A simple way of understanding some exact results on critical phenomena in non-homogeneous and finite Ising systems II

By means of a continuum approximation introduced in a preceding paper, we present a more transparent treatment of the critical phenomena in the specific heat of a two-dimensional Ising model with a “layer” type spatial inhomogeneity of its exchange coupling. This calculational procedure (which will be tested by comparison with an earlier exact treatment by Becker and Hahn) enables us to develop a physical picture of cooperative effects in real (spatially non-homogeneous) systems with quenched impurities. To be more concrete: By examination of the one-particle eigenvalues and eigenfunctions of the transfer matrix, it becomes apparent how the critical behaviour is dominated by i) “local” correlation lengths, describing the spatial decay of correlations within homogeneous subdomains of the system and becoming large near certain temperatures which indicate transitions from disorder to spin alignment within each of these domains, and ii), by the spatial average of the inverse of these local decay lengths, termed “global” reciprocal correlation length, which describes correlations between different subdomains on a scale on which these domains look like fluctuating “block spins”. This “global” decay length diverges at a true phase transition of the system as a whole. Correspondingly, for each sort of subdomain a sharp, but finite logarithmic peak in the specific heat and, as long as the layering of the inhomogeneous Ising system is periodic, a true logarithmic singularity are found.