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Journal of Physics A Mathematical and General

The Ising model in a random boundary field

John Cardy

Published Nov 21, 1991• 24 citations
Publisher

Abstract

Random magnetic fields acting on the boundary of a system are relevant if eta /sub ///(1. For the case of the two-dimensional Ising model they are marginal, giving rise to calculable logarithmic corrections to boundary quantities near the bulk critical point. These considerations may be important in the study of finite-size effects in the critical behaviour of adsorbed systems.

DOI
10.1088/0305-4470/24/22/003

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Random magnetic fields acting on the boundary of a system are relevant if eta /sub ///(1. For the case of the two-dimensional Ising model they are marginal, giving rise to calculable logarithmic corrections to boundary quantities near the bulk critical point. These considerations may be important in the study of finite-size effects in the critical behaviour of adsorbed systems.
Using previous results from boundary conformal field theory and integrability, a phase diagram is derived for the two-dimensional Ising model at its bulk tri-critical point as a function of boundary magnetic field and boundary spin-coupling constant. A boundary tri-critical point separates phases where the spins on the boundary are ordered or disordered. In the latter range of coupling constant, there is a non-zero critical field where the magnetization is singular. In the former range, as the temperature is lowered, the boundary undergoes a first-order transition while the bulk simultaneously undergoes a second-order transition.
A review is given of recent work on the ordinary surface critical behaviour of systems in two dimensions. Several models of interest in statistical mechanics are considered: Potts model, percolation, Ising clusters, ZN-model, O(n) model and polymers. Numerical results for surface exponents, obtained by suitable finite size scaling extrapolations, are discussed in the light of recent advances based on the conformal invariance approach. Surface exponents are often seen as important tests of conformal invariance predictions. In other cases these exponents provide important information for a location of the problem within the classification schemes offered by the conformal approach, and a determination of its universality class. A relevant example of the first aspect is the study of the q-state Potts model with q near 4, for which an analytical study of logarithmic scaling corrections is needed to achieve a successful test. The latter point of view applies, e.g., to the more controversial cases of polymers at the theta point and critical Ising clusters. Emphasis is put on the importance of an integrated study of both bulk and surface properties. Relevant issues, like the possible existence of analytical expressions for the indices in particular model families, or of general relationships between bulk and surface exponents, are critically discussed. The new problem of critical behaviour at fractal boundaries is also considered for random (RW) and self-avoiding walks (SAW). From the numerical analysis of this problem remarkable universalities of the surface exponents seem to emerge, which, in the case of SAW’s, are still far from being understood.