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arXiv:cond-mat/9904408 · Journal of Physics A Mathematical and General

High-precision Monte Carlo study of the 3DXY-universality class

M Hasenbusch, T Török

Published Aug 26, 1999• 113 citations
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Abstract

We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.

DOI
10.1088/0305-4470/32/36/301

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We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.
We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional L×L×LL\times L \times LL×L×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)α=-0.01293(48)α=−0.01293(48) and ν=0.67098(16)ν=0.67098(16)ν=0.67098(16). Our estimate for the correlation-length exponent ννν, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate νexp=0.6709(1)ν_{\rm exp}=0.6709(1)νexp​=0.6709(1) at the superfluid transition of 4^44He in a microgravity environment.
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 Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)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=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-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 improve the theoretical estimates of the critical exponents for the three-dimensional XY\mathrm{XY}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 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 study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. In addition to the XY model, we study the three-dimensional two-component ϕ 4 model on the simple cubic lattice. The parameter of the ϕ 4 model is chosen such that leading corrections to scaling are small. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R ϒ = 0.411(2), R B = 2.83(1), R ξ + = 0.3562(10) and R ξ − = 0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the λ-transition of 4 He.