The Information Geometry of the One-Dimensional Potts Model

Abstract

In various statistical-mechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, $β$, and the external field variable, $h$, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, ${\cal R}$, of this metric can be calculated explicitly in the thermodynamic limit and is found to be ${\cal R} = 1 + \cosh (h) / \sqrt{\sinh^2 (h) + \exp (- 4 β)}$. This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field ``critical point'' of the model. In this note we calculate ${\cal R}$ for the one-dimensional $q$-state Potts model, finding an expression of the form ${\cal R} = A(q,β,h) + B (q,β,h)/\sqrt{η(q,β,h)}$, where $η(q,β,h)$ is the Potts analogue of $\sinh^2 (h) + \exp (- 4 β)$. This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee-Yang edge.