Inhomogeneous Systems with Unusual Critical Behaviour

TL;DR

This paper reviews the critical behavior of inhomogeneous systems with various geometries and defects, highlighting unusual phenomena like stretched exponential decay and local first order transitions, supported by theoretical and exact results.

Contribution

It provides a comprehensive analysis of how shape and defects influence critical phenomena, including new insights into marginal and relevant perturbations in inhomogeneous systems.

Findings

Local exponents depend on parameters in marginal cases

Unusual stretched exponential behavior observed with relevant defects

Exact results obtained for certain Ising models

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.