Generalizing the Tomboulis-Yaffe Inequality to SU(N) Lattice Gauge Theories and General Classical Spin Systems

TL;DR

This paper generalizes the Tomboulis-Yaffe inequality from SU(2) to SU(N) lattice gauge theories and classical spin systems, providing theoretical bounds on mass gaps and illustrating the results with solvable models and conjectures.

Contribution

It extends the inequality to broader systems using reflection positivity and explores implications for mass gaps, including a conjecture for the triangular Ising model.

Findings

Inequality guarantees non-zero mass gap in systems insensitive to boundary conditions.

Explicit illustration of the theorem in solvable models.

Proposed conjecture for off-axis mass gap of the triangular Ising model.

Abstract

We extend the inequality of Tomboulis and Yaffe in SU(2) lattice gauge theory (LGT) to SU(N) LGT and to general classical spin systems, by use of reflection positivity. Basically the inequalities guarantee that a system in a box that is sufficiently insensitive to boundary conditions has a non-zero mass gap. We explicitly illustrate the theorem in some solvable models. Strong coupling expansion is then utilized to discuss some aspects of the theorem. Finally a conjecture for exact expression to the off-axis mass gap of the triangular Ising model is presented. The validity of the conjecture is tested in multiple ways.