R\'enyi-like entanglement probe of the chiral central charge

TL;DR

This paper introduces a new entanglement measure, _{\u03b1,eta}, for 2D gapped quantum systems that generalizes the modular commutator and can reveal the chiral central charge from ground state wave functions.

Contribution

It proposes a Re9nyi-like entanglement probe _{\u03b1,eta} for gapped 2D systems, providing analytic expressions and a practical measurement approach for the chiral central charge.

Findings

_{\u03b1,eta} has a universal value related to the chiral central charge.

Analytic expressions derived for non-interacting fermions and string-net models.

Permutation operator representation enables numerical and experimental measurement.

Abstract

We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by $\omega_{\alpha,\beta}$, is parameterized by two positive real numbers $\alpha, \beta$, and can be seen as a ``R\'enyi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for $\omega_{\alpha,\beta}$ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that $\omega_{\alpha,\beta}$ takes a universal value related to the chiral central charge. For integer values of $\alpha$ and $\beta$, our quantity $\omega_{\alpha,\beta}$ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring $\omega_{\alpha,\beta}$ in numerical simulations and potentially, experiments.