Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code
The research investigates topological order in a 3D toric code under generic magnetic fields by introducing two diagnostics: Topological Entanglement Entropy ($\gamma$) and a Wilson-loop correlation ($f_W$). While $\gamma$ shows quantization, $f_W$ is shown to be a robust channel invariant that distinguishes the true topological phase (where $f_W=1$) from trivial product states (where $f_W=0), providing a complete phase classification.