Modular commutator in gapped quantum many-body systems

TL;DR

This paper explores the properties of the modular commutator in gapped quantum systems, linking it to topological invariants like the chiral central charge, and demonstrates its consistency through numerical calculations on lattice models.

Contribution

It establishes theoretical connections of the modular commutator with various quantum information measures and proves its invariance across gapped domain walls, supported by numerical evidence.

Findings

Modular commutator relates to conditional mutual information and modular flow.

Gapped domain walls do not change the modular commutator.

Numerical calculations on lattice models support the theoretical formula.

Abstract

In 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%.