Symmetry-resolved entanglement in many-body systems

TL;DR

The paper addresses how entanglement in many-body quantum systems can be decomposed into symmetry sectors associated with conserved charges. It introduces a geometric replica-trick approach that threads a space-time Aharonov-Bohm flux through an $n$-sheet Riemann surface, encoding $s_n(\alpha)=\mathrm{Tr}(\rho_A^n e^{i\alpha \hat{N}_A})$. In 1+1D conformal field theory, the authors derive a general expression for the scaling of $s_n(\alpha)$ with the composite twist field $\mathcal{T}_{\mathcal{V}}$, yielding $\Delta_n(\alpha)=\frac{c(n-n^{-1})}{24}+\frac{\Delta_{\mathcal{V}}}{n}$ and $s_n(\alpha)\sim L^{-\frac{c}{6}(n-n^{-1})} L^{-2(\Delta_{\mathcal{V}}+\bar{\Delta}_{\mathcal{V}})/n}$. They map the symmetry-resolved entropy to sector probabilities; show for $U(1)$ charge the variance scales as $\langle \Delta N_A^2\rangle=\frac{K\ln L}{\pi^2}$ and the sector entropies scale as $\mathcal{S}(N_A)\sim \sqrt{\ln L}$ (or $\mathcal{O}(1)$ when extra spin conservation is present); verify numerically for free and interacting fermions, spin chains, and parafermions, and outline experimental measurement routes. The work provides a practical route to measure symmetry-resolved entanglement and deepens understanding of how symmetries shape quantum correlations.

Abstract

Similarly to the system Hamiltonian, a subsystem's reduced density matrix is composed of blocks characterized by symmetry quantum numbers (charge sectors). We present a geometric approach for extracting the contribution of individual charge sectors to the subsystem's entanglement measures within the replica trick method, via threading appropriate conjugate Aharonov-Bohm fluxes through a multi-sheet Riemann surface. Specializing to the case of 1+1D conformal field theory, we obtain general exact results for the entanglement entropies and spectrum, and apply them to a variety of systems, ranging from free and interacting fermions to spin and parafermion chains, and verify them numerically. We find that the total entanglement entropy, which scales as $\ln L$, is composed of $\sqrt{\ln L}$ contributions of individual subsystem charge sectors for interacting fermion chains, or even $\mathcal{O} (L^0)$ contributions when total spin conservation is also accounted for. We also explain how measurements of the contribution to the entanglement from separate charge sectors can be performed experimentally with existing techniques.

Figures (3)

• Figure 1: (Color online) (a) An example of a 3-sheet Riemann surface geometry with an inserted space-time Aharonov-Bohm flux $\alpha$. (b) A generic many body wavefunction is a superposition of subsystem charge states.
• Figure 2: (Color online) Charge distribution $P(N_A)$ and charge-sector contributions to entanglement entropy $\mathcal{S}(N_A)$ in a subsystem of $L=10000$ sites of an infinite half-filled tight-binding chain, computed numerically (dots) and analytically (continuous lines).
• Figure 3: (Color online) Integrated density of the entanglement spectrum $n(\lambda, N_A)$ for the same system in Fig. 2 computed numerically (discontinuous lines) and analytically (continuous lines) for various particle numbers $N_A$. Numerically we used the 24 closest-to-zero single particle eigenvalues of the entanglement Hamiltonian to build the highest many-body eigenvalues of $\rho_A$.