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Physical review. B, Solid state

Ising model with magnetic field and the lattice gas

Gerald Dennis Mahan, Francisco H. Claro

Published Aug 1, 1977• 54 citations
Publisher

Abstract

Approximate solutions in two and three dimensions are given for the Ising model with magnetic field, and the corresponding lattice gas. Results were obtained using nonlinear renormalization-group methods with periodic cell clusters. The two-dimensional lattices are square, honeycomb, and triangular. The three-dimensional lattices are sc, bcc, and fcc.

DOI
10.1103/physrevb.16.1168

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Approximate solutions in two and three dimensions are given for the Ising model with magnetic field, and the corresponding lattice gas. Results were obtained using nonlinear renormalization-group methods with periodic cell clusters. The two-dimensional lattices are square, honeycomb, and triangular. The three-dimensional lattices are sc, bcc, and fcc.
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.
We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization M(H,T)M(H,T)M(H,T). The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of M(H,T)M(H,T)M(H,T) converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for H=OH=OH=O, which reproduces exactly the critical-point behavior of the full Ising model.