| Summary: | 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 χ_{F} of the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result for D=1 that expresses the logL 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 χ_{F}. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to the D+1 order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional to χ_{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.
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