGraph.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{2cm}\n\\subsection*{The Boson-Lattice construction}\n\\addcontentsline{toc}{section}{The Boson-Lattice construction}\n\nWhen trying to conceive a construction for anyon models, \nI had always clear that the Abelian models $\\mathbb{Z}_n$ could serve as appropriate building blocks.\nAs for the principle of assembly, I had in mind some kind of procedure that would make copies of elementary anyon models indistinguishable, though I did not have a befitting language to articulate the idea of {\\em indistinguishability} of anyon models.\n\nThe first key idea came through a visual abstraction. The graph language helped me to represent the Abelian model $\\mathbb{Z}_n$ by a particle in a one-dimensional lattice. Triggered by this idea the principle of {\\em bosonization} naturally arose as an appropriate way to implement the notion of indistinguishability.\n\nThere are exponentially many ways to define a graph in a boson lattice system. To define the Boson-lattice graph as the connectivity graph of Fock states was a guess. A guess that happened to be right.\n\nThe Boson-Lattice construction represents the collapse of two languages into one. It reveals that the mathematical language to describe anyon models can be the same as the one describing boson-lattice systems. It provides us with a physically meaningful language (the one of bosons, Fock states and tunneling Hamiltonians) to describe the mathematical properties (fusion rules and braiding rules) of anyon models.\nI believe this physical language can help us  in our way to fill the explanatory gap between the mathematical and the physical sides of topological orders.\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{BOLATable1.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{1cm}\n\\subsection*{The construction works}\n\\addcontentsline{toc}{section}{The construction works}\nThe Boson-Lattice construction provides an orderly systematic way to construct anyon models. By assembling different numbers of identical building blocks (by varying the number of bosons $k$) and by changing their size (varying the number of lattice sites $n$) the construction succeeds in generating towers of well known tabulated anyon models.\n\nThe construction reveals a skeleton for the space of anyon models. It tells us that towers of more and more complex anyon models can be constructed by assembling identical anyon models and making them indistinguishable.\n\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{BOLATable2.jpg}\n\\end{center}\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{3cm}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{IntroGraph.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\subsection*{Conformal and topological quantum field theories}\n\\addcontentsline{toc}{section}{Conformal and topological quantum field theories}\nI have focused in this work on the construction of {\\em modular} anyon models, for which corresponding conformal field theories and topological quantum field theories exist.\nIt would be very interesting to delineate a graph of correspondences between the Boson-Lattice construction and conformal and topological quantum field theories. \n\nHow are known concepts in Chern-Simons theory and conformal field theory expressed in the Boson-Lattice language? \nMight the Boson-Lattice construction bring light into our understanding of the anatomy of conformal field theories and topological quantum field theories?\n\nThe Boson-Lattice construction can also serve to build up non-modular anyon models, for which no conformal field theories exist.\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{3cm}\n\\begin{center}\n\\includegraphics[width=0.73\\textwidth]{BOLAGeneral.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\subsection*{Generalizations of the Boson-Lattice construction}\n\\addcontentsline{toc}{section}{Generalizations of the Boson-Lattice construction}\nThe Boson-Lattice construction can be enriched by adding internal degrees of freedom to the bosons participating. For example, we can consider bosons with color or spin, or also change the dimensionality of the lattice. In this way, anyon models corresponding to non-chiral topological orders, such as {\\em quantum doubles}, can be generated.\n\nMoreover, the construction can be extended to fermions. By considering Fermion-Lattice graphs other towers of anyon models are built up.\n\nAdditionally, other anyon models beyond $\\mathbb{Z}_n$ can be used as initial building blocks at the first level of the construction.\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2.5cm}\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{ClosuresAnyonCondensation.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{3cm}\n\\subsection*{Anyon condensation}\n\\addcontentsline{toc}{section}{Anyon condensation}\n\nIn light of the Boson-Lattice construction, the reverse process of disintegrating anyon models into simpler pieces acquires an enlightened perspective.\n\nWithin the Boson-Lattice picture, anyon condensation is a condensation of actual bosons. Moreover, the concept of topological symmetry breaking corresponds to actual symmetry breaking in the Boson-lattice system.\n\nI find extremely interesting to investigate how anyon condensation is precisely described in the Boson-Lattice language, and, moreover, whether the Boson-Lattice construction can guide us to develop a systematic framework to describe anyon condensation, for which no fully general description is known.\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{4.5cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{ClosuresAnsatz.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{1.5cm}\n\\subsection*{A dual construction at the physical level}\n\\addcontentsline{toc}{section}{}\n\nEspecially interesting to me is the prospect of developing a dual construction at the microscopic physical level. I believe the Boson-Lattice construction for anyon models can inspire the blueprint of a construction of many-body wave functions and Hamiltonians for the corresponding topological orders.\n\nA dual bosonization principle of assembly could be used to orderly build up complex topologically ordered many-body wave functions from elementary ones.\nSuch systematic framework would reveal a dual anatomy in the phase space of topologically ordered systems.\n\nIn previous work \\cite{Paredes1,Paredes2,Paredes3}, I have proposed an Ansatz for the systematic construction of non-Abelian topologically ordered states by symmetrizing identical copies of simpler states like Abelian topological states. This type of construction is known to be the one behind non-Abelian quantum Hall states \\cite{NA1,NA4,NA5,Barkeshli1,Barkeshli2,Barkeshli3}. I have revealed its potential to describe the structure of general non-Abelian topological states, describing, for example, other families of interesting non-Abelian states such as quantum doubles or string-net condensates.\n\nHow does the notion of indistinguishability of wave functions match the one of indistinguishability of anyon models?\nIn which way is the symmetrization of many-body wave functions connected to the Boson-Lattice construction?\n\nI believe the Boson-lattice construction can also guide the systematic construction of parent Hamiltonians for topological states.\nBoson-Lattice graphs can encode the physical ingredients (local degrees of freedom, interactions between them) of the corresponding topological order. \nThey constitute dual entities able to simultaneously embody the mathematical and physical parts of topological orders.\n\nIt would be alluring to investigate the connections between the many-body wave functions and Hamiltonians emerging from the Boson-Lattice construction and those of seminal topological systems and models, such as {\\em fractional quantum Hall systems} \\cite{NA1,NA2,NA3,NA4,NA5,NA6,NA7}, {\\em quantum loop models} \\cite{Kitaev2, QLM1, QLM2, QLM3}, and {\\em string-net models} \\cite{SNC1,SNC2,SNC3,SNC4}.\n\n\n\n\\newpage\n\\thispagestyle{empty}\n\\vspace*{2cm}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{GravityDuality.jpg}\n\\end{center}\n\n\\newpage\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{Closures and openings}\n\\rfoot{\\thepage}\n\\vspace*{3.5cm}\n\\subsection*{Duality between anyon models and gravity}\n\\addcontentsline{toc}{section}{Duality between anyon models and gravity}\n\nI have drawn a connection between anyon models and curved-space geometries.\nWithin the Boson-Lattice approach an anyon model corresponds to a system of bosons in a lattice whose metric has been curved.\nAbelian models are associated with flat geometries, whereas non-Abelian models correspond to curved ones. \n\nThis connection is extremely attractive to me. What type of space-geometries do anyon models correspond to? \nHow can time be included into the theory?\n\nI  believe that this connection anticipates a beautiful duality between anyon models and gravity, which I feel compelled to unveil.\n\n\n\\newpage\n\\thispagestyle{empty}\n\n\\vspace*{4cm}\n\nAcknowledgment of DFG funding:\n\nThis work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 277974659 via Research Unit FOR 2414.\n\n\n\\newpage\n\\thispagestyle{empty}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{References}\n\\cfoot{\\thepage}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig1.pdf}\n\\caption{Comparison of (a) One-Anatomy-One-Network, (b) Multiple-Anatomy-One-Network, and (c) Our Proposed Multiple-Anatomy-Parameterized-Network paradigm.}\n\\label{fig_intro}\n\\end{figure*}\n\n\\IEEEPARstart{M}{agnetic} Resonance Images (MRIs), which can provide detailed anatomical information without invasion, are widely used for various image-guided therapies. As a typical compressed-sensing \\cite{donoho2006compressed} based inverse problem, reconstructing high-quality MRIs from undersampled k-space signals plays an important role in accelerating the imaging process \\cite{liang2009accelerating,huang2011efficient}. Recently, deep convolutional neural networks (CNNs) have proven soaring successes in efficiently reconstructing high-quality de-aliased anatomy MRIs \\cite{liang2020deep,knoll2020deep} with their stronger ability of automatically learning high-level anatomical representation, wherein most efforts \\cite{wang2016accelerating,schlemper2017deep,qin2018convolutional,hammernik2018learning,aggarwal2018modl,lee2018deep,sun2018compressed,zheng2019cascaded,yan2020neural,sriram2020end,jun2021joint} follow the one-anatomy-one-network (OAON) fashion, i.e., each expert network is trained and evaluated for specific anatomy, as shown in Fig. \\ref{fig_intro} (a). However, training multiple independent networks for different anatomies greatly increases the deployment difficulty since tens of anatomies are examined with MRIs in clinical practice. Moreover, apart from inefficiency in deployment, such convention ignores the shared de-aliasing knowledge across various anatomies, which can benefit each other for reconstruction. \n\nTherefore, it is natural to ask: \\textbf{whether a deep network can simultaneously learn from mixed anatomy data for all-round MRI reconstruction?} Intuitively, one naive solution is to adopt a multiple-anatomy-one-network (MAON) fashion that directly introduces all the data from various anatomies to train an all-round network, as shown in Fig. \\ref{fig_intro} (b). Unfortunately, despite the existence of the shared de-aliasing knowledge and better generalization ability, we reveal that the exclusive knowledge across different anatomies (e.g., unique intensity distributions and structure\/location priors) can mislead the network and deteriorate specific reconstruction targets, yielding overall decreased reconstruction quality. \n\nMost relevantly, Liu et al. \\cite{liu2021universal} introduced a universal undersampled MRI reconstruction network trained by distilling knowledge from pre-trained expert models into one universal network and the anatomy-specific instance normalization was proposed to adapt the various intensity distributions for multiple anatomies. Given the knowledge distilling as a  complicated process where pre-trained expert models are still required, here we lean to disentangle knowledge via different parameterized learners. In terms of the anatomy-specific normalization, we make a step further than \\cite{liu2021universal} to use the parameterized learners for capturing anatomy-specific information with a novel unified framework, where anatomy-specific normalization can be regarded as one of the possible implementations.\n\nIn this study, we propose to seek common ground while reserving differences across different anatomies by redesigning a novel deep MRI reconstruction framework constructed with anatomy-shared and anatomy-specific parameterized learners, named as multiple-anatomy-parameterized-network (MAPN) framework shown in Fig. \\ref{fig_intro} (c). Particularly, such design echos the multi-domain visual classification learning \\cite{rebuffi17,rebuffi18} and allows the MRI reconstruction of each anatomy to enjoy both (i) the flourishing shared de-aliasing knowledge by training the primary anatomy-shared learners with mixed anatomies, and (ii) the exclusive knowledge reserved for it in the tailored anatomy-specific learners. In fact, the OAON and MAON fashion are the special cases of our MAPN where there exist either anatomy-specific or anatomy-shared learners. Under our proposed unified framework, we explore four types of anatomy-specific learners on top of our framework, including anatomy-specific normalization \\cite{liu2021universal}, channel attention, series learners and paralleled learners, and integrated them into two different MRI reconstruction networks. Comprehensive experiments on brain, knee and cardiac MRI datasets demonstrate the effectiveness of our new ``all-round\" network with various insightful findings. The contributions of this work can be summarized as follows:\n\n\\begin{itemize}\n    \\item We unify current one-anatomy-one-network and multiple-anatomy-one-network solutions under a novel multiple-anatomy-parameterized-network framework where we can both seek common ground and reserve differences across anatomies.\n    \\item We implement four possible designs of the compact anatomy-specific parameterized learners as the early attempts under the proposed framework and three of them show the ability to enhance multiple anatomy collaborative learning with two different reconstruction networks, providing insightful findings on how to better capture anatomy-specific information with our framework.\n    \\item Extensive experiments are performed on brain, knee and cardiac MRI datasets demonstrating the effectiveness of our strategy for improving different anatomy reconstruction quality with different reconstruction networks.\n\\end{itemize}\n\n\n\\section{Related Work}\n\n\\subsection{Deep Learning Based Undersampled MRI Reconstruction}\nSince Wang et al.\\cite{wang2016accelerating} firstly introduced deep networks for undersampled MRI reconstruction, researchers have developed various approaches which can be roughly categorized into the data-driven \\cite{wang2016accelerating,fastMRI,zhu2018image,yang2017dagan,lee2018deep}, the model-driven \\cite{hammernik2018learning,aggarwal2018modl,akccakaya2019scan,schlemper2017deep,qin2018convolutional,jun2021joint,zhang2021dual}. Among these works, DCCNN \\cite{schlemper2017deep} has gained much attention with a series of variants \\cite{sun2018compressed,zheng2019cascaded,yan2020neural,sriram2020end,jun2021joint} where researchers explored how to design neural networks for better representation learning. However, these methods all followed the OAON fashion as mentioned above, and there only existed a few previous works discussing the anatomy generalization ability in MRI reconstruction. Yan et al. \\cite{yan2020neural} used neural architecture search technology to search different network architectures for knee and brain, proving that different anatomies prefer different architectures. Transfer learning in MRI reconstruction is discussed in \\cite{ARSHAD202196, LV2021104504,dar2020transfer}, where researchers pre-trained a network on a large pool of data and then transferred the learned weights for one anatomy reconstruction by fine-tuning, while our aim is to learn and reconstruct multiple anatomies simultaneously in one network. Guo et al. \\cite{guo2021multi} introduced the distribution shifting problem of MRIs from different institutions and proposed a federated learning-based solution but only for brain MRI reconstruction tasks. Few attempts were made to explore how different anatomies can be reconstructed by one network. \n\nAs mentioned above, Liu et al. \\cite{liu2021universal} made the first step to introduce a universal undersampled MRI reconstruction network with knowledge distilling and the anatomy-specific instance normalization strategy. In our framework, we lean to disentangle knowledge via different parameterized learners without the need of difficult knowledge distilling process. Since the idea of anatomy-specific instance normalization can be regarded as a setting of the anatomy-specific learners, we re-implemented \\cite{liu2021universal} as the anatomy-specific normalization settings without knowledge distilling to make a fair comparison with other settings in our experiments. \n\n\\subsection{Multi-domain Visual Classification Learning}\n\nOur method is closely related to recent works in multi-domain visual classification learning \\cite{bilen2017universal,rebuffi17,rebuffi18,mallya2018piggyback}, where researchers aim to train a single network to perform image classification tasks across different domains (animals, flowers, digits, ...). Bilen and Vedaldi \\cite{bilen2017universal} used parameters in batch and instance normalization layers to model different domains for task-specific classification. Rebuffi et al. extended \\cite{bilen2017universal} in \\cite{rebuffi17,rebuffi18} to parameterize the standard residual network architecture into a network family to capture different domains. Mallya et al. \\cite{mallya2018piggyback} proposed to adapt a single network to different classification tasks by masking weights of different convolution kernels. \n\nDistinct from the above works, our aim is to make MRI reconstruction model jointly and better learn from multiple anatomies data and we make the early attempt to introduce the idea in \\cite{rebuffi17,rebuffi18} for building an all-round MRI reconstruction model with a parameterized network where the compact anatomy-specific learners focus on exclusive anatomy prior while the major anatomy-shared learners enjoy common de-aliasing knowledge. Our experiments demonstrate that such setting can help different anatomies collaborate with each other in the simultaneous learning process.\n\n\\section{Methodology}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig2.pdf}\n\\caption{Illustration of (a) Anatomy parameterization reconstruction framework for multiple anatomy collaborative learning. Our framework can be generalized to both data-driven and model-driven models. (b) Details of four anatomy parameterization settings. (c) Notes on shading boxes. (d) Anatomy-specific parameterization example with paralleled learners setting (d-i) and its anatomy-shared variant (d-ii) for the ablation study.}\n\\label{fig_framework}\n\\end{figure*}\n\nWithout loss of generalization ability, we study multiple anatomy undersampled MRI reconstruction problems under single-coil reconstruction scenario following \\cite{sun2018compressed,zheng2019cascaded,liu2021universal}. We firstly introduce the problem formulation and two baseline solutions: the data-driven U-Net \\cite{ronneberger2015u} based reconstruction model \\cite{fastMRI,yang2017dagan,lee2018deep} and the model-driven deep cascaded network (DCCNN) based reconstruction model \\cite{schlemper2017deep,sun2018compressed,zheng2019cascaded,yan2020neural,liu2021universal}. Then, we illustrate the details of the proposed MAPN solution with four anatomy-specific learner designs. The overall framework is shown in Fig. \\ref{fig_framework}.\n\n\\subsection{Problem Formulation and Baseline Solutions}\n\nThe aim of MRI reconstruction is to recover the fully-sampled MR image $x$ from the under-sampled k-space signals $s$, which can be formulated as:\n\\begin{equation}\n\\label{equ:1}\ns = F_{u} x,\n\\end{equation}\nwhere $F_{u}$ is the under-sampling Fourier encoding matrix and $x$ can be obtained by solving the following inverse problem:\n\\begin{equation}\n\\label{equ:2}\n\\min_x \\lVert s - F_{u} x\\rVert_2^2 + \\mathcal{R}(x),\n\\end{equation}\nwhere $\\mathcal{R}(x)$ is the regularization term. By integrating CNN-derived constraints into Eq.~\\ref{equ:2}, this problem can be formulated as:\n\\begin{equation}\n\\label{equ:3}\n\\min_x  \\lVert s - F_{u} x\\rVert_2^{2} + \\lambda \\lVert x - \\mathcal{N} \\left( x , \\omega )\\right \\rVert^2,\n\\end{equation}\nwhere $\\mathcal{N}$ represents the deep CNN network with learn-able weights $\\omega$.\n\nFor data-driven solutions, researchers usually solve Eq.~\\ref{equ:3} by directly learning a mapping between between paired under-sampled MRI $x_u$ and fully-sampled MRI $x$. Typically, U-Net \\cite{ronneberger2015u} is adapted in \\cite{fastMRI,yang2017dagan,lee2018deep} as $\\mathcal{N}$ and we have: \n\\begin{equation}\n\\label{equ:u}\ny = \\mathcal{N}( x_u , \\omega).\n\\end{equation}\nThen, $\\mathcal{N}$ is trained to reconstruction $x$ from $x_u$ by minimizing the objective function $\\mathcal{L}$ as: \n\\begin{equation}\n\\mathop{{\\min}}\\limits_{{\\omega}} \\mathcal{L} (\\mathcal{N}(x_{u}, \\omega)-x),\n\\end{equation}\nwhere $\\mathcal{L}$ is the pixel-wise loss, e.g., L1 or L2 loss.\n\nFor model-driven solutions, researchers tend to regard $\\lVert s - F_{u} x\\rVert_2^2$ as the data consistence (DC) term between the image and k-space data, then unroll Eq.~\\ref{equ:3} with the alternating minimization steps \\cite{schlemper2017deep,sun2018compressed,zheng2019cascaded,yan2020neural,liu2021universal}. Schlemper et al. \\cite{schlemper2017deep} firstly proposed to use the data consistency (DC) modules and re-formulated Eq.~\\ref{equ:3} by cascading $n$ sub-networks as the following two steps:\n\\begin{equation}\n\\label{equ:k}\ny^{(n)} = \\mathcal{N}^{(n)}( x^{(n)} , \\omega^{(n)} ),\n\\end{equation}\n\\begin{equation}\n\\label{equ:m}\nx^{(n+1)} = \\arg\\min_x  \\lambda \\lVert s_u - F_{u} x\\rVert_2^{2} + \\lVert x - y^{(n)}\\rVert^2,\n\\end{equation}\nwhere $x^{(0)}=x_u$. The problem of Eq. \\ref{equ:m} is given with a closed-form solution with the DC layer, which can be regarded as a plug-in module in the deep network to align current reconstructed k-space of $y^{(n)}$ with the ground truth sampled signals in $s$, given $\\lambda \\rightarrow \\infty$. The problem of Eq. \\ref{equ:k} is similar to Eq. \\ref{equ:u}. A super network $\\mathcal{N}$ is built with $n$-fold cascaded sub-CNNs $\\mathcal{N}^{(n)}$ and DC modules, and trained by minimizing the objective function $\\mathcal{L}$ to solve Eq. \\ref{equ:k}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig3.pdf}\n\\caption{Illustration of Anatomy Parameterization based U-Net \\cite{fastMRI} (a) and DCCNN \\cite{schlemper2017deep} (b) for multiple anatomy collaborative learning.}\n\\label{fig_unet}\n\\end{figure*}\n\nFor better understanding, we visualize U-Net based reconstruction network in Fig.\\ref{fig_unet} (a), where we follow the setting in \\cite{fastMRI} with 4 down\/up-sampling operations, and DCCNN based reconstruction network in Fig.\\ref{fig_unet} (b) with our proposed anatomy parameterization strategy. At a higher level, we can regard the U-Net based reconstruction solution as a sub-problem of the DCCNN based one, and U-Net \\cite{ronneberger2015u} can be directly used as the sub-CNNs $\\mathcal{N}^{(n)}$ in DCCNN. Here, we independently treat these two solutions to demonstrate the generalization ability of our anatomy parameterization framework. What's more, in original DCCNN \\cite{schlemper2017deep}, the researchers only use plain convolutional layers to build their network. Comparing to \\cite{schlemper2017deep}, the U-Net includes 4 down-sampling blocks with convolution layers and 4 up-sampling blocks with de-convolution layers. For simplicity and to keep consistent with DCCNN, we replace all the convolution layers in U-Net with anatomy-shared\/specific learners and leave the de-convolution layers as anatomy-shared learners.\n\n\n \nNotably, most existed works follow the OAON setting, where one independent $\\mathcal{N}$ is trained from scratch and only tested for one target reconstruction anatomy. In this work, we benchmark above baseline solutions with multiple anatomy reconstruction task.\n\n\n\\subsection{Multiple Anatomy Parameterized Framework}\nRevisiting Eq. \\ref{equ:u} and Eq. \\ref{equ:k} for multiple-anatomy reconstruction scenario, it is a natural idea to disentangle $\\omega$ with the whole network $\\mathcal{N}$ for the shared and exclusive knowledge via rewriting Eq. \\ref{equ:u} and Eq. \\ref{equ:k} as:\n\\begin{equation}\n\\label{equ:f}\ny = \\mathcal{N}( x_u, \\omega_s, \\omega_{e^i}),\n\\end{equation}\nwhere $\\omega_s$ is trained with all anatomies to learn shared de-aliasing knowledge and $\\omega_{e^i}$ is from a parametric set $\\Omega_e=\\{\\omega_{e^1},\\omega_{e^2},...,\\omega_{e^n}\\}$ in which each element is corresponding to a specific anatomy. Examples of detailed implementations of $\\omega_s$ and $\\Omega_e$ are visualized in Fig. \\ref{fig_framework} (d) for better understanding, termed as anatomy-shared\/specific parameterization. With Eq. \\ref{equ:f} as the basis of our MAPN framework, we then need to answer: \\textbf{how to properly design $\\Omega_e$?}\n\nThe design of $\\Omega_e$ needs to consider the following two criteria: (1) $\\omega_{e^i}$ should be more efficient compared to $\\omega_s$, otherwise it brings in more learning parameters; (2) the $\\omega_{e^i}$ should be easy to plug into and out from the network with $\\omega_s$ such that they can be easily switched for different organs and enjoy the shared knowledge. \n\nWith the above two criteria, when we further investigate the convolution blocks shown in Fig. \\ref{fig_framework}\n(b) PN0, we can find that an intuitive setting is to use the $3\\times3$ convolutional layers as $\\omega_{s}$, while the common Batch Normalization (BN) \\cite{ioffe2015batch} layers can be regarded as an anatomy-specific setting of $\\omega_{e^i}$ shown in Fig. \\ref{fig_framework}\n(b) PN1 noted by Anatomy-specific Normalization strategy. It is obvious that MRIs of different anatomies have significant feature distribution shifting in terms of means and variances. Given the input feature map as $m$, the normalization layers utilized to calibrate the feature map can be formulated as:\n\\begin{equation}\nn = \\gamma \\cdot \\frac{ m - \\mu(m)}{\\sigma(m)} + \\beta,\n\\end{equation}\nwhere scale weight $\\gamma$, shift bias $\\beta$ in BN layers. To make BN layers better adapt to one specific anatomy with its unique intensity distribution, one independent set of normalization layers is assigned with unique learning parameters $[\\gamma, \\beta]$ by an anatomy switch operation in the network for each anatomy, as indicting in Fig. \\ref{fig_framework} (d-i). It is reminded that the anatomy-specific instance normalization used in \\cite{liu2021universal} is similar to PN1 setting with $[\\mu(m), \\sigma(m)]$ calculated differently and a complicated knowledge distillation step. With a fair comparison protocol comparing to following other settings, we think PN1 is our enhanced re-implemented version of \\cite{liu2021universal}, given codes\/models of \\cite{liu2021universal} are not open source. \n\nWith Anatomy-specific Normalization as a good starting point, another direction to design $\\Omega_{e}$  is to plug some additive compact tailored anatomy-specific structure into the convolution block for exclusive knowledge learning. We propose to use the channel attention mechanism, which was introduced in Squeeze-and-Excited (SE) Network \\cite{hu2018squeeze}, to model anatomy knowledge by re-calibrating channel-wise feature responses with fully-connected layers shown in Fig. \\ref{fig_framework} (b) PN2 noted by Anatomy-specific Channel Attention strategy. \n\nSo far, the previous two solutions do not change the major graphic structure of the network. In order to leverage features from different levels and inspired by successes in multi-domain natural image classification \\cite{rebuffi17,rebuffi18}, we use $1\\times1$ convolutional layers as residual connections to adjust features from different levels and then obtain the setting in Fig. \\ref{fig_framework} (b) PN3 and PN4, namely Anatomy-specific Series and Parallel Learners, correspondingly, according to where the learners locate. The $1\\times1$ convolutional operations are chosen due to parameter efficiency. \n\nThe above four settings are explored as possible implementations under our proposed framework to prove its effectiveness and how to design the optimal anatomy-specific learners remains an open problem. We explore these implementations to shed the light in a new direction for multiple anatomy collaborative reconstruction. \n\n\n\n\\section{Experiments}\n\\label{sec:exp}\n\\subsection{Implementation Details}\n\\subsubsection{Environment} All the experiments were carried out on an NVIDIA GeForce RTX 3090 GPU with the Pytorch.\n\\subsubsection{Dataset Preprocessing and Evaluation Metrics} We conducted our experiments on three public MRI datasets of different anatomies with Institutional Review Board approval: \\textbf{Knee}: we randomly sampled 800 knee MR slices from 25 subjects in the fastMRI dataset \\cite{fastMRI} provided by New York University School of Medicine. \\textbf{Brain}: we randomly sampled 800 brain MR slices from 50 subjects in the fastMRI dataset \\cite{fastMRI}. \\textbf{Cardiac:} we randomly sampled 800 cardiac MR slices from 90 cardiac subjects in the ACDC dataset \\cite{bernard2018deep} obtained from the University Hospital of Dijon. Noteworthily, we removed some slices where no biological structure exists to make three anatomies have equal slices for fairness. It is reminded that the above three datasets are acquired with different \nprotocols, and we follow the setting in \\cite{liu2021universal} to treat them as the same single-coil slice-level MR image for reconstruction.\nFollowing \\cite{sun2018compressed,zheng2019cascaded,liu2021universal}, we adopted these data pre-processing steps: the MRIs were cropped into $320 \\times 320$, then normalized in magnitude, and clipped to [-6, 6] with zero phases to simulate 2-channel complex-valued images. \nAll the models were trained with under-sampled 1D Cartesian masks and evaluated with  $4\\times$ and $6\\times$ under-sampled reconstruction tasks.\nWe report PSNR and SSIM \\cite{hore2010image} between reconstructed and target fully-sampling MRIs as the quantitative metrics. The final results are calculated on a 5-fold cross-validation setting by dividing the above three datasets according to subjects with a ratio of 4:1 for the training and validation.\n\\subsubsection{Network Setting} To implement DCCNN reconstruction network, we used 5 stacked convolutional blocks to build each sub-CNN and cascaded the sub-CNNs and DC modules for 5 times to build the baseline DCCNN. Each block includes cascading Conv.-BN-LeakyReLU operations, as shown in Fig. \\ref{fig_framework} (b) PN0, with 64 channels. We followed the setting in \\cite{fastMRI} to build a U-Net based reconstruction network but modified the input and output as 2 channels to keep consistent with the DCCNN. To perform anatomy parameterization settings, we replaced all the convolutional operations, i.e. PN0, in U-Net\/DCCNN by PN1-4, as shown in Fig. \\ref{fig_framework} (b).\n\n\\subsubsection{Training Strategies} To implement the OAON baselines, we trained U-Net\/DCCNN based reconstruction models independently on three datasets for 50 epochs. \nTo implement the multiple-anatomy-related baseline, we fed a batch of data from the same anatomy in a round-robin fashion in each iteration for $3\\times50=150$ epochs as we used 3 anatomies in this study.\nFor the MAPN methods, the anatomy-specific learners were optimized only once in each iteration for their target anatomy, while the anatomy-shared $3\\times3$ convolution layers were optimized in every mini-batch across anatomies. We also adopted a warm-up strategy for anatomy parameterized learners to dig out exclusive knowledge: we loaded and fixed the weight of $3\\times3$ convolution layers obtained from the MAON baseline for the first 30 epochs to better initialize the additive anatomy-specific modules, then the whole network was fully optimized for the next 120 epochs. The L1 loss was adopted for supervision and the Adam optimizer with the same learning rate of 0.01 was adopted for all the methods.\n\n\\subsection{Experiments of U-Net based Reconstruction} \n\nHere, we report the reconstruction performance of the anatomy parameterization based U-Net as shown in Fig. \\ref{fig_unet} (a). \n\n\\subsubsection{Quantitative Results} We summarized the quantitative results in Table. \\ref{tab_3}, which can be divided into three parts under $4\\times$ and $6\\times$ under-sampling ratio: (i) OAON setting where knee\/brain\/cardiac expert models are trained independently, (ii) MAON setting where all-round models are trained on the mixture of three datasets, (iii) four different MAPN settings. Though the overall performance with $6\\times$ under-sampling ratio is lower than that with $4\\times$ ratio, findings are quite consistent between them, suggesting the robustness of proposed method. Intuitively, we can find that MRI reconstruction results can not benefit from direct multiple anatomy data mixed training. Expert for brain reconstruction with $4\\times$ ratio, the knee\/brain\/cardiac expert models all outperform the MAON model although with unsatisfied anatomy generalization ability. Our proposed MAPN solutions can learn from different anatomy to better reconstruct MRIs compared to the naive MAON solution. By integrating anatomy-specific batch normalization (PN1), we can find that the network can enjoy the shared knowledge learned in $3\\times3$ convolutional layers to outperform the MAON baseline in all three tasks but can not provide better results than OAON experts in terms of the cardiac reconstructions. After integrating anatomy-specific learners with PN2-4 settings, the MAPN network can reconstruct all three anatomies better than both OAON expert models and MAON baselines from collaborative learning. Notably, the PN2 setting achieves comparable results with PN4 settings in U-Net based reconstruction.\n\n\\subsubsection{Qualitative Results} We also visualize reconstruction performance of all settings in U-Net with $4\\times$ ratio in Fig. \\ref{fig_comparison_u}. Similar to the qualitative results, we can find that networks with our parameterized learners can reconstruct images with clearer details and reduced error maps.\n\n\\begin{table*}\n\\caption{Quantitative reconstruction results of U-Net based reconstruction on 5-fold cross validation of different anatomy parameterization strategies with the metric of PSNR(dB) and SSIM(\\%). `Und. Rat.' represents Under-sampling Ratio. `K',`B', and `C' are short for `Knee', `Brain' and `Cardiac' at the 3rd column for compactness. `OAON' is short for `One-Anatomy-One-Network'. `MAON' is short for `Multiple-Anatomy-One-Network'. `MAPN' is short for `Multiple-Anatomy-Parameterized-Network'. The best results are addressed in bold.}\n\\label{tab_3}\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{c|c|c|c|ccc|ccc|ccc}\n\\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Und.\\\\ Rat.\\end{tabular}}&\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Learning\\\\ Fashion\\end{tabular}} & \\multirow{2}{*}{Model} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Training\\\\ Anatomy\\end{tabular}} & \\multicolumn{3}{c|}{Knee Reconstruction}                                             & \\multicolumn{3}{c|}{Brain  Reconstruction}                                            & \\multicolumn{3}{c}{Cardiac  Reconstruction}                                          \\\\ \\cline{5-13} \n                   &                 &                        &                          & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        \\\\ \\hline \\hline\n\\multirow{9}{*}{4x}&\\multirow{3}{*}{OAON}                                                       & U-Net                 & K                        & \\multicolumn{1}{c|}{34.82} & \\multicolumn{1}{c|}{82.09} & \\textcolor{red}{0.51\/2.24}   & \\multicolumn{1}{c|}{34.45} & \\multicolumn{1}{c|}{87.24} & \\textcolor{blue}{-2.11\/-3.92} & \\multicolumn{1}{c|}{34.99} & \\multicolumn{1}{c|}{85.74} & \\textcolor{blue}{-1.56\/-3.90} \\\\  \\cline{3-13} \n                                            &                              & U-Net                  & B                        & \\multicolumn{1}{c|}{31.12} & \\multicolumn{1}{c|}{70.22} & \\textcolor{blue}{-3.19\/-9.63} & \\multicolumn{1}{c|}{36.50} & \\multicolumn{1}{c|}{90.83} & \\textcolor{blue}{-0.06\/-0.33}   & \\multicolumn{1}{c|}{33.57} & \\multicolumn{1}{c|}{83.85} & \\textcolor{blue}{-2.98\/-5.79} \\\\ \\cline{3-13} \n                                                    &                        & U-Net                  & C                        & \\multicolumn{1}{c|}{33.32} & \\multicolumn{1}{c|}{75.64} & \\textcolor{blue}{-0.99\/-4.21} & \\multicolumn{1}{c|}{35.54} & \\multicolumn{1}{c|}{89.36} & \\textcolor{blue}{-1.02\/-1.80} & \\multicolumn{1}{c|}{37.17} & \\multicolumn{1}{c|}{90.74} & \\textcolor{red}{0.62\/1.10}   \\\\ \\cline{2-13} \n&MAON                                                       & U-Net & K+B+C               & \\multicolumn{1}{c|}{34.31} & \\multicolumn{1}{c|}{79.85} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{36.56} & \\multicolumn{1}{c|}{91.16} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{36.55} & \\multicolumn{1}{c|}{89.64} & \\multicolumn{1}{c}{0\/0}  \\\\ \\cline{2-13} \n&\\multirow{4}{*}{MAPN}                                                       & U-Net+PN1 \\cite{liu2021universal}                   & K+B+C                    & \\multicolumn{1}{c|}{34.86} & \\multicolumn{1}{c|}{81.91} & \\textcolor{red}{0.55\/2.06}   & \\multicolumn{1}{c|}{37.00} & \\multicolumn{1}{c|}{91.50} & \\textcolor{red}{0.44\/0.34}   & \\multicolumn{1}{c|}{37.18} & \\multicolumn{1}{c|}{90.67} & \\textcolor{red}{0.63\/1.03}  \\\\ \\cline{3-13} \n                                          &                                  & U-Net+PN2                    & K+B+C                    & \\multicolumn{1}{c|}{35.30} & \\multicolumn{1}{c|}{\\textbf{83.18}} & \\textcolor{red}{0.99}\/\\textcolor{red}{\\textbf{3.33}}  & \\multicolumn{1}{c|}{37.59} & \\multicolumn{1}{c|}{\\textbf{92.65}} & \\textcolor{red}{1.03\/\\textbf{1.49}} & \\multicolumn{1}{c|}{37.29} & \\multicolumn{1}{c|}{90.83} & \\textcolor{red}{0.74\/1.19}  \\\\ \\cline{3-13} \n                                        &                                    & U-Net+PN3                    & K+B+C                    & \\multicolumn{1}{c|}{35.22} & \\multicolumn{1}{c|}{82.77} & \\textcolor{red}{0.91\/2.92}   & \\multicolumn{1}{c|}{37.37} & \\multicolumn{1}{c|}{91.93} & \\textcolor{red}{0.81\/0.77}   & \\multicolumn{1}{c|}{37.28} & \\multicolumn{1}{c|}{90.83} & \\textcolor{red}{0.73\/1.19}   \\\\ \\cline{3-13} \n                                        &                                    & U-Net+PN4                    & K+B+C                    & \\multicolumn{1}{c|}{\\textbf{35.36}} & \\multicolumn{1}{c|}{83.15} & \\textcolor{red}{\\textbf{1.05}\/3.30}   & \\multicolumn{1}{c|}{\\textbf{37.66}} & \\multicolumn{1}{c|}{92.10} & \\textcolor{red}{\\textbf{1.10}\/0.94}   & \\multicolumn{1}{c|}{\\textbf{37.30}} & \\multicolumn{1}{c|}{\\textbf{90.86}} & \\textcolor{red}{\\textbf{0.75}\/\\textbf{1.22}}   \\\\ \\hline \\hline\n\n\\multirow{9}{*}{6x}&\\multirow{3}{*}{OAON}                                                       & U-Net                 & K                        & \\multicolumn{1}{c|}{33.58} & \\multicolumn{1}{c|}{77.65} & \\textcolor{red}{0.27\/1.32}   & \\multicolumn{1}{c|}{32.92} & \\multicolumn{1}{c|}{84.21} & \\textcolor{blue}{-1.25\/-3.34} & \\multicolumn{1}{c|}{33.70} & \\multicolumn{1}{c|}{83.24} & \\textcolor{blue}{-0.89\/-2.96} \\\\  \\cline{3-13} \n                                            &                              & U-Net                  & B                        & \\multicolumn{1}{c|}{30.70} & \\multicolumn{1}{c|}{68.15} & \\textcolor{blue}{-2.61\/-8.18} & \\multicolumn{1}{c|}{34.29} & \\multicolumn{1}{c|}{87.83} & \\textcolor{red}{0.12\/0.28}   & \\multicolumn{1}{c|}{32.67} & \\multicolumn{1}{c|}{81.76} & \\textcolor{blue}{-1.92\/-4.44} \\\\ \\cline{3-13} \n                                                    &                        & U-Net                  & C                        & \\multicolumn{1}{c|}{32.90} & \\multicolumn{1}{c|}{74.02} & \\textcolor{blue}{-0.41\/-2.31} & \\multicolumn{1}{c|}{34.04} & \\multicolumn{1}{c|}{87.11} & \\textcolor{blue}{-0.13\/-0.44} & \\multicolumn{1}{c|}{35.46} & \\multicolumn{1}{c|}{88.11} & \\textcolor{red}{0.87\/1.91}   \\\\ \\cline{2-13}\n  \n&MAON                                                       & U-Net & K+B+C               & \\multicolumn{1}{c|}{33.31} & \\multicolumn{1}{c|}{76.33} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{34.17} & \\multicolumn{1}{c|}{87.55} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{34.59} & \\multicolumn{1}{c|}{86.20} & \\multicolumn{1}{c}{0\/0}  \\\\ \\cline{2-13} \n\n&\\multirow{4}{*}{MAPN}                                                       & U-Net+PN1 \\cite{liu2021universal}                    & K+B+C                    & \\multicolumn{1}{c|}{33.66} & \\multicolumn{1}{c|}{77.76} & \\textcolor{red}{0.35\/1.43}   & \\multicolumn{1}{c|}{34.78} & \\multicolumn{1}{c|}{88.66} & \\textcolor{red}{0.61\/1.11}   & \\multicolumn{1}{c|}{35.25} & \\multicolumn{1}{c|}{87.71} & \\textcolor{red}{0.66\/1.51}  \\\\ \\cline{3-13} \n                                          &                                  & U-Net+PN2                    & K+B+C                    & \\multicolumn{1}{c|}{\\textbf{34.25}} & \\multicolumn{1}{c|}{\\textbf{79.30}} & \\textbf{\\textcolor{red}{0.94\/2.97}}  & \\multicolumn{1}{c|}{\\textbf{35.86}} & \\multicolumn{1}{c|}{\\textbf{90.44}} & \\textbf{\\textcolor{red}{1.69\/2.89}} & \\multicolumn{1}{c|}{\\textbf{35.80}} & \\multicolumn{1}{c|}{\\textbf{88.58}} & \\textbf{\\textcolor{red}{1.21\/2.38}}  \\\\ \\cline{3-13} \n                                        &                                    & U-Net+PN3                    & K+B+C                    & \\multicolumn{1}{c|}{33.93} & \\multicolumn{1}{c|}{78.47} & \\textcolor{red}{0.62\/2.14}   & \\multicolumn{1}{c|}{35.08} & \\multicolumn{1}{c|}{89.17} & \\textcolor{red}{0.91\/1.62}   & \\multicolumn{1}{c|}{35.53} & \\multicolumn{1}{c|}{88.21} & \\textcolor{red}{0.94\/2.01}   \\\\ \\cline{3-13} \n                                        &                                    & U-Net+PN4                    & K+B+C                    & \\multicolumn{1}{c|}{34.12} & \\multicolumn{1}{c|}{79.01} & \\textcolor{red}{0.81\/2.68}   & \\multicolumn{1}{c|}{35.66} & \\multicolumn{1}{c|}{{89.91}} & \\textcolor{red}{1.49\/2.36}   & \\multicolumn{1}{c|}{35.64} & \\multicolumn{1}{c|}{88.40} & \\textcolor{red}{1.05\/2.20}   \\\\\n\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig4.pdf}\n\\caption{Qualitative comparison of different anatomy parameterization U-Net reconstruction settings with $4\\times$ under-sampling ratio. Original MRIs are normalized to [0, 1] for visualization and their intensity distributions are shown in the first column. From the second column, reconstructed image (top) and its absolute error map (bottom) are shown. PSNR (dB) is listed at the right bottom of each error map with the highest one highlighted in red. Values $>0.1$ in error maps are clipped to 0.1 for better visual comparison. Green boxes address where significant performance difference can be observed.}\n\\label{fig_comparison_u}\n\\end{figure*}\n\n\\subsection{Experiments of DCCNN based Reconstruction}\n\nHere, we report the reconstruction performance of the anatomy parameterization based DCCNN as shown in Fig. \\ref{fig_unet} (b). Experiments show that DCCNN's performance is superior to U-Net in MRI reconstruction, which is consistent with previous works \\cite{sun2018compressed,zheng2019cascaded,yan2020neural}. Therefore, we perform more ablation experiments with anatomy parameterization based DCCNN.\n\n\\subsubsection{Quantitative Results} The quantitative results are summarized in Table. \\ref{tab_1} with the same structure as Table. \\ref{tab_3}. Similar to the results of U-Net based reconstruction, we can conclude that networks trained with MAON setting are confused by different anatomies and can not reconstruct MRI as better as the corresponding OAON experts. Our proposed MAPN solutions can learn from different anatomy to better reconstruct MRIs compared to the naive MAON solution or the MAON model with more learning parameters (DCCNN+PN4) - will be discussed later in the following ablation study. Most differently, anatomy-specific channel attention learners (PN2) fail to dig out useful information in most cases in DCCNN based reconstruction. We assume that this is due to the different network architectures between DCCNN and U-Net. Since U-Net's encoder will double the size of feature map after pooling while DCCNN's feature map size is kept constant, the anatomy-specific channel attention may have a better chance in capturing useful information and thus generates better results. The optimal reconstruction performance is obtained by anatomy-specific parallel learners (PN4), which also performs robustly in U-Net based reconstruction results.\n\n\\begin{table*}\n\\caption{Quantitative reconstruction results of DCCNN based reconstruction on 5-fold cross validation of different anatomy parameterization strategies with the metric of PSNR(dB) and SSIM(\\%). `Und. Rat.' represents Under-sampling Ratio. `K',`B', and `C' are short for `Knee', `Brain' and `Cardiac' at the 3rd column for compactness. `OAON' is short for `One-Anatomy-One-Network'. `MAON' is short for `Multiple-Anatomy-One-Network'. `MAPN' is short for `Multiple-Anatomy-Parameterized-Network'. The best results are addressed in bold.}\n\\label{tab_1}\n\\centering\n\\scalebox{0.9}{\n\\begin{tabular}{c|c|c|c|ccc|ccc|ccc}\n\\hline\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Und.\\\\ Rat.\\end{tabular}}&\\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Learning\\\\ Fashion\\end{tabular}} & \\multirow{2}{*}{Model} & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Training\\\\ Anatomy\\end{tabular}} & \\multicolumn{3}{c|}{Knee Reconstruction}                                             & \\multicolumn{3}{c|}{Brain  Reconstruction}                                            & \\multicolumn{3}{c}{Cardiac  Reconstruction}                                          \\\\ \\cline{5-13} \n                   &                 &                        &                          & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        & \\multicolumn{1}{c|}{PSNR}  & \\multicolumn{1}{c|}{SSIM}  & $\\Delta$ PSNR\/SSIM        \\\\ \\hline \\hline\n\\multirow{9}{*}{4x}&\\multirow{3}{*}{OAON}                                                       & DCCNN                 & K                        & \\multicolumn{1}{c|}{36.08} & \\multicolumn{1}{c|}{84.76} & \\textcolor{red}{0.29\/1.05}   & \\multicolumn{1}{c|}{36.54} & \\multicolumn{1}{c|}{90.26} & \\textcolor{blue}{-2.14\/-2.68} & \\multicolumn{1}{c|}{37.73} & \\multicolumn{1}{c|}{90.76} & \\textcolor{blue}{-1.62\/-2.37} \\\\  \\cline{3-13} \n                                            &                              & DCCNN                  & B                        & \\multicolumn{1}{c|}{33.45} & \\multicolumn{1}{c|}{76.14} & \\textcolor{blue}{-2.34\/-7.57} & \\multicolumn{1}{c|}{39.19} & \\multicolumn{1}{c|}{93.38} & \\textcolor{red}{0.51\/0.44}   & \\multicolumn{1}{c|}{36.99} & \\multicolumn{1}{c|}{88.94} & \\textcolor{blue}{-2.36\/-4.19} \\\\ \\cline{3-13} \n                                                    &                        & DCCNN                  & C                        & \\multicolumn{1}{c|}{34.75} & \\multicolumn{1}{c|}{80.47} & \\textcolor{blue}{-1.04\/-3.24} & \\multicolumn{1}{c|}{37.74} & \\multicolumn{1}{c|}{91.79} & \\textcolor{blue}{-0.94\/-1.15} & \\multicolumn{1}{c|}{40.30} & \\multicolumn{1}{c|}{94.25} & \\textcolor{red}{0.95\/1.12}   \\\\ \\cline{2-13} \n&\\multirow{2}{*}{MAON}                                                       & DCCNN & K+B+C               & \\multicolumn{1}{c|}{35.79} & \\multicolumn{1}{c|}{83.71} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{38.68} & \\multicolumn{1}{c|}{92.94} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{39.35} & \\multicolumn{1}{c|}{93.13} & \\multicolumn{1}{c}{0\/0}  \\\\ \\cline{3-13} \n                                         &                                   & DCCNN+PN4 & K+B+C                                                                            & \\multicolumn{1}{c|}{35.49} & \\multicolumn{1}{c|}{82.91} & \\textcolor{blue}{-0.30\/-0.80}               & \\multicolumn{1}{c|}{38.25} & \\multicolumn{1}{c|}{92.61} & \\textcolor{blue}{-0.43\/-0.33}               & \\multicolumn{1}{c|}{38.82} & \\multicolumn{1}{c|}{92.41} & \\textcolor{blue}{-0.53\/-0.72} \\\\ \\cline{2-13} \n\n&\\multirow{4}{*}{MAPN}                                                       & DCCNN+PN1 \\cite{liu2021universal}                   & K+B+C                    & \\multicolumn{1}{c|}{35.95} & \\multicolumn{1}{c|}{84.33} & \\textcolor{red}{0.16\/0.62}   & \\multicolumn{1}{c|}{38.91} & \\multicolumn{1}{c|}{93.19} & \\textcolor{red}{0.23\/0.25}   & \\multicolumn{1}{c|}{39.57} & \\multicolumn{1}{c|}{93.36} & \\textcolor{red}{0.21\/0.23}  \\\\ \\cline{3-13} \n                                          &                                  & DCCNN+PN2                    & K+B+C                    & \\multicolumn{1}{c|}{35.75} & \\multicolumn{1}{c|}{84.14} & \\textcolor{blue}{-0.04}\/\\textcolor{red}{0.43}  & \\multicolumn{1}{c|}{38.39} & \\multicolumn{1}{c|}{92.82} & \\textcolor{blue}{-0.29\/-0.12} & \\multicolumn{1}{c|}{38.92} & \\multicolumn{1}{c|}{92.61} & \\textcolor{blue}{-0.43\/-0.52}  \\\\ \\cline{3-13} \n                                        &                                    & DCCNN+PN3                    & K+B+C                    & \\multicolumn{1}{c|}{36.31} & \\multicolumn{1}{c|}{85.35} & \\textcolor{red}{0.52\/1.64}   & \\multicolumn{1}{c|}{39.86} & \\multicolumn{1}{c|}{93.92} & \\textcolor{red}{1.18\/0.98}   & \\multicolumn{1}{c|}{40.31} & \\multicolumn{1}{c|}{94.25} & \\textcolor{red}{0.96\/1.12}   \\\\ \\cline{3-13} \n                                        &                                    & DCCNN+PN4                    & K+B+C                    & \\multicolumn{1}{c|}{\\textbf{36.50}} & \\multicolumn{1}{c|}{\\textbf{85.82}} & \\textbf{\\textcolor{red}{0.71\/2.11}}   & \\multicolumn{1}{c|}{\\textbf{40.15}} & \\multicolumn{1}{c|}{\\textbf{94.32}} & \\textbf{\\textcolor{red}{1.47\/1.38}}   & \\multicolumn{1}{c|}{\\textbf{40.49}} & \\multicolumn{1}{c|}{\\textbf{94.45}} & \\textbf{\\textcolor{red}{1.14\/1.32}}   \\\\ \\hline \\hline\n\n\\multirow{9}{*}{6x}&\\multirow{3}{*}{OAON}                                                       & DCCNN                 & K                        & \\multicolumn{1}{c|}{34.13} & \\multicolumn{1}{c|}{78.77} & \\textcolor{red}{0.19\/0.78}   & \\multicolumn{1}{c|}{33.78} & \\multicolumn{1}{c|}{85.47} & \\textcolor{blue}{-1.39\/-3.06} & \\multicolumn{1}{c|}{35.02} & \\multicolumn{1}{c|}{86.18} & \\textcolor{blue}{-1.17\/-2.61} \\\\  \\cline{3-13} \n                                            &                              & DCCNN                  & B                        & \\multicolumn{1}{c|}{32.12} & \\multicolumn{1}{c|}{71.49} & \\textcolor{blue}{-1.82\/-6.50} & \\multicolumn{1}{c|}{35.64} & \\multicolumn{1}{c|}{89.34} & \\textcolor{red}{0.47\/0.81}   & \\multicolumn{1}{c|}{34.78} & \\multicolumn{1}{c|}{85.07} & \\textcolor{blue}{-1.41\/-3.72} \\\\ \\cline{3-13} \n                                                    &                        & DCCNN                  & C                        & \\multicolumn{1}{c|}{33.39} & \\multicolumn{1}{c|}{75.95} & \\textcolor{blue}{-0.55\/-2.04} & \\multicolumn{1}{c|}{34.72} & \\multicolumn{1}{c|}{87.61} & \\textcolor{blue}{-0.45\/-0.92} & \\multicolumn{1}{c|}{36.85} & \\multicolumn{1}{c|}{90.05} & \\textcolor{red}{0.66\/1.26}   \\\\ \\cline{2-13}\n  \n&MAON                                                       & DCCNN & K+B+C               & \\multicolumn{1}{c|}{33.94} & \\multicolumn{1}{c|}{77.99} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{35.17} & \\multicolumn{1}{c|}{88.53} & \\multicolumn{1}{c|}{0\/0}  & \\multicolumn{1}{c|}{36.19} & \\multicolumn{1}{c|}{88.79} & \\multicolumn{1}{c}{0\/0}  \\\\ \\cline{2-13} \n\n&\\multirow{4}{*}{MAPN}                                                       & DCCNN+PN1 \\cite{liu2021universal}                    & K+B+C                    & \\multicolumn{1}{c|}{34.05} & \\multicolumn{1}{c|}{78.48} & \\textcolor{red}{0.11\/0.49}   & \\multicolumn{1}{c|}{35.45} & \\multicolumn{1}{c|}{89.07} & \\textcolor{red}{0.28\/0.54}   & \\multicolumn{1}{c|}{36.48} & \\multicolumn{1}{c|}{89.29} & \\textcolor{red}{0.29\/0.50}  \\\\ \\cline{3-13} \n                                          &                                  & DCCNN+PN2                    & K+B+C                    & \\multicolumn{1}{c|}{34.11} & \\multicolumn{1}{c|}{78.71} & \\textcolor{red}{0.17}\/\\textcolor{red}{0.72}  & \\multicolumn{1}{c|}{35.23} & \\multicolumn{1}{c|}{88.83} & \\textcolor{red}{0.06\/0.30} & \\multicolumn{1}{c|}{36.10} & \\multicolumn{1}{c|}{88.66} & \\textcolor{blue}{-0.09\/-0.13}  \\\\ \\cline{3-13} \n                                        &                                    & DCCNN+PN3                    & K+B+C                    & \\multicolumn{1}{c|}{34.28} & \\multicolumn{1}{c|}{79.24} & \\textcolor{red}{0.34\/1.25}   & \\multicolumn{1}{c|}{36.17} & \\multicolumn{1}{c|}{90.21} & \\textcolor{red}{1.00\/1.68}   & \\multicolumn{1}{c|}{36.87} & \\multicolumn{1}{c|}{90.06} & \\textcolor{red}{0.68\/1.27}   \\\\ \\cline{3-13} \n                                        &                                    & DCCNN+PN4                    & K+B+C                    & \\multicolumn{1}{c|}{\\textbf{34.47}} & \\multicolumn{1}{c|}{\\textbf{79.75}} & \\textbf{\\textcolor{red}{0.53\/1.76}}   & \\multicolumn{1}{c|}{\\textbf{36.51}} & \\multicolumn{1}{c|}{\\textbf{90.77}} & \\textbf{\\textcolor{red}{1.34\/2.24}}   & \\multicolumn{1}{c|}{\\textbf{37.07}} & \\multicolumn{1}{c|}{\\textbf{90.39}} & \\textbf{\\textcolor{red}{0.88\/1.60}}   \\\\\n\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{fig5.pdf}\n\\caption{Qualitative comparison of different anatomy parameterization DCCNN reconstruction settings with $4\\times$ under-sampling ratio. Original MRIs are normalized to [0, 1] for visualization and their intensity distributions are shown in the first column. From the second column, reconstructed image (top) and its absolute error map (bottom) are shown. PSNR (dB) is listed at the right bottom of each error map with the highest one highlighted in red. Values $>0.1$ in error maps are clipped to 0.1 for better visual comparison. Green boxes address where significant performance difference can be observed.}\n\\label{fig_comparison_bn}\n\\end{figure*}\n\n\\subsubsection{Qualitative Results} We also visually compare the reconstruction performance of all settings in DCCNN with $4\\times$ ratio in Fig. \\ref{fig_comparison_bn}. We can also find that the visual quality of final reconstructed MRIs are improved with our strategies.\n\n\\begin{figure*}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth]{fig6.pdf}}\n  \\caption{Visualization of learned parameters in the 3rd CNN of DCCNN+PN4 model. The average weight values in BN of 5 blocks are shown in (a), and the average weight values of $1\\times1$ convolutional layers are reported in (b). The weights in additive learning modules are anatomy-sensitive. The weights of BN can re-scale different distributions for different anatomies with simple value shifting. The weights of anatomy-specific $1\\times1$ convolutional layers vary more significantly between each other and help to improve the final results.}\n  \\label{fig_visp}\n\\end{figure*}\n\n\\begin{table}\n\\caption{Comparison of learning parameter number between different anatomy parameterization settings with DCCNN. Note that the parameter sum of MAON+DCCNN+PN4 is not exactly equal to MAPN+DCCNN+PN4 due to anatomy-shared BN settings.}\n\\label{tab_2}\n\\centering\n\\scalebox{0.8}{\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\begin{tabular}[c]{@{}c@{}}Learning\\\\ Fashion\\end{tabular} & Model     & \\begin{tabular}[c]{@{}c@{}}Anatomy-shared \\\\ Parameters (K)\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Anatomy-specific \\\\ Parameters (K) \\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Three Antomy\\\\ Param. Sum (K)\\end{tabular} \\\\ \\hline\nOAON                                                       & DCCNN     & 0.00                                                                 & 569.64                                                                & 1708.92                                                           \\\\ \\hline\n\\multirow{2}{*}{MAON}                                      & DCCNN     & 569.64                                                              & 0.00                                                                   & 569.64                                                           \\\\ \\cline{2-5} \n                                                           & DCCNN+PN4 & 757.80                                                              & 0.00                                                                   & 757.80                                                           \\\\ \\hline\n\\multirow{4}{*}{MAPN}                                      & DCCNN+PN1 \\cite{liu2021universal}  & 564.48                                                              & 5.16                                                                  & 579.96                                                           \\\\ \\cline{2-5} \n                                                           & DCCNN+PN2 & 564.48                                                              & 87.10                                                                 & 825.78                                                           \\\\ \\cline{2-5} \n                                                           & DCCNN+PN3 & 564.48                                                              & 89.68                                                                 & 833.52                                                           \\\\ \\cline{2-5} \n                                                           & DCCNN+PN4 & 564.48                                                              & 67.88                                                                 & 768.12                                                           \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\\subsubsection{Comparison on the Increment of Learning Parameters} There is a remaining question that whether the improvement is brought by the increment of learning parameters. We re-draw the MAPN+PN4 in Fig.\\ref{fig_framework} (d-i) as MAON+PN4 (d-ii), where 3 paralleled $1 \\times 1$ convolutional layers are added, and we trained the DCCNN+PN4 with the MAON setting with the quantitative results reported in Table. \\ref{tab_1} with $4\\times$ ratio. Here, we do not use 3 paralleled BN layers because such setting seems to be non-sense. It can be concluded that MAON+DCCNN+PN4 results in worse reconstruction than MAPN+DCCNN+PN4, suggesting that the performance gain achieved by MAPN+PN4 is not due to more parameters nor the parallel convolution with different receptivity fields. The learning parameter numbers of all the involved anatomy parameterization settings are summarized in Table. \\ref{tab_2}. For a fair comparison, OAON setting needs 3 expert models with $3 \\times 569.64$K = $1708.92$K parameters in total to outperform the MAON baseline with $569.64$K parameters for three anatomy reconstruction tasks. While our MAPN+PN4 models contains $768.12$K parameters, which can save $>50\\%$ parameters with better results than OAON. The MAON+PN4 model has $757.80$K parameters, which are not exactly equal to MAPN+PN4 without 3 paralleled BN in every convolution blocks. \n\n\\subsubsection{Comparison on the Inference Time} We also evaluate the inference time for different methods, and PN1-4 show little difference from the original DCCNN due to PN1-4 are parameter-efficient. It takes $<0.6s$ to reconstruct per slice for all the methods with an NVIDIA GeForce RTX 3090 GPU and Pytorch backend.\n\n\\subsubsection{Visualization of the Unique Information Captured by Anatomy-specific Learners} Here, we try to discuss what the simple anatomy-specific learners (BN and $1\\times1$ convolution layers) capture for better reconstruction. However, since MRI reconstruction is a low-level signal reconstruction task, to visualize and explain the feature maps inside the network remains an unsolved problem itself. So, we visualize the final learned parameters in the anatomy-specific learners.\n\nFor simplicity, we focus on the parameters in DCCNN+PN4, which achieves the best performance in our experiments. Without loss of generalization ability, we visualize the learned parameters of the 3rd CNN by the following strategy: we calculate and present the average value of weights in anatomy-specific BN and $1 \\times 1$ convolutional operations across 5 blocks, and show them in Fig.\\ref{fig_visp} (a) and (b). \n\nWe can find that the weights in additive learning modules are anatomy-sensitive. The weights of BN can re-scale different distributions for different anatomies with simple value shifting. The weights of anatomy-specific $1\\times1$ convolutional layers vary more significantly between each other compared to anatomy-specific BN layers, indicating that paralleled $1\\times1$ learners do learn something more unique than BN layers. As a result, the integrated paralleled learners can then effectively guide the reconstruction for their targeted anatomy and finally contribute to the overall improvement. \n\n\n\\section{Conclusion and Discussion}\nIn this work, we focus on an under-explored problem that how to make different anatomy to collaborate with each other in one network for better MRI reconstruction results. We reformulate the common deep MRI reconstruction network with anatomy-shared and anatomy-specific learners and build a novel Multiple-Anatomy-Parameterized-Network framework. Under our proposed framework, we explore four possible implementations of anatomy-specific learners and evaluate their performance on three anatomies with U-Net or DCCNN based reconstruction networks. Experimental results prove that our framework can help to boost the final performance in multiple anatomy reconstruction tasks in general for different networks. It is surprising that anatomy-specific learners built with simple BN layers and $1 \\times 1$ convolution can capture useful information, which is evident in not only the reconstruction results but also the visualization of learned parameters inside these learners. However, under our proposed MAPN framework, how to design the optimal anatomy-specific learners is still an open problem, and we believe this work sheds the light in a new direction towards more robust MRI reconstruction for multiple anatomy scene.\n\nOur work is closely related to previous works \\cite{yan2020neural} and \\cite{liu2021universal}. In \\cite{yan2020neural}, researchers found that the optimal network architectures obtained via neural architecture searching are different for brain and knee reconstruction tasks. However, it is difficult and impractical to design or search an anatomy-specific architecture for every anatomy. In this work, we solve this problem by disentangling the whole network into anatomy-share and anatomy-specific parts and using the flexible anatomy-specific learners to model unique knowledge instead of changing the entire architecture. We also reveal that the same anatomy-specific learners, especially PN2, perform differently when they are integrated into different reconstruction network architectures (U-Net and DCCNN). Based on this observation, one potential future extension of our work is to search the architectures of the flexible anatomy-specific learners via neural architecture searching. Another important work towards a universal MRI reconstruction is \\cite{liu2021universal}, where the researchers also used the anatomy-specific normalization settings (PN1). However, they mainly focused on using knowledge distillation in DCCNN to enhance the final performance, where well pre-trained expert networks are required before training the universal model. In contrast, our framework does not require any pre-trained expert models, and we further enhance anatomy-specific normalization with more advanced designed learners. Experiments show that our settings can robustly handle different networks.\n\nDespite the good results, current experiments are still early attempts of our proposed framework, and there exist some limitations. First, current experiments are only conducted in single-coil MRI reconstruction scenario for demonstration purpose. As multi-coil MRI reconstruction \\cite{sriram2020grappanet,jun2021joint} is more popular in clinical practice, extending this work to multi-coil reconstruction will be an important future work. Since we have evaluated our framework with DCCNN \\cite{schlemper2017deep}, it is easy to extend DCCNN \\cite{aggarwal2018modl} to MoDL \\cite{aggarwal2018modl} framework by modifying the network slightly and using conjugate gradient method to solve the Eq. \\ref{equ:m} for multi-coil MRIs. Another limitation lies in the objective loss function. In this work, we only adopted L1 loss to supervise the learning process of deep networks for a fair and initial comparison without considering the real distribution of the noise. It will be an important future work to introduce more robust objective loss functions \\cite{yang2017dagan,cheng2021learning,zhou2022accelerating} to supervise the designed learners for better modeling the reconstruction process in practice.\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank the fastMRI \\cite{fastMRI} project for making their codes and datasets accessible online.\n\n\\bibliographystyle{IEEEtran.bst}\n\\input{main.bbl}\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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