Local-Channel Stability of Classical $\mathbb Z_2$ Lattice Gauge Models

Abstract

We establish sufficient conditions for two-way local quantum-channel stability in classical $\mathbb Z_2$ lattice gauge models with spins on $m$-dimensional cells of a $d$-dimensional lattice. Each interaction multiplies the spins on the boundary of an $(m+1)$-cell. This family, defined on the full spin Hilbert space without a Gauss-law constraint, includes the Ising model at $m=0$ and the magnetic sector of the toric code at $m=1$. The low-temperature proof combines two ingredients: locally indistinguishable flat sectors and extended magnetic defects of dimension $d-m-1$. For $1\le m\le d-2$, sufficiently weak local classical perturbations at sufficiently low positive temperature preserve equivalence to the canonical flat ensemble. No internal or gauge symmetry is required of the perturbation. For weak noncommuting perturbations, we prove the same conclusion in a smaller sufficient window by reducing the diagonal probabilities to a weak classical interaction and recovering the coherences with a local channel. The conversion channels use product ancillary inputs, have polylogarithmic causal radius, and achieve any prescribed inverse-power whole-system trace-norm error. Sufficiently high-temperature states are instead two-way equivalent to a product state. Independent finite-dimensional onsite systems can be included with weak local couplings. An optimized covariance between two distant neighborhoods of parallel $m$-tori is nonincreasing under one-way local conversion and invariant under two-way conversion. Its values $\mathcal F_m=1$ and $0$ distinguish the proved low- and high-temperature domains without a protecting symmetry. These sufficient domains do not determine the full phase diagram.