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Physica Scripta

On the helicity modulus, the critical temperature and Monte Carlo simulations for the two-dimensional XY-model

Peter Olsson, Petter Minnhagen

Published Feb 1, 1991• 38 citations
Publisher

Abstract

A lattice size dependence of the helicity modulus for the two-dimensional XY-model is inferred by aid of the Coulomb gas-vortex analogy and an explicit form of this dependence is obtained from a set of self-consistent equations for the Coulomb gas. High precision Monte Carlo simulations in conjunction with this inferred lattice size dependence is used to determine the critical temperature. Some other recent results in the literature are discussed in the light of the present investigation.

DOI
10.1088/0031-8949/43/2/016

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A lattice size dependence of the helicity modulus for the two-dimensional XY-model is inferred by aid of the Coulomb gas-vortex analogy and an explicit form of this dependence is obtained from a set of self-consistent equations for the Coulomb gas. High precision Monte Carlo simulations in conjunction with this inferred lattice size dependence is used to determine the critical temperature. Some other recent results in the literature are discussed in the light of the present investigation.
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.
Two-dimensional generalized XY spin model on a triangular lattice is studied by means of Monte-Carlo simulations. The critical temperatures of Berezinskii-Kosterlitz-Thouless (BKT) phase transition are obtained by the method of helicity modulus. It is found that the results are consistent with those obtained by other methods. The vortex density and the vortex-antivortex pair formation energy are also obtained. The result shows that the critical temperature decreases with the increase of the generalization parameter q . While the vortex-antivortex pair formation energy increases with the increase of q when q>1 .
Monte Carlo simulations are performed for the spin-\textonehalf{} XY\mathrm{XY}XY model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature Tc{T}_{c}Tc​, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.
A Monte Carlo simulation of the helicity modulus and the specific heat of the classical XY\mathrm{XY}XY model in two dimensions is presented. In particular, we have investigated the dependence of these quantities with the size of the Monte Carlo sample. The height of the specific-heat peak is found to saturate as the infinite-volume limit is approached. The peak height of the derivative of the helicity modulus is found to grow rapidly with the increase in the size of the system and is located only 7% below the temperature at which the specific heat has the maximum.
The two-dimensional S=1/2S=1/2S=1/2 XY model is investigated with an extensive quantum Monte Carlo simulation. The helicity modulus is precisely estimated through a continuous-time loop algorithm for systems up to 128×128128 \times 128128×128 near and below the critical temperature. The critical temperature is estimated as TKT=0.3427(2)JT_{\rm KT} = 0.3427(2)JTKT​=0.3427(2)J. The obtained estimates for the helicity modulus are well fitted by a scaling form derived from the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.
We study the classical XY (plane rotator) model at the Kosterlitz–Thouless phase transition. We simulate the model using the single-cluster algorithm on square lattices of a linear size up to L = 2048. We derive the finite-size behaviour of the second moment correlation length over the lattice size ξ 2nd / L at the transition temperature. This new prediction and the analogous one for the helicity modulus ϒ are confronted with our Monte Carlo data. This way β KT = 1.1199 is confirmed as inverse transition temperature. Finally, we address the puzzle of logarithmic corrections of the magnetic susceptibility χ at the transition temperature.