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

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 (Jₓ) in the x direction and antiferromagnetic interaction (Jy) in the y direction. The phase diagram in the T-H plane is obtained for the particular case Jₓ=Jy. 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=Jy∕Jₓ, and in the r→0 limit we have found, using BP and EFT, the approximate form kBTN∕JₓA∕ln(1∕r) in accordance with the exact result of Onsager.