Entanglement Measure Response to Modular Flow and Chiral Topological Phases

Abstract

Recent years have witnessed significant progress in the entanglement-based characterization of quantum phases of matter. The primary objects of interest are the reduced density matrix and its associated entanglement Hamiltonian. As intrinsic properties of a quantum state, these quantities theoretically determine all experimentally accessible local observables. In this work, we investigate the response of two entanglement measures to the real-time dynamics driven by the entanglement Hamiltonian--a process known as modular flow. We demonstrate that our results can be unified into a single generating function, $\langle\rho_{AB}^\alpha \mathrm{e}^{\lambda {Q}_{AB}}\mathrm{e}^{\mu{Q}_{BC}}\rho_{BC}^\beta\rangle$. This function is of independent interest as it represents a generalization of the recently proposed R\'enyi modular commutator. In appropriate limits, this function yields the response of R\'enyi entropy and its charged version, which we find to be uniquely determined by chiral topological invariants, specifically the chiral central charge and the Hall conductance. Our analytical findings are validated through two independent approaches: (i) free fermion systems using the real-space Chern number formula, and (ii) an effective field theory treatment that regularizes the entanglement cut via chiral conformal field theory. Both methods yield consistent results.