Free Energy Analysis of Spin Models on Hyperbolic Lattice Geometries

TL;DR

This paper explores how the free energy and phase transition temperatures of spin models are influenced by the curvature of hyperbolic lattices, revealing geometric effects on thermodynamic properties and potential implications for AdS-CFT correspondence.

Contribution

It derives recurrence relations for free energy on non-Euclidean lattices and analyzes the relation between free energy, curvature, and phase transitions, highlighting the unique role of Euclidean geometry.

Findings

Free energy is minimized in Euclidean flat geometry.

Free energy and transition temperatures reflect lattice curvature.

Results suggest geometric influence on thermodynamic behavior.

Abstract

We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy normalized per lattice site of various multistate spin models in the thermal equilibrium on distinct non-Euclidean surface lattices of the infinite sizes. Whereas the free energy is calculated numerically by means of the Corner Transfer Matrix Renormalization Group algorithm, the radius of curvature has an analytic expression. Two tasks are considered in this work. First, we search for such a lattice geometry, which minimizes the free energy per site. We conjecture that the only Euclidean flat geometry results in the minimal free energy per site regardless of the spin model. Second, the relations among the free energy, the radius of curvature, and the phase transition temperatures are analyzed. We found out that both the free energy and the phase transition temperature inherit the structure of the lattice geometry and asymptotically approach the profile of the Gaussian radius of curvature. This achievement opens new perspectives in the AdS-CFT correspondence theories.