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 $D+1$ 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.

中文翻译：

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费米液体中的拓扑多方纠缠

我们证明了费米海的拓扑结构$D$维费米气体反映在多方纠缠表征中$D+1$在一点相遇的区域。对于奇数$D$我们引入了多方互信息并表明它表现出${\mathrm{日志}}^D\mathrm{大号}$作为系统规模函数的散度$\mathrm{大号}$具有与欧拉特性成比例的通用系数${\chi }_F$费米海的。这为费米气体提供了一个众所周知的结果的推广$D=1$表示$\mathrm{日志}\mathrm{大号}$就中心电荷而言的二分纠缠熵的发散$C$表征共形场论。对于甚至$D$我们引入了一个电荷加权纠缠熵，它在粒子-空穴变换下明显是奇怪的。我们表明，相应的电荷加权互信息表现出相似的${\mathrm{日志}}^D\mathrm{大号}$散度成正比${\chi }_F$. 我们的分析将在没有交互的情况下多方互信息的普遍行为与$D+1$阶等时间密度相关函数，我们展示了它在长波长极限中表现出的普遍行为，与${\chi }_F$. 我们的分析结果基于复制方法。此外，我们对电荷加权互信息进行了数值研究$D=2$这证实了分析理论的几个方面。最后，我们在复制理论中考虑了相互作用的影响。我们证明了$D=3$这${\mathrm{日志}}^3\mathrm{大号}$拓扑互信息的发散不受弱短程相互作用的干扰，尽管对于$D=2$电荷加权互信息受到扰动。因此，对于$D=3$多方互信息提供了一个强大的分类，可以区分不同的拓扑费米液相。