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

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.