Journal of the Korean Physical Society

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

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

Published Feb 1, 2012• 2 citations

Abstract

The critical properties of the three-dimensional XY model are studied in large-scale Monte-Carlo simulations by using a worm algorithm for the integer-current representation of the model. In large systems, singular behavior of the internal energy around the critical point is observed, and the finite-size scaling analyses of the superfluid stiffness lead us to find the critical inverse temperature K c = 0.45416(1) and the correlation length critical exponent ν = 0.6708(5).

DOI: 10.3938/jkps.60.581

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Journal of the Korean Physical Society

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

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

Published Feb 1, 2012• 2 citations

Publisher

Abstract

DOI: 10.3938/jkps.60.581

Related papers

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Massimo Campostrini B, Martin Hasenbusch B, Andrea Pelissetto C +2 more

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.

arXiv:cond-mat/0010360Physical review. B, Condensed matterMay 2, 2001386 citations

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.

We present a Monte Carlo study of the Bose glass to superfluid transition in the three-dimensional Bose-Hubbard model. Simulations are performed on the classical

(3+1)

dimensional link-current representation using the geometrical worm algorithm. Finite-size scaling analysis (on lattices as large as $16\ifmmode\times\else\texttimes\fi{}16\ifmmode\times\else\texttimes\fi{}16\ifmmode\times\else\texttimes\fi{}512$ sites) of the superfluid stiffness and the compressibility is consistent with a value of the dynamical critical exponent

z=3

, in agreement with existing scaling and renormalization group arguments that

z=d

. We find also a value of $\ensuremath{\nu}=0.70(12)$ for the correlation length exponent, satisfying the relation $\ensuremath{\nu}\ensuremath{\ge}2∕d$. However, a detailed study of the correlation functions, $C(r,\ensuremath{\tau})$, at the quantum critical point are not consistent with this value of

z

. We speculate that this discrepancy could be due to the fact that the correlation functions have not reached their true asymptotic behavior because of the relatively small spatial extent of the lattices used in the present study.

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.