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.

1 aKim, Isaac, H.1 aShi, Bowen1 aKato, Kohtaro1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2110.0693201471nas a2200157 4500008004100000245005900041210005800100260001400158490000800172520102100180100001901201700001501220700001801235700002301253856003701276 2022 eng d00aModular commutator in gapped quantum many-body systems0 aModular commutator in gapped quantum manybody systems c8/26/20220 v1063 aIn 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%.

1 aKim, Isaac, H.1 aShi, Bowen1 aKato, Kohtaro1 aAlbert, Victor, V. uhttps://arxiv.org/abs/2110.10400