Topological disorder parameter: A many-body invariant to characterize gapped quantum phases

We introduce a many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in 2 + 1 dimensions. TDP is defined as the constant correction that appears in the ground-state expectation value of a partial symmetry transformation applied to a connected spatial region M, the absolute value of which scales generically as exp(-alpha l + gamma) where l is the perimeter of M and gamma is the TDP. Motivated by a topological quantum field theory interpretation of the operator, we show that e(gamma). can be related to the quantum dimension of the symmetry defect, and provide a general formula for. when the entanglement Hamiltonian of the topological phase can be described by a (1 + 1)-dimensional conformal field theory (CFT). A special case of TDP is equivalent to the topological Renyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase. We then investigate several examples of lattice models of topological phases, both analytically and numerically, in particular when the assumption of having a CFT edge theory is not satisfied. We also consider an example of partial translation symmetry in Wen's plaquette model and show that the result can be understood using the edge CFT. Our results establish an alternative tool to detect quantum topological order.

Automated summary

We introduce a new many-body topological invariant, called the topological disorder parameter (TDP), to characterize gapped quantum phases with global internal symmetry in 2 + 1 dimensions. Inspired by a topological quantum field theory interpretation, we establish a formula relating the TDP to the quantum dimension of the symmetry defect. We validate the effectiveness and applicability of the TDP through analytical and numerical investigations of several lattice models of topological phases.