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Physical review. B, Condensed matter

Testing a simple criterion for the critical temperature

Ivo Souza, Carlos Lobo, J. Seixas

Published Mar 1, 1997• 5 citations
Publisher

Abstract

We investigate the applicability of a simple criterion proposed by Wosiek [Phys. Rev. B 49, 15 023 (1994)] for locating analytically critical temperatures; we apply it to Ising-like spin models with anisotropic interactions, more than one type of interaction between the spins, and multiple-spin interactions. The predicted critical temperatures only agree with the known exact results or with Monte Carlo simulations for very particular, highly symmetrical values of the coupling constants.

DOI
10.1103/physrevb.55.6356

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We investigate the applicability of a simple criterion proposed by Wosiek [Phys. Rev. B 49, 15 023 (1994)] for locating analytically critical temperatures; we apply it to Ising-like spin models with anisotropic interactions, more than one type of interaction between the spins, and multiple-spin interactions. The predicted critical temperatures only agree with the known exact results or with Monte Carlo simulations for very particular, highly symmetrical values of the coupling constants.
We have found a simple criterion which allows for the straightforward determination of the order-disorder critical temperatures. The method reproduces exactly known results for the two-dimensional Ising, Potts, and Z(N5) models. It also works for the Ising model on the triangular lattice. For systems that are not self-dual, our proposition remains an unproven conjecture. It predicts ${\mathrm{\ensuremath{\beta}}}_{\mathit{c}}$=0.2656. . . for the two coupled layers of Ising spins. The critical temperature of the three-dimensional Ising model is related to the free energy of the two-layer Ising system.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Abstract Analytic predictions have been derived recently by Dohm and Wessel (2021 Phys. Rev. Lett. 126 060601) from anisotropic φ 4 theory and conformal field theory for the amplitude F c of the critical free energy of finite anisotropic systems in the two-dimensional Ising universality class. These predictions employ the hypothesis of multiparameter universality. We test these predictions by means of high-precision Monte Carlo (MC) simulations for F c of the Ising model on a square lattice with isotropic ferromagnetic couplings between nearest neighbors and with an anisotropic coupling between next-nearest neighbors along one diagonal. We find remarkable agreement between the MC data and the analytical prediction. This agreement supports the validity of multiparameter universality and invalidates two-scale-factor universality as F c is found to exhibit a nonuniversal dependence on the microscopic couplings of the scalar φ 4 model and the Ising model. Our results are compared with the exact result for F c in the three-dimensional φ 4 model with a planar anisotropy in the spherical limit. The critical Casimir amplitude is briefly discussed.