Understand
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)d.
- 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 $\gamma$ when the entanglement Hamiltonian of the topological phase can be described by a (1+1)d conformal field theory (CFT).
- A special case of TDP is equivalent to the topological R\'enyi entanglement entropy when the symmetry is the cyclic permutation of the replica of the gapped phase.