Thermodynamic Stability of Classical $\mathbb Z_2$ Lattice Gauge Models

Abstract

Finite-temperature phases are traditionally studied through spontaneous symmetry breaking, but symmetry-breaking fields can remove their distinction, as in the Ising model. This motivates the search for stable nontrivial thermal phases without symmetry protection. For classical $\mathbb Z_2$ lattice gauge models with spins on $m$-dimensional cells in a fixed spatial dimension $d$, we prove joint analyticity of the thermodynamic free-energy density in temperature and perturbation strength at sufficiently low positive temperatures under sufficiently weak local perturbations, whenever $1\le m\le d-2$. Perturbations need not preserve internal or gauge symmetries and may include noncommuting quantum interactions. A separate high-temperature expansion yields an analytic region reaching infinite temperature at any fixed finite perturbation strength. These results establish thermodynamic stability and support the conjecture that low- and high-temperature regimes remain separated by a thermal transition without symmetry protection.