The Fermi sea in a metal is a topological object characterized by an integer topological invariant called the Euler characteristic, ¿F. In this talk we will argue that for a 2D Fermi gas ¿F is reflected in a quantized frequency dependent non-linear 3 terminal conductance that generalizes the Landauer conductance in D=1. We will critically address the roles of electrical contacts and Fermi liquid interactions, and we will propose experiments on 2D Dirac materials, such as graphene, using a triple point contact geometry. We will go on to show that for a D dimensional Fermi gas, ¿F is also reflected in the multipartite entanglement characterizing D+1 regions that meet at a point. This generalizes a well-known result that relates the bipartite entanglement entropy of a 1+1D conformal field theory to its central charge c. We will argue that for an interacting 3D Fermi liquid, ¿F distinguishes distinct topological Fermi liquid phases.
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