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Communications in Theoretical Physics

Critical Point of the Honeycomb Antiferromagnetic Ising Model in a Nonzero Magnetic Field: An Analytical Analysis

Ya-Li Song, Jin Min Yang, Gong Ru Lu

Published Jun 1, 1992• 0 citations
Publisher

Abstract

In this paper we determined the critical point of the antiferromagnetic Ising model in a nonzero magnetic field on the honeycomb lattice by an analytical method—the generalized cumulant expansion with the mean field hypothesis. By calculating the magnetization to the fourth order correction, we get encouraging results when compared with the known numerical results by finite size analysis.

DOI
10.1088/0253-6102/17/4/425

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In this paper we determined the critical point of the antiferromagnetic Ising model in a nonzero magnetic field on the honeycomb lattice by an analytical method—the generalized cumulant expansion with the mean field hypothesis. By calculating the magnetization to the fourth order correction, we get encouraging results when compared with the known numerical results by finite size analysis.
In this paper we present highly accurate numerical results of the determination of the critical point of the antiferromagnetic Ising model in a nonzero magnetic field for the honeycomb lattice, including the critical fugacity of a nearest-neighbor exclusion lattice gas. We compute the correlation length of the Ising model using a transfer matrix approach, and locate the critical point from the data on the correlation lengths using finite-size analysis. For the purpose of a maximum numerical accuracy, the analysis is carried out by taking transfer matrices proceeding in two perpendicular directions of the lattice.
By extending the method due to Muller-Hartmann and Zittartz (1977), the authors calculate the interface tension for the nearest-neighbour antiferromagnetic Ising model on the honeycomb lattice at finite magnetic fields. The calculation reproduces exact interface tension at zero field. From the zero point of the interface tension they derive a compact analytic form of the 'critical frontier' in the (H, T) plane (H: magnetic field, T: temperature), which agrees excellently with the recent 'quasi-analytical' result presented by Wu, Wu and Blote (1989).
A closed-form expression is proposed for the critical frontier, or the critical line, of the antiferromagnetic Ising model on the honeycomb lattice in a nonzero external magnetic field. We formulate the Ising model as an 8-vertex model and identify the critical frontier as a locus invariant under a generalized weak-graph transformation. In its simplest form the locus is an expression containing two unknown constants which we determine numerically. The resulting critical frontier lies very close to results of a finite-size analysis.
Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.