arXiv:1211.0780

High-precision Monte Carlo study of the three-dimensional XY model on GPU

Ti-Yen Lan, Yun-Da Hsieh, Ying-Jer Kao

Published Nov 5, 2012• 3 citations

Abstract

We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional $L\times L \times L$ cubic lattice using the graphics processing unit (GPU). By the combination of Metropolis single-spin flip, over-relaxation and parallel-tempering methods, we simulate systems up to L=160. Performing the finite-size scaling analysis, we obtain estimates of the critical exponents for the three-dimensional XY universality class: $α=-0.01293(48)$ and $ν=0.67098(16)$. Our estimate for the correlation-length exponent $ν$, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate $ν_{\rm exp}=0.6709(1)$ at the superfluid transition of $^4$He in a microgravity environment.

DOI: 10.48550/arxiv.1211.0780

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arXiv:1211.0780

High-precision Monte Carlo study of the three-dimensional XY model on GPU

Ti-Yen Lan, Yun-Da Hsieh, Ying-Jer Kao

Published Nov 5, 2012• 3 citations

arXiv HTML PDF

Abstract

We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional

L\times L \times L

cubic lattice using the graphics processing unit (GPU). By the combination of Metropolis single-spin flip, over-relaxation and parallel-tempering methods, we simulate systems up to L=160. Performing the finite-size scaling analysis, we obtain estimates of the critical exponents for the three-dimensional XY universality class:

α=-0.01293(48)

and

ν=0.67098(16)

. Our estimate for the correlation-length exponent

ν

, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate

ν_{\rm exp}=0.6709(1)

at the superfluid transition of

^4

He in a microgravity environment.

DOI: 10.48550/arxiv.1211.0780

Related papers

Theoretical estimates of the critical exponents of the superfluid transition in He4 by lattice methods

Massimo Campostrini B, Martin Hasenbusch B, Andrea Pelissetto C +1 more

We improve the theoretical estimates of the critical exponents for the three-dimensional $XY$ universality class that apply to the superfluid transition in $^{4}\mathrm{He}$ along the $\ensuremath{\lambda}$ line of its phase diagram. We obtain the estimates $\ensuremath{\alpha}=\ensuremath{-}0.0151(3)$, $\ensuremath{\nu}=0.6717(1)$, $\ensuremath{\eta}=0.0381(2)$, $\ensuremath{\gamma}=1.3178(2)$, $\ensuremath{\beta}=0.3486(1)$, and $\ensuremath{\delta}=4.780(1)$. Our results are obtained by finite-size scaling analyses of high-statistics Monte Carlo simulations up to lattice size $L=128$ and resummations of 22nd-order high-temperature expansions of two improved models with suppressed leading scaling corrections. We note that our result for the specific-heat exponent $\ensuremath{\alpha}$ disagrees with the most recent experimental estimate $\ensuremath{\alpha}=\ensuremath{-}0.0127(3)$ at the superfluid transition of $^{4}\mathrm{He}$ in a microgravity environment.We improve the theoretical estimates of the critical exponents for the three-dimensional $XY$ universality class that apply to the superfluid transition in $^{4}\mathrm{He}$ along the $\ensuremath{\lambda}$ line of its phase diagram. We obtain the estimates $\ensuremath{\alpha}=\ensuremath{-}0.0151(3)$, $\ensuremath{\nu}=0.6717(1)$, $\ensuremath{\eta}=0.0381(2)$, $\ensuremath{\gamma}=1.3178(2)$, $\ensuremath{\beta}=0.3486(1)$, and $\ensuremath{\delta}=4.780(1)$. Our results are obtained by finite-size scaling analyses of high-statistics Monte Carlo simulations up to lattice size $L=128$ and resummations of 22nd-order high-temperature expansions of two improved models with suppressed leading scaling corrections. We note that our result for the specific-heat exponent $\ensuremath{\alpha}$ disagrees with the most recent experimental estimate $\ensuremath{\alpha}=\ensuremath{-}0.0127(3)$ at the superfluid transition of $^{4}\mathrm{He}$ in a microgravity environment.

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We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class. We fine alpha=-0.0146(8), gamma =1.3177(5), nu =0.671 55(27), eta =0.0380(4), beta =0.3485(2), and delta =4.780(2). We observe a discrepancy with the most recent experimental estimate of alpha; this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class. We fine alpha=-0.0146(8), gamma =1.3177(5), nu =0.671 55(27), eta =0.0380(4), beta =0.3485(2), and delta =4.780(2). We observe a discrepancy with the most recent experimental estimate of alpha; this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.

Jan 1, 20010 citations

Critical behavior of the three-dimensional XY universality class

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We improve the theoretical estimates of the critical exponents for the three-dimensional $\mathrm{XY}$ universality class. We find $\ensuremath{\alpha}=\ensuremath{-}0.0146(8),$ $\ensuremath{\gamma}=1.3177(5),$ $\ensuremath{\nu}=0.67155(27),$ $\ensuremath{\eta}=0.0380(4),$ $\ensuremath{\beta}=0.3485(2),$ and $\ensuremath{\delta}=4.780(2).$ We observe a discrepancy with the most recent experimental estimate of $\ensuremath{\alpha};$ this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.We improve the theoretical estimates of the critical exponents for the three-dimensional $\mathrm{XY}$ universality class. We find $\ensuremath{\alpha}=\ensuremath{-}0.0146(8),$ $\ensuremath{\gamma}=1.3177(5),$ $\ensuremath{\nu}=0.67155(27),$ $\ensuremath{\eta}=0.0380(4),$ $\ensuremath{\beta}=0.3485(2),$ and $\ensuremath{\delta}=4.780(2).$ We observe a discrepancy with the most recent experimental estimate of $\ensuremath{\alpha};$ this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.

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Monte Carlo study of an improved clock model in three dimensions

Martin Hasenbusch B

We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with ${Z}_{8}$ symmetry. At the transition, with increasing length scale, $O(2)$ symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to $L=512$ . The field variable requires less memory and the updates are faster than for a model with $O(2)$ symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional $XY$ -universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with ${Z}_{8}$ symmetry. At the transition, with increasing length scale, $O(2)$ symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to $L=512$. The field variable requires less memory and the updates are faster than for a model with $O(2)$ symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional $XY$-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.

arXiv:1910.05916Physical review. B./Physical review. BDec 26, 2019107 citations

Critical properties of the three-dimensional XY model in large-scale Monte-Carlo simulations

Inho Jeon, Jong-Geun Shin, Min-Chul Cha

Journal of the Korean Physical SocietyFeb 1, 20122 citations

We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional

L\times L \times L

α=-0.01293(48)

and

ν=0.67098(16)

. Our estimate for the correlation-length exponent

ν

, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate

ν_{\rm exp}=0.6709(1)

at the superfluid transition of

^4

He in a microgravity environment.

We improve the theoretical estimates of the critical exponents for the three-dimensional

XY

universality class that apply to the superfluid transition in

^{4}\mathrm{He}

along the $\ensuremath{\lambda}$ line of its phase diagram. We obtain the estimates $\ensuremath{\alpha}=\ensuremath{-}0.0151(3)$, $\ensuremath{\nu}=0.6717(1)$, $\ensuremath{\eta}=0.0381(2)$, $\ensuremath{\gamma}=1.3178(2)$, $\ensuremath{\beta}=0.3486(1)$, and $\ensuremath{\delta}=4.780(1)$. Our results are obtained by finite-size scaling analyses of high-statistics Monte Carlo simulations up to lattice size

L=128

and resummations of 22nd-order high-temperature expansions of two improved models with suppressed leading scaling corrections. We note that our result for the specific-heat exponent $\ensuremath{\alpha}$ disagrees with the most recent experimental estimate $\ensuremath{\alpha}=\ensuremath{-}0.0127(3)$ at the superfluid transition of

^{4}\mathrm{He}

in a microgravity environment.

We improve the theoretical estimates of the critical exponents for the three-dimensional

\mathrm{XY}

universality class. We find $\ensuremath{\alpha}=\ensuremath{-}0.0146(8),$ $\ensuremath{\gamma}=1.3177(5),$ $\ensuremath{\nu}=0.67155(27),$ $\ensuremath{\eta}=0.0380(4),$ $\ensuremath{\beta}=0.3485(2),$ and $\ensuremath{\delta}=4.780(2).$ We observe a discrepancy with the most recent experimental estimate of $\ensuremath{\alpha};$ this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.

{Z}_{8}

symmetry. At the transition, with increasing length scale,

O(2)

symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to

L=512

. The field variable requires less memory and the updates are faster than for a model with

O(2)

symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional

XY

-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.