Topological Multipartite Entanglement in a Fermi Liquid

TL;DR

This paper reveals how the topology of the Fermi sea influences multipartite entanglement in Fermi gases across different dimensions, establishing universal divergence behaviors linked to topological invariants and analyzing the impact of interactions.

Contribution

It introduces a universal framework connecting Fermi sea topology to multipartite entanglement and extends known 1D results to higher dimensions using the replica method.

Findings

Multipartite mutual information diverges as log^D L with a coefficient proportional to the Euler characteristic.

Charge-weighted entanglement entropy exhibits similar divergence and odd symmetry under particle-hole transformation.

Weak short-range interactions do not perturb the topological divergence in 3D, confirming robustness of the classification.

Abstract

We show that the topology of the Fermi sea of a $D$-dimensional Fermi gas is reflected in the multipartite entanglement characterizing $D+1$ regions that meet at a point. For odd $D$ we introduce the multipartite mutual information, and show that it exhibits a $\log^D L$ divergence as a function of system size $L$ with a universal coefficient that is proportional to the Euler characteristic $\chi_F$ of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for $D=1$ that expresses the $\log L$ divergence of the bipartite entanglement entropy in terms of the central charge $c$ characterizing a conformal field theory. For even $D$ we introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation. We show that the corresponding charge-weighted mutual information exhibits a similar $\log^D L$ divergence proportional to $\chi_F$. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the $D+1$'th order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to $\chi_F$. Our analytic results are based on the replica method. In addition we perform a numerical study of the charge-weighted mutual information for $D=2$ that confirms several aspects of the analytic theory. Finally, we consider the effect of interactions perturbatively within the replica theory. We show that for $D=3$ the $\log^3 L$ divergence of the topological mutual information is not perturbed by weak short-ranged interactions, though for $D=2$ the charge-weighted mutual information is perturbed. Thus, for $D=3$ the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.