Information Geometry, One, Two, Three (and Four)

Although the notion of entropy lies at the core of statistical mechanics, it is not often used in statistical mechanical models to characterize phase transitions, a role more usually played by quantities such as various order parameters, specific heats or susceptibilities. The relative entropy induces a metric, the so-called information or Fisher–Rao metric, on the space of parameters and the geometrical invariants of this metric carry information about the phase structure of the model. In various models the scalar curvature, ℛ, of the information metric has been found to diverge at the phase transition point and a plausible scaling relation postulated. For spin models the necessity of calculating in non-zero field has limited analytic consideration to one-dimensional, mean-field and Bethe lattice Ising models. We extend the list somewhat in the current note by considering the one-dimensional Potts model, the two-dimensional Ising model coupled to two-dimensional quantum gravity and the three-dimensional spherical model. We note that similar ideas have been applied to elucidate possible critical behaviour in families of black hole solutions in four space-time dimensions.