Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

TL;DR

This paper investigates the analyticity and duality properties of free energy in complex few-body Ising models with extensive homology, revealing phase transition behaviors and critical point multiplicities on hyperbolic tilings.

Contribution

It introduces bounds on free energy analyticity for dual Ising models with extensive homology, and demonstrates multiple critical points on hyperbolic plane tilings.

Findings

Existence of low-temperature weak-disorder region where defects do not affect free energy.

Analyticity of free energy at high and low temperatures proven.

Critical points with different boundary conditions are shown to differ on hyperbolic tilings.

Abstract

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds), such that each model can be obtained from the dual of the other after freezing $k$ spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with $k$ being the rank of the first homology group. Our focus is on the case where $k$ is extensive, that is, scales linearly with the number of bonds $n$. Flipping any of these additional spins introduces a homologically non-trivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects have no effect on the free energy density $f(T)$ in the thermodynamical limit, and of a high-temperature region where in the ferromagnetic case an extensive homological defect does not affect $f(T)$. We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting $f(T)$ at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all $\{f,d\}$ tilings of the hyperbolic plane, where $df/(d+f)>2$. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, $T_c^{(\mathrm{f})}<T_c^{(\mathrm{w})}$.