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Physical review. B, Condensed matter

Surface critical behavior of the smoothly inhomogeneous Ising model

Theodore W. Burkhardt, Ihnsouk Guim

Published Jan 1, 1984• 30 citations
Publisher

Abstract

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large $m$, $m$ being the distance from the edge. The case $A<0$ of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when $A>0$. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.

DOI
10.1103/physrevb.29.508

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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 .
A semi-infinite nearest-neighbor square Ising system whose couplings at a distance lll from the boundary differ from homogeneity by an amount $\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}$ is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$, ${\ensuremath{\beta}}_{1}$, ${\ensuremath{\gamma}}_{11}$, and ${\ensuremath{\delta}}_{11}$ display rich nonuniversal behavior as a function of the amplitude AAA. For AAA below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk ($l=\ensuremath{\infty}$) is critical.
An explicit formula for the boundary magnetization of a two-dimensional Ising model with a strip of inhomogeneous interactions is obtained by means of a transfer matrix mean-field method introduced by Lipowski and Suzuki. There is clear numerical evidence that the formula is exact. By taking the limit where the width of the strip approaches infinity and the interactions have well-defined bulk limits, I arrive at the boundary magnetization for a model which includes the Hilhorst-van Leeuwen model. The rich critical behavior of the latter magnetization is thereby rederived with little effort.
A semi-infinite nearest-neighbour square Ising lattice is investigated whose couplings J(l) at a distance l from the boundary differ from homogeneity by an amount j(l)-J(infinity) approximately -A/l. On the basis of the Pfaffian method the critical behaviour at the surface of this system is investigated. The exponents eta 11 , nu 11 , beta 1 , gamma 11 and delta 11 all display rich non-universal behaviour as a function of the amplitude A. For A below a critical value there is a spontaneous surface magnetisation when the bulk (l = infinity) is critical and an asymmetry between the exponents on either side of the critical point.