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Physical review. B, Condensed matter

Monte Carlo determination of the critical temperature for the two-dimensional XY model

Hans Werner Weber, Petter Minnhagen

Published Apr 1, 1988• 231 citations
Publisher

Abstract

The critical temperature for the two-dimensional XY model on a square lattice is determined to within a few tenths of a percent by combining Monte Carlo simulations with a lattice size scaling relation.
DOI
10.1103/physrevb.37.5986

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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 method of critical index calculation is proposed. The method is based on matching the temperature values of two different size lattices by finding the optimized solution of the equation t'=bTY{\mathit{b}}_{\mathit{T}}^{\mathit{Y}}bTY​t, where t and t' are the reduced temperatures of the lattices, b is the ratio of lattice sizes, and YT{\mathit{Y}}_{\mathit{T}}YT​ is the thermal critical index. The values of t and t' are obtained from the fourth cumulant of the order parameter, which satisfies the relation Qr{\mathit{Q}}_{\mathit{r}}Qr​(t,L)=Qr{\mathit{Q}}_{\mathit{r}}Qr​(t'L') around the critical point. By using this method, the critical indices of the two-dimensional Ising model and the (2+1)-dimensional lattice gauge theory are calculated. The advantage of the method is that one can obtain the critical point and the critical indices with high accuracy by performing a relatively small number of Monte Carlo iterations.
The properties of phase transition and the lower-temperature phase of the plane rotator model are studied in the square lattice with Monte Carlo method. Physical results thus obtained for this model are quite similar to those of the two-dimensional quantum XY-model,1) that is, the susceptibility diverges and the specific heat does not diverge at the critical point. Furthermore the size-dependence of physical quantities and scaling relations among some temperature-dependent indices are also investigated.