Anisotropic Ising model in a magnetic field: Effective-field theory analysis

Abstract

The anisotropic two-dimensional Ising model in the presence of a magnetic field is studied within two different approaches: the effective-field theory (EFT) with correlation and the Bethe-Peierls (BP) approximation. The model consists of ferromagnetic interaction $({J}_{x})$ in the $x$ direction and antiferromagnetic interaction $({J}_{y})$ in the $y$ direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case ${J}_{x}={J}_{y}$. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of $H∕J∊[0,2]$, with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of $r={J}_{y}∕{J}_{x}$, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.