Inhomogeneous systems with unusual critical behaviour

Abstract

The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.

Nonuniversal and anomalous critical behavior of the contact process near an extended defect

“…(1) have been studied before in two-dimensional Ising [34], Gaussian [35] and directed walk models [36], as well as in the mean-field approximation [37]; for a review, see Ref. [38]. According to exact results the local critical behavior in these problems is in agreement with the relevance-irrelevance criterion in Eq.…”

“…The local critical behavior of the contact process at an extended surface defect seems to be similar to that of the planar Ising model with an extended surface defect, for which analytical results exist [34,[38][39][40]. In that model, in the marginal case s = 1/ν = 1, the spatial spin cor- In the contact process, the quantity which is analogous to G (r) is the density autocorrelation function C(t 2 − t 1 ) in the steady state.…”

“…The singular behavior of the surface order parameter can be calculated through a scaling theory [38]. For enhanced surface couplings, A > 0, the survival probability (surface order parameter) stays finite at the bulk critical point and it satisfies the scaling relation:…”

Critical adsorption and Casimir torque in wedges and at ridges

“…The opening angle is therefore a marginal variable in a renormalization transformation, and may enter into the expressions for the exponents. The same will happen for all other scale-invariant shapes of the boundaries [25]. However, for a given opening angle, the values of the critical exponents are expected to be universal and independent of microscopic details.…”

Random transverse Ising spin chain and random walks

“…At the very critical point the appropriate way to describe the position dependent physical quantities is to use density profiles rather then bulk and surface observables. For a number of universality classes much is known about the spatially inhomogeneous behavior, in particular in two-dimensions, where conformal invariance provides a powerful tool to study various geometries [22].…”