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Zeitschrift für Naturforschung A

Spatial Variations of Spontaneous Magnetization in a Layered Ising Model

K. Ränder

Published Dec 1, 1976• 8 citations
Publisher

Abstract

Abstract The extrema of the position-dependent spontaneous magnetization in a periodically layered two-dimensional 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 d-3/2exp (-d/ξ₁˂ where d is the distance to the inhomogeneity and ξ₁˂ the correlation length in the underlying homogeneous lattice. The connection of this decay law to the asymptotic decay of correlations is dicussed.

DOI
10.1515/zna-1976-1204

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Abstract The extrema of the position-dependent spontaneous magnetization in a periodically layered two-dimensional 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 d-3/2exp (-d/ξ₁˂ where d is the distance to the inhomogeneity and ξ₁˂ the correlation length in the underlying homogeneous lattice. The connection of this decay law to the asymptotic decay of correlations is dicussed.
We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance mmm from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large mmm is examined in detail. For y<1y<1y<1 the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For y=1y=1y=1, A>AcA>{A}_{c}A>Ac​ (Ac{A}_{c}Ac​ being a positive critical value), and for y<1y<1y<1, A>0A>0A>0, there is a spontaneous surface magnetization at the bulk critical temperature.
The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are obtained both for the ferromagnetic and paramagnetic regions of coupling constant. The peculiarities caused by the finite size of the system are discussed. The asymptotic behaviour of the corresponding quantities as functions of size is calculated. Some generalization of the obtained result is conjectured.
The self-consistent approach to the two-dimensional Ising model with quenched random bonds is extended to the full lattice theory described by four real fermions. A calculation of the averaged spin-spin correlation function for large separation of the spins in the disorder-dominated phase indicates an exponential decay of this quantity and therefore a vanishing spontaneous magnetization. The corresponding correlation length is proportional to 1/ eta 2 , where eta denotes the order parameter of the new phase introduced by Ziegler (1990).
Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am -y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v -1 , a surface magnetic phase at the bulk critical temperature for y<v -1 , A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v -1 .