Modular Commutators in Conformal Field Theory

Abstract

The modular commutator is a recently discovered multipartite entanglement measure that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in $1+1$ dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped $(2+1)$-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.

6+24 pages, 5+12 figures

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• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Numerical investigations of the extensive entanglement Hamiltonian in quantum spin ladders
• Unveiling chiral states in the XXZ chain: Finite-size scaling probing symmetry-enriched $c=1$ conformal field theories
• Many-body systems with spurious modular commutators
• Petz map recovery for long-range entangled quantum many-body states
• Chiral Virasoro algebra from a single wavefunction
• Conformal geometry from entanglement
• Holographic multipartite entanglement structures in IR modified geometries
• Boundary and bulk notions of transport in the AdS${}_3$/CFT${}_2$ correspondence
• Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order
• Rényi-like entanglement probe of the chiral central charge
• Geometric additivity of modular commutator for multipartite entanglement