New Multipartite Entanglement Measures for Topologically Ordered States

We address the challenge of identifying 2+1-dimensional topologically ordered phases directly from their ground-state wavefunctions. In our first result, we introduce a family of bulk multipartite entanglement measures that extract the invariants ∑_a​d_a²θ_a^r for any r≥2, where ​d_a and θ_a​ are the quantum dimension and topological spin of anyon a, respectively. These quantities are obtained as expectation values of permutation operators acting between 2r replicas of the wavefunction, with distinct permutations applied on four regions of the plane. The resulting measures provide a refined diagnostic of topological order, capturing information beyond conventional probes such as the topological entanglement entropy.

For chiral topological phases, additional subtleties arise due to the gravitational anomaly of the edge theory. In this case, we show that the anomaly itself enables new multipartite measures that extend the known "modular commutator" to Renyi-like measures that extract the chiral central charge and the quantum Hall conductance using a finite number of replicas of the wavefunction. Together, these results allow for almost-complete characterization by of the topological phase by measurements of the ground-state wavefunction.