Topological order and long-range entanglement
Some many-body phases cannot be characterized by local symmetry-breaking order parameters.
$$S(A)=\alpha|\partial A|-\gamma+\cdots$$
What this means
Some many-body phases cannot be characterized by local symmetry-breaking order parameters.
Deep dive: mathematical and physical mechanism
Topological order features long-range entanglement, fractional excitations, and topology-dependent degeneracy. The constant $\gamma=\ln\mathcal D$ is topological entanglement entropy in suitable 2D gapped phases, where $\mathcal D$ is total quantum dimension.
How it appears in quantum phenomena
Toric-code ground states and fractional Hall liquids.
Assumptions, caveats and limitations
Extracting $\gamma$ numerically requires controlling finite-size and boundary effects.
Connect the ideas
For context, see the Interacting Topological Phases topic guide and the reading references.