Fractals, phase transitions and criticality

Abstract

Analogies between behaviour in fractal media and at phase transitions suggest deeper connections. These arise from the scale invariance of configurations at continuous phase transitions, making fractal view-points useful there and making scaling techniques necessary in both areas. These concepts, inter-relations, and techniques are illustrated with the percolation problem and the Ising spin system, which provide simple examples of geometrical and thermal transitions respectively. Diluted magnets involve both geometrical and thermal effects and, when prepared at concentrations near the percolation threshold, exhibit the influence of geometrical self-similarity on such varied processes as cooperative thermal behaviour and dynamics. Phase transitions in these systems are briefly discussed. The anomalous dynamical behaviour of self-similar systems is introduced, and illustrated by the critical dynamics of Heisenberg and Ising magnets diluted to the percolation threshold. These involve linear (spin wave), and nonlinear (activated) dynamical processes on the underlying fractal percolation structure, leading in the linear case to the magnetic analogue of ‘fracton’ dynamics and in the case of activated dynamics to a highly singular critical behaviour.