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