Symmetry-Twisted Multipartite Entanglement Order Parameters for 2D SPTs

By virtue of their local featurelessness, it has been a longstanding challenge to construct order parameters that detect symmetry protected topological phases (SPTs) in lattice systems in spatial dimensions higher than one. We fill in this gap and propose a pair of non-local order parameters that distinguish all bosonic SPTs protected by on-site Abelian symmetries in 2D lattice systems by effectively simulating their flux response on non-trivial spacetime manifolds. The inputs for our order parameters are the reduced density matrix of the wavefunction on a disk and its protecting symmetries -- unlike previous proposals, we do not make use of additional (e.g. crystalline) symmetries, nor do we explicitly insert a flux. Our order parameters are given by expectation values of partial symmetry and permutation operations acting on certain numbers of replicas of the system, with the permutations acting on different subregions of a multipartition of the disk. In this sense, these order parameters detect multipartite correlations; indeed, one of these order parameters is a symmetry-twisted version of a Renyi multi-entropy, a recently proposed multipartite entanglement quantity. Our scheme readily extends to higher dimensions and motivates entanglement order parameters for topological orders supporting anyons, suggesting that multipartite entanglement is a distinguishing feature of topological phases.