Spatial Variations of Spontaneous Magnetization in a Layered Ising Model

Abstract

The extrema of the position-dependent spontaneous magnetization in a periodically layered twodimensional Ising model are calculated exactly. Their asymptotic behaviour for infinite width of the surrounding homogeneous sublayer is given. The perturbations caused by the neighbouring sublayers on this extremum in a very thick sublayer are shown to be decoupled. Thus the asymptotic decay of the magnetization far from a single layer-shaped inhomogeneity can be inferred from the quoted asymptotics of an extremum, and it is found to be exp( -d/£i) where d is the distance to the inhomogeneity and ff the correlation length in the underlying homogeneous lattice. The connection of this decay law to the asymptotic decay of correlations is dicussed.

Ising Models with Holes: Crossover Behavior

“…Consequently, the generating function in ( 26) becomes (22), reproducing the 2-d behavior again as it should. More generally there is a crossover for finite m: If α is expressed in terms of t ∝ T /T c − 1 as in [11], then α −j is exponentially small when m|t| is large, and the system behaves as two-dimensional, otherwise it acts one-dimensional.…”

“…As m increases, one needs to calculate more and more roots. We have shown that in the limit m → ∞, the generating function becomes the well-known square-root function (22) for the correlation function of Onsager's 2-d Ising model. Therefore, it would be very difficult, but most interesting, to study the scaling behavior of these correlations in the limit, m → ∞ and T → T c , where T c is Onsager's critical temperature given by (1).…”