Critical Exponent for the Density of Percolating Flux

Abstract

This paper is a study of some of the critical properties of a simple model for flux. The model is motivated by gauge theory and is equivalent to the Ising model in three dimensions. The phase with condensed flux is studied. This is the ordered phase of the Ising model and the high temperature, deconfined phase of the gauge theory. The flux picture will be used in this phase. Near the transition, the density is low enough so that flux variables remain useful. There is a finite density of finite flux clusters on both sides of the phase transition. In the deconfined phase, there is also an infinite, percolating network of flux with a density that vanishes as $T \rightarrow T_{c}^{+}$. On both sides of the critical point, the nonanalyticity in the total flux density is characterized by the exponent $(1-α)$. The main result of this paper is a calculation of the critical exponent for the percolating network. The exponent for the density of the percolating cluster is $ ζ= (1-α) - (φ-1)$. The specific heat exponent $α$ and the crossover exponent $φ$ can be computed in the $ε$-expansion. Since $ζ< (1-α)$, the variation in the separate densities is much more rapid than that of the total. Flux is moving from the infinite cluster to the finite clusters much more rapidly than the total density is decreasing.