An Invariant Between Hyperbolic Surfaces and Lattice Spin Models

TL;DR

This paper reveals a connection between hyperbolic surface partition functions and boundary Ising models, linking geometric structures with statistical physics and number theory, suggesting a new invariant bridging these fields.

Contribution

It introduces a novel invariant connecting hyperbolic surfaces and lattice spin models, expanding the understanding of geometric and physical systems through this relationship.

Findings

Partition function of hyperbolic surfaces relates to boundary Ising models

Connection supported by Poincare's Uniformization and Patterson-Sullivan's Theorem

Potential implications for phase transitions and quantum chaos

Abstract

In this succinct note, it is showed that a partition function of equivalent classes of hyperbolic surfaces can be connected to an Ising model located on the boundary of the Poincare disc, as hinted by Poincare's Uniformization theorem and Patterson-Sullivan's Theorem. Keywords: Hyperbolic spaces, Schottky groups, Ising models, locations of Lee-Yang Zeros, non-trivial zeros of Riemann zeta function, phase transition, and quantum chaos.