Droplet theory of correlations in ordered phases

Abstract

The explicit droplet theory of low-dimensional phase transitions, developed recently, is extended to yield a description of the pair correlation function for the universality class of q-state Potts models in the ordered phase and in zero external field. It is shown that, in d=1+ epsilon dimensions, the short distance behaviour of the correlation function and in particular, the exponent eta , are controlled by nested nearly spherical droplets. In contrast, it is argued that the spatial dependence of the large distance behaviour, in d=2 or above, is controlled by highly anisotropic droplets, thus illuminating the Widom relation, sigma xi ^d-1 approximately= constant, linking the dimensionless surface tension sigma and the bulk correlation length xi . In the particular case of d=2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings, in the spirit of recent independent, arguments by Abraham and by Fisher: a non-Ornstein-Zernike correlation function prefactor, and the result of sigma xi =¹/₂ follow in accord with exact results for the q=2 (Ising) case, and with implications for the q to 1 (percolation) problem.