Square Ising ferromagnetic and antiferromagnetic lattices in a magnetic field: A new perturbation approach

Abstract

The incorporation of the magnetic interaction energy $B$ into an effective-exchange parameter in the transfer matrix has been used to probe the behavior of the square ferromagnetic and antiferromagnetic Ising lattices with nearest-neighbor interactions $\ensuremath{\epsilon}$ in finite magnetic fields. The ferromagnetic-specific-heat capacity $C(B,T)$ is proportional to $\ensuremath{-}\mathrm{ln}(\frac{B}{\ensuremath{\epsilon}})$ at the zero-field critical temperature $T$, when $\frac{B}{\ensuremath{\epsilon}}\ensuremath{\ll}1$, in agreement with a previous analysis. The temperature ${T}_{m}$ at which $C(B,T)$ attains a maximum is initially depressed below ${T}_{c}$, while exhibiting a quadratic dependence on $B$. In larger fields, ${T}_{m}$ turns around and is elevated, where now it is directly proportional to $B$. Qualitatively, this pattern exhibited by $C(B,T)$ also is found with the mean-field approximation. The specific-heat capacity for the antiferromagnetic lattice essentially retains its zero-field form through $\frac{B}{|\ensuremath{\epsilon}|}<2$ and exhibits a logarithmic divergence at the Neel temperature ${T}_{N}(B)$, as does the magnetic susceptibility when $B>0$. The ${T}_{N}(B)$ obtained through this method, over the range where the method is thought to be reliable, is in excellent agreement with previous work. Finally, some general features which are applicable to the three-dimensional case are noted.