Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code

Abstract: We study the finite-temperature topological order of the three-dimensional $\mathbb{Z}_2$ toric code in a generic magnetic field, where every higher-form symmetry is explicitly broken and can at most be emergent. We show perturbatively, and confirm by large-scale quantum Monte Carlo, that the topological entanglement entropy stays quantized at $γ= \ln 2$ throughout the topological phase -- at finite temperature and under the symmetry-breaking field alike -- and collapses to $0$ across the thermal transition, a quantization protected geometrically by the Bianchi identity rather than by any exact symmetry of the system. The plateau $γ= \ln 2$ is, however, not invariant under quasi-local channels: a constant-depth channel can generate this identical quantized value from a trivial product state. We therefore introduce the decoded Wilson-loop correlation $f_W$, which quantizes to $1$ in the topological phase and $0$ in the trivial phase as $L\to\infty$ and, unlike $γ$, is a quasi-local-channel invariant -- a robust topological invariant of the mixed state.