Rényi-like entanglement probe of the chiral central charge

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 $\ddot{I}{‰}_{\hat{I}\pm ,{\hat{I}}^2}$, is parameterized by two positive real numbers $\hat{I}\pm ,{\hat{I}}^2$, and can be seen as a ``Rényi-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 $\ddot{I}{‰}_{\hat{I}\pm ,{\hat{I}}^2}$ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that $\ddot{I}{‰}_{\hat{I}\pm ,{\hat{I}}^2}$ takes a universal value related to the chiral central charge. For integer values of $\hat{I}\pm$ and ${\hat{I}}^2$, our quantity $\ddot{I}{‰}_{\hat{I}\pm ,{\hat{I}}^2}$ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring $\ddot{I}{‰}_{\hat{I}\pm ,{\hat{I}}^2}$ in numerical simulations and potentially, experiments.

15 pages, 6 figures