Probability density functions for collective coordinates in Ising-like systems

Abstract

This paper describes a study of the behaviour of 'block' coordinates, measuring the spatial average, over a block of side L, of the local scalar ordering coordinates in a system undergoing a continuous phase transition. A general renormalisation group argument suggests that, in the limit in which both L and xi (the correlation length) are large compared with the lattice spacing, the block coordinate probability density function tends to a universal form, Pinfinity dependent on the ratio L/ xi . The function Pinfinity is calculated explicitly for dimension d=1. It is controlled by the statistical mechanics of kinks; intra-cluster ripples (phonons in the structural phase transition context) contribute identifiable corrections to the large L, xi limit. In the critical limit, L/ xi to 0, Pinfinity tends to a fixed point form P* with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* m with an extreme order-disorder character. Wilson's recursion formula is used to determine the form of P* for d=3 and d=2; the latter result is substantiated by a calculation based on exactly established properties of the planar Ising model. In d=3Pinfinity is singly-peaked; in d=2 it has a strongly double-peaked character reminiscent of the d=1 system at low temperatures. The results are used to illuminate the nature of short-range order at criticality and the character of the collective excitation spectrum in the critical region, with particular reference to scattering experiments near structural phase transitions.