Topologists can differentiate between bagels and pretzels by simply counting holes in each bread. The number of holes, formally described by the Euler characteristic, is a topological invariant insensitive to smooth deformation of the shape and size of an object. In condensed matter physics, we study an analogue of pastry, the Fermi sea. Nature provides a variety of exotic topology, e.g. noble metals like copper have a Fermi sea that resembles a pretzel with 4 holes! So does this type of topology have any significant consequences?

In this talk, I will address this question by introducing physical quantities that measure the Euler characteristic of a D-dimensional Fermi sea. Particularly, I will highlight a universal behavior in the equal-time density correlation function, which is proportional to the Euler number in the long wavelength limit. I will explain how this connects to entanglement entropy in real space, and demonstrate that the finite-size scaling of the multipartite mutual information elegantly captures the topology of Fermi sea in momentum space. This generalizes the Calabrese-Cardy formula for bipartite entanglement entropy in 1D Luttinger liquid, and provides a new perspective to connect between topology and entanglement in gapless phases. Concepts introduced here may also be useful for developing experimental probes of Fermi sea topology.