A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real-valued in any local product basis.

%B Phys. Rev. Lett. %V 128 %P 176402 %8 4/28/2022 %G eng %U https://arxiv.org/abs/2110.06932 %N 17 %R 10.1103/PhysRevLett.128.176402 %0 Journal Article %J Physical Review B %D 2022 %T Modular commutator in gapped quantum many-body systems %A Isaac H. Kim %A Bowen Shi %A Kohtaro Kato %A Victor V. Albert %XIn arXiv:2110.06932, we argued that the chiral central charge -- a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system -- can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.7%.

%B Physical Review B %V 106 %8 8/26/2022 %G eng %U https://arxiv.org/abs/2110.10400 %R 10.1103/physrevb.106.075147