Modular commutator in gapped quantum many-body systems

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%.

Knots and entanglement

“…Because the 2d reference state obtained this way allows a gapped boundary, the system cannot be chiral. From the recent perspective [12], this manifests in the fact that the modular commutator is zero.…”

“…These axioms, which can be stated in the approximate case, can be useful for the study of 2d chiral systems with a bulk energy gap and gapless edge modes, whose thermal Hall conductance is quantized [8,9] according to its chiral central charge [10]. In fact, with related techniques, a formula for the chiral central charge, in terms of a single bulk ground state wave function, has been argued and tested [11,12]. No rigorous derivation of this formula is known, at the moment, which calls for new innovation of concepts and techniques of the framework.…”

From Entanglement Generated Dynamics to the Gravitational Anomaly and Chiral Central Charge

“…We are interested in the response of the entanglement entropy. This idea has been widely used in the study of holography [7][8][9] and recently discussed in condensed matter systems [10]. Since the subregion where the modular flow is applied and that where the entanglement is measured must have an overlap to give a non-trivial answer, we denote the subregions as AB and BC, respectively, with B being the overlapping part.…”