Quantum mechanics

Theoretical physics

Pure mathematics

Lie algebra

Operating system

Computer science

Extrapolation

Modular design

Physics

Quantum entanglement

Central charge

Mathematics

Mathematical analysis

Quantum

Conformal map

Charge (physics)

Lie conformal algebra

Commutator

arXiv (Cornell University)

In Phys. Rev. Lett. 128, 176402 (2022), 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 $0.7%$.

Modular commutator in gapped quantum many-body systems