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, $\langleρ_{AB}^α\mathrm{e}^{λ{Q}_{AB}}\mathrm{e}^{μ{Q}_{BC}}ρ_{BC}^β\rangle$. This function is of independent interest as it represents a generalization of the recently proposed Rényi modular commutator. In appropriate limits, this function yields the response of Rényi 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.

Figures (5)

• Figure 1: The geometric setting. (a): A large but finite disk $ABC$ is tripartited. (b): Modular flow causes chiral motion of entanglement modes (red lines) in chiral topological phases.
• Figure 2: Free fermion systems. (a): A large but finite disk ABC is tripartited, with its complementary region labeled by D. Four red shaded regions are triple contact points. (b): An infinite plane is tripartited. In both cases, only the topology is essential and a specific shape does not matter.
• Figure 3: Numerical verification of the generating function $\Omega_{\alpha,\beta,\lambda,\mu}$ for the QWZ model on a $L=32$ lattice with subsystem dimensions $L_{ABC}=16$. With charge parameters fixed at $\lambda=2$ and $\mu=1$, we show the ratio $\arg(\Omega)_{\text{num}} / \arg(\Omega)_{\text{ref}}$ for different replica indices $\alpha$ and $\beta$. The reference phase $\arg(\Omega)_{\text{ref}}$ is obtained from the analytical expression in Eq. \ref{['eq:result']} by setting $c_{-} = 2\pi\sigma_{xy} = 1$. The convergence of the ratio across different $M$-values confirms the universal scaling predicted by our theory.
• Figure 4: (a): Tripartited region $ABC$ with four triple points $v_1,v_2,v_3,v_4$. (b): regularization surface $Y$ is cut into four 3-punctured spheres $Y_v$, $v=1,2,3,4$.
• Figure 5: Example for $\Sigma_v$ with $m=3$, $n=2$. Red/green lines denotes $\lambda$/$\mu$-flux. For simplicity, we don't draw geodesic lines. We notice there is an ambiguity in choosing $\pm\lambda$ or $\pm\mu$ in $\theta_{vv^\prime}$, but both choices give the same $\theta_{vv^\prime}^2$ results.