The Information Geometry of the Spherical Model

Abstract

Motivated by previous observations that geometrizing statistical mechanics offers an interesting alternative to more standard approaches,we have recently calculated the curvature (the fundamental object in this approach) of the information geometry metric for the Ising model on an ensemble of planar random graphs. The standard critical exponents for this model are alpha=-1, beta=1/2, gamma=2 and we found that the scalar curvature, R, behaves as epsilon^(-2),where epsilon = beta_c - beta is the distance from criticality. This contrasts with the naively expected R ~ epsilon^(-3) and the apparent discrepancy was traced back to the effect of a negative alpha on the scaling of R. Oddly,the set of standard critical exponents is shared with the 3D spherical model. In this paper we calculate the scaling behaviour of R for the 3D spherical model, again finding that R ~ epsilon^(-2), coinciding with the scaling behaviour of the Ising model on planar random graphs. We also discuss briefly the scaling of R in higher dimensions, where mean-field behaviour sets in.