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

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 at fields up to half the zero-temperature critical values, that the topological entanglement entropy stays quantized at $\gamma =\ln 2$ 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 $\gamma =\ln 2$ 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 $f_W$ – the connected correlator of Wilson loops read out after error correction – which quantizes to $1$ in the topological phase and $0$ in the trivial phase as $L\to \mathrm{\infty }$. Unlike $\gamma$, $f_W$ is a quasi-local-channel invariant: it is pinned to $0$ on every quasi-local-channel image of a product state and to $1$ in the topological phase, so no quasi-local channel carries the trivial phase to the topological one, and a fortiori no two-way equivalence connects them – a robust topological invariant of the mixed state.