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arXiv:cond-mat/0202161 · Physical review. B, Condensed matter

Probability-changing cluster algorithm for two-dimensional XY and clock models

Yusuke Tomita, Yutaka Okabe

Published Apr 12, 2002• 91 citations
arXivHTMLPDF

Abstract

We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff's idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) $\mathrm{XY}$ and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),$ we determine the KT transition temperature and the decay exponent $\ensuremath{\eta}$ as ${T}_{\mathrm{KT}}=0.8933(6)$ and $\ensuremath{\eta}=0.243(4)$ for the 2D $\mathrm{XY}$ model. We investigate two transitions of the KT type for the 2D q-state clock models with $q=6,8,12$ and confirm the prediction of $\ensuremath{\eta}{=4/q}^{2}$ at ${T}_{1},$ the low-temperature critical point between the ordered and $\mathrm{XY}$-like phases, systematically.

DOI
10.1103/physrevb.65.184405

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We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff's idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) XY\mathrm{XY}XY and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, \ensuremathξ\ensuremath∝exp(c/T/TKT\ensuremath−1),\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),\ensuremathξ\ensuremath∝exp(c/T/TKT​\ensuremath−1​), we determine the KT transition temperature and the decay exponent \ensuremathη\ensuremath{\eta}\ensuremathη as TKT=0.8933(6){T}_{\mathrm{KT}}=0.8933(6)TKT​=0.8933(6) and \ensuremathη=0.243(4)\ensuremath{\eta}=0.243(4)\ensuremathη=0.243(4) for the 2D XY\mathrm{XY}XY model. We investigate two transitions of the KT type for the 2D q-state clock models with q=6,8,12q=6,8,12q=6,8,12 and confirm the prediction of \ensuremathη=4/q2\ensuremath{\eta}{=4/q}^{2}\ensuremathη=4/q2 at T1,{T}_{1},T1​, the low-temperature critical point between the ordered and XY\mathrm{XY}XY-like phases, systematically.
We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L=65536, we investigate the Kosterlitz-Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β_{KT} is 1.11996(6). We estimate the exponent to specify the multiplicative logarithmic correction, -2r, and precisely reproduce the theoretical prediction -2r=1/8.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the 8-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ = 1 and different values of variance: from σ2 = 0.1 to σ2 = 3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. Our results show that, a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.
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 have examined the critical properties of the q=6 clock (vector Potts) model in two dimensions through Monte Carlo simulations. The model was investigated on L\ifmmode\times\else\texttimes\fi{}L square lattices of size L=4 to L=72 with periodic boundary conditions. We found an intermediate XY-like phase between a low-temperature ordered phase and a high-temperature disordered phase. The phase transitions occur at kB{k}_{B}kB​T1{T}_{1}T1​/J=0.68\ifmmode\pm\else\textpm\fi{}0.02 and kB{k}_{B}kB​T2{T}_{2}T2​/J=0.92\ifmmode\pm\else\textpm\fi{}0.01 and are of the Kosterlitz-Thouless type. The susceptibility diverges in the intermediate phase and the exponent \ensuremath{\eta} varies between 0.100 at T1{T}_{1}T1​ and 0.275 at T2{T}_{2}T2​.
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.