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Physical review. B, Condensed matter

Monte Carlo studies of the finite three-dimensionalXY-model helicity modulus

K. K. Mon

Published Aug 1, 1991• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.2643

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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.
The critical scaling of the helicity modulus of the classical O(3) 3d3d3d Heisenberg ferromagnet is studied directly. Monte Carlo methods that impose either an antiperiodic boundary condition or a finite twist of definite handedness across otherwise periodic boundaries in one lattice direction are used to measure scale-dependent enthalpy variations in a simple cubic lattice at the ferromagnetic critical temperature. Finite-size scaling is then used to determine the critical exponents vE{v}_{E}vE​ and vF{v}_{F}vF​ for helicity and, by evaluating three independent hyperscaling-linked pairs of $\ensuremath{\nu}$ and $\ensuremath{\alpha}$, to test hyperscaling for this model. It is observed that antiperiodic boundary conditions in particular constrain the lattice to have a nonzero topological charge, establishing a connection between topological charge and helicity in the model.
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 the response of a two-dimensional classical XY model to a finite (noninfinitesimal) twist of the boundary conditions. We use Monte Carlo simulations to evaluate the free energy difference between periodic and twisted-periodic boundary conditions and find deviations from the expected quadratic dependence on the twist angle. Consequently, the helicity modulus (spin stiffness) shows a nontrivial dependence on the twist angle. We show that the deviation from the expected behavior arises because of the mixing of states with opposite chirality which leads to an additional entropy contribution in the quasi-long-range ordered phase. We give an improved prescription for the numerical evaluation of the helicity modulus for a finite twist, and we discuss the spontaneous breaking of the chiral symmetry for the antiperiodic boundary conditions. We also discuss applications to discrete spin systems and some experimental scenarios where boundary conditions with finite twist are necessary.