The Information Geometry of the Spherical Model

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 α = −1, β = 1/2, γ = 2 and we found that the scalar curvature, R, behaves as ǫ−2,where ǫ = βc−β is the distance from criticality. This contrasts with the naively expected R ∼ ǫ−3 and the apparent discrepancy was traced back to the effect of a negative α 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 ∼ ǫ−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. 1 The Information Geometry of Spin Models The idea of endowing the space of parameters with a metric and geometrical structure has been borrowed from parametric statistics [1] and employed to some effect in statistical mechanics [2, 3, 4, 5, 6, 7, 8]. The approach seems to be particularly fruitful for a spin model in field where the parameters are β, the inverse temperature, and h, the external field. In this case the (Fisher-Rao) metric is simply given by Gij = ∂i∂jf , (1) where f is the reduced free energy per site and ∂i = (∂/∂β, ∂/∂h). For such a metric the scalar curvature may be calculated as R = − 1 2G2 ∣