Extracting topological spins from bulk multipartite entanglement

We address the problem of identifying a 2+1d topologically ordered phase using measurements on the ground-state wavefunction. For non-chiral topological order, we describe a series of bulk multipartite entanglement measures that extract the invariants $\sum_a d_a^2 \theta_a^r$ for any $r \geq 2$, where $d_a$ and $\theta_a$ are the quantum dimension and topological spin of an anyon $a$, respectively. These invariants are obtained as expectation values of permutation operators between $2r$ replicas of the wavefunction, applying different permutations on four distinct regions of the plane. Our proposed measures provide a refined tool for distinguishing topological phases, capturing information beyond conventional entanglement measures such as the topological entanglement entropy. We argue that any operator capable of extracting the above invariants must act on at least $2r$ replicas, making our procedure optimal in terms of the required number of replicas. We discuss the generalization of our results to chiral states.