Layered inhomogeneous Ising models with frustration on a square lattice

Abstract

Generalizing a previous treatment we study inhomogeneous Ising models on a square lattice. In particular, we consider models with a layered structure, i.e. translational invariance in the horizontal direction. Otherwise, couplings are allowed to be of arbitrary strength and sign. The partition function and free energy are calculated excatly for a random coupling distribution of finite period. The phase transition is universally of Ising type. The critical temperature can be calculated from a remarkably simple formula and depends only on the mean couplings, averaged over the period. Generally ground state properties do not reflect the singular structure of the free energy at finite temperature. This is shown by a special example.