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+{"text":"\\section{Introduction}\n\n \\citet{blandford:1979} suggested that the flat-spectrum\n  radio cores seen in many active galactic nuclei are the optically\n  thick parts of conical plasma jets. Very Long Baseline\n  Interferometry (VLBI) has indeed revealed a jet-like geometry in\n  many flat-spectrum compact radio cores. Interestingly, the center of\n  our Galaxy also hosts a flat-spectrum core, called Sgr A*, and it\n  has been suggested that it may be associated with a relativistic\n  downsized jet from a starving supermassive black hole (BH)\n  (\\citealt{falcke:1993,falcke:2000,markoff:2007}). Unfortunately,\n  scattering by interstellar electrons smears out the source\n  structure. Nonetheless, the presence of a relativistic outflow in\n  Sgr~A* is strengthened by the size-frequency relation\n  \\citep{bower:2004} and by a 20-minute time-lag between flares in the 43\n  and 22 GHz light curves \\citep{yusef:2006}, which can be well explained\n  by a jet model \\citep{falcke:2009}.\n\n At millimeter wavelengths, the spectrum of Sgr A* peaks in the\n  so-called submillimeter bump \\citep{falcke:1998}, which can be\n  modeled by synchrotron emission arising from a radiatively\n  inefficient accretion flow onto a BH (RIAFs; first applied\n  to Sgr~A* by \\citealt{narayan:1995}). Over the past years, the RIAF\nmodel has progressed from a simple semi-analytical model to more\ncomplex time-dependent general-relativistic\n  magnetohydrodynamic (GRMHD) models. In these models relativistic\njets are often produced \\citep[e.g.,][]{beckwith:2008}. The\nsynchrotron emission from a GRMHD-RIAF can now be calculated with a\nhigh degree of precision using general relativistic radiative-transfer\ncodes (e.g.,\n\\citealt{broderick:2009a,dolence:2009,dexter:2009,roman:2012}).\nAdvances in numerical modeling, together with less scattered, mm-wave\nVLBI measurements of Sgr A*, provide an opportunity to put tight constraints\non jet-plus-disk models for BHs (\\citealt{broderick:2009b,dexter:2012}).\n\nSo far, GRMHD-RIAF models have not naturally reproduced the radio\nspectrum of Sgr A* --- with or without a jet. This may be partly\nbecause of the uncertainty in how to treat plasma temperatures. The\ndynamics of the plasma around the BH is sensitive to the\ntemperature of protons $T_{\\rm p}$, whereas the radio synchrotron spectra\ndepend on the temperature of electrons $T_{\\rm e}$.  In a collisionless\nplasma system, such as Sgr A*, the strength of $e$--$p$ coupling is\nunknown. Modeling the effect of the $e$--$p$ energy exchange is very\nsimplified or not considered in typical MHD simulations. In a\nstandard approach one often assumes that $T_{\\rm p}\/T_{\\rm e}=const$ in the entire\nsimulation domain (e.g., \\citealt{moscibrodzka:2009,moscibrodzka:2012}\nconsider $T_{\\rm p}\/T_{\\rm e}=1,3$, and 10). The constant $T_{\\rm p}\/T_{\\rm e}$ everywhere\nsuppresses emission from a GRMHD outflow because the outflow from\nnear the event horizon naturally must have much lower densities than\nthe inflow.\n\nIn this work, we relax the assumption of constant\n  $T_{\\rm p}\/T_{\\rm e}$. Indeed, early RIAF disk models assumed a high $T_{\\rm p}\/T_{\\rm e}$ to\naccount for the submillimeter bump, while early jet\n  models assumed a rather high $T_{\\rm e}$ to reproduce the radio emission of\nSgr~A*. Only later $T_{\\rm p}\/T_{\\rm e}$ was decreased in RIAF models to allow for\nlower accretion rates imposed by Faraday rotation measurements\n(\\citealt{BowerFalckeWright2005a,marrone:2007}). \n\n\\citet{yuan:2002} pointed out that the electron temperature in the\ndisk should be lower than that of the jet by a factor of ten to make\nboth disk and jet contribute to the emerging spectrum.  Here, we adopt\na similar approach. The use of a higher $T_{\\rm e}$ in the GRMHD jets is\nmotivated by the presence of several physical processes that are\nenhanced in the outflows and may cause stronger heating, such as\nstronger plasma magnetization, stronger shearing motion in the jet\nsheath, and shocks. All these processes are observed in the\nGRMHD simulations of jets (Brinkerink et al., in prep.).\n\nThe paper is organized as follows. In Sect.~\\ref{model}, we briefly describe\nthe GRMHD model of the BH accretion flow, the radiative transfer\ntechnique, and the electron temperature parameterization used to compute\nspectra and images of the jet-plus-disk system. We present and discuss the new\nresults in Sects.~\\ref{results} and~\\ref{discussion},\nrespectively.\n\n\\begin{table}[t]\n\\caption{List of radiative-transfer models.$^*$preferred $T_{\\rm p}\/T_{\\rm e}$ in the  disk.}\n\\label{tab:1}      \n\\centering                 \n\\begin{tabular}{c c c c c c c }   \n\\hline\\hline               \n $i \\mathrm{[\\degr]}$&$\\Theta_{\\rm e,j}$ & $^*(T_{\\rm p}\/T_{\\rm e})_{\\rm d}$ & $n_{\\rm e,0} {\\rm [cm^{-3}]}$& $B_{\\rm 0} {\\rm [kG]}$ & $\\dot{M} [\\rm M_\\sun yr^{-1}]$\\\\\n\\hline                        \n $90$ & $30$ & $15-20$ & $7\\times 10^8$ & $3.6$ &$10^{-7.3}$\\\\\n $60$ & $30$ & $15-20$ & $7\\times 10^8$ & $3.6$ &$10^{-7.3}$\\\\\n $30$ & $30$ & $10-15$ & $4\\times 10^8$ & $2.7$ &$10^{-7.6}$\\\\\n\\hline                     \n\\end{tabular}\n\\end{table}\n\n\\section{Model description}\\label{model}\n\nTo investigate the radiative properties of the jet--disk--BH triad, \nwe split the numerical modeling into three steps: \n(1) we computed the evolution of the GRMHD flow onto a BH;\n(2) we rescaled the dynamical model to Sgr~A*; \nand (3) we computed synchrotron spectra and images of the system.\n\nThe accretion-flow evolution was calculated by using the axisymmetric\nGRMHD code HARM-2D \\citep{gammie:2003}. The simulation's initial\nconditions and its computational grid are similar to those adopted in\n\\citet{moscibrodzka:2009} (and references therein).  The initial\ncondition is a weakly magnetized torus in orbit around a Kerr black\nhole with $a_*=0.94$, where $a_*$ is the BH angular momentum.\nThe inner and outer radius of the initial torus are $6R_g$ and\n$42R_g$ ($R_g=GM_{BH}\/c^2$ and $M_{BH}\\equiv$ BH mass), respectively.\nThe difference between the previous \\citep{moscibrodzka:2009} and the\ncurrent model is the size of the computational domain. Here we extended\nthe outer boundary to $r = 1000 R_g$. This large computational\ndomain is necessary to model the radio spectrum from $1-10^4$ GHz. The\nsimulation was evolved until $t_{final}=4000 R_g\/c$, which corresponds\nto 16 orbital rotations at pressure maximum of the initial disk.\n\nAs the simulation advances in time, the magnetorotational instability (MRI)\nturns the smooth torus into a turbulent accretion flow. The turbulence\ntransports angular momentum of the gas outward and inner portions of the disk\nfall toward the BH horizon. During the simulation, the accretion flow\nproduces a variety of outflows. A low-density strongly magnetized\nrelativistic outflow develops above the BH poles\n(\\citealt{blandford:1977,mckinney:2004}), the inner accretion disk launches a\nmildly relativistic outflow, and subrelativistic winds are produced by the\nouter regions of the disk. The latter outflow is formed because the\nouter-disk plasma gains an excess angular momentum from the accreting matter. \n\nFig.~\\ref{fig:maps} shows the overall structure of the GRMHD\njet--disk--BH model.  We defined the jet produced by the\naccretion flow as an {\\em unbound} gas outflowing with a minimum bulk\nvelocity $\\beta_{min}=0.2$~\\footnote{$\\beta $ is the velocity measured in\na normal observer frame.}. Everything beyond the jet region we\nrefer to as an accretion disk\/flow\/wind. In Fig.~\\ref{fig:maps}, the\njet region is separated from the accretion flow by a solid contour.\nOur formal definition of the jet indicates that it has two\ncomponents. The first component is a strongly magnetized nearly empty\nfunnel where $B^2\/\\rho_0 > 0.1$, hereafter called jet spine\n(the region within the dashed contour  in\nFig.~\\ref{fig:maps}).  The second\ncomponent is the mildly relativistic outflow along the funnel\n  wall (the region between the dashed and solid contours in\n  Fig.~\\ref{fig:maps}), hereafter called jet sheath. The choice of\n$\\beta_{min}=0.2$ was arbitrary, but proved to be a relatively robust\nthreshold to separate jet and disk winds, which exhibit a sharp\nboundary (in the jet sheath $\\left<\\beta\\right>=0.4$). With\n$\\beta_{min}=0.2$ the simulation grid resolves the jet sheath with a\nfew points. The mass-outflow rate in the jet sheath can be as high as\n20\\% of the average mass-inflow rate onto the BH. Hence,\nradiation from the jet region is dominated by the jet sheath.\n\n\\begin{figure*}[htb]\n\\includegraphics[angle=-90,scale=0.85]{fig1.ps}\n\\caption{\nOverall structure of the flow (upper half of the\nsimulation domain). Panels from left to right show maps of the density, magnetic\nfield strength, gas temperature, and bulk speed of the gas.  The first\ntwo are given in\ndimensionless numerical code units.  Solid and dashed contours illustrate our\njet and disk definitions, which are introduced in Sect.~\\ref{model}. The solid line\nseparates the jet from the disk, and the dashed line separates the jet spine\n from jet sheath.}\\label{fig:maps}\n\\end{figure*}\n\nThe dynamical models of the accreting BH are scale-free, but\nthe radiative transfer models are not.  We rescaled the numerical model\nto the Sgr~A* system.  We fixed the BH mass ($M_{\\mathrm BH}=4.5\n\\times 10^6 {\\rm M_{\\sun}}$, \\citealt{ghez:2008}) and distance (D =\n8.4 kpc, \\citealt{gillessen:2009}). The remaining model parameters\nare the source inclination $i$, the density scaling constant $n_{e,0}$,\nand the electron temperature $T_e$. The magnetic-field strength is\ngiven by multiplying the dimensionless $B$ field by $B_{\\rm 0} = c\n\\sqrt{4 \\pi m_p n_{e,0}}$. We chose $n_{\\rm e,0}$ so that the radio\nflux (at $\\nu=1-100{\\rm GHz}$) matched the observed data points. \n $T_{\\rm e}$ was parameterized as follows. In the disk we have\n  $(T_{\\rm p}\/T_{\\rm e})_d = const$, where $T_{\\rm p}$ is given by the dynamical model and\n  $(T_{\\rm p}\/T_{\\rm e})_d$ is a free parameter. In the jet we have $\n  T_{e,j} = 30 m_e c^2\/k$, independently of the proton temperature.\nThe last assumption was motivated by results presented in the next\nsection and models of isothermal jets. However, as explained in the\nnext section, the value of $T_{\\rm e,j}$ is similar to the average\n$T_{\\rm p}$ found in the GRMHD model in the jet region, that is,\n$(T_{\\rm p}\/T_{\\rm e})_j\\sim1$.\n\nSpectral energy distributions (SEDs) \nand images of the plasma around the BH are produced by a 3D\ngeneral relativistic ray--tracing radiative-transfer code. In the\nradiative-transfer \ncomputations, we assumed that the plasma has a Maxwellian energy\ndistribution. We used an independent numerical radiative-transfer scheme as\ndescribed for instance in \\citet{noble:2007}.\n\n\n\\section{Results}~\\label{results}\n\n\\begin{figure}[htb]\n\\includegraphics[angle=-90,scale=0.23]{fig2a.ps}\n\\includegraphics[angle=-90,scale=0.23]{fig2b.ps}\n\\includegraphics[angle=-90,scale=0.23]{fig2c.ps}\n\\caption{\nTime- and $\\theta$-angle-averaged (see Eq.~\\ref{eq:avg}) profiles of $n_{\\rm e}$,\n$B$, and $\\Theta_e=kT_{\\rm e} \/m_e c^2$ (assuming $T_p\/T_e=1$) in the jet and in the\ndisk. The dotted lines correspond to the quantities measured in the turbulent\naccretion flow, and the solid lines are the averaged profiles of quantities\nmeasured in the jet region.  The solid green lines are the profiles measured\nin the jet without the empty jet spine (i.e. excluding zones where\n$B^2\/\\rho_{\\rm 0} > 0.1$).  The plasma density and magnetic-field strength are shown\nin the numerical code units.} \\label{fig:prof}\n\\end{figure}\n\nWe analyzed the radial structure of inflows and outflows produced in the\nsimulation. \nFig.~\\ref{fig:prof} shows profiles of three quantities measured\nin the accretion flow (dotted lines), in the jet spine (solid black lines), and in\nthe jet sheath (solid green lines). The time- and shell- volume-averaged\nradial profiles of $n_{\\rm e}$, $B$ and the dimensionless electron temperature,\n$\\Theta_{\\rm e}=kT_{\\rm e} \/m_ec^2$, were calculated using the following definition:\n\\begin{equation}\n\\left< q(r)\\right> = \\frac{1}{\\Delta t}\\int_{t_{min}}^{t_{max}} \n\\frac{\\int_0^{2\\pi}  \\int_0^{\\pi}  q(r, \\theta, \\phi , t)\\sqrt{-g} d\\theta d\\phi }\n      {\\int_0^{2\\pi} \\int_0^{\\pi} \\sqrt{-g} d\\theta d\\phi } dt, \\label{eq:avg}\n\\end{equation}\nwhere $q$ is a scalar quantity, $\\sqrt{-g}$ is the determinant of the metric,\n$t_{min}=3500 \\mathrm{M}$, and $\\Delta t=t_{max}-t_{min}=500 \\mathrm{M}$.\n\nThe radial profiles of the three quantities show approximately power-law\nshapes. The density in the accretion flow decreases with radius as $n_{\\rm e} \\sim\nr^{-3}$. The steep power-law dependence is an artifact caused by the adopted\ninitial conditions (small size of the torus).  The density in the jet\ndecreases with radius as $n_{\\rm e} \\sim r^{-2}$.  Close to the BH, the\ndensity of the disk is about 100 times higher than the density in the\noutflow. The magnetic fields, both in the inflow and in the outflow, decrease\nwith distance as $B\\sim r^{-3\/2}$.  The protons in the accretion flow are near\ntheir virial temperature, $T_{\\rm p} \\sim r^{-1}$. The temperature of the gas in the\njet is approximately constant between $5R_g$ (where the jet starts) and $r\\sim\n100R_g$, and decreases for $r>100R_g$.  For $r \\le 100R_g$, the gas\ntemperature is $\\left<\\Theta_e\\right>=30$.  The break in the temperature\npower-law at $r\\approx 100 R_g$ may be caused by decollimation of the jet\n(because of the small disk size) and\/or by a poor numerical resolution of the\nmodel at large radii.\n\nInterestingly, density $n_{\\rm e}$ and $B$-fields in the outflow decrease with\nradius in a similar manner as the $n_{\\rm e}$ and $B$ fields in the semi-analytical\nrelativistic jet models by \\citet{falcke:2000}. Their model, which produces a\nflat radio SED, assumes that the temperature of electrons (or, to be more\nprecise, the electron energy distribution function) is almost constant along the\njet. Hence, we simply adopted an isothermal\njet with $\\Theta_e=30$,\nin our radiative-transfer models, which corresponds to \n$(T_{\\rm p}\/T_{\\rm e})_j\\sim1$ up to $r\\la 100 R_g$. Outside the jet $T_{\\rm e}$ is\ngiven by $(T_{\\rm p}\/T_{\\rm e})_d=const>1$.\n\nThe radiative-transfer model parameters are given in Table~\\ref{tab:1}. \nFig.~\\ref{fig:sed} shows time-averaged SEDs of the jet-plus-disk system. In Fig.~\\ref{fig:sed}, three\nlines in each panel correspond to models with various $(T_{\\rm p}\/T_{\\rm e})_d=10,15$, and $20$.\n\n\nIndeed, the jet spectrum is nearly flat, whereas the disk produces a hump at\nsubmillimeter wavelengths. The two components together are able to reproduce\nthe spectrum remarkably well. At this point, we do not intend to conduct a full\nstatistical exploration of the parameter space and, therefore, there is no strong\nreason to favor any specific parameter combination yet. However, it is already\nclear that the jet is crucial for filling in the low-frequency radio spectrum.\n\nAs predicted in \\citet{yuan:2002}, the disk $T_{\\rm e}$ has to be low and $(T_{\\rm p}\/T_{\\rm e})_d\n\\sim 10-20$ to smoothly connect jet and disk emission. The spectral\nshape at lower frequencies depends weakly on the observing inclination angle.\nFor $i=30\\degr$, the slope of the SED is somewhat shallower\n($\\alpha_{\\nu}\\sim0$) than the slope seen under higher\ninclinations ($\\alpha_{\\nu}\\sim 0.3$). This dependence is in line with\nprevious analytical estimates \\citep{falcke:1999}.\n\nAre size and structure of Sgr~A* also consistent with the model?  Scattering\nis, of course, a major problem and the relative orientation between scattering\ndisk and jet axis adds another free parameter. We approached this here by\nscatter-broadening our model images for one arbitrary orientation and\ncomparing them to the available data \\citep{falcke:2009}. Given the very\nlimited structural information we have -- a size in east-west direction --\nthis is the best we can do for now.\n\nWe proceeded as follows: Our radiative-transfer model computes images of the\nflow at $\\lambda=3,7$ and $13$ millimeters. The synthetic images are then\nconvolved with the symmetric scattering Gaussian profile with a $FWHM=1.309\n\\lambda^2$ \\citep{bower:2006}. Next, the size of the scatter-broadened image\nis measured by computing the eigenvalues of the matrix formed by taking the\nsecond angular moments of the image (the principal axis lengths). This method\nyields an accurate size of the emitting region if the brightness distribution\nis Gaussian-like and is therefore most accurate for strongly scattered images.\n\nIn the bottom-right panel of Fig.~\\ref{fig:sed}, we compare the measured\nmajor-axis \nsizes of Sgr~A* at $\\lambda=0.3-2\\rm{cm}$ with those predicted by the\nmodel. We did not extend the model size to $\\lambda=1.3mm$, because here the\nsimulated image is highly non-Gaussian and a single major-axis size is\nmeaningless. Furthermore the VLBI data were measured on a single, very narrow triangle\nof baselines only. We also cannot extend our model sizes to $\\lambda >\n2\\rm{cm}$ because the size of the scattered image becomes larger than our\ncomputational domain.  However, in the region around $\\lambda \\sim\n0.7\\rm{cm}$, where the sizes are most reliably determined, the major-axis\nsizes are of the correct order. Moreover, the images look Gaussian-like despite\nthe underlying jet-structure.\n\n\n\\begin{figure}\n\\includegraphics[scale=0.4,angle=-90]{fig3a.ps}\n\\includegraphics[scale=0.4,angle=-90]{fig3b.ps}\n\\includegraphics[scale=0.4,angle=-90]{fig3c.ps}\n\\includegraphics[scale=0.41,angle=-90]{fig3d.ps}\n\\caption{\nSEDs computed for parameters given in table~\\ref{tab:1}.  SEDs are time\naveraged. Each line corresponds to a model with a  different temperature ratio in\nthe accretion disk, $(T_{\\rm p}\/T_{\\rm e})_{d}=10,15$ and 20), and various inclinations.  The\nred dashed line is the 'best-bet' model from \\citet{moscibrodzka:2009}, which\nassumes $i\\approx90\\degr$ and $T_{\\rm p}\/T_{\\rm e}=3$ in the entire computational\ndomain. The data points are the same as those used in\n\\citet{moscibrodzka:2009}. The bottom-right panel shows the sizes of the GRMHD jet\nmodel {\\em including scatter-broadening}.  Open symbols are Sgr~A* measured sizes from\n\\citet{bower:2006}.}\\label{fig:sed}\n\\end{figure}\n\n\n\\section{Summary}\\label{discussion}\n\nWe have reanalyzed the structure of inflows (disk) and outflows (jets)\nproduced in some GRMHD models of accreting BH \\citep{gammie:2003}.  We\npointed out that the time-averaged radial profiles of plasma density and\nB-fields in the jet-like outflows are similar, though not exactly identical,\nto those used in semi-analytic jet models that readily explain the radio\nSED of Sgr~A* \\citep{falcke:2000,yuan:2002}\nbased on a scaled-down \\citet{blandford:1979} model combined with a radiatively\ninefficient accretion flow or RIAF.  Consequently, we were able for the first\ntime to relatively easily reproduce the radio spectrum and size of Sgr A* at\nmm-waves with a standard GRMHD model by simply allowing for different\nelectron heating in jet and disk.\n\nIn particular, we found that the flat radio spectrum of jets is indeed\nreproduced by the simulations when we kept the electron temperature\nconstant along the jet, as assumed in the \\citet{blandford:1979}\nmodel. In the inner parts the jet is well-described by a single-temperature \nplasma with a proton-to-electron temperature ratio on\nthe order unity. To reproduce the overall spectrum, however, the electron\ntemperature in the disk needs to be lower than in the jet by at least\nan order of magnitude to avoid strong absorption and huge emission at\nsubmillimeter waves.  In turn, this requires the presence of a\ntwo-temperature plasma in the accretion flow, with a high\nproton-to-electron temperature ratio $(\\sim10-20)$. This is expected because \ntwo-temperature plasmas were actually postulated in the earliest models for RIAFs\n\\citep{narayan:1995}. Interestingly, $T_{\\rm p}\/T_{\\rm e}=10-20$ is consistent\n  with results of local (shearing box) simulations of collisionless\n  plasma in which kinetic effects are included \\citep{sharma:2007}.\n\n In the jet models, $\\dot{M}$ is about 20 times higher than \n  $\\dot{M}\\approx2\\times10^{-9} \\rm M_\\sun yr^{-1}$ in the '$T_{\\rm p}\/T_{\\rm e}=3$ everywhere'\n  best-bet model from \\citet{moscibrodzka:2009}.  Our\n  $\\dot{M}=4.5\\times10^{-8} \\rm M_\\sun yr^{-1}$ (see Table 1) is consistent with\n  $\\dot{M}=6\\times10^{-8} \\rm M_\\sun yr^{-1}$ found in models where the BH is fed by\n  stellar winds (\\citealt{roman:2010}, but see\n  \\citealt{quataert:2004}), and with estimates by\n  \\citet{sharma:2007} based on local collisionless plasma models. \nA higher $\\dot{M}$ may also change\nexpectations for the recently discovered cloud-like object G2 that moves\ntoward Sgr~A* \\citep{gillessen:2012}, which is expected to be accreted\nonto the BH soon. For a higher pre-impact accretion rate the increase\nin $\\dot{M}$ due to the cloud would be much less dramatic than predicted\n(e.g., in \\citealt{moscibrodzka:2012} or \\citealt{sadowski:2013}).\n\n Summarizing, the electron distribution function in GRMHD models\n  is a free function that can vary with space and time. We showed that\n  a {\\it natural} modification of this distribution function produces \n  SEDs that fit the observations well.  The\nre-heating (or re-acceleration) of electrons in the jet might be due\nto effects that were described in the GRMHD models, such as shear,\nstrong magnetization, or shocks in the jet sheath, but this requires a more detailed\ninvestigation.  We conclude that the exact nature of the electron\nheating in jets and disks deserves more attention in the future.\n\n\n\\begin{acknowledgements} \nWe thank C. Gammie and J. Dexter for comments.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s:i}\nConstructing efficient signal filters is a fundamental problem\nin signal processing with a vast literature; see, e.g.,\\\nrecent papers \\cite{phj12,6778405,6319316,mrs2009,ckh2014,6957555} and references there. A filter \ncan be described by a transformation $F$, often non-linear, of an input signal, \nrepresented by a vector $x$, into a filtered signal, represented by a vector  $F(x)$.   \nWe~revisit some classical constructions of filters aimed at signal noise reduction, \nwith the emphasis on bilateral filter, popular in image denoising \\cite{1042377,4587843,Zhang2014299,M13}. \nThe goal of the filter is signal smoothing, reducing a high oscillatory additive noise. \nThe smoothing can be achieved by averaging, which can be \ntypically interpreted as a low-pass filter, minimizing the \ncontribution in the filtered signal of highly oscillatory modes, \ntreated as eigevectors of a graph Laplacian;\nsee, e.g., \\cite{Chung97a}. \n\nIt is desirable to preserve edges in the ideal noise-free signal, \neven at the costs of an increased PSNR. \nEdge-conscious filters detect, often implicitly, the locations of the edges \nand attempt using less aggressive or anisotropic averaging at these locations. \nFully automatic edge detection in a noisy signal is difficult, \ntypically resulting in non-linear filters, i.e. where \nthe filtered vector $F(x)$ depends non-linearly on the input vector $x.$ \nHowever, it can be assisted by a guiding signal, \nhaving the edges in the same locations as in the ideal signal; see,\ne.g.,\\ \\cite{6319316,4359322,5539896}. \n\nGraph signal processing, introducing eigenvectors of the graph Laplacian as\nnatural extensions of the Fourier bases, \nsheds new light at image processing; see, e.g.,  \\cite{leo_book,shuman_signal_2013,Sunil_GlobalSip13,gadde2013bilateral}.\nIn~\\cite{TKMV14}, graph-based filtering of noisy images is performed by directly computing a projection of the image to be filtered onto a lower dimensional Krylov subspace of the normalized graph Laplacian, constructed using nonnegative graph weights determined by distances between image data corresponding to image pixels. We extend the construction of the graph Laplacian to the case, where some weights can be negative, radically departing from the traditional assumption.\n\n\\newpage\n\\section{Preliminaries}\\label{s:p}\nLet us for simplicity first assume that the guiding signal, denoted by $y$,\nis available and can be used to reliably detect the locations of the edges\nand, most importantly, to determine the edge-conscious \\emph{linear} transformation (matrix)\n$F_y$ such that the action of the filter $F(x)$ is given by the following \nmatrix-vector product $F_y x = F(x).$ Having a specific construction of the guided filter \nmatrix $F_y$ as a function of~$y$, one can define a self-guided non-linear filter, \ne.g.,\\ as $F_x x$, which can be applied iteratively, starting with the \ninput signal vector $x_0$ as follows, $x_{i+1}=F(x_i),\\, i=0,1,\\ldots,m$;\ncf., e.g., \\cite{Sarela:2005:DSS:1046920.1058110}.\n\nSimilarly, an iterative application of the linear guided filter can be used, \nmathematically equivalent to applying the powers of the square matrix $F_y$, i.e.\n$x_m=\\left(F_y\\right)^m x_0$, thus naturally called the \\emph{power method},\nwhich is an iterative form of PCA; see, e.g.,\\cite{NIPS1998_1491,Rossi2015}.\nTo avoid a re-normalization of the filtered signal, it is convenient to \nconstruct the matrix $F_y$ in the form $F_y=D_y^{-1}W_y$, where entries of the square matrix \n$W_y$ are called \\emph{weighs}. The matrix $D_y$ is diagonal, made of row-sums of the \nmatrix $W_y$, which are assumed to be non-zero. Thus, $D_y^{-1}W_y$ multiplied by a column-vector of ones, gives again the column-vector of ones. \n\nLet us further assume that the matrix $W_y$ is symmetric and that all the entries (weighs) in $W_y$ are nonnegative. \nFor signal denoising, the following observations are the most important. The right eigenvector $v_1$ of the matrix $D_y^{-1}W_y$ with the eigenvalue $\\mu_1=1$ is trivial, just made of ones, only affecting the signal offset. \nSince the iterative matrix $F_y=D_y^{-1}W_y$ is diagonalizable, the power method gives \n\\begin{equation}\nx_m=\\left(F_y\\right)^m x_0 = \\Sigma_j \\; \\mu_j^m \\left(v_j^T D x_0\\right) v_j,\n\\label{eq:power}\n\\end{equation} \nwhere $1=|\\mu_1|\\geq|\\mu_2|\\geq\\ldots$ are the eigenvalues of the matrix $D_y^{-1}W_y$ corresponding to the eigenvectors $v_j$ scaled such that $v_i^T D v_j=\\delta_{ij}$. \nThe power method, according to~\\eqref{eq:power}, \nsuppresses contributions of the eigenvectors corresponding to\nthe smallest eigenvalues. Thus, the matrix $W_y$ needs to be constructed in such a way \nthat these eigenvectors represent the noisy part of the input signal, while \nthe other eigenvectors are edge-conscious; cf. anisotropic diffusion \\cite{PM87, PM90, GraphSP_HeatKernel:2008}.\n\nLet us introduce the guiding Laplacian $L_y=D_y-W_y$ and \nnormalized  Laplacian $D_y^{-1}L_y=I-D_y^{-1}W_y$ matrices. \nIn~\\cite{TKMV14}, the power method \\eqref{eq:power} is replaced with a projection of the image vector $x$ \nto be denoised onto a lower dimensional Krylov subspace of the guiding normalized graph Laplacian $D_y^{-1}L_y$ and implemented, e.g., using the Conjugate Gradient (CG) method; see, e.g., \\cite{Hestenes&Stiefel:1952,G97,823968}. \n\n\\newpage\n\\section{Motivation} \\label{s:m}\nOne of the most popular edge-preserving denoising filters is the bilateral filter (BF), see, e.g.,\\ \\cite{TM98, PKTD09} and references there, which takes the weighted average of the nearby pixels.\nThe weights  $w_{ij}$ may depend on spatial distances and signal data similarity, e.g.,\\\n\\begin{equation}\\label{eq2}\n w_{ij}=\\exp\\left(-\\frac{\\left\\|p_i-p_j\\right\\|^2}{2\\sigma_d^2}\\right)\n \\exp\\left(-\\frac{\\left\\|y[i]-y[j]\\right\\|^2}{2\\sigma_r^2}\\right),\n\\end{equation}\nwhere $p_i$ denotes the position of the pixel $i$, the value $y[i]$ is the signal intensity, and $\\sigma_d$ and $\\sigma_r$ are filter parameters.\nTo simplify the presentation and our arguments, we further assume that \nthe signal is scalar on a one-dimensional uniform grid, setting \nwithout loss of generality the first multiplier in \\eqref{eq2} to be $1$, \nand that the weights $w_{ij}$ are computed only for the nearest neighbors and set to zero otherwise.\n\nLet us start with a constant signal, where $y[i]-y[j]=0$. \nThen, $w_{i-1\\, i}=w_{i\\, i}= w_{i\\, i+1}=1$ and the graph\nLaplacian $L_y=D_y-W_y$ is a tridiagonal matrix that has nonzero entries \n$1$ and $-1$ in the first row, $-1$ and $1$ in the last row, and \n$[-1\\,\\; 2\\, -1]$ in every other row. This is a standard three-point-stencil finite-difference \napproximation of the negative second derivative of functions \nwith homogeneous Neumann boundary conditions, i.e., vanishing \nfirst derivatives at the end points of the interval. Its eigenvectors \nare the basis vectors of the discrete cosine transform; see the first five \nlow frequency eigenmodes (the eigenvectors corresponding to the smallest eigenvalues) of  $L_y$\nin Figure \\ref{fig:1}.  \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{1}\n\\caption{Discrete cosine transform low frequency modes.}\n\\label{fig:1}\n\\end{figure}\nAs can be seen in Figure~\\ref{fig:1}, all smooth low frequency eigenmodes \nturn flat at the end points of the interval, due to the Neumann conditions. \n\nThe key observation is that\nthe Laplacian row sums in the first and last rows vanish for \\emph{any} signal,\naccording to the standard construction of the graph Laplacian,\nno matter what formulas for the weights are being used!\nThus, \\emph{any} low pass filter \nbased on low frequency eigenmodes of the graph Laplacian flattens \nthe signal at the end points. \n\nLet us now use formula \\eqref{eq2} for a piece-wise constant\nguiding signal $y$ with the jump large enough to result in a small value  \n$w_{i\\, i+1}=w_{i+1 \\, i}$ for some index~$i$. The first five vectors of the \ncorresponding Laplacian are shown in Figure \\ref{fig:2}.  \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{2}\n\\caption{Edge-preserving low frequency eigenmodes.}\n\\label{fig:2}\n\\end{figure}\nAll the plotted in Figure \\ref{fig:2} \nvectors are aware of the jump, representing an edge in our \none-dimensional signal~$y$, but they are also all flat on both sides of the edge! \nSuch a flatness is expected to appear for any guiding signal $y$ giving \na small value $w_{i\\, i+1}=w_{i+1 \\, i}$.\n\nThe presence of the flatness in the low frequency modes of the graph Laplacian $L_y$ on both sides of the edge in the guiding signal $y$ is easy to explain. When the value $w_{i\\, i+1}=w_{i+1 \\, i}$ is small relative to other entries, the matrix  $L_y$ becomes \nnearly block diagonal, with two blocks, which approximate graph Laplacian matrices \nof the signal $y$ restricted to sub-intervals of the signal domain \nto the left and to the right of the edge. \n\nThe low frequency eigenmodes\nof the graph Laplacian $L_y$ approximate combinations of the \nlow frequency eigenmodes of the graph Laplacians on the sub-intervals.\nBut each of the low frequency eigenmodes of the graph Laplacian on the sub-interval\nsuffers from the flattening effect on both ends of the sub-interval, as explained above. \nCombined, it results in the flatness in the low frequency modes of the graph Laplacian $L_y$\non both sides of the edge.\nFor denoising, the flatness of the vectors determining the low-pass filter \nmay have a negative effect for self-guided denoising even of piece-wise constant signals,\nif the noise is large enough relative to the jump in the signal, as shown in Section \\ref{s:eef}.\n\nThe attentive reader notices that method~\\eqref{eq:power} is based on $D_y^{-1}W_y$, related to the \\emph{normalized} graph Laplacian $D_y^{-1}L_y$, not the Laplacian $L_y$ used in our arguments above. Although the diagonal matrix\n$D_y$ is not a scalar identity, and so the eigenvectors of $D_y^{-1}L_y$,\nnot plotted here, and of $L_y$ are different, the difference is not qualitative enough to noticeably change the figures and invalidate our explanation. \n \n\\section{Negative weights in spectral graph partitioning and for signal edge enhancing}\n\\label{s:n}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{3}\n\\caption{Edge-enhancing low frequency eigenmodes, small negative.}\n\\label{fig:3}\n\\end{figure}\nThe low frequency eigenmodes of the graph Laplacian play a fundamental role \nin spectral graph partitioning, which is one of the most popular \ntools for data clustering; see, e.g.,\\ \\cite{Shi:2000:NCI:351581.351611,ng2002spectral,k2003}.\nA limitation of the conventional spectral clustering approach is embedded in its definition based on the weights of graph, which must be nonnegative, e.g.,\\ based on a distance measuring relative similarities of each pair of points in the dataset. For the dataset representing values of a signal, e.g.,\\ pixel values of an image, formula \\eqref{eq2} is a typical example of \ndetermining the nonnegative weights, leading to the graph adjacency matrix $W_y$ with nonnegative entries, as assumed in Section \\ref{s:p} and in all existing literature. \n\nIn applications, data points may represent feature vectors or functions, allowing the use of correlation for their pairwise comparison. The correlation can be negative, or,  more generally, points in the dataset can be dissimilar, contrasting each other.\nIn conventional spectral clustering, the only available possibility to handle such a case is to replace the anticorrelation, i.e. negative correlation, of the data points with the uncorrelation, i.e. zero correlation. The replacement changes the corresponding negative entry in the graph adjacency matrix to zero, to enable the conventional spectral clustering to proceed, but nullifies a valid comparison. \n\nA common motivation of spectral clustering comes from analyzing a mechanical vibration model in a spring-mass system, where the masses that are tightly connected have a tendency to move synchronically in low-frequency free vibrations; e.g.,~\\cite{Park20143245}. Analyzing the signs of the components corresponding to different masses of the low-frequency vibration modes of the system allows one to determine the clusters. The mechanical vibration model may describe conventional clustering when all the springs are pre-tensed to create an attracting force between the masses. However, one can pre-tense some of the springs to create repulsive forces!\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{4}\n\\caption{Edge-enhancing low frequency eigenmodes, more negative.}\n\\label{fig:4}\n\\end{figure}\nIn the context of data clustering formulated as graph partitioning, that corresponds to negative entries in the adjacency matrix. The negative entries in the adjacency matrix are not allowed in conventional graph spectral clustering. Nevertheless, the model of mechanical vibrations of the spring-mass system with repulsive springs remains valid, motivating us to consider \nthe effects of negative graph weights. \n\nIn the spring-mass system, the masses, which are attracted, would move together synchronically in the same direction in low-frequency free vibrations, while the masses, which are repulsed, have the tendency to move synchronically in the opposite direction. \nUsing negative, rather than zero, weights at the edge of the guiding signal $y$ \nfor the purposes of the low-pass filters thus \nis expected to repulse the flatness of low frequency eigenmodes\nof the graph Laplacian $L_y$ on the opposite sides of the edge of the signal $y$, \nmaking the low frequency eigenmodes to be edge-enhancing, rather than just edge-preserving; \ncf. \\cite{Durand:2002:FBF:566654.566574} on sharpening. \n\nFigures \\ref{fig:3} and \\ref{fig:4} demonstrate the effect of edge-enhancing, as a proof of concept.\nBoth Figures \\ref{fig:3} and \\ref{fig:4} display the five eigenvectors for the five smallest eigenvalues of the same tridiagonal graph Laplacian as that corresponding to Figure \\ref{fig:2}\nexcept that the small positive entry of the weights $w_{i\\, i+1}=w_{i+1 \\, i}$ for the same $i$  \nis substituted by $-0.05$ in Figure \\ref{fig:3} and by $-0.2$ in Figure \\ref{fig:4}. \nThe previously flat around the edge eigenmodes in Figure \\ref{fig:2}  \nare repelled in opposite directions on the opposite sides of the edge in Figures \\ref{fig:3} and~\\ref{fig:4}. \n\nNegative weights require caution, since even small changes \ndramatically alter the behaviors of the low frequency eigenmodes around the \nedge, as seen in Figures \\ref{fig:3} and \\ref{fig:4}. Making the \nnegative value more negative, we observe by comparing Figure \\ref{fig:3} to Figure \\ref{fig:4}\nthat the leading eigenmode, displayed using the blue color in both figures, corresponding to the smallest nonzero eigenvalue forms a narrowing layer around the \nsignal edge, while other eigenmodes become less affected by the change\nin the negative value. \n\n\\section{Edge-enhancing filters}\\label{s:eef}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{zero}\n\\caption{Edge-preserving filtering.}\n\\label{fig:zero}\n\\end{figure}\n\nIn this section, as a proof of concept, we numerically test the proposed edge-enhancing filters\non a toy one-dimensional example using the classical nonlinear self-guiding BF and \na guided (by a noiseless signal) BF\naccelerated with a conjugate gradient (CG-BF) method, as suggested in \\cite{TKMV14}. \nThe specific CG algorithm used in our tests is as described in Algorithm~\\ref{alg:CG}.\n\n\\begin{algorithm}\n{Input: signal vector to be filtered $x_0$, matrices $D_y$ and $L_y$} \\\\\n{$r_0=-L_y x_0$}\\\\\n    \\For{$k=0,1,\\ldots,m-1$}{\n    $s_k = D_y^{-1} r_k$\\\\\n    \\eIf{$k=0$}{\n      $p_0 = s_0$\\\\\n    }\n    {\n    $p_k = s_k + \\beta_k p_{k-1}$, where \\\\\n    $\\displaystyle \\beta_k = \\frac{(s_k,r_k)}{(s_{k-1},r_{k-1})}$ \n    }\n    $q_k=L_y p_k$\\\\\n    $\\displaystyle \\alpha_k = \\frac{(s_k,r_k)}{(p_k,q_k)}$\\\\\n    $x_{k+1} = x_k + \\alpha_k p_k$\\\\\n    $r_{k+1} = r_k - \\alpha_k q_k$\\\\\n    }\n    {Output: filtered vector $x_m$}\n    \\caption{Conjugate Gradient Guided Filter}\n    \\label{alg:CG}\n\\end{algorithm}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\linewidth]{negative}\n\\caption{Edge-enhancing filtering.}\n\\label{fig:negative}\n\\end{figure}\nThe noise is additive Gaussian, and the noisy signal is displayed using grey dots.\nThe nonzero weights are computed by \\eqref{eq2} with $\\sigma_d=0.5$ and $\\sigma_1=0.1$ \nonly for $j=i-1, \\, i, \\, i+1$, resulting in tridiagonal matrices $W$ and $L$. \nBF is self-guided, with  $W$ and $L$ recomputed on every iteration\nusing the current approximation $x_k$ to the final filtered signal $x_m$.\nCG-BF uses the fixed  nonzero weights computed also by \\eqref{eq2}, but for the \nnoiseless signal $y$ resulting in the fixed tridiagonal matrices $W_y$ and $L_y$. \nThe number of iterations in BF, 100, and CG-BF, 15, is tuned to match the errors.\nFormula \\eqref{eq2} puts ones on the main diagonal of $W$, thus for \nsmall positive or negative $w_{i\\, i+1}=w_{i+1 \\, i}$ the matrix $D$ is well-conditioned. \n\nFigure \\ref {fig:zero} demonstrates the traditional approach with nonnegative weighs.\nWe observe, as discussed in Section~\\ref{s:m}, flattening at the end points. \nMoreover, there is noticeable edge smoothing in all corners,\ndue to a large noise and relatively small number of signal samples, despite of the use of the edge-preserving \nformula \\eqref{eq2}. \nWe set tuned negative graph weights $-2\\times 10^{-3}$, $-10^{-3}$, $-10^{-8}$ \nfor $i=100,\\, 250$, and $350$ correspondingly,\nto obtain \nFigure \\ref{fig:negative}, which shows dramatic improvements both in terms of PSNR \nand edge matching, compared to  Figure \\ref {fig:zero}. \n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe proposed novel technology of negative graph weights allows \ndesigning edge enhancing filters, as explained theoretically and\nshown numerically for a synthetic example. \nOur future work concerns determining the optimal negative weights, testing the concept \nfor image filtering, and exploring its advantages in spectral data clustering using correlations. \n\n\\newpage\n\\IEEEtriggeratref{17}\n\n\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe hallmark of topological states of matter is the exact quantization of a physical observable in terms of a conserved quantity, the topological invariant~\\cite{thouless_quantized_1982,thouless_quantization_1983}.\nA paradigmatic example is the quantum Hall conductance, which is quantized as integer (or fractional) multiples of $e^2\/h$ with a precision exceeding one part in a billion~\\cite{klitzing_new_1980,laughlin_quantized_1981}.\nMoreover, this quantization is robust against perturbations, i.e., it persists in the presence of disorder, defects, impurities, or imperfections of the experimental sample.\nThis led to an extremely precise definition of the electrical resistance standard and experimental determination of the finite-structure constant~\\cite{klitzing_quantum_2017}.\n\nA topologically equivalent state is the Thouless pump~\\cite{thouless_quantization_1983,niu_towards_1990,shindou_quantum_2005,fu_time_2006,wei_anomalous_2015,roux_quasiperiodic_2008,wang_topological_2013,marra_fractional_2015,marra_fractional_2017,matsuda_two-dimensional_2019}, which can be engineered, e.g., with ultracold atoms~\\cite{lewenstein_ultracold_2007,bloch_many-body_2008,zhang_topological_2018,cooper_topological_2019} in a superlattice created by the superposition of two optical lattices with different wavelengths~\\cite{nakajima_topological_2016,lohse_thouless_2016,taddia_topological_2017,das_realizing_2019}.\nWhen the superlattice is adiabatically and periodically varied in time $t$, the charge pumped through the atomic cloud is quantized in terms of the topological invariant, i.e., the Chern number~\\cite{thouless_quantization_1983}.\nHowever, the charge is quantized only when the duration of the pumping process is an integer multiple of the full adiabatic cycle, and deviations from the quantized value are linear in time. \nIn this sense, the quantization of the pumped charge is not exact:\nThis constitutes a fundamental hindrance to the realization of metrological standards.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig1.png}%\n\\setlength{\\belowcaptionskip}{-14pt}\\setlength{\\abovecaptionskip}{2pt}\n\t\\caption{%\nThe superposition of two stationary lattices in a tilted direction produces a quasiperiodic one-dimensional lattice when $\\alpha=\\lambda_\\mathrm{S}\/(\\lambda_{\\mathrm{L}}\\cos{\\theta})$ is an irrational number.\n}\n\t\\label{fig1}\n\\end{figure}\n\nIn this Rapid Communication, we will show that the quantization of the pumped \\emph{current} can be indeed realized by Thouless pumps in the \\emph{quasiperiodic} regime and, most importantly, that this quantization is exact.\nIn ultracold atomic systems, quasiperiodicity~\\cite{kraus_topological_2012-1,kraus_topological_2012-2,kraus_quasiperiodicity_2016,ozawa_topological_2019,valiente_super_2019,kuno_disorder-induced_2019,yao_critical_2019} is realized using a superposition of two optical lattices with incommensurate lattice constants, i.e., their ratio $\\alpha$ is an irrational number.\nIn this regime, the translational symmetry is completely broken, the familiar concept of Brillouin zone (BZ) becomes ill-defined, and the usual definition of the Chern number as an integral of the Berry curvature breaks down.\nIn order to consider a realistic experimental setup, we will derive an effective tight-binding (TB) model describing an atomic gas in a bichromatic potential~\\cite{das_realizing_2019,roux_quasiperiodic_2008}, which coincides with a generalized Aubry-Andr\\'{e}-Harper-Hofstadter (AAHH) model~\\cite{harper_single_1955,hofstadter_energy_1976,aubry_analyticity_1980,hatsugai_energy_1990,osadchy_hofstadter_2001,hatsuda_hofstadters_2016,ikeda_hofstadters_2018} with an extra spatially dependent tunneling term.\nFurthermore, we will operatively define the Chern number by taking the limit of an ensemble of periodic and topologically equivalent states which progressively approximate quasiperiodicity. \nIn this limit, the Bloch bands and Berry curvatures become asymptotically flat, as already known~\\cite{kraus_topological_2012-1,harper_perturbative_2014}.\nFinally, we describe the experimental fingerprint of the quasiperiodic topological state, which reveals itself in the charge transport and adiabatic evolution of the center of mass of the atomic cloud.\nWhereas in the commensurate (nonquasiperiodic) case the current is not constant and the pumped charge is quantized only at exact multiples of the pumping cycle, we find that the quasiperiodic nontrivial state is characterized by a steady and topologically quantized pumping current, independently from the duration of the pumping process.\nMost importantly, we find that this quantization is exact up to exponentially small corrections, it is robust against perturbations which do not break the symmetries of the system, and does not depend on the details of the model considered.\nThis exact quantization is a direct consequence of quasiperiodicity, and may contribute to a more accurate definition of current standards~\\cite{kaneko_review_2016}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig2.png}%\n\\setlength{\\belowcaptionskip}{-4pt}\\setlength{\\abovecaptionskip}{2pt}\n\t\\caption{%\nEnergy spectra of the TB Hamiltonian \\eqref{Hk} calculated for $V=J$ and $K=0.25J$. \nThe large central gap has Chern number $C=1$ and is topologically equivalent to the RM model ($\\alpha=1\/2$).\nFor $K\\to0$ (not shown) the central gaps close at $\\alpha=p\/q$ for $q$ even.\nDifferent color shades correspond to different Chern numbers.\n}\n\t\\label{fig2}\n\\end{figure}\n\nExperimentally, Thouless pumps are realized by ultracold Fermi gases loaded into dynamically controlled bichromatic lattices~\\cite{nakajima_topological_2016,lohse_thouless_2016}.\nUsing a tilted setup~\\cite{nakajima_disorder_2020,matsuda_two-dimensional_2019} as in \\cref{fig1}, two sets of counterpropagating laser beams produce two standing waves with wavelengths $\\lambda_\\mathrm{S}$ and $\\lambda_\\mathrm{L}>\\lambda_\\mathrm{S}$ which intersect at an angle $\\theta$.\nFor an atomic cloud confined in the $x$ direction, the total dipole potential is\n\\begin{equation}\nV(x,\\phi)= \n V_\\mathrm{S}\\cos^2\\left(\\frac{\\pi x}{d_\\mathrm{S}}\\right)\n+ \nV_\\mathrm{L}\\cos^2\\left(\\frac{\\pi x}{d_\\mathrm{L}}-\\frac\\phi2 \\right),\n\\label{CH}\n\\end{equation}\nwhere \n$d_\\mathrm{S}=\\lambda_\\mathrm{S}$ and $d_\\mathrm{L}=\\lambda_\\mathrm{L}\\cos{\\theta}$ are respectively the short and long lattice constants,\n$V_\\mathrm{S,L}$ the lattice depths, and $\\phi$ the phase difference between the two lattices, which varies \nin time as $\\phi=\\nu t$ with instantaneous frequency $\\nu$.\nThe commensuration $\\alpha=d_\\mathrm{S}\/d_\\mathrm{L}=\\lambda_\\mathrm{S}\/(\\lambda_\\mathrm{L}\\cos{\\theta})$ between the two lattices is controlled by the tilting angle $\\theta$.\nWe assume a deep lattice regime $V_\\mathrm{S}> E_r$ (here, $E_r=h^2\/(8 M d_\\mathrm{S}^2)$ is the recoil energy of the short lattice~\\cite{bloch_many-body_2008}).\nIf $V_\\mathrm{S}> V_\\mathrm{L}$, the continuum Hamiltonian ${\\cal H}=p^2\/2M +V(x,\\phi)$ can be discretized using localized states at the short lattice minima and treating the long lattice as a perturbation~\\cite{roux_quasiperiodic_2008}.\nThis leads to an effective low-energy TB Hamiltonian corresponding to a generalized Harper equation which reads\n\\begin{align}\n&\n[-J \n\\!+\\! \n2K \\alpha\\sin{(\\pi\\alpha)} \\cos(2\\pi\\alpha (n +1) \\!-\\! \\phi) ] (\\psi_{n-1} \\!+\\! \\psi_{n+1})\n\\,+\\nonumber\\\\\n&\\qquad\n+ 2V\\cos(2\\pi\\alpha (n+1\/2)- \\phi) \\psi_{n} = E \\psi_{n}.\n\\label{HHe}\n\\end{align}\nThis is a generalization of the AAHH model, which includes an extra site-dependent tunneling term $K\\propto V_\\mathrm{L}$.\nMoreover, for $\\alpha=1\/2$ (staggered field), \\cref{HHe} reduces to the Rice-Mele (RM) model~\\cite{rice_elementary_1982,rice_mele_shen_topological_2017}\n\\begin{align}\n&\n [-J \\!-\\! K (-1)^n \\cos\\phi ] (\\psi_{n-1} \\!+\\! \\psi_{n+1})\n\\nonumber\\\\\n&\\qquad\n+ 2V(-1)^n\\sin\\phi \\,\\psi_{n} = E \\psi_{n},\n\\label{eq:RM}\n\\end{align}\nwhich has an energy gap $\\Delta E_\\mathrm{RM}=4\\min(|J|,|V|,|K|)$.\n\nIn the commensurate case, i.e., $\\alpha=p\/q$ with $p, q$ integer coprimes, one can verify that \\cref{CH,HHe} are invariant up to translations $n\\to n+q$, and consequently the superlattice unit cell has length $q d_\\mathrm{S}$.\nIn momentum space,\n\\begin{align}\n&\\qquad\nH= \n\\sum_k -2J\\cos{k}\\, c_k^\\dag c_k\n+\n e^{\\ii (\\pi \\alpha-\\phi)}\n \\nonumber\\\\\\times\n &\n \\left[\n V \\!+\\! 2K \\alpha\\sin{(\\pi\\alpha)}\n\\cos{(k\\!+\\!\\pi\\alpha)}\n\\right]\\!\nc^\\dag_{k} c_{k+2\\pi\\alpha}\n\\!+\\! \\text{H.~c.},\n\\label{Hk}\n\\end{align}\nwhere $k$ is restricted to the first BZ $[0,2\\pi\/q]$.\n\\Cref{fig2} shows the energy spectra of the TB model, which are a deformed version of the Hofstadter butterfly~\\cite{hofstadter_energy_1976,avila_ten_2009}.\nIndeed, whereas the Hofstadter butterfly ($K=0$) is symmetric with respect to the transformations $\\alpha\\to1-\\alpha$ and $E\\to -E$ (corresponding to $k\\to k+\\pi$), the spatially dependent tunneling term breaks these symmetries.\nFor small $K$, one can assume that the intraband gaps remain open for $K\\to0$ and are thus homeomorphic to the gaps of the Hofstadter butterfly.\nThus, the intraband gaps are topologically nontrivial with Chern number $C\\neq0$ satisfying the diophantine equation $p C \\equiv j \\mod q$ (analogously to the Hofstadter butterfly $K=0$).\nUnlike the original Hofstadter butterfly, the energy spectra is gapped at $E=0$ for $\\alpha=p\/q$ with $q$ even.\nIntraband gaps with low Chern numbers are generally wide and remain open for a broad range of the commensuration $\\alpha$.\nIn particular, the large central gap in \\cref{fig2} is open for any value of $\\alpha$ and is topologically equivalent to the RM model: \nIt can be continuously deformed into $\\alpha\\to1\/2$, where \\cref{HHe} reduces to \\cref{eq:RM}.\n\n\nIn the commensurate case, assuming homogeneously populated bands below the Fermi level $E_\\mathrm{F}$ and at zero temperature, the total charge pumped during an adiabatic evolution $\\phi\\to \\phi+ 2\\pi $ is quantized and equal to the Chern number $C$ of the filled Bloch bands~\\cite{thouless_quantization_1983}\n$\nQ=C=\n(1\/2\\pi)\n\\int_{\\phi}^{\\phi+2\\pi} \\dd \\phi\n\\int_\\mathrm{0}^{2\\pi\/q} \\dd k\n\\Omega\n$.\nHere, \n$\\Omega\n=\\sum_i\n\\Theta(E_\\mathrm{F}-E_i)\n\\omega_i\n$ \nis the total Berry curvature at the Fermi level $E_\\mathrm{F}$, \nwith\n$\\Theta(E)$ the Heaviside step function and \n$\\omega_i\n=2\\Im\n\\braket{\\partial_\\phi u_i | \\partial_k u_i}$\nthe Berry curvature of the $i$-th band, defined in terms of the Bloch wavefunctions \n$\\ket{\\psi_{i}(k,x)}=e^{\\ii k x}\\ket{u_i(k,x)}$.\nMoreover, the current \n$\nI=\\partial_\\phi Q=(1\/2\\pi)\n\\int_\\mathrm{0}^{2\\pi\/q} \\dd k\n\\Omega\n$\nis not quantized and not constant during the pumping process, oscillating around an average value \n$\\langle I \\rangle=\n\\langle\\Omega\n\\rangle\/{q}$ \nwith maximum variation\n$\\delta I \\leq \n\\delta\\Omega\n\/{q}$ \nwhere $\\delta\\Omega\n=\\max \\Omega\n-\\min \\Omega\n$.\n\nDue to translational invariance, Hamiltonian \\eqref{Hk} is periodic in the momentum $k\\to k+2\\pi\/q$, but not in the phase since $H(\\phi+2\\pi\/q)\\neq H(\\phi)$.\nOne can show that a phase shift $\\phi\\to\\phi+2\\pi m\/q$ in \\cref{CH,HHe} is equivalent to a translation $n\\to n-c$, where $c$ satisfies the diophantine equation $p c\\equiv m\\mod q$.\nThus, the Hamiltonian is ``unitarily'' periodic~\\cite{marra_fractional_2015,marra_fractional_2017} in the phase $\\phi$ up to lattice translations, i.e., it is periodic up to unitary transformations (translations),\n\\begin{equation}\n\\label{translation}\n H(\\phi+2\\pi m\/q)= {T}^{-c} H(\\phi) {T}^{c}, \n\\end{equation}\nwhere ${T}$ is defined by $T V(x,\\phi)T^{-1}= V(x+d_\\mathrm{S},\\phi)$.\nConsequently, energies and Berry curvatures are periodic in the phase $\\phi\\to\\phi+2\\pi\/q$, and the pumped charge at well-defined fractions of the pumping cycle $\\Delta\\phi=2\\pi m\/q$ is quantized as fractions of the Chern number~\\cite{marra_fractional_2015,marra_fractional_2017}\n$\nQ=m C\/q=\n(1\/2\\pi)\n\\int_{\\phi}^{\\phi+2\\pi m\/q} \\dd \\phi\n\\int_\\mathrm{0}^{2\\pi\/q} \\dd k\\,\n\\Omega\n$.\nMoreover, the energy bands, Berry curvatures, and  total Berry curvature become flat in the limit of large denominators $q$.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1\\columnwidth]{Fig3.pdf}%\n\\setlength{\\belowcaptionskip}{-4pt}\\setlength{\\abovecaptionskip}{2pt}\n\t\\caption{%\nPumped charge and current for the central gap with Chern number $C=1$ calculated with the continuous model (a,b) and the effective TB model (c,d) respectively.\nDifferent curves correspond to successive rational approximations of $\\alpha=1\/\\Phi^2$, corresponding to tilting angles $\\theta=\\arccos{(1\/(4\\alpha))}$ in the range between $\\ang{60}$ and $\\ang{49}$ for typical laser wavelengths $\\lambda_\\mathrm{S}=\\SI{266}{nm}$ and $\\lambda_\\mathrm{L}=\\SI{1064}{nm}$.\nThe pumped charge is quantized as $m C\/q$ for pumping periods $\\Delta\\phi=2\\pi m\/q$.\nIn the limit $\\alpha_n\\to\\alpha$, the charge has a linear dependence $Q=\\Delta \\phi C_\\alpha\/2\\pi$.\nThe current shows large fluctuations but becomes steady for large denominators $q$, reaching its quantized value $I_\\alpha=C\/2\\pi$.\nWe use $V_\\mathrm{S}=2 E_r$, $V_\\mathrm{L}=0.5 E_r$.\n(e)\nThe current approaches its quantized value exponentially as $\\delta I=|I-I_\\alpha|\\propto\\exp(-q\/\\xi)$.\nDifferent data sets correspond to $V=J, 1.25 J$ and $K=0.25 J, 0.375 J, 0.5 J$.\n}\n\t\\label{fig3}\n\\end{figure}\n\nWe now consider the incommensurate quasiperiodic case $\\alpha\\in\\mathbb R-\\mathbb Q$.\nEvery irrational number $\\alpha$ can be written uniquely as an infinite continued fraction~\\cite{continued_fractions_hardy} \n$\\alpha=[a_0; a_1, a_2, \\,\\ldots ] =\na_0+\n1\/(a_1 + \n1\/(a_2 + \n\\dots))$\nwith $a_i$ integers.\nSuccessive approximations obtained by truncating the continued fraction representation $\\alpha_n=[a_0; a_1, a_2, \\,\\ldots, a_n]$ are rational numbers, and converge to $\\alpha$.\nWe thus consider the ensemble of Hamiltonians $H^{(\\alpha_n)}$ describing commensurate systems with $\\alpha_n= p_n\/q_n=[a_0; a_1, a_2, \\,\\ldots, a_n]$.\nWe assume that the insulating gap at the Fermi level remains open, such that the Hamiltonians $H^{(\\alpha_n)}$ are topologically equivalent.\nAs the denominator $q_n$ increases for $n\\to\\infty$, the BZ $[0,2\\pi\/q_n]$ shrinks and becomes ill-defined in the quasiperiodic limit.\nThus, the usual definition of the Chern number as an integral of the Berry curvature in the BZ needs to be reformulated.\nHowever, energy bands and Berry curvatures become constant in the quasiperiodic limit ($q_n\\to\\infty$).\nHence, if the gap remains open for $\\alpha_n\\to\\alpha$, the Berry integral converges for $n\\to\\infty$, and we can define the Chern number in the quasiperiodic limit as\n\\begin{equation}\nC_\\alpha=\\!\\!\\lim_{\\alpha_n\\to\\alpha}\n\\frac1{2\\pi} \\!\n\\int_{0}^{2\\pi} \\!\\!\\!\\! \\!\\!\\! \\dd \\phi \\!\n\\int_\\mathrm{0}^{{2\\pi}\/{q_n}} \\!\\!\\! \\!\\!\\! \\dd k\\,\n\\Omega^{(\\alpha_n)} \n=\\!\\!\n\\lim_{\\alpha_n\\to\\alpha}\n\\frac{2\\pi}{q_n}\\, \\Omega^{(\\alpha_n)} \n.\n\\label{chern}\n\\end{equation}\nIn this limit, the Chern number is simply proportional to the total Berry curvature, which diverges asymptotically as $\\Omega^{(\\alpha_n)}\\sim q C\/2\\pi$.\nMoreover, since the total Berry curvature is flat, the charge pumped during adiabatic transformations, for any initial and final values of the phase $\\phi\\to\\phi+\\Delta\\phi$, becomes\n\\begin{equation}\n\\label{charge}\nQ_\\alpha=\n\\!\\!\\lim_{\\alpha_n\\to\\alpha}\n\\frac1{2\\pi} \\!\n\\int_{\\phi}^{\\phi+\\Delta\\phi} \\!\\!\\!\\! \\!\\!\\! \\dd \\phi \\!\n\\int_\\mathrm{0}^{{2\\pi}\/{q_n}} \\!\\! \\dd k\\,\n\\Omega^{(\\alpha_n)} \n=\n\\frac{\\Delta\\phi}{2\\pi} C_\\alpha,\n\\end{equation}\nwhereas the instantaneous charge current becomes\n\\begin{equation}\n\\label{current}\nI_\\alpha=\n\\!\\!\\lim_{\\alpha_n\\to\\alpha}\n\\frac1{2\\pi}\n\\int_\\mathrm{0}^{{2\\pi}\/{q_n}} \\!\\! \\dd k\\,\n\\Omega^{(\\alpha_n)} \n=\n\\frac{C}{2\\pi}\n\\end{equation}\nIn the quasiperiodic limit, the pumped charge becomes linear in the phase difference $\\Delta\\phi$, whereas the current $I=\\partial_\\phi Q$ becomes constant and proportional to the Chern number. \nNotice that, in order to observe the effects of quasiperiodicity, the system size $L$ must be larger than the unit cell $q d_\\mathrm{S}$.\nIn this sense, the limit $\\alpha_n\\to\\alpha$ corresponds to the infinite-size limit $L\\to\\infty$.\n\nThese effects are robust against perturbations which do not break translational symmetry.\nIn fact, adding a perturbation $\\lambda V$ in \\cref{translation}, one can verify that the perturbed Hamiltonian satisfies\n\\begin{equation}\n\\label{brokentranslation}\n H'(\\phi+2\\pi m\/q)= {T}^{-c} (H'(\\phi)+c \\lambda [T,V] T^{-1}) {T}^{c}.\n\\end{equation}\nIf translational symmetry is unbroken, this equation reduces to \\cref{translation}.\nIn this case, energy levels and Berry curvatures are still periodic and become flat in the quasiperiodic limit, and the current remains quantized.\nHowever, if $[V,T]\\neq0$, from \\cref{brokentranslation} one can expect polynomial corrections $O(\\lambda)$ to the energy levels and Berry curvatures.\nThus, spatial disorder is expected to break down the exact quantization of the current.\nHowever, disorder is usually negligible in optical lattices, contrarily to solid state systems.\n\n\nWe now consider the continuous Hamiltonian $\n{\\cal H}=\n{p^2}\/{2M}\n+V(x,t)\n+ V_\\mathrm{T} x^2\n$\ndescribing an ultracold atomic cloud in a bichromatic potential, confined by a shallow harmonic trap $\\propto V_\\mathrm{T}$.\nThe pumped current $I=\\partial_\\phi Q=\\partial_t Q\/\\nu$ is related to a simple physical observable, i.e., the center of mass of the atomic cloud.\nThe variation of the center of mass $\\langle x (t)\\rangle=(1\/N) \\sum_{i=1}^j \\int_{-\\infty}^\\infty |\\Psi_i(x,t)|^2 x \\dd x$ is proportional to the pumped charge~\\cite{marra_fractional_2015,wang_topological_2013}, i.e.,\n$Q=\\rho [\\langle x (t+\\Delta t)\\rangle - \\langle x (t)\\rangle]$,\nwhere $\\rho=j\/(q d_\\mathrm{S})$ is the number of atoms $j$ per unit cell.\nAssuming the number of filled bands to be $j \\equiv p C \\mod{q}$, the total length of a cloud of $N$ atoms is given by $N\/j$ unit cells (of length $q d_\\mathrm{S}$).\nHence the number of atoms $N$ must be multiple of the filling factor $j$, and the system length $L$ must be tuned such that\n\\begin{equation}\n\\label{condition}\nd_\\mathrm{S}\\frac{N}{L} \\equiv \\alpha C\\mod q.\n\\end{equation}\nMoreover, in order to minimize thermal and nonadiabatic effects, one should consider a filling factor $j=p$  corresponding to the large central gap in \\cref{fig2} with Chern number $C=1$.\nThis gap $\\Delta E$ has the same order of magnitude for a wide range of values of the commensuration $\\alpha$, including $\\alpha=1\/2$ where the system is equivalent to the RM model, i.e., $\\Delta E \\approx \\Delta E_\\mathrm{RM}$.\nThis fixes the temperature and timescales to $T<\\Delta E_\\mathrm{RM}\/k_\\mathrm{B}$ and $\\nu<\\Delta E_\\mathrm{RM}\/\\hbar$.\nNote that the RM quantum pump has been already realized experimentally~\\cite{nakajima_topological_2016,lohse_thouless_2016}. \nNote also that the experimental errors in measuring the center of mass can be reduced by averaging over a large number of cycles~\\cite{nakajima_topological_2016,lohse_thouless_2016}.\n\n\nFigure~\\ref{fig3} shows the pumped charge $Q$ and the current $I=\\partial_\\phi Q$ obtained by calculating the center of mass of the continuous system in the adiabatic limit and, alternatively, using the effective TB Hamiltonian \\eqref{Hk}.\nDifferent curves correspond to successive rational approximations of $\\alpha=1\/\\Phi^2\\in \\mathbb{R}-\\mathbb{Q}$, where $\\Phi$ is the golden ratio.\nWe tune the trapping potential such that the length $L$ satisfies \\cref{condition}.\nThe pumped charge is quantized as integer fractions of the Chern number $(m\/q) C$ for well-defined fractions of the pumping period $\\Delta\\phi=2\\pi m\/q$.\nFor increasing denominators $q$, the pumped charge is approximately $Q=\\Delta \\phi C_\\alpha\/2\\pi$, whereas the current approaches its quantized value $I_\\alpha=C_\\alpha\/2\\pi$ for $\\alpha_n\\to\\alpha$.\n\n\nHence, the pumped current $I_\\alpha$ in the quasiperiodic limit is quantized and equal to the Chern number (in elementary units).\nWe will now determine the asymptotic behavior of the current approaching the quasiperiodic limit.\nFor $K=0$, \\cref{Hk} reduces to the AAHH model: \nIn this case, it has been shown numerically and perturbatively~\\cite{harper_perturbative_2014} that the total Berry curvature takes the form \n$\n\\Omega^{(p\/q)}\\approx F+G e^{- q\/\\xi} [\\cos{(q k)}+\\cos{(q \\phi)}]\n$.\nIt is reasonable to extrapolate this result also to $K\\neq0$.\n\\Cref{chern} gives $F=q C\/2\\pi$, whereas $G\\propto q^2$~\\cite{harper_perturbative_2014}.\nHence, the flattening of the total Berry curvature is exponential, and  \n$\\delta\\Omega^{(q)}\\approx g q^2 e^{- q\/\\xi}$  asymptotically for large $q$, where $g>0$ is a constant.\nThus, the current approaches its quantized value as\n\\begin{equation}\n\\delta I=|I-I_\\alpha|\\lesssim\ng q \\\ne^{- q_n\/\\xi}\n\\approx\n\\frac{g \\ e^{- \\sfrac{1}{\\xi \\sqrt{D|\\alpha-\\alpha_n|}}}}{\\sqrt{D |\\alpha-\\alpha_n|}},\n\\label{scaling}\n\\end{equation}\nwhere $|\\alpha-\\alpha_n|\\sim 1\/{D q_n^2}$ with $D<\\sqrt{5}$, due to the Dirichlet's approximation theorem and Hurwitz's theorem~\\cite{continued_fractions_hardy}.\nThus, \\cref{scaling} describes the scaling behavior of the current in the quasiperiodic limit, in terms of the difference $|\\alpha-\\alpha_n|$ between the irrational commensuration $\\alpha$ and its successive rational approximations $\\alpha_n=p_n\/q_n$.\nThe denominator $q_n$ determines the length scale $L_n=q_n d_\\mathrm{S}$ where the effects of quasiperiodicity become relevant.\nConsequently, \\cref{scaling} mandates that corrections to the quantized value of the current are exponentially small in the system size $L$.\nThis is a distinctive fingerprint of topological quantization, and is analogous to the case of, e.g., the quantum Hall effect, where corrections to the quantized conductance are exponentially small in the linear dimensions of the system~\\cite{niu_quantum_1987,exponentially_small_topological_thouless_1998}.\n\\Cref{fig3}(e) shows the variations $\\delta I$ calculated numerically via \\cref{current} using the effective TB Hamiltonian \\eqref{Hk}.\nAs expected, the current approaches its quantized value $I_\\alpha=C_\\alpha\/2\\pi$ exponentially for $\\alpha_n\\to\\alpha$.\n\n\nIn summary, we have shown how a quasiperiodic and topologically nontrivial Thouless pump can be realized by an atomic gas confined in a quasiperiodic optical lattice, which is a superposition of two harmonic potentials with incommensurate periodicities.\nThis system is characterized by a topological invariant defined as the limit of the Chern numbers of an ensemble of topologically equivalent and periodic Hamiltonians.\nThe distinctive fingerprint of this quasiperiodic and topologically nontrivial state is the exact quantization of the current, which is a consequence of the flattening of the Bloch bands and of the Berry curvatures.\nThis exact quantization is measurable in a typical experimental setting of ultracold atomic gases in optical lattices, and may open new perspectives for a more accurate definition of current standards.\n\n\\begin{acknowledgments}\nP.~M. thanks Yoshihito Kuno, Michael Lohse, Shuta Nakajima, Yoshiro Takahashi, and Nobuyuki Takei for useful discussions. \nThe work of P.~M. is supported by the Japan Science and Technology Agency (JST) of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), JST CREST Grant~No.~JPMJCR19T2, by the (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities ``Topological Science'' (Grant No.~S1511006), and by JSPS Grant-in-Aid for Early-Career Scientists (Grant No.~20K14375).\nThe work of M.~N.~is partially supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) Grants No.~16H03984 and No.~18H01217 and by a Grant-in-Aid for Scientific Research on Innovative Areas ``Topological Materials Science'' (KAKENHI Grant No.~15H05855) from MEXT of Japan.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\baselineskip \n 24pt\nThe folding model with realistic effective nucleon-nucleon interactions\nhas given a good insight into the nucleus-nucleus interaction (see e.g. a \nrecent review by Brandan and Satchler~\\cite{Bra96}). However,  \nwhen a light ion interacts with other ions there is also a\npossibility of the breakup of the projectile into two or more\nfragments. If the breakup channel is strong, \nit will affect not only the imaginary potential but also the\nreal one. This leads to a dynamical polarisation potential\n(DPP) which has to be added to the real potential obtained by the folding\nmodel. The DPP due to the breakup process can,  \nfor example, be estimated from the adiabatic model proposed \nby Johnson and Soper~\\cite{John70}, or by more sophisticated\ncontinuum discretised coupled channels techniques~\\cite{Sak86}. The DPP \nis required to describe the elastic scattering of\nthe weakly bound projectiles like $^{6,7}$Li and $^{9}$Be in the folding\nmodel with reasonable renormalisation factors ~\\cite{Bra96}.\n\nExperimental studies have shown that the breakup probabilities\nincrease drastically with energy even for tightly bound projectiles \n~\\cite{Wu78, Bud78}. For example, the cross section for the  \nbreakup of the $\\alpha$ particle increases by, at least, an order of\nmagnitude as the beam energy is varied from 65 MeV to\n140 MeV~\\cite{Mei83,Koo79}. At higher beam energies ($\\geq$ 140 MeV)\nthe breakup cross section can be as large as 25$\\%$\nof the total reaction cross section. Thus the DPP could be required\nalso for tightly bound projectiles for beam energies above 30 MeV\/A.\n\nIn a recent report~\\cite{Ing96} one of us investigated the contributions\nto the $\\alpha$-particle optical potential from the ($\\alpha,^{3}$He) \nbreakup reaction on $^{62}$Ni target at the incident energy of 172.5 MeV.\nThe calculations proceed in two steps. First \nthe breakup probabilities are calculated within a theory\nwhich is formulated in the frame-work of the post form distorted-wave\nBorn-approximation (PFDWBA).\nThis theory has been found to reproduce the experimental data on the\nbreakup of light projectiles extremely well~\\cite{Shyam84}.  \nIn the second step, these probabilities are fitted to generate the breakup\npart of the optical potential which is assumed to consist of two parts; one\nwhich is entirely due to the breakup of the projectile and the another\nindependent of it. It was found in ~\\cite{Ing96} that breakup   \ncontributes substantially  to the optical potential.\nHowever, in ~\\cite{Ing96}, the DPP for the alpha elastic scattering was \nobtained by fitting only to the $(\\alpha,^3$He) breakup channel.\nFor a complete determination of the DPP due to the breakup process \nthe breakup probabilities for all the $\\alpha$-breakup channels, namely, \n($\\alpha$,p), ($\\alpha$,n), ($\\alpha$,d), \n($\\alpha,^{3}$He) and ($\\alpha$,t) should be calculated and fitted.\n\nBefore taking up this rather ambitious task, which nobody has investigated\nso far, we considered it worthwhile to perform calculations for the\nsimple system of the deuteron to test the method in detail. In this\ncase there are only two breakup channels, $(d,p)$ and $(d,n)$ and\nthe experimental data on the deuteron breakup (see e.g.~\\cite{Mat80})\nis quite comprehensive.  In this paper, we investigate the scattering\nof 56 MeV deuterons from $^{51}$V. The breakup probabilities for the\n$(d,p)$ and $(d,n)$ channels are calculated for this particular case.\nThese are fitted to determine the DPP due to the breakup process. \n\nIn sections 2 we describe the calculation of the breakup  \nprobabilities in the optical model. The calculation of the same within the\nPFDWBA theory of breakup is presented in section 3.\nThe results for the DPP obtained by fitting the total and elastic breakup\nprobabilities are discussed in sections 4.  Our conclusions are\npresented in section 5.\n \n\\section{Calculations of breakup probabilities from the optical model.}\n\nThe calculations are based on the same principles as in Ref.~\\cite{Ing96}.\nWe assume that the optical potential $V(r)$ is known from the phenomenological\nanalyses of elastic scattering data and solve the radial \nSchr\\\"odinger equation\n\\begin{equation}\n \\frac{d^{2}y_{\\ell}(r)}{dr^{2}} + [ k^{2}- U(r) -\\frac{\\ell(\\ell+1)}{r^{2}}]\n\\:y_{\\ell}(r) = 0,\n\\end{equation}\nwhere $k$ is the wavenumber and $U(r)$ is related to the optical \npotential $V(r)$ through $U(r)=(2m\/\\hbar^{2})V(r)$.\nThe solutions for the radial wavefunctions, $y_{\\ell}(r)$, are\n normalised according to\n\\begin{equation}\ny_{\\ell}(r) = e^{i\\delta_{\\ell}}[ \\cos\\delta_{\\ell} F_{\\ell}(kr) +\n\\sin\\delta_{\\ell} G_{\\ell}(kr)],\n\\end{equation}\nwhere $F_{\\ell}$ and $G_{\\ell}$ are the regular and irregular Coulomb \nfunctions.\nWith this normalisation the partial wave amplitudes, $T_{\\ell}$, are written \nin terms of the phase-shifts $\\delta_{\\ell}$ as \n\\begin{equation}\nT_{\\ell}= e^{i\\delta_{\\ell}} \\sin\\delta_{\\ell}\n\\end{equation}\nThe partial wave amplitudes may also be calculated from the relation\n\\begin{equation}\nT_{\\ell}=-k^{-1}\\int_{0}^{\\infty}F_{\\ell}(kr)U(r) y_{\\ell}(r) dr\n\\end{equation}\n\nNow we define the optical potential due to breakup as $U_{bu}(r)$ and write the \nfull potential $U(r)$ as $(U(r) - U_{bu}(r)) + U_{bu}(r)$. With this\ndefinition, Eq. 1 can be recast as\n\\begin{equation}\n \\frac{d^{2}y_{\\ell}(r)}{dr^{2}} + [ k^{2}- (U(r)-U_{bu}(r)) - U_{bu}(r)\n -\\frac{\\ell(\\ell+1)}{r^{2}}]\\: y_{\\ell}(r) = 0\n\\end{equation}\nA discussion of this  problem may be found in \nRef.~\\cite{RodTha,Wat69}.\n\nBy solving the Schr\\\"odinger equation for the \"bare\" potential $U(r)-U_{bu}(r)$, \n\\begin{equation}\n \\frac{d^{2}v_{\\ell}(r)}{dr^{2}} + [ k^{2}- (U(r)-U_{bu})\n -\\frac{\\ell(\\ell+1)}{r^{2}}]\\: v_{\\ell}(r) = 0,\n\\end{equation}\nwe obtain the radial wavefunctions $v_{\\ell}(r)$, the phase shifts\n$\\delta'_{\\ell}$, and the partial wave amplitudes, $T'_{\\ell}$.\n\nThe relation between the partial wave amplitudes are \n\\begin{equation}\n T_{\\ell} = T'_{\\ell} +T^{bu}_{\\ell}\n\\end{equation}\n\nThe partial wave amplitudes for breakup are thus given by the difference\nbetween the partial wave amplitudes obtained for the potentials \n$U(r)$ and $U(r)-U_{bu}(r)$,\nrespectively. These can also be calculated from \nthe formula (see e.g.~\\cite{RodTha,Wat69})\n\n\\begin{equation}\nT^{bu}_{\\ell}=-k^{-1}\\int_{0}^{\\infty}v_{\\ell}(kr)U_{bu}(r) y_{\\ell}(r) dr\n\\end{equation}\n\nIt should be noted that the integral in Eq. 8 contains the radial wavefunctions \n$v_{\\ell}$, obtained with the bare potential $U(r)-U_{bu}(r)$, as well as \nthe radial wavefunctions\n$y_{\\ell}$, obtained with the full potential $U(r)$. The assumption implicite\ntherein is that the bare potential cannot lead to the breakup reaction.\n\nThe breakup probability for a certain $\\ell$-value \nis determined from the partial wave amplitudes \n$T^{bu}_{\\ell}$ according to\n\\begin{equation}\n\\mathrm{Breakup \\ probability} = |T^{bu}_{\\ell}|^{2}\n\\end{equation}\n\nThese breakup probabilities are compared with those calculated within the\nPFDWBA theory of breakup reactions\n(discussed in the next section) to determine the breakup potential,\n$U_{bu}$. In our procedure, this potential (with certain ${\\it a priori}$\nassumed form) is varied to reproduce the breakup probabilities calculated\nwithin PFDWBA. It may be remarked here that when the potential $U_{bu}(r)$ is \nvaried, new values of $T'_{\\ell}$ have to be calculated from the solution \nof eq. 6, whereas the values of $T_{\\ell}$ remain unchanged. \n\nIt is interesting to note that the reaction cross section in the elastic\nscattering is known from the solutions of eq. (1). Furthermore the total \nbreakup cross section can be calculated from the breakup probabilities.\nEven then, the reaction cross section for the bare potential\n$U(r)-U_{bu}(r)$, will depend on the shape of the potential,  \n$U_{bu}(r)$. \n\nIn Ref.~\\cite{Ing96}, the peripheral dominance of the breakup process\nmade us parametrize the breakup potential as a Gaussian with the addition\nof a Woods-Saxon form factor. In this work, however, this preconceived\nview is avoided and a Fourier-Bessel (FB) expansion \n\\begin{equation}\nU_{bu}(r)= \\sum_{n=1}^{N} a_{n} \\frac{sin(n \\pi r)}{R},\n\\end{equation}\nis used for both the real as well as imaginary parts of the breakup potential.\nThe radius R was kept fixed at 18 fm in all calculations.\n\n\\section{Microscopic calculations of breakup probabilities.}\n\n\nIn the theory of the breakup reaction ($d \\rightarrow p + n$ ) formulated\nwithin the framework of the post form distorted-wave Born-approximation,\nthe probability of breakup  $P_{\\ell_a}^{b-up(d,p)}$ is defined\nby ~\\cite{shyam80}\n\\begin{eqnarray}\n\\sigma_{total}^{b-up}(d,p) & = & \\int d\\Omega_p dE_p \\frac{d^2 \\sigma(d,p)}\n                             {d\\Omega_p dE_p} \\nonumber \\\\\\\n                         & = & \\frac{\\pi}{k_d^2} \\sum_{\\ell_d} (2\\ell_d + 1)\n\t\t               P_{\\ell_d}^{b-up(d,p)},\n\\end{eqnarray}\nwhere $\\frac{d^2\\sigma(d,p)}{d\\Omega_p dE_p}$ is the double differential \ncross section for the inclusive breakup reaction (d,p), which is the sum\nof the elastic and inelastic breakup modes. These are given by\n\\begin{eqnarray}\n\\frac{d^2\\sigma(elastic)}{d\\Omega_p dE_p} & = & \\frac{\\mu_d \\mu_p \\mu_n}\n                              {(2\\pi)^5 \\hbar^6}\n                              \\frac{k_p k_n}{k_d}\n                              \\sum_{\\ell_n m_n}\\mid T_{\\ell_n m_n}\n                              \\mid^2,\n\\end{eqnarray}\nwhere the T-matrix $T_{\\ell_n m_n}$ is  \n\\begin{eqnarray}\nT_{\\ell_n m_n} & = &D_{0} \\int d^3 r \\chi_{p}^{(-)*}({\\bf{k_p}}, \n                     \\frac{A}{A+1}{\\bf{r}})\n                     \\frac{\\chi_{\\ell_n}}{k_nr} Y_{\\ell_n m_n}(\\hat r) \n                     \\chi_{d}^{(+)}( {\\bf{k_d,r}}) \\Lambda(r) P(r)\n\\end{eqnarray}\n$D_{0}$ is the well known zero range constant for the\n$d \\rightarrow p + n $ vertex.\n$\\Lambda(r)$ and $P(r)$ are the finite range and nonlocality correction\nfactors respectively. $\\chi^{\\pm}$ are the optical model wave functions\nin the respective channels with k's being the corresponding wave vectors.\n                \nThe inelastic breakup cross section is given by              \n\\begin{eqnarray}\n\\frac{d^2\\sigma(inelastic)}{d\\Omega_p dE_p} & = & \\frac{\\mu_d \\mu_p \\mu_n}\n                              {(2\\pi)^5 \\hbar^6}\n                              \\frac{k_p k_n}{k_d}\n       \\sum_{\\ell_n m_n}(\\sigma_{\\ell_n}^{reaction}\/\\sigma_{\\ell_n}^{elastic})\n                              \\mid T_{\\ell_n m_n}-T_{\\ell_n m_n}^0 \\mid^2\n\\end{eqnarray}\nIn this equation $\\sigma_{\\ell_n}^{reaction}$ and $\\sigma_{\\ell_n}^{elastic}$\nare the reaction and elastic scattering cross sections for the neutron - target    \nsystem corresponding to the partial wave $\\ell_n$ respectively. The $T$ matrix\n$T_{\\ell_n m_n}^{0}$ is defined in the same way as Eq. (13) with the\nelastics scattering wave function\n$\\chi_{\\ell_n}$ being replaced by the spherical Bessel function. \nIt may be noted\nthat Eq. (14) includes contributions from all the inelastic\nchannels of the neutron + target system ( see e.g ~\\cite{Shyam84} for complete detail ). \n\nThe angular integration in Eq. (13) is performed by introducing the partial \nwave expansion of the distorted waves and using the orthogonality of the \nspherical harmonics. The resulting slowly converging radial integrals are\nevaluated very effectively by following the contour integration technique\nof Vincent and Fortune ~\\cite{vin70}.\n\nWe require the optical potentials in the incident and outgoing channels\nas input in our calculations. In the results presented in this paper, the\ndeuteron optical potentials were taken from the global sets given by\nDaehnick, Childs and Vrcelj ~\\cite{dae80} whereas the potentials of \nBecchetti and Greenlees ~\\cite{becc69} were used in the neutron and\nproton channels. \n\nThe total breakup probability $P_{\\ell_d}$ is defined as following:\n\\begin{eqnarray}\nP_{\\ell_d}^{b-up,d} & = & P_{\\ell_d}^{b-up(d,pn)}(elastic) + \n                          P_{\\ell_d}^{b-up(d,p)}(inelastic) + \\nonumber \\\\\n                    &   &   P_{\\ell_d}^{b-up(d,n)}(inelastic)  \n\\end{eqnarray}                    \nIn Fig. ~\\ref{fig:figa} we show the results for the breakup \nprobability for the deuteron incident on a $^{51}$V target at the beam energy\nof 56 MeV. We can see that the $(d,p)$ and $(d,n)$ breakup probabilities\nare similar in shape and absolute magnitude. The elastic breakup\nprobability is much smaller and shows a different behaviour as a function\nof $\\ell_d$.\n\nAs discussed in Ref. ~\\cite{Shyam84} the total cross section can also \nbe expressed as  \n\\begin{equation}\n\\sigma_d^{breakup} = 2 \\pi \\int db\\; b\\; P_{\\ell_{d}}^{b-up,d}\n\\end{equation}\nwhere the impact parameter, $b$, is related to the angular momentum, $\\ell_d$,\nand the wavenumber $k_d$, through $b = (\\ell_d+ 1\/2)\/k_d$.\n\nThe cross sections for the inelastic (d,p), inelastic (d,n) and  \nelastic (d,pn) breakup processes were found to be 290 mb, 214 mb, and 122 mb\nrespectively. This leads to a total breakup cross section of 627 mb.\nIt may be noted that our total (d,p) breakup cross section \n(which is the sum of inelastic (d,p) and elastic (d,pn) cross sections)\nis 412 mb which is in  \nin reasonable agreement with the measured value of 481 mb reported by Matsuoka\net al. ~\\cite{Mat80}.\n\n\n\\section{Results and discussions}\n\nThe fitting procedure was the same way as that described in Ref. \n~\\cite{Ing96}. However, the imaginary part of the breakup potential,\n$U_{bu}$, was assumed to be less than that of the full potential $U(r)$, so\nthat ($U(r)-U_{bu}(r)$) was absorptive for all the radii. The \nerrors associated with the calculated breakup\nprobabilities in the optical model (Eq. (9) ) were about\n10$\\%$. In the fitting procedure, 14 coefficients ( $a^{n}$ as defined in \nEq. 10) were varied simultaneously for each potential to get \na minimum in the $\\chi^{2}$. \n\nIn the calculations for ($\\alpha-^{3}$He) breakup ~\\cite{Ing96} it was found\nthat the gross properties of the breakup probabilities could be reproduced\nwith a purely real as well as a purely imaginary breakup potential. An\nimaginary part of the \npotential was needed for fitting them at only small $\\ell$-values.\nHowever, in case of the deuteron we found it impossible to reproduce \nthe breakup probabilities ($P_{\\ell_d}$) without a complex breakup potential.\nOne of the reasons for this could be the fact that the breakup probabilities in the \nthis case are considerably larger than those for the\n($\\alpha-^{3}$He) reaction.  Fig. ~\\ref{fig:figb} shows the best fit\nobtained with a purely real breakup potential whereas Fig.~\\ref{fig:figc} \nwith a purely imaginary one. As can be seen from these figures, the imaginary\npotential is very important for reproducing the breakup probabilities at\nsmall $\\ell$-values, while the real part is required to reproduce these\nfor the grazing partial waves and beyond. This suppports the observation made \nin ~\\cite{Ing96b} that the real potential, due to non eikonal effects,\nincreases the absorption at large radii and enhances strongly the peripheral\ncollisions. \n\nThe best fit result obtained with a complex potential is shown in\nFig. ~\\ref{fig:figd}.  Both potentials include 14 terms in the FB expansion.\nIt can be seen that the breakup probabilities calculated within\nPFDWBA (shown in  Fig. ~\\ref{fig:figa}), are reproduced very well.\nThe imaginary part of the breakup potential is found to be \nvery strong. This is not surprising since the inelastic interactions\n(which could cause the breakup of the projectile) are \ntaken into account by the imaginary part of $U(r)$. These will automatically \nbe included in the imaginary part of $U_{bu}$. In the folding model\ncalculations of the elastic scattering where the imaginary part of the\npotential is phenomenologically determined, it is mainly the real potential  \nwhich is of interest. Therefore, our result suggest that the inclusion \nof the potential, $U_{bu}$, in these calculations will lead to a strong\nmodification to the real part of the folding model potentials.\n\nYabana et al ~\\cite{Yab92} have investigated the effects of breakup for\nthe elastic scattering of 80 MeV deuterons from $^{58}$Ni. \nBoth real and imaginary parts of the DPP obtained by these authors\nare weaker than those obtained in our work. It\nmay be remarked here that, these authors have neglected the Coulomb \nbreakup process, which can be quite large for the deuteron target\ninteraction~\\cite{Baur76, Shyam84}. In our work, on the other hand, this is \nincluded on the same footing as the nuclear breakup. Moreover, our breakup\nprobabilities lead to the cross sections which agree well with the\nexperimental data on the deuteron breakup.  \n\nWe should, however, stress that, there are (as in optical model \ncalculations) ambiguities in the breakup potentials obtained by us. For         \ninstance, in Fig. ~\\ref{fig:fige} we show the results obtained \nwhen the number of terms in the real potential is decreased from 14 to 4. \nThe fit to the PFDWBA breakup probabilities is\nstill reasonable. This therefore, makes it difficult to arrive at a \ndefinite conclusion about the shapes of the DPP. Nevertheless, it is \ninteresting to note that the real part of the DPP obtained in this way\nis still larger than those of Yabana et al.~\\cite{Yab92}\n\nNext we discuss the breakup probability for the elastic breakup ( a \nprocess in which the target nucleus remains in the ground state).\nIn theories like the one suggested by Bertsch, Brown and Sagawa\n~\\cite{Ber89} the reaction cross section in nucleus-nucleus collisions\nare calculated from collisions between nucleon-nucleon pairs in \nthe projectile and target nuclei. In such approaches it is necessary to\ninclude a correction due to the elastic breakup of the projectile.\nWe term the corresponding potential as the \"dissociation\" potential \n($U_{dis}$) to separate it from the potential $U_{bu}$ defined earlier.\nIt would also be worthwhile to study how the dissociation affects\nthe optical potential and the reaction cross section. $U_{dis}$ is obtained\nby fitting (in the same way as described above) the elastic breakup\nprobabilites as calculated by PFDWBA theory \n(see Fig. ~\\ref{fig:figa}). We stress that these include both \nCoulomb and nuclear breakup as well as their interference terms.  \n\n\nThe results obtained with a purely real $U_{dis}$ are shown in\nFig. ~\\ref{fig:figf}. The fits to the elastic breakup probabilities are \nsatifactory for the grazing partial waves. However, those for the \nlower partial waves are poorer. On the other hand attempts to fit\nthem with a purely imaginary $U_{dis}$ resulted\nin a very bad agreement.  Therefore we reduced the real potential shown \nin Fig. ~\\ref{fig:figf} and tried to reproduce the \ndata by a variation of the imaginary potential. The results from this\nsearch is shown in Fig. ~\\ref{fig:figg} and it is \nevident that the maximum is very badly reproduced.\nHowever, when both the real and imaginary potentials are varied\na good fit to the elastic breakup probabilities are obtained, even if\nthe shape of the imaginary part of $U_{br}$ so \nobtained looks somewhat unusual. Figure ~\\ref{fig:figh}  shows\none of the best fit potentials obtained in this way.\n\nAs discussed in section 2, the partial wave\namplitude for the potential $U(r)$ is given by the sum of the partial \nwave amplitudes the potentials,$U(r)-U_{bu}(r)$ and $U_{bu}(r)$.\nThis is not true for the reaction cross sections calculated from each \npotential separately, since these are sensitive to interference effects.\nThe reaction cross section for the full potential is 1571 mb.\nWith the real potential shown in Fig. ~\\ref{fig:figf} the reaction\ncross section for the potential, $U(r)-U_{bu}(r)$, is found to be 1405 mb.\nThe difference (166 mb) is somewhat larger than the value (122 mb) obtained\nfor the total elastic breakup cross section using Eq. (11).\nWith the complex potential shown in Fig. ~\\ref{fig:figh}, the\nreaction cross section without breakup potential\ndoes not decrease, instead it goes to a value of 2173 mb.\nThese examples show the importance of including the effects of the\ndissassociation in the optical model. The inteference effects are\nimportant and have to be treated correctly.\n\nOur calculations, therefore, indicate that the dissassociation of the deuteron\nshould give a substantial contribution to the dynamical polarisation\npotential. The assumption that the optical potential, obtained from a fit\nto elastic scattering data, include effects of dissassociation and \ninelastic breakup in a correct way must be questioned. \nThere are examples, as in Refs. ~\\cite{shyam80}, where the absorption due\nto breakup is even larger than the total absorption predicted by the optical\nmodel for large $\\ell$-values.\nSince the elastic breakup is considerably larger than the inelastic breakup\nfor large $\\ell$-values, this indicates that the optical \nmodels should be modified with a contribution of the dissassociation \nprocess, which may not necessarily have an effect on the angular\ndistributions for elastic scattering.  \n\n\\section{Conclusion}\n\nIn conclusion, our study of the effects of the breakup on the optical \npotential for the elastic scattering of 56 MeV deuterons from $^{51}V$ \nshows that breakup gives a substantial contribution to this potential. \nWe found that the strong enhancement of the peripheral collisions requires that\nthis contribution has a real part. Thus the folding model calculations should\ninclude a dynamical polarisation potential.\n\nThe investigations of the effects of the dissassociation also indicate\na contribution to the dynamical polarisation potential. However,\nthe conventional shapes used for the optical potential do\nnot account for the dissassociation process in a correct way. Therefore the \ninvestigation should be repeated with more sophisticated optical model \npotentials with different shapes.   \n\nWe plan to calculate breakup probabilities\nfor all breakup channels for $\\alpha$-particles. We believe that these \ntogether will give total breakup probalities of the same order as those\nof the deuterons studied here, at least for energies above 40 MeV\/A. \nWe see no reason why the breakup potential for $\\alpha$-particles\nshould be purely imaginary when it is complex for deuterons.\nTherefore we believe that all light particles require dynamical\npolarisation potentials in folding model calculations at higher\nenergies and that the energy dependence of effective interactions,\nwhich presently reproduce $\\alpha$-particle scattering, \nshould to be modified. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nEquilibrium and dynamic properties of vortex matter in highly anisotropic, \nlayered, high-temperature superconductors are known~\\cite{review} \nto be strongly affected by the presence\nof pinning. Effects of random columnar pinning produced by heavy-ion bombardment\nhave been studied theoretically~\\cite{NV93,Radz95}, experimentally~\\cite{Banerjee04} \nand numerically~\\cite{TG03,NH04,usprl,us2,us3} for the situation in which both\nthe columnar pins and the magnetic field are perpendicular to the layers. In\nthis geometry, if the areal concentration of \ncolumnar pins exceeds that of vortex lines\n(i.e. for $B_\\phi > B$, where $B_\\phi$ is the matching field and $B$ is the magnetic induction),\nthe vortex system exhibits a continuous Bose glass (BoG) \nto vortex liquid (VL) transition~\\cite{NV93} as the temperature is increased. If, on the\nother hand, the relative pin concentration $c \\equiv B_\\phi\/B$ is substantially\nsmaller than unity, then\nthe vortex system exhibits a first-order transition~\\cite{Radz95,Banerjee04} \nbetween a high-temperature VL and a \nlow-temperature BoG phase that has a polycrystalline structure~\\cite{Banerjee04,usprl,us2} \nwith grain boundaries\nseparating crystalline domains of different orientations. The VL into which the BoG melts\nhas the characteristics of an ``interstitial liquid'' in which some of the vortices remain\nlocalized at the columnar pins, producing solid-like regions around them, whereas the \nremaining, interstitial vortices form liquid-like regions. The pinned vortices delocalize\nat a ``depinning crossover'' that occurs at a higher temperature. Numerical \nstudies~\\cite{NH04,usprl,us2} also indicate the occurrence of a topologically ordered\nphase, analogous to the Bragg glass (BrG) phase~\\cite{natt,GLD95} in systems with random point pinning,\nat low temperatures if the relative concentration of columnar pins is sufficiently small. \n\nThe question of how the behavior described above is modified in the case of tilted\ncolumnar pinning, where the magnetic field is tilted away from the direction of the columnar\npins, has also received considerable attention in the past. Theoretical studies~\\cite{NV93,Hwa93}\nhave considered the geometry in which the columnar pins are perpendicular to the layers and\nthe applied magnetic field makes an angle $\\theta$ with the normal to the layers. These \nstudies predict that for $B < B_\\phi$, the effects of the correlated nature \nof the columnar pins become less\npronounced as the angle $\\theta$ is increased. Specifically, the vortex lines are predicted to\nremain locked to the columnar pins if $\\theta$ is sufficiently small, producing a ``transverse\nMeissner effect''. For larger values of the angle $\\theta$, the vortices hop from one columnar\npin to the next one, forming a staircase structure. As $\\theta$ is increased further, the \ndirectional effect of columnar pinning is lost and the vortex lines follow the field \ndirection. The low-temperature BoG phase phase persists for small values of $\\theta$, but\ndisappears as $\\theta$ is increased beyond a critical value. \nSome of these theoretical predictions have been verified in experiments~\\cite{Ammor04}.\n\nThe behavior of vortex systems with tilted columnar pinning and $B > B_\\phi$ ($c<1$) has been\ninvestigated recently in experiments~\\cite{zeldov07} and simulations~\\cite{zeldov07,gl07}.  \nThe experiments were performed on a sample of ${\\rm Bi_2Sr_2CaCu_2O_8}$\n(BSCCO) with a small concentration\nof random columnar pins tilted at an angle of 45$^\\circ$ from the normal ($z$-direction) to the\ncopper oxide layers. The magnitude and direction of the applied magnetic field $\\bf H$ were \nvaried and the location of the BoG to interstitial VL transition in the $H_z$ versus\ntemperature ($T$) plane was determined for several values of the tilting angle $\\theta$ between  \nthe directions of the magnetic induction $\\bf B$ and the columnar pins. The values of $\\bf H$\nconsidered in the experiment were such that the number density of pancake vortices on the\nlayers (determined by $B_z$) is higher than that of the columnar pins. The main result of\nthis experiment is that the temperature at which the BoG to VL transition occurs for a fixed\nvalue of $H_z$ is  {\\it independent} of the tilt angle $\\theta$. The temperature at which\nthe inhomogeneous VL (called ``vortex nanoliquid'' in Ref.\\onlinecite{zeldov07}) crosses over\nto the depinned, homogeneous liquid was also found to be independent of $\\theta$ for a fixed\nvalue of $H_z$. The simulations were performed for a fixed number density of pancake vortices\non the layers (fixed $B_z$) and different orientations of the columnar pins, keeping the\nnumber density of pinning centers on each layer fixed at a value lower than that of \npancake vortices. Both Josephson and electromagnetic interactions between pancake vortices\non different layers were included in the simulations. The results of the simulations were\nfound to be consistent with the experimental observation that the locations of the   \nthermodynamic transitions are independent of the angle between the columnar pins and\nthe applied field if the number densities of pancake vortices and pinning centers on each\nlayer (i.e. the values of $B_z$ and $B_\\phi \\cos\\psi$ where $\\psi$ is the angle between\nthe layer normal and the direction of the columnar pins) are held fixed.\n\nThese results are surprising because tilting the columnar pins away from the direction of\nthe layer normal introduces ``frustration'' in the system in the following sense. If the\npinning potential of each pinning center is sufficiently strong and the temperature sufficiently\nlow (these conditions are satisfied in the experiment and simulation described above), then\nnearly all the pinning centers on each layer would be occupied by pancake\nvortices. For columnar pins perpendicular to the layers,\nthe pinned vortices on different layers would then be\naligned directly on top of one another. This alignment of the pancake vortices \nin the direction of\nthe layer normal minimizes both the Josephson and electromagnetic interactions between\nvortices on different layers. However, if the columnar pins are tilted away from the layer\nnormal, then the pinned pancake vortices on different layers would not be aligned directly on top\nof one another, thereby increasing the energy associated with the interlayer\ninteractions of these vortices. For $B_z > B_\\phi \\cos\\psi$ (the case considered in \nRefs.~\\onlinecite{zeldov07,gl07}), interstitial pancake vortices that are not localized at \npinning centers can relieve this frustration to some extent by forming a staircase-like\nstructure in which they remain aligned in the direction of the layer\nnormal for a few layers and then shift in the direction of the tilt. This, however, would\nincrease the energy associated with the interaction of pancake vortices on the same layer\nbecause the positions of the interstitial vortices relative to those of the pinned ones,\nwhich shift in the direction of the tilt by a constant amount as one goes from one \nlayer to the next one, would not be optimal on all the layers. Thus, tilting the columnar\npins away from the direction of the layer normal should increase the frustration arising\nfrom the competition between the interaction of the vortices with the pinning centers and\nthe intervortex interactions. This should have a measurable effect on the transition \ntemperatures unless the energy associated with interlayer interactions among \nthe pancake vortices is  negligibly\nsmall compared to the other energy scales (the intralayer interactions and the pinning \nenergy) of the problem. Since increased frustration tends to lower the temperature at which\nan ordering transition occurs, the transition temperatures of the vortex system are expected\nto decrease as the tilting angle is increased from zero.\n\nTo shed some light on this problem, we have studied the structural and thermodynamic\nproperties of a system of pancake vortices in a strongly anisotropic, layered superconductor\nin the presence of tilted columnar pinning, using a mean-field, free-energy based numerical\nmethod developed in our earlier studies~\\cite{usprl,us2,us3,prbv,dv06,dv07} of vortex matter with\ndifferent kinds of pinning. In this method, the free energy of a system of pancake vortices\ninteracting among themselves and with pinning centers is written as a functional of the \ntime-averaged local areal density of the vortices. Only the electromagnetic interaction \nbetween pancake vortices on different layers is considered. Different phases, represented by\ndifferent local minima of the free energy,  \nare obtained by numerically minimizing the free energy, starting from different initial\nconfigurations of the local density. In this description,\na first order phase transition between two phases corresponds to a\ncrossing of the free energies of two distinct minima representing the two\nphases. Here, we use parameters appropriate for  BSCCO\nand fix the areal density of pancake vortices at a value corresponding to\n$B_z$ = 2 kG\nfor the component of the magnetic induction normal to the layers. \nThis corresponds to the experimental situation where the applied magnetic field $\\bf H$\nis in the $z$-direction and its magnitude is such that $B_z$ equals 2 kG.\nWe\nconsider different concentrations of columnar pinning centers, keeping their areal density \nsmaller than that of the pancake vortices, so that the relative pin concentration \n$c \\equiv B_\\phi \\cos\\psi\/B_z$\nis much smaller than unity. The columnar nature of the pins is modeled\nby repeating the positions of the pinning centers on successive layers with a constant\nshift in the case  of tilted pins.  We then compare the results obtained for tilted pins \nwith different\ntilting angles with those obtained for the same in-plane\nconfiguration of pinning centers, but without\nany tilt (without any shift for pins oriented in the direction\nof the layer normal) to analyze the effects of tilting the columnar pins. The main results of our study\nare summarized below.\n\nThe structural and thermodynamic properties of the systems with\ntilted columnar pins are found to be very similar to those found in our earlier\nstudies~\\cite{usprl,us2} of vortex systems in which both the magnetic field and a  \nsmall concentration of random columnar pins are perpendicular to the layers. Specifically,\nfor small values of the relative pin concentration $c$ defined above, we find,  at \nlow temperatures, two distinct \nminima of the free energy. At both these minima, nearly all the pinning \ncenters are occupied by vortices, and both the pinned and the interstitial vortices form\nlines that are tilted in the direction of the columnar pins. The degree of alignment of the \nvortices in the direction of the tilt is nearly perfect. \nOne of these two minima corresponds to the BoG phase in which the vortices on each layer\nexhibit substantial short-range translational and bond-orientational order, but topological\ndefects such as dislocations are present in small concentrations. The other minimum is\nalmost perfectly crystalline over the length scale of our finite samples and exhibits features\ncharacteristic of the topologically ordered BrG phase of systems with weak point pinning. \nAt temperatures close to the melting temperature of the vortex system without any pinning,\nthe BoG phase is the thermodynamically stable one with lower free energy. As the temperature\nis decreased, the free energy of the more ordered phase crosses that of the BoG phase at a first\norder phase transition, so that the BrG-like phase becomes the thermodynamically\nstable one at low temperatures. The minimum representing the BoG phase evolves continuously to the\nhigh-temperature, depinned VL as the temperature is increased -- we do not find a \nfirst-order transition to the VL for the pin concentrations considered in this study. Using a\ncriterion based on percolation of liquid-like regions~\\cite{us2,us3}, we define a crossover\ntemperature for the transformation of the BoG to the interstitial VL. This crossover\noccurs at a temperature higher than the melting temperature of the vortex lattice in pristine\nsamples without any pinning. For larger values of\nthe relative pin concentration $c$, the low-temperature BrG-like phase is absent and \nonly the crossover between the BoG and VL phases is found.\n\nAlthough the general behavior found for tilted columnar pins is qualitatively similar to\nthat of systems with ``vertical'' columnar pins normal to the layers, \na detailed comparison between the results\nfor the same vortex system with tilted and vertical columnar pins with the same \nin-plane arrangement\nof the pinning centers\nreveals, in contrast with\nsome previous studies,\\cite{zeldov07,gl07}\na  significant differences between the two cases. First, the temperature of the \nfirst-order transition between the BrG and BoG phases for small values of $c$\nis found to be lower by over 5\\% (about one degree) in the case of tilted pins. The temperature of the\nBoG to VL crossover for tilted pins is also decreased by a similar amount from \nthat for vertical columnar pins. Thus, the expected reduction in the transition temperatures\ndue to increased frustration in the tilted pin case is observed in our calculation. Second, the\ndegree of localization of the pancake vortices, measured by the heights of the local density\npeaks that represent vortex positions at the free-energy minima, is always slightly lower\nwhen the pins are tilted. This is true for both the vortices trapped at the pinning centers\nand the interstitial ones. This is a consequence of the additional tilting-induced \ncompetition between the pinning potential\nand interlayer vortex interactions mentioned above. This competition makes the pinning \ncenters less effective in trapping vortices\nand reduces the extent of in-plane order by decreasing the degree of localization of the \ninterstitial vortices. \n\nThe rest of the paper is organized as follows. The model considered  and the numerical\nmethods used in our study are described in section~\\ref{methods}. The results obtained \nin this study are described in\ndetail in section~\\ref{results}. We conclude in  section~\\ref{summary} \nwith a discussion of our main results in the context\nof those of earlier studies.\n\n  \n\\section{Model and Methods}\n\\label{methods}\n\n\n\nThe general method we use here is that of minimizing a mean-field free energy\nfunctional with respect to the time-averaged local vortex density $\\rho_n(\\bf\nr)$, where $\\bf r$ is a two dimensional vector denoting a location\nin the layer $n$. The free energy includes both intrinsic and pinning\nterms:\n\\begin{equation}\nF[\\rho]=F_{RY}[\\rho]+F_p[\\rho].\n\\label{fe}\n\\end{equation}\nFor the first term, we take the Ramakrishnan-Yussouff\\cite{ry} form: \n\n\\begin{widetext}\n\\begin{equation}\n\\beta F_{RY}[\\rho] = \\sum_{n}\\int{d {\\bf r}\\{\\rho_n({\\bf r})\n\\ln (\\rho_n({\\bf r})\/\\rho_0)-\\delta\\rho_n({\\bf r})\\} }\n -(1\/2)\\sum_m \\sum_n \\int{d {\\bf r} \\int {d{\\bf r}^\\prime\nC_{mn}({|\\bf r}-{\\bf r^\\prime|}) \\delta \\rho_m ({\\bf r}) \\delta\n\\rho_n({\\bf r}^\\prime)}} ,\n\\label{ryfe}\n\\end{equation}\n\\end{widetext}\nwhere $\\beta$ is\nthe inverse temperature and the integrals\nare two-dimensional. This free energy is defined with respect\nto that of a  vortex liquid with uniform density  \n$\\rho_0 = B_z\/\\Phi_0$ where $B_z$ is the component of the\nmagnetic induction in the direction ($z$ direction) normal to the layers \nand $\\Phi_0$  the superconducting flux quantum.\nIn the above expression,\n$\\delta \\rho_n ({\\bf r})\\equiv \\rho_n({\\bf r})-\\rho_0$ is the\ndeviation of  ${\\rho_n(\\bf r})$ from $\\rho_0$ and \n$C_{mn}(r)$ is the direct pair correlation\nfunction of the layered vortex liquid~\\cite{hansen}\nat density $\\rho_0$. \nThis static correlation function depends on the layer separation $|m-n|$\nand on the distance $r$ in the layer plane,\nand it\ncontains all the required information about the interactions in the system.\nAs in previous work on \ncolumnar pins\\cite{usprl,us2,us3} normal to the layers, we use here the $C_{mn}(r)$ obtained from a \ncalculation~\\cite{menon1} via the \nhypernetted chain approximation~\\cite{hansen} for parameter values appropriate\nfor  the layered material BSCCO.  \nWithin these premises, two material parameters enter\nthe calculations: the London penetration depth $\\lambda(T)$ and the\ndimensionless parameter $\\Gamma$:\n\\begin{equation}\n\\Gamma = \\beta d \\Phi^2_0\/8 \\pi^2 \\lambda^2(T).\n\\label{gamma}\n\\end{equation}\nwhere $d$ is the interplanar distance. We will\ntake here values appropriate\nto BSCCO; thus $d=15 \\AA$.\n\nThe second term in the right side of Eq.~(\\ref{fe}) is the pinning term and\nwe write it in the form:\n\\begin{equation}\nF_p[\\rho]= \\sum_n \\int{d {\\bf r} V^p_n({\\bf r}) [\\rho_n({\\bf r})-\\rho_0] },\n\\label{pinpot1}\n\\end{equation}\nwhere the pinning potential $V^p_n({\\bf r})$ is computed by summing over\nthe positions ${\\bf R}_{j,n}$ of the $j$th pinning center in the $n$th plane:\n\\begin{equation}\nV_n^p({\\bf r})=\\sum_{j} V_0(|{\\bf r}-{\\bf R}_{j,n}|),\n\\label{pinpot}\n\\end{equation}\nThe potential $V_0$ corresponding to a single pinning center\nis taken to be of the usual truncated parabolic form:\\cite{daf}\n\\begin{equation}\n\\beta V_0(r)=-\\alpha\\Gamma[1-(r\/r_0)^2]\\Theta(r_0-r)\n\\label{single}\n\\end{equation}\nwhere $r_0$ is the range. In terms of our unit of length $a_0$, defined\nby $\\pi a_0^2 \\rho_0=1$, we take $r_0=0.1 a_0$. For the strength $\\alpha$,\nwhich is a dimensionless number, we take the value ($\\alpha=0.05$) at\nwhich\\cite{prbv} each pinning center pins slightly less than\none vortex in the temperature range studied. This is the same value used\nin the previous  studies\\cite{usprl,us2,us3} of vertical columnar pins. \nThe number of vortices is\ndetermined by $B_z$ and we will consider here a fixed value $B_z$ =2 kG. \nAs in the numerical studies of Refs.~\\onlinecite{zeldov07,gl07}, the magnitude\nand direction of the applied magnetic field $\\bf H$ do not appear explicitly\nin our calculation. The situation we consider here may be realized experimentally\nby applying a magnetic field in the $z$-direction and adjusting its magnitude\nto yield the value of 2 kG for the $z$-component of the magnetic induction \n$\\bf B$ in the superconductor.  \nThe pinning columns make an angle $\\psi$\nwith the $z$ direction.  The relative pin\nconcentration $c$ is (equivalently with the definition given\nabove) the ratio of the number $N_p$ of columnar pins to the number\n$N_v$ of vortices in the system.\n\nTo study the phase diagram we discretize the position variable and\nnumerically minimize the free energy with respect to the discrete set\nof variables $\\rho_{n,i}$ where the index $i$ denotes a position in the $n$\nlayer of the discretized triangular lattice. We have\n$\\rho_{n,i} \\equiv \\rho_n({\\bf r_i}) A_0$  where\n$A_0= h^2 \\sqrt 3\/2$ is the area of the in-plane computational\ncell of lattice constant $h$. \nThe computational lattice is of size $N^2 \\times N_L$.    \nAs in previous work\\cite{usprl,us2,us3,prbv,dv06,dv07\nwe take $h=a\/16$ where $a = 1.99 a_0$ is the equilibrium value\\cite{prbv} of the lattice\nconstant of the system in the absence of pinning at the chosen value of\n$B_z$. The minimization procedure we use\\cite{cdo} ensures the non-negativity\nof the variables $\\rho_{n,i}$.\n\n\nThere are some computational issues in solving this problem which must\nbe explained here. We wish to consider the case where the pin concentration\n$c$ is much smaller than unity.\nWe also want to consider values of the tilt angle $\\psi$ in the reasonable\nexperimental range. The value of $N$ must be large enough so that the number\nof vortices present is not too small. The value $N_L$ of the number\nof layers in the computational lattice has to  be\\cite{dv06} at least several\nhundred. There are of course computational limitations: in our recent\\cite{dv07}\nwork on point pinning the total number of \ncomputational lattice sites attainable was $N_C=N^2N_L=2^{23}$.\nBut the main problem here is that the periodic boundary conditions in the $z$\ndirection impose, computationally, an effective ``quantization condition''\non the values of  $\\psi$ that can be used and, indirectly,\non the range of $c$ that can be studied. This occurs\nfor the following reason: implementation\nof  periodic boundary conditions is only possible\nif, after $N_L$ layers,\nthe pinning potential repeats itself. Assume that the potential due\nto one of the tilted columnar pins is such\nthat after an integer number $n$ of layers it has shifted horizontally\nby another integer $m$ of in-plane computational lattice sites. \nThe two  integers $n$ and $m$ determine the tilt angle via \n$\\tan \\psi =(m h\/nd)$. In order to implement the \nperiodic boundary conditions in the $z$ direction, the total horizontal\nshift (in units of $h$)  after $N_L$ layers, which is $(N_L \/n)m$,\nhas to equal $N$ so that $(N_L\/n)=(N\/m)$. Thus one also has \n$\\tan \\psi = (N h\/N_Ld)$. This implies, since $h\/d\\approx 70\/15$\nfor the chosen value of $B_z$, \nthat one needs\na large value of $N_L$ in order to keep $\\psi$ from being too large. But\none cannot increase $N_L$ arbitrarily, since the total number of computational\nsites $N_C$ must remain \nwithin feasible bounds. The value of $N_L$ must nevertheless be taken\nas large as possible, but, given $N_C$ and $N_L$, one must still have a\nnumber of vortices $N_v=(N\/16)^2$\nlarge enough. One has to note also that the\nvalue of $N_v$ puts a lower bound on the values of $c$ that can be studied,\nsince after all one cannot put less than one pin\nin the system.  Thus a complicated series\nof compromises must be made to optimize the parameter\nvalues for which data are obtained.\n\nWith the above in mind, the data presented here have been obtained with\n$N_L=1024$. Two values of $N$ have been used: most of\nthe data have been obtained for $N=96$ (which means\n$N_C=2^{20}3^2> 2^{23}$) and additional results\nwill be presented for $N=128$ ($N_C=2^{24}$). In the first case $N_v=36$\nand we have taken $N_p=4$ or $c=1\/9$ while  in the second case \n$N_v=64$ and we have taken\n$N_p=8$ and the somewhat larger concentration $c=1\/8$. The\nnumber of vortices in our samples is larger that that used in other computational\nwork.\\cite{gl07} At $N=96$ therefore, we have $\\tan \\psi=0.437$ while at \n$N=128$ we have a larger angle, $\\tan \\psi=0.583$. \n\n\\section{Results}\n\\label{results}\n\nWe can now discuss the results obtained using the methods described above. \nThe accuracy of these procedures has been repeatedly discussed in previous \nwork\\cite{usprl,us2,us3} and this issue and other technicalities need not be further elaborated\nupon here. \nThe iteration process continues until the system reaches a local free energy \nminimum. The structure of the system at that minimum is then inferred by \nanalyzing the vortex density structure, i.e., the set of variables $\\{\\rho_{n,i}\\}$. One \nneeds some initial condition to start the minimization procedure. If one starts\nwith perfectly disordered initial conditions, ($\\delta \\rho_{n,i}\\equiv 0$) and \none quenches to a sufficiently high temperature, one obtains a disordered \nminimum structure. The resulting values of\n$\\{\\rho_{n,i}\\}$ can then be used as the initial condition set at a nearby\n$T$. Ordered structures can be then obtained upon  cooling the system to a \nlower $T$ sufficiently slowly. Ordered states can also be obtained by using a \ncrystalline structure (we take that which minimizes\nthe pinning energy with \nrespect to all the symmetry operations of the lattice) as the\ninitial configuration. These ordered \nconfigurations can then be warmed up and of course, they eventually become\ndisordered. In general, the ordered configurations are to be identified, as\nwe will explain below, with \nBrG states while the disordered ones are BoG at lower $T$, becoming eventually \nliquid upon warming. At certain temperatures, more than one local \nminimum may be found, and the values of the free energy then establish which is \nthe stable configuration and which are only metastable.\n\n\\subsection{Structure of minima at $c=1\/9$}\n\nWe have studied three random pin configurations at $c=1\/9$, $N=96$ (as explained \nabove). The behavior for all three configurations is extremely consistent.\nFor each pin configuration, we have studied also, for comparison purposes,\nthe behavior of the system with the same pin configuration in the top layer but\nwith the pinning columns being normal to the layers, that is, parallel to\nthe $z$ crystal axis, instead of tilted ($\\psi=0$). \nIn so doing, we consider the same pin configuration\nat the same value\nof $N$ to avoid sample to sample variation or\nfinite size effects tainting the comparison.\\cite{old}\nThe value\nof $N_z$ is immaterial for vertical columns, since the problem should be quasi \ntwo-dimensional in this case, but we have explicitly verified that the results \ndo not change\nwhen $N_z$ is reduced from 1024 to eight. \n\nIt is important and very useful\nto visualize the structure of the free energy minima from\nthe values of the variables $\\{\\rho_{n,i}\\}$. One \nway of doing so is by considering  the {\\it vortex lattice} itself, as opposed to\nthe computational lattice. From the $\\{\\rho_{n,i}\\}$ set\nof values, we can locate the position of a vortex at site $i$ in the $n$ layer \nif the value of $\\rho_{n,i}$ at that site is larger that the value of \n$\\rho_{n,j}$ at any site $j$ within a distance $a\/2$ of site $i$.\nThe position of \nthese locations can then be directly plotted. This allows a clear visualization of \nthe arrangement of the vortices at different minima of the free energy.\n\nWe first address the question of the degree of alignment of the vortices along the\ntilted columnar pins. In our samples, the pin locations shift in the $x$-direction \nby $N=96$ spacings of the computational lattice across $N_L=1024$ layers. Therefore, \nthere is a shift of 3 spacings of the computational lattice after every 32 layers.\nIf the vortices are aligned with the columnar pins, then their positions would\nalso shift by 3 spacings of the computational lattice after every 32 layers. We \ncan check whether this happens by showing in the same plot the vortex positions\non layers $n_l = k+32l$ where $k$ is an arbitrary integer between 1 and 32 and \n$l=0,1,...,31$. To compensate for the expected shift due to the presence of the tilted\ncolumns, we shift the vortex positions on layer $n_l$ by $3l$ spacings of the \ncomputational lattice in the negative $x$-direction. Then, the plotted vortex positions\nafter the shifts on all the different layers, $l=0,1,...,31$, for any $k$ should lie on top of\none another if the vortices are aligned with the tilted pins. In Fig.\\ref{fig1}, we\nshow two such plots for two distinct local minima of the free energy at $T=17.8$K.\nAs discussed below (see Fig.~\\ref{fig6}) in detail,\nthese two minima correspond to the BoG (top panel)\nand BrG (bottom panel) states at a temperature close to the transition temperature\nat which their free energies cross. We emphasize that each plot shows the vortex\npositions on 32 different layers, corresponding to  $l=0,1,...,31$, shifted appropriately\nto compensate for a tilt in the direction of the columnar pins. The dots\nin the plot are the pin positions. All other symbols are\nvortex lattice sites, the precise meaning of their shapes and colors\nis explained below. The vortex positions\non these different layers are found to fall directly on top of one another after the\nshifts in both panels of Fig.~\\ref{fig1}, so that only\none symbol per site can be seen. This observation indicates that the vortices\nare almost perfectly aligned in the tilt direction.\n\n\\begin{figure}\n\\includegraphics [scale=0.5]{fig1a.eps}\n\\includegraphics [scale=0.5] {fig1b.eps}\n\\caption{(Color online) Vortex lattice structure for the \nBoG (top panel) and BrG (bottom panel) phases. \nThe temperature is $17.8 K$ where both\nthe phases are locally stable.\nEach plot shows vortex positions on 32 different \nlayers, appropriately shifted to compensate for a tilt along the pinning \ncolumns (see text). Dots represent pin\npositions, all other symbols are vortex positions. \nThe Voronoi analysis (see text) \nof the vortex structure is  shown\nby the symbol shape\n(and color).  The (black) circles represent ordinary six-fold\ncoordinated sites, (blue) triangles: 5-fold coordinated, (red) \nsquares: 7-fold coordinated. \n}\n\\label{fig1}\n\\end{figure}\n\nTo examine the degree of alignment of the vortices with the tilted pins for other\nvalues (not multiples of 32) of layer separation, we consider the quantity $d(n)$\nwhich is defined as\nthe average distance between a vortex site and its nearest neighbor \nin an adjacent plane separated by $n$ layers. This is plotted in Fig.~\\ref{fig2} as a \nfunction the separation $n$ between planes for the same\nBrG and BoG minima and temperature as in Fig.~\\ref{fig1}.\nIf the vortex lines are perfectly tilted, then, from the geometrical\nconsiderations in Sec.~\\ref{methods} and the numerical\nvalues given there, it follows that \na plot of $d(n)$ vs. $n$ should be a straight line with slope $s=(Nh)\/N_L \\simeq 0.01165 a_0$\nfor smaller values of $n$. Departure from a straight line is to be expected if\n$n$ exceeds the value for which $d(n)$ reaches a value close to $a_0$, since\n$a_0$ is (as previously mentioned) approximately\nhalf of the average spacing $a$ \nbetween nearest-neighboring vortices on a layer. This is because for such \nlarger values of $n$, the vortex\nin layer $(n+m)$ that has the smallest lateral separation from a vortex on layer $m$ is {\\it not}\nthe one located at a position shifted by $ns$ in the direction of the tilt from the position \nof the vortex in layer $m$. Thus, since $d(n)$ measures the smallest lateral separation \nbetween two vortices\nlocated on planes separated by $n$ layer spacings, the linear increase of $d(n)$ with $n$ should\nbe observed only for $d(n) \\lesssim a_0$ or \n$n \\lesssim a_0\/s \\approx 86$.\nOne can see from Fig.~\\ref{fig2} that\na straight line with the expected slope fits the results perfectly well in the relevant\n$n$ range and this, together with the argument in the previous paragraph shows that\nas stated in the Introduction, the \nvortex lines are indeed \nnearly perfectly tilted along the direction of the pinning columns. \nThis behavior is a consequence of the dominance of the pinning energy over interlayer\nvortex interactions for the realistic parameter values used in our calculation.\n\n\\begin{figure}\n\\includegraphics [scale=0.5] {fig2.eps}\n\\caption{(Color online) Distance $d(n)$ between a lattice point and\nit nearest neighbor in an  adjacent plane (see text) plotted\nvs layer separation $n$ for the BrG and BoG phases at $T=17.8 K$.\nThe circles are our results for the BoG minimum, the (blue) solid line \nrepresents the results for the BrG minimum, and the (red) dashed straight line \nshows the result expected for \n$d(n) \\lesssim a_0$ (see text)\nfor perfect alignment with the tilted columnar pins.}\n\\label{fig2}\n\\end{figure}\n\nTurning now to the structure\nin the $xy$ plane, we have analyzed the structure of the vortex arrangement in each plane\nby means of a Voronoi construction. A Voronoi construction in any lattice \nis performed by dividing it into cells, one cell\nper lattice point, each cell consisting of the region \nof space which is closest to a certain lattice point than to any other. For\na crystalline lattice, this is the Wigner-Seitz cell. In general, the \nnumber of sides of the Voronoi cell surrounding a lattice point \nis the number of neighbors of the lattice\npoint. The Voronoi analysis then reflects directly the defect structure.\nThe use of different symbols in Fig.~\\ref{fig1} is\nmeant to show examples of such Voronoi plots for the shifted lattice. \nWe see that from the point of view of the Voronoi construction there is a\ncontrast \nbetween the two cases shown, at the same $T=17.8 K$ where two phases are locally stable\nand  have approximately the same free\nenergy. The state in  the top panel contains a considerable number of defects,\nas can be seen by the adjacent site pairs with five or seven neighbors,\nwhile the state in the bottom panel contains none. Hence the first state\ncan at least tentatively \nbe identified as a BoG state while  the phase in the bottom\npanel, which in the spatial scale of the computation looks like\na perfect crystal, can be identified as a BrG with a more\nordered structure than the BoG. \n\nOne can alternatively describe the structure and verify the above\nidentifications by studying the density \ncorrelation functions. It is straightforward to extract from the \nvortex positions the in-plane angularly averaged two-point correlation function \n$g(r)$ of the vortex positions,\ndefined as\n\\begin{equation}\ng(r)= \\frac{A}{N_L N_v (N_v-1)} \\frac{\\sum_n \\sum_{i\\ne j} m(n,i) m(n,j) f_{ij}(r,\\Delta r)}\n{2 \\pi r \\Delta r},\n\\label{gofr}\n\\end{equation}\nwhere $m(n,i) = 1$ if the computational lattice site $i$ on layer $n$ corresponds to a vortex\nposition (i.e. if the local density peaks at this computational lattice site), and $m(n,i)=0$\notherwise, $A$ is the area of the sample in the $xy$-plane, and $f_{ij}(r,\\Delta r) =1$ if the distance\nbetween the lattice sites $(n,i)$ and $(n,j)$ lies between $r$ and $r+\\Delta r$ (we use $\\Delta r=0.2 a_0$ in\nour calculation), and $f_{ij}(r,\\Delta r)=0$ otherwise. The normalization of\n$g(r)$ is\nsuch that it should approach unity in the large-$r$ limit if there is no long-range\ntranslation order in the planes. For a perfect triangular lattice, the first 5 peaks of $g(r)$\nshould occur at $r=1.99a_0$, $3.44a_0$, $3.98a_0$, $5.26a_0$ and $5.97a_0$. This $g(r)$ is {\\it different}\nfrom the more familiar pair distribution function that measures the two-point \ncorrelation of the local density. In particular, information about the degree of\nlocalization of the local density peaks corresponding to the vortex\npositions is not contained in $g(r)$ because only the positions\nof these peaks are used in its calculation. \nExamples of $g(r)$ are plotted in Fig.~\\ref{fig3} for the\nsame two cases as in Fig.~\\ref{fig1}. Again, we see the contrast\nbetween the two cases. Although the relatively small size of the system precludes\nstudying the very long $r$ behavior, one can see that the correlation function\nfor the state which in Fig.~\\ref{fig1} exhibited\nno defects \nhas a more ordered structure \n(higher and better defined peaks at the values of $r$ for which sharp peaks are\nexpected for a triangular lattice) than the\none we tentatively identified as a BoG state based on the Voronoi constructions\nof Fig.~\\ref{fig1}. Thus, this analysis if $g(r)$  confirms the identifications made\nbased on direct visualization and the Voronoi construction.\n\n\\begin{figure}\n\\includegraphics [scale=0.5] {fig3.eps}\n\\caption{(Color online) Angularly averaged in-plane correlation function $g(r)$.\nResults for the same states and $T$ as in Fig.~\\ref{fig1} are shown.\nThe (black) line and circles are for the more ordered (BrG) state and the (blue) line\nand triangles are for the BoG. }\n\\label{fig3}\n\\end{figure}\n\nNext, in Figure \\ref{fig4} we consider a measure of the order as a\nfunction of temperature. There are a number of ways in which one can define an\n``order parameter'' and here we choose the value of $g(r)$ at its first $r>0$\npeak. This quantity, which we call $g_{max}$, \nis plotted as a function of $T$ for the same configuration\npresented in the previous figures. We\ndo this for both the BrG phase and\nthe BoG one. As we shall see below in the discussion associated with\nFig.~\\ref{fig6}, the BrG does not exist, even as a metastable, state for $T>17.6$\nand the same holds for the BoG at $T<17.0$, hence the ranges plotted. We see\nthat this quantity decreases with $T$ in either case but that it is considerably\nlarger in the BrG than in the BoG, as one would expect. At $T\\approx 17.5 K$ where, as\nwe shall see below, the free energies of the two states cross, there is a marked\ndiscontinuity in the equilibrium value of $g_{max}$. The nearly constant value of $g_{max}$\nfor the BoG phase at temperatures higher than 17.8K is a reflection of the above\nmentioned fact that the $g(r)$ considered here does not take into account \nthe broadening of the local density peaks with increasing temperature.\n\n\\begin{figure}\n\\includegraphics [scale=0.5] {fig4.eps}\n\\caption{ The quantity $g_{max}$ (see text)  used\nas a measure of the order parameter,\nplotted as a function of $T$ for both the BoG and\nthe BrG phases. Triangles: ordered (BrG)  state, Circles: BoG.}\n\\label{fig4}\n\\end{figure}\n\n\nWe end this section with a comparison of the structures of the BoG and BrG minima\nobtained for the same in-plane pin configuration, but for tilted pins in one case and \nvertical pins\nin the other case. In Fig.~\\ref{fig5}, we show vortex position plots similar to those in \nFig.\\ref{fig1} except that no Voronoi analysis is performed.\nThe top panel shows the data for the BoG phase at $T=18.4$K and the bottom panel shows the\nresults for the BrG phase at $T=17.8$K. It is clear from these plots that the in-plane\nstructure for tilted and vertical pins are very similar in both the BrG and BoG phases.\nThe degree of alignment with the pins is also found to be very similar for tilted and \nvertical pins. There are differences, however, between the vertical and tilted cases, as\nwe will see below.\n\n\\begin{figure} \n\\includegraphics [scale=0.5]{fig5a.eps}\n\\includegraphics [scale=0.5] {fig5b.eps}\n\\caption{(Color online) Comparisons of \nthe in-plane structures of BoG and BrG minima obtained (see text)\nfor tilted and vertical pins at the same\ntemperature and same in-plane pin configuration. Top panel: BoG phase at $T=18.4$K,\n(red) circles: vertical pins, (blue) triangles: tilted pins. Bottom panel: same\nfor the BrG phase at $T=17.8$K.}\n\\label{fig5}\n\\end{figure}\n\n\\subsection {Free energy and phase transitions}\nThe minimization procedure yields, of course, the value of the free energy at\neach local minimum. By considering the free energy values as a function of $T$\nthe possible phase transitions in the system can be studied. \nIn Fig.~\\ref{fig6} we show typical results at $c=1\/9$. The main plot is for the\ntilted case with $N=96$ in which case, as explained above, $\\psi=0.41$. The\nfree energy per vortex is plotted as a function of temperature. At high\ntemperatures only one state is stable. The corresponding free energy is plotted\nas the (red) crosses. By analyzing the results\nat each $T$  as explained in the previous\nsubsection, we find that this state is disordered, a  BoG. It exists down to\n$T=17.0$K, where, as one can see in  the figure, it becomes unstable to the other\nstate. This other state, the free energy of which is denoted by the (green) \n$\\times$ signs connected by dotted lines, is \nfound in the same way to be the BrG state. At temperatures\nin the range $17 K\\le T \\le 18.2 K$ both states can be found, one being of course\nonly metastable. The crossing of the free energies occurs at $T \\approx 17.6 K$  where\ntherefore a {\\it first order} transition occurs, as seen  by the difference in slopes\nof the free energy and the discontinuity of the order parameter in Fig.~\\ref{fig4}.\n\nThe insert shows, in a reduced temperature range, similar results for the same\npin configuration but at $\\psi=0$ (vertical pins). We see that in that case the\nfirst order transition occurs near $T=18.6 K$, about one degree higher than in the\ntilted case. This one degree shift occurs\nfor all pin configurations investigated at this value\nof $c$: although the values of the individual transition temperatures\nshow some sample-to-sample variation, the one degree shift\nalways occurs. We see then that for $c=1\/9$ increasing the angle $\\psi$ leads to a\nnotable decrease of the temperature at which the  BrG transforms to the BoG.\n\n\n\\begin{figure}\n\\includegraphics [scale =0.6]{fig6.eps}\n\\caption{(Color online) Free energy vs temperature. \nThe data points are the\nresults, the lines join the data points. The (green) $\\times$\nsymbols are for the BrG state and the (red) plusses for\nthe BoG. The main plot shows the free energy per\nvortex for a tilted configuration at $c=1\/9$ (see text). The inset shows the\nsame data, in a restricted region, for the same configuration but vertical\npinning lines.}\n\\label{fig6}\n\\end{figure}\n\nAt higher temperatures the BoG crosses over to an interstitial liquid phase. As\nwe have seen in the vertical pin case\\cite{usprl,us2,us3} this transition coincides with the\nonset of percolation of the liquid phase. The determination of this transition is\nshown in Fig.~\\ref{fig7}. The quantity plotted there is the fraction $f$ of the liquid-like\nlocal density peaks as a function of temperature. A vortex lattice site is assumed to\nbe liquid-like if\\cite{us2} the local value of $\\rho_{n,i}$ does not exceed\n$3\\rho_0$ (excluding of course the pinning sites). This fraction of liquid-like sites\nis small at lower $T$ and it rises rapidly up to temperatures higher than the first order\ntransition. Then it flattens somewhat and it crosses the value of $1\/2$ (the threshold\nvalue for site percolation on a triangular lattice) at a higher\ntemperature $T\\approx 18.4$K. We take this to be the temperature of crossing over\nfrom the BoG to the IL region. In Fig.~\\ref{fig7} results are also plotted for the\nvertical pins case. The percolation crossover is found to occur at a slightly higher\ntemperature, $T \\approx 19.0$K, for vertical pins.\n\n\\begin{figure}\n\\includegraphics [scale=0.5] {fig7.eps}\n\\caption{Fraction $f$ of liquid-like sites for BoG minima, plotted as a function\nof $T$. The triangles are for the tilted pin system, the circles for\nthe same configuration but with vertical pins.}\n\\label{fig7}\n\\end{figure}\n\nThere are some additional\nnoteworthy differences between the tilted pin results and the results\nfor vertical pins.\nWe have already seen that the\ntransition temperatures from BrG to BoG (Fig.~\\ref{fig6})\nand from Bog to IL (Fig.~\\ref{fig7}) \nare  higher for vertical pins.\nIn the Fig.~\\ref{fig6} plots one can also observe\nthat the difference in the slopes at the crossing, which is a measure of the\nlatent heat per vortex, is smaller in the vertical pin\ncase, as compared to the tilted situation. \nAn additional difference is plotted in\nFig.~\\ref{fig8}. There we plot, in  a semilog scale, the local density peak height  \nas a function of coordinate in the $x$ direction, for both the tilted case\n(plotted with lines ending with dots) and the vertical one (triangles). This is done in one panel\nat $T=18.4 K$ in the BoG phase and at $T=17.8 K$ (BrG) in the other panel. It is\nstriking that in both cases the peak heights for vertical pins are always\nhigher.\n\nThe free energy per vortex\nis somewhat {\\it lower} (at the same $T$) in the vertical case at lower values\nof $T$ but the difference becomes negligible at sufficiently high temperatures\nwhere the stable state is the BoG in both cases.  This occurs, we think,\nfor the following reason: in Fig.~\\ref{fig8} and similar data, the integrated\nvortex densities with values close to unity correspond to pin locations, indicating that the\npins are almost fully occupied by vortices. Therefore the portion of the free energy arising\nfrom inter-plane electromagnetic interactions will tend to be higher in the tilted case.\nHowever, Fig.~\\ref{fig8} also shows that the smaller peak densities away\nfrom the pinning columns are also higher for vertical pinning columns. This\nmeans that the density distribution in the vertical case is more localized,\nwhich is consistent with the higher transition temperature. At or above the\nmelting temperature of the pure vortex system without pins,\na more localized\ndensity distribution will tend to have larger contributions to the free energy\narising from entropy and in-plane interactions. At temperatures  above $18.4 K$\nthe free energy gain arising from the lower localization in the tilted case\nbasically cancels the free energy cost from the inter-plane vortex interaction.\n\n\\begin{figure}\n\\includegraphics [scale=0.5] {fig8a.eps}\n\\includegraphics [scale=0.5] {fig8b.eps}\n\\caption{ (Color online) Peak height vs position for $c=1\/9$, comparing  vertical and tilted\n($\\tan \\psi=0.437$) cases. Results at $\\psi=0$ are shown as the (red) triangles and\nthose of the tilted case by the (blue) dots and impulses. The top panel is for $T=18.4 K$\nand BoG states while the bottom one is at $T=17.8 K$ (BrG).}\n\\label{fig8}\n\\end{figure}\n\n\n\\subsection{Results at $c=1\/8$}\n\nWe have also studied a somewhat higher concentration, $c=1\/8$ at $N=128$. This\ncorresponds to a somewhat larger tilt angle, $\\tan \\psi= 0.583$. Results for\nthe obtained equilibrium structure are given in Fig.~\\ref{fig9}. The top panel\nof this  figure\nshows the vortex lattice structure along with the results of a Voronoi analysis\n(completely analogous to Fig.~\\ref{fig1}).\nDespite the very low value of the temperature ($T=16.8 K$) we find that a good number\nof defects remain and that the structure is the same as the BoG one in the top\npanel of Fig.~\\ref{fig1}. This is confirmed in the bottom panel of Fig.~\\ref{fig9}\nwhere we plot, at the\nsame $T$, the correlation function $g(r)$ as in Fig.~\\ref{fig3}. We see \n(compare with Fig.~\\ref{fig3}) that\nthe correlation function structure is  of the BoG type. This remains the situation\ndown to the lowest temperatures reached ($T=15.2 K$). The free energy per vortex is not\ntoo different from that in the $c=1\/9$ case (Fig.~\\ref{fig5}) but \nno instability to a more ordered state is found, down to \nthe lowest $T$ attained. It is possible to obtain\nBrG like structures\nby quenching to low\ntemperatures with initial conditions corresponding to a crystal: \nwe have done so by quenching to $T \\le 16.0 K$  but the\nresulting free energy values are considerably higher than those for the BoG at\nthe same $T$. Thus\nin this case only the BoG\nis found as an equilibrium state. \n\nWe conclude that at these values of $c$ and $\\psi$ no transition\nto a BrG occurs except possibly at much lower temperatures. We have also studied the\nsame pin configuration, at this value of $N$, for vertical pins. We have again found no\nBoG to  BrG transition upon cooling. We conclude then that the change in $c$, not the different\nvalue of $\\psi$, is responsible for the different behavior found in\nthe two cases studied here.\nThe high sensitivity of the possible BoG to BrG  transition to $c$  should not\ncome as a surprise. In previous work (see in particular Fig.~1 of\nRef.~\\onlinecite{us3}) for vertical columns and a much larger\nvalue of $N$, where because the problem is quasi two-dimensional\nwe were able to map the phase diagram in the $(T,c)$ plane\nat constant field, we found that the line in\nthe $(T,c)$ plane separating the BrG from the BoG, while nearly vertical at \nsmall $c$, eventually curves sharply and then becomes nearly horizontal,\nreflecting a very strong dependence of the transition temperature on $c$ and\nleading in fact to the disappearance of this first order transition at somewhat\nlarger $c$.  This is quite consistent with what we find here.\nThere are however small quantitative differences\nwith the results of Ref.~\\onlinecite{us3}. Here,\nwe find that the BrG phase is still present at low temperatures for $c=1\/9$, whereas \nRef.~\\onlinecite{us3} reported this phase  absent for $c > 1\/32$. \nThe transition and crossover \ntemperatures found here are also slightly different from the values reported in our\nearlier work. We believe that this\nis due to the large difference between the sizes of the systems considered.\nSince the system with vertical columnar pins is effectively two-dimensional, \nit was possible to study\nmuch larger systems (with $N_v=4096$,  about 100 times larger than those\nconsidered here) in our earlier studies. The smallness of the system size in\nthe present study makes the results quantitatively less reliable: this is clear from the observed\nsample-to-sample variations of the transition and crossover temperatures.\nOur earlier results  obtained for much larger samples, \nwould be more reliable for vertical pins. \nThe purpose of considering vertical pins in the present work was to make\na direct comparison with the behavior for tilted columnar pins without having to worry about\nsample to sample variations or finite size effects.\n\n\\begin{figure}\n\\includegraphics[scale=0.5] {fig9a.eps}\n\\includegraphics[scale=0.5] {fig9b.eps}\n\\caption{Analysis of the structure of a BoG minimum for a tilted pin configuration at\n$N=128$, $c=1\/8$, $T=16.8 K$. In the top panel we have shown the results of \na Voronoi analysis. The symbols\nmean the same as in Fig.~\\ref{fig1}. In the bottom panel we have $g(r)$ plotted in\nthe same way as in Fig.~\\ref{fig3}.}\n\\label{fig9}\n\\end{figure}\n\n\n\\section{Summary and discussions} \n\\label{summary}\n\nOur detailed \ncomparison of the results  for the thermodynamic behavior of the \nvortex system in the presence\nof a dilute array of tilted columnar pinning centers \nreveals significant quantitative differences between this system\nand a smilar system with  vertical pinning columns, normal to the layers, in the same\nin-plane configuration.   \nThe thermodynamic behavior of the tilted pins system is \nhowever qualitatively similar to that\nfound in our earlier studies~\\cite{usprl,us2,us3} of the vortex system with  \ncolumnar pins perpendicular to the layers. In both cases, all the pins are occupied \nby vortices if the relative concentration $c$ of the pinning centers is small. In the\ntilted case, we find that the interstitial vortices are well-aligned in the tilt direction.\nIf the relative pin concentration is low ($c=1\/9$), the low-temperature phase exhibits the\ncharacteristics of a Bragg glass. As the temperature is increased, this phase transforms, via\na first order transition, to a more disordered BoG phase which crosses over to\nan interstitial liquid at a slightly higher temperature. For a higher pin concentration\n($c=1\/8$), the Bragg glass phase is absent and the system exhibits only the crossover \nfrom the low-temperature BoG to the interstitial liquid phase as the temperature\nis increased. This is qualitatively similar to what occurs in the vertical pins case.\n\nQuantitatively, \nthe temperatures at which the\ntransition from the BrG phase to the BoG phase and the crossover from the BoG to the interstitial\nliquid occur are found to be appreciably higher (by about one degree, or over 5\\%) \nin the vertical pin case. The degree\nof localization of the vortices in the low temperature, solid-like phases is also significantly\nhigher for vertical pins. We attribute these differences to the ``frustration'' in the tilted case, \narising from a competition between the interlayer vortex interaction, which is minimized when the\npancake vortices on different layers are stacked in the vertical direction, and the pinning energy\nwhich is minimized when the vortices are aligned in the direction of the tilt. This competition\nalso makes the free energy in the tilted case slightly higher than that for vertical pins at\nlow temperatures, as we have seen. \nThese physical effects of tilting the columnar pins away from the layer\nnormal should be observable in experiments.\n\n\nThe differences we find between the results for vertical and tilted pins seem to contradict\nsome  experimental~\\cite{zeldov07} studies\nwhich concluded that the thermodynamic behavior of the vortex system is independent of the angle\nbetween the magnetic field and the tilt direction if the areal densities of pancake vortices and \npinning centers on each layer are held fixed. It is important to understand the reasons for this\napparent disagreement. In the experiment of\nRef.~\\onlinecite{zeldov07}, the effects of changing the angle between the magnetic field and the \ndirection of columnar pins were explored by changing the field direction for a sample with \ncolumnar pins tilted by\n45$^\\circ$ from the layer normal. This is not the same as the situation considered in our study.\nIn an isotropic superconductor, the individual directions of the field and the columnar pins are\nnot important: the behavior of the vortex system is determined by the angle between the two\ndirections. But for highly anisotropic layered materials such as high-$T_c$\nsuperconductors, the directions of both the field and the columnar pins \nare important. The experiment of Ref.~\\onlinecite{zeldov07} did not present any \ncomparison between the results obtained for the two cases considered in our study:\none in which the pins are tilted away from the layer normal, and the other in which the pins\nare perpendicular to the layers, but the areal densities of the pins and pancake vortices\non each layer are the same as those in the first case. Since the measurements for different \norientations of the field were carried out for the same sample with the columnar pins tilted\naway from the direction of the layer normal, the frustration effects mentioned above, \narising from the competition between interlayer interactions and pinning, were present in all the\nmeasurements. In contrast, these frustration effects are not present in one of the cases\n(vertical pins) considered in our study. Thus there is no real contradiction.\nIn view of our results, an experiment that makes \na comparison between the thermodynamic behavior in \nthe two cases considered in our study would be very interesting.\n\nIt is more difficult to understand the reason for the difference between our results \nand those of Langevin simulations~\\cite{zeldov07,gl07} performed on systems very similar to those\nconsidered in our study. The simulations described in these papers were carried out\nfor both vertical and tilted columnar pins, keeping the areal densities of pinning centers and\npancake vortices fixed. Both electromagnetic and Josephson interactions between pancake vortices \non different layers were included. Since both these interactions prefer\nvortices on different layers to stack up in the direction of the layer normal, the frustration\narising from the competition between these interactions and the pinning potential for tilted\ncolumnar pins is expected to be stronger in these simulations in comparison to that in our \nstudy which considers only the electromagnetic interaction. However, these simulations did not\nfind any significant difference between the results for vertical and tilted pins. This disagreement\nwith the results of our study may be a consequence of differences in \nsystem parameters. The values of $c$ used in the simulations ($c=0.35$ and $0.5$) are substantially higher\nthat those (1\/9 and 1\/8) considered here. A large concentration of pinning centers\nhas the effect of reducing the \nrelative importance of the interlayer interactions by making the pinning energy the dominant\nterm in the total energy of the vortex system. In fact, it is argued in \nRefs.~\\onlinecite{zeldov07,gl07} that the cost in Josephson and electromagnetic \nenergies due to the tilting of the vortices is negligibly small compared to the gain in pinning \nenergy for the parameters used in the simulations. If this is so, then it is not surprising that\nthe simulations did not find any difference between the thermodynamic behavior for tilted and\nvertical pins. It is also possible that the  simulations are not \nsufficiently accurate to capture the \nfairly small differences between the results for the two cases\nfound in our study. The relatively small size of the simulated systems ($N_v=36$, \nand a number of layers $N_L=200$,\nwhich is substantially smaller than that considered in our study) implies that there \nwould be large fluctuations in the quantities measured in the simulations. This would lead to\nsubstantial uncertainties in the determination of transition temperatures -- it is well-known that\nit is very difficult to determine transition temperatures accurately from simulations of\nsmall systems. \nThe authors mention in Ref~.\\onlinecite{gl07} that \ntheir simulation is not accurate enough to determine transition temperatures with an accuracy of\n$1 K$. Since the differences between the transition and crossover temperatures for tilted and\nvertical pins found in our study are of the order of $1 K$, these differences would not be\ndetected in the simulation.\nSome of the detailed comparisons between the results for the \ntwo cases, shown in Fig.~8 of Ref.~\\onlinecite{gl07}, are actually in agreement with our observations.\nFor example, it is shown in panel (c) of Fig.~8 of  Ref.~\\onlinecite{gl07} that the mean-square displacement\nof the vortices from their equilibrium positions is slightly higher in the tilted case. This is\nvery similar to the results shown in Fig.~\\ref{fig8} above.\nWe expect that the other differences between the results for vertical\nand tilted columnar pins found in our study will also be observed in simulations if the measurements\nare done with sufficient accuracy at the same values of $c$ and other relevant parameters.\n\n\n\n\n\\begin{acknowledgments}\nThis work was supported  in part by NSF (OISE-0352598) and by\nDST (India).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}