A topological signature of multipartite entanglement

Topology and entanglement promise to play a prominent role in understanding non-local properties of many-body quantum systems. Here, we present a novel proposal to relate the two and study multipartite entanglement by collecting statistics of topological invariants of hard-core bosons worldline configurations in two-dimensional lattices. We build our approach upon the path-integral formulation of the density matrix in the limit of zero temperature, and consider worldline configurations, i.e. collections of particle paths, as geometric braids with a certain topological structure. We support our proposal by studying checkerboard and stripe solids, superfluid phases, and $\mathbb{Z}_2$ topologically ordered phases by means of unbiased quantum Monte Carlo simulations. We find that topological invariants can be used to differentiate among ground-states with different entanglement properties. More specifically, we are able to differentiate between standard insulators and $\mathbb{Z}_2$ topologically ordered phases. We also find that certain non-local probes based on topological invariants can signal the insulating to superfluid phase transition.