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Physical review. B, Condensed matter

Critical behavior of the two-dimensionalXYmodel

Rajan Gupta, CLIVE F. BAILLIE

Published Feb 1, 1992• 196 citations
Publisher

Abstract

We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat ${\mathit{C}}_{\mathit{v}}$ for the XY model. The simulations are done on ${64}^{2}$, ${128}^{2}$, ${256}^{2}$, and ${512}^{2}$ lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, ${\mathit{T}}_{\mathit{c}}$=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.

DOI
10.1103/physrevb.45.2883

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We report two sets of high-precision Monte Carlo simulations of the two-dimensional XY model in Villain's formulation on large square lattices, employing the single-cluster update algorithm. In one set of simulations we use improved estimators to study the correlation length \ensuremath{\xi} and susceptibility \ensuremath{\chi} in the high-temperature phase. On a 1200\ifmmode\times\else\texttimes\fi{}1200 lattice this allows measurements up to \ensuremath{\xi}\ensuremath{\approxeq}140 with statistical errors less than 0.25%. We judge quantitatively the advantage of using improved estimators, estimate autocorrelation times, and compare the numerical efficiency of various definitions of the correlation length. From least-square fits to these data we find clear support for the exponential divergence predicted by Kosterlitz and Thouless. The other set of simulations is performed in the vicinity of the transition point. Here we apply finite-size-scaling theory to obtain an estimate for the exponent \ensuremath{\eta} at criticality.
Using two sets of high-precision Monte Carlo data for the two-dimensional XY model in the Villain formulation on square L\ifmmode\times\else\texttimes\fi{}L lattices, the scaling behavior of the susceptibility \ensuremath{\chi} and correlation length \ensuremath{\xi} at the Kosterlitz-Thouless phase transition is analyzed with emphasis on multiplicative logarithmic corrections (ln L${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the finite-size scaling region and (ln \ensuremath{\xi}${)}^{\mathrm{\ensuremath{-}}2\mathrm{r}}$ in the high-temperature phase near criticality, respectively. By analyzing the susceptibility at criticality on lattices of size up to 5122{512}^{2}5122 we obtain r=-0.0270(10), in agreement with recent work of Kenna and Irving on the finite-size scaling of Lee-Yang zeros in the cosine formulation of the XY model. By studying susceptibilities and correlation lengths up to \ensuremath{\xi}\ensuremath{\approx}140 in the high-temperature phase, however, we arrive at quite a different estimate of r=0.0560(17), which is in good agreement with recent analyses of thermodynamic Monte Carlo data and high-temperature series expansions of the cosine formulation.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.
We present detailed results for the susceptibility $\ensuremath{\chi}$, correlation length $\ensuremath{\xi}$, and specific heat ${c}_{\ensuremath{\nu}}$ for the XY\mathrm{XY}XY model. The best fit to $\ensuremath{\chi}$ and $\ensuremath{\xi}$ data is obtained by use of the Kosterlitz-Thouless scaling form with $\ensuremath{\nu}=0.500(1)$, but $\ensuremath{\eta}$ shows considerable deviation from \textonequarter{} down to T=1.03T=1.03T=1.03. The critical temperature is estimated to be Tc=0.898(2){T}_{c}=0.898(2)Tc​=0.898(2). The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices with use of an overrelaxed algorithm which decorrelates as $\ensuremath{\tau}\ensuremath{\approx}0.15{\ensuremath{\xi}}^{1.2}$. Similar reduction in critical slowing down is anticipated for all continuous-spin models.
Results of Monte Carlo simulations of the ferromagnetic classical XXZ model on a square lattice are reported for interactions of the out of plane spin components \ensuremath{\lambda} varying between 0 and 0.99. For all the value of \ensuremath{\lambda} the signature of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition was observed, and the corresponding critical temperature was located both by BKT fits and finite-size scaling methods. Data for in-plane and out of plane susceptibility, correlation length, specific heat, and internal energy are given in a broad range of temperature and for lattice sizes ranging from 32\ifmmode\times\else\texttimes\fi{}32 to 256\ifmmode\times\else\texttimes\fi{}256.