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 $n^{1/3}$ which leads to significant corrections already for small densities. Our results place strong constraints on lattice extrapolations of dilute Fermi gas properties.

Figure 1

Comparison between different lattice DOSs under investigation: semicircular (full red line), lattice gas 1 (dashed blue line), and lattice gas 2 (dashed-dotted green line). For reference, the DOSs per lattice site $l^3{D}_{\text{free}}$ for free fermions in three-dimensional space is shown as a dotted black line. We set $\hslash =l=m=1$.

Figure 2

Trajectories with constant $\eta$ in $\left(\right|U|,n)$ plane of attractive Hubbard model for $\eta =-10,-1,-0.1$ (BCS side), $\eta =0$ (unitarity), and $\eta =0.1,1,10$ (BEC side). The results shown concern the semicircular DOS (see Sec. 2) but a similar picture is expected for the other cases.

Figure 3

Rescaled chemical potential $\tilde{\mu }$, gap $\tilde{\Delta }$, and ground-state energy $\xi$ for the semicircular DOS case along the unitary trajectory ($\eta =0$) as a function of density $n$.

Figure 4

Rescaled chemical potential $\tilde{\mu }$, gap $\tilde{\Delta }$ and ground-state energy $\xi$ at unitarity ($\eta =0$) as a function of the lattice dilution parameter $k_{F}l$ for the case of semicircular DOS (red circles), lattice gas 1 (blue squares), and lattice gas 2 (green diamonds).

Figure 5

Rescaled chemical potential $\tilde{\mu }$, gap $\tilde{\Delta }$, and ground-state energy $\xi$ at unitarity ($\eta =0$) as a function of the lattice effective dilution parameter $k_F{R}_e$ for the case of semicircular DOS (red circles), lattice gas 1 (blue squares), and lattice gas 2 (green diamonds).

Figure 6

Relative deviation $D^{\Delta }$ (upper panel) and $D^{\mu }$ (lower panel) of the rescaled parameters from Leggett values as a function of the density $n$ (log scale) for different values of the gas parameter $\eta$ ($\eta =-2,-1,0,0.6,1,2$).

Figure 7

Asymptotic slope of lattice corrections to chemical potential $a_{\mu }^{{\text{LG}}_1}$ and to gap parameter $a_{\Delta }^{{\text{LG}}_1}$ for ${\text{LG}}_1$ model as a function of inverse gap parameter $\eta$ in BCS-BEC crossover. The inset shows a magnified view of the the region around the unitary point.