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, $\beta$, 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, $\mathcal{R}$, of this metric can be calculated explicitly in the thermodynamic limit and is found to be $\mathcal{R}=1+\cosh (h)/\sqrt{{\sinh }^2(h)+\exp (-4\beta )}$. 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 $\mathcal{R}$ for the one-dimensional $q$-state Potts model, finding an expression of the form $\mathcal{R}=A(q,\beta ,h)+B(q,\beta ,h)/\sqrt{\eta (q,\beta ,h)}$, where $\eta (q,\beta ,h)$ is the Potts analogue of ${\sinh }^2(h)+\exp (-4\beta )$. 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.

Comments: 9 pages + 4 eps figures