Lattice approaches to dilute Fermi gases: Legacy of broken Galilean invariance

In the dilute limit, the properties of fermionic lattice models with short-range attractive interactions converge to those of a dilute Fermi gas in continuum space. We investigate this connection using mean-field theory, and we show that the existence of finite lattice spacing has consequences down to very small densities. In particular, we show that the reduced translational invariance associated with the lattice periodicity has a pivotal role in the finite-density corrections to the universal zero-density limit. For a parabolic dispersion with a sharp cutoff, we provide an analytical expression for the corrections, and we find that the unavoidable cutoff contributes at leading-order to the corrections to the relevant observables. In a generic lattice we find a universal power-law behavior n1/3 which leads to significant corrections already for small densities. Our results place strong constraints on lattice extrapolations of dilute Fermi gas properties.