Topological density correlations in a Fermi gas

A Fermi gas of noninteracting electrons, or ultracold fermionic atoms, has a quantum ground state defined by a region of occupancy in momentum space known as the Fermi sea. The Euler characteristic chi F of the Fermi sea serves to topologically classify these gapless fermionic states. The topology of a D -dimensional Fermi sea is physically encoded in the D + 1 point equal time density correlation function. In this work, we first present a simple proof of this fact by showing that the evaluation of the correlation function can be formulated in terms of a triangulation of the Fermi sea with a collection of points, links, and triangles and their higher dimensional analogs. We then make use of the topological D + 1 point density correlation to reveal universal structures of the more general M point density correlation functions in a D -dimensional Fermi gas. Two experimental methods are proposed for observing these correlations in D = 2. In cold atomic gases imaged by quantum gas microscopy, our analysis supports the feasibility of measuring the third -order density correlation, from which chi F can be reliably extracted in systems with as few as around 100 atoms. For solid-state electron gases, we propose measuring correlations in the speckle pattern of intensity fluctuations in nonlinear x-ray scattering experiments.