Thermodynamic Curvature and the Widom Ridge in Interacting Spin Systems

TL;DR

This paper introduces a geometric approach to thermodynamics in the Ising model, linking curvature to fluctuations, and identifies the Widom line as a curvature ridge indicating maximal response.

Contribution

It develops a geometric formulation of thermodynamic response, revealing how curvature depends on control variables and identifying the Widom line as a curvature ridge.

Findings

Curvature depends on the choice of control variables and is non-zero in the (eta,h) manifold.

The curvature field exhibits a ridge extending from the critical point into the supercritical regime.

The Widom line corresponds to a locus of maximal thermodynamic response as a geometric feature.

Abstract

We develop a geometric formulation of thermodynamic response in the classical Ising model by defining a curvature field over the control manifold spanned by inverse temperature $\beta$ and magnetic field $h$. We show that the existence of nontrivial curvature depends sensitively on the choice of control variables: while the $(J,h)$ manifold at fixed temperature is integrable and exhibits zero curvature, the $(\beta,h)$ manifold supports a finite curvature field arising from variations of the statistical ensemble. This curvature is given by a mixed derivative of the free energy and can be expressed directly as the covariance between energy and magnetization fluctuations. We evaluate the curvature field using Monte Carlo sampling and demonstrate that it develops a pronounced ridge structure extending from the critical point into the supercritical regime. This identifies the Widom line as a geometric feature of control space, corresponding to a locus of maximal thermodynamic response. More generally, the formulation provides a direct connection between geometric thermodynamics, critical phenomena, and experimentally accessible observables, and suggests that thermodynamic curvature may be probed through measurements of work performed under cyclic driving protocols.