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Physical review. B, Condensed matter

Finite-size scaling study of the three-dimensional classicalXYmodel

Yinghong Li, Stephen Teitel

Published Nov 1, 1989• 105 citations
Publisher

Abstract

The three-dimensional classical XY model on a cubic lattice has been studied using Monte Carlo simulations. The finite-size scaling method of the phenomenological renormalization group has been used to calculate the critical exponents \ensuremath{\nu}, v, and \ensuremath{\beta} of the correlation length, helicity modulus, and order parameter. Good agreement with series-expansion results is obtained.

DOI
10.1103/physrevb.40.9122

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The three-dimensional classical XY model on a cubic lattice has been studied using Monte Carlo simulations. The finite-size scaling method of the phenomenological renormalization group has been used to calculate the critical exponents \ensuremath{\nu}, v, and \ensuremath{\beta} of the correlation length, helicity modulus, and order parameter. Good agreement with series-expansion results is obtained.
The critical properties of the three-dimensional XY model are investigated through measurements of the two-point correlation functions via a worm-algorithm Monte Carlo calculation, which automatically crunches the correlation functions while tracing worms, in large systems. This method has the advantage of directly measuring the order parameter, as well as the correlation length, the susceptibility, and their associated critical exponents. From the finite-size scaling behavior, we find 2 β/ν = 1.033(5) and γ/ν = 1.967(5), confirming the hyperscaling relations.
The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above Tc{\mathit{T}}_{\mathit{c}}Tc​. The critical temperature is determined to be Tc{\mathit{T}}_{\mathit{c}}Tc​=0.892 13(10). The temperature dependence of the characteristic length right above Tc{\mathit{T}}_{\mathit{c}}Tc​ is also obtained and found to be in agreement with recent results from high-temperature series expansions.
A new logarithmic term in the finite-size helicity modulus has been predicted recently for the three-dimensional classical XY-model universality class under nonperiodic boundary conditions. These theoretical ideas have important implications for the analysis of superfluid-density measurements in confined geometries. We present numerical results from Monte Carlo simulations of the helicity modulus for the three-dimensional classical XY model on a finite simple-cubic lattice, with pinned-surface-spin boundary conditions. Our data are consistent with the presence of an additional logarithmic term and an estimate for the amplitude is given.
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 have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to 1003{100}^{3}1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.