Modular Commutators in Conformal Field Theory

TL;DR

This paper derives a universal formula for the modular commutator in 1+1D conformal field theories, linking it to chiral central charge and conformal cross ratio, with applications to quantum Hall states and holography.

Contribution

It provides the first universal expression for the modular commutator in conformal field theories and connects it to geometric duals in AdS/CFT correspondence.

Findings

The modular commutator depends only on chiral central charge and conformal cross ratio.

The formula matches numerical simulations for quantum Hall states.

A geometric dual involving crossing angles of Ryu-Takayanagi surfaces is proposed.

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.