The Information Geometry of the One-Dimensional Potts Model

TL;DR

This paper explores the geometric structure of the parameter space in the one-dimensional Potts model, extending previous work on the Ising model by calculating the scalar curvature and analyzing its divergence at critical points.

Contribution

It derives an explicit expression for the scalar curvature of the parameter space in the 1D Potts model, generalizing the geometric analysis from the Ising case and examining its divergence behavior.

Findings

Scalar curvature diverges at the critical point in the real parameter space.

The curvature expression involves a Potts-specific function analogous to the Ising case.

Analytic continuation reveals additional divergences at the Lee-Yang edge.

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