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%.

15+3 pages, 19 figures, Expanded version of arXiv:2110.06932

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• From entanglement generated dynamics to the gravitational anomaly and chiral central charge
• Modular Commutators in Conformal Field Theory
• Entanglement in tripartitions of topological orders: a diagrammatic approach
• Extracting higher central charge from a single wave function
• Generalized Real-space Chern Number Formula and Entanglement Hamiltonian
• Multipartite entanglement in two-dimensional chiral topological liquids
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Multipartite edge modes and tensor networks
• Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders
• Many-body systems with spurious modular commutators
• Petz map recovery for long-range entangled quantum many-body states
• Anyon Quantum Dimensions from an Arbitrary Ground State Wave Function
• Conformal geometry from entanglement
• Edge Theories for Anyon Condensation Phase Transitions
• The c-d conjecture
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• Detection of 2D SPT Order with Partial Symmetries
• Entanglement groups
• Knots and entanglement
• Geometric additivity of modular commutator for multipartite entanglement