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.

arXiv comment6+24 pages, 5+12 figures

References in corpus (36)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Anyons in an exactly solved model and beyond
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• The Black Hole in Three Dimensional Space Time
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Entanglement Entropy and Quantum Field Theory
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Geometry of the 2+1 Black Hole
• Towards a derivation of holographic entanglement entropy
• Reduced density matrices and entanglement entropy in free lattice models
• Notes on Some Entanglement Properties of Quantum Field Theory
• The (2+1)-Dimensional Black Hole
• Relative entropy equals bulk relative entropy
• Chiral Gravity in Three Dimensions
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Quantized Thermal Transport in the Fractional Quantum Hall Effect
• Entanglement hamiltonians in two-dimensional conformal field theory
• Energy flow in non-equilibrium conformal field theory
• Monotonicity of quantum relative entropy revisited
• Momentum polarization: an entanglement measure of topological spin and chiral central charge
• Holographic entanglement entropy and gravitational anomalies
• Extremal Surface Barriers
• A modular toolkit for bulk reconstruction
• Light Rays, Singularities, and All That
• Chiral central charge from a single bulk wave function
• Modular commutator in gapped quantum many-body systems
• Fusion rules from entanglement
• Modular Flow as a Disentangler
• Localization length exponent in two models of quantum Hall plateau transitions
• From entanglement generated dynamics to the gravitational anomaly and chiral central charge
• Criticality of two-dimensional disordered Dirac fermions in the unitary class and universality of the integer quantum Hall transition
• Anomalies of the Entanglement Entropy in Chiral Theories
• On Local and Integrated Stress-Tensor Commutators
• Quantum Information Theory of the Gravitational Anomaly
• The action of HRT-areas as operators in semiclassical gravity

Cited by in corpus (19)

• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Symmetry-enforced many-body separability transitions
• From entanglement generated dynamics to the gravitational anomaly and chiral central charge
• Entanglement in tripartitions of topological orders: a diagrammatic approach
• Extracting higher central charge from a single wave function
• Multipartite edge modes and tensor networks
• 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