arXiv:1210.6116 · Journal of the Physical Society of Japan

Large-Scale Monte Carlo Simulation of Two-Dimensional Classical XY Model Using Multiple GPUs

Yukihiro Komura, Yutaka Okabe

Published Oct 12, 2012• 80 citations

Abstract

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.

DOI: 10.1143/jpsj.81.113001

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arXiv:1210.6116 · Journal of the Physical Society of Japan

Large-Scale Monte Carlo Simulation of Two-Dimensional Classical XY Model Using Multiple GPUs

Yukihiro Komura, Yutaka Okabe

Published Oct 12, 2012• 80 citations

arXiv HTML PDF

Abstract

DOI: 10.1143/jpsj.81.113001

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\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.

The two-dimensional

S=1/2

XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to

128 \times 128

near and below the critical temperature. The critical temperature is estimated as

T_{\rm KT} = 0.3427(2)J

. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

{\mathrm{L}}_{0}

), which is inferred from the solution of the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

We study the two-dimensional generalized XY model that depends on an integer

q

by the Monte Carlo method. This model was recently proposed by Romano and Zagrebnov. We find a single Kosterlitz-Thouless (KT) transition for all values of

q

, in contrast with the previous speculation that there may be two transitions, one a regular KT transition and another a first-order transition at a higher temperature. We show the universality of the KT transitions by comparing the universal finite-size scaling behaviors at different values of

q

without assuming a specific universal form in terms of the KT transition temperature

T_{\rm KT}