Finite-density corrections to the unitary Fermi gas: A lattice perspective from dynamical mean-field theory

We investigate the approach to the universal regime of the dilute unitary Fermi gas as the density is reduced to zero in a lattice model. To this end we study the chemical potential, superfluid order parameter and internal energy of the attractive Hubbard model in three different lattices with densities of states (DOSs) which share the same low-energy behavior of fermions in three-dimensional free space: a cubic lattice, a “Bethe lattice” with a semicircular DOS, and a “lattice gas” with parabolic dispersion and a sharp energy cutoff that ensures the normalization of the DOS. The model is solved using dynamical mean-field theory, that treats directly the thermodynamic limit and arbitrarily low densities, eliminating finite-size effects. At densities on the order of one fermion per site the lattice and its specific form dominate the results. The evolution to the low-density limit is smooth and it does not allow to define an unambiguous low-density regime. Such finite-density effects are significantly reduced using the lattice gas, and they are maximal for the three-dimensional cubic lattice. Even though dynamical mean-field theory is bound to reduce to the more standard static mean field in the limit of zero density due to the local nature of the self-energy and of the vertex functions, it compares well with accurate Monte Carlo simulations down to the lowest densities accessible to the latter.

Figure 1

(Color online) Comparison between the three different DOS’s with parameters such that the low-energy behavior coincides with that of free electrons, as discussed in the text: semicircular DOS (dot-dashed red line), cubic three-dimensional lattice (dashed green line), and lattice gas (solid blue line). For reference the DOS for free fermions in three-dimensional space is shown as a dotted black line. Energies are expressed in units of the half-bandwidth $D$ of the semicircular DOS, the DOS per site is measured in units of $1/D$.

Figure 2

(Color online) Static mean-field results for the dimensionless rescaled chemical potential $\mu /E_F$ and superconducting order parameter $\Delta /E_F$ as a function of density of particles $n$ (dimensionless). The dashed blue line is for the cubic lattice, the red solid line for the semicircular DOS, and the dot-dashed green line is for the lattice gas. Horizontal arrows mark Leggett’s results in the zero-density limit.

Figure 3

(Color online) Particle density $n$ (solid blue line) and superfluid order parameter $\Delta$ (red dashed line) as a function of the chemical potential $\mu$ for a semicircular DOS. The inset shows the low-density region in logarithmic scale, which shows that $n\propto {\mu }^{3/2}$ and $\Delta \propto \mu$. $\Delta$ and $\mu$ are measured in units of $D$, $n$ is dimensionless.

Figure 4

(Color online) Ground-state energy density (total energy divided by the density) as a function of the density $n$. We plot the total-energy density $e$ (blue solid line), the kinetic contribution $k$ (red dashed line), the absolute value of the potential-energy contribution $\left|v\right|$ (dot-dashed green line). The inset shows the low-density region in logarithmic scale. Energies are expressed in units of $D$.

Figure 5

(Color online) Dimensionless renormalized chemical potential as a function of density. Comparison between static MF and DMFT (marked by open dots). Blue dashed line is for the cubic lattice, solid red line for the semicircular DOS, and dot-dashed green line for the lattice gas.

Figure 6

(Color online) Dimensionless renormalized internal energy as a function of density. Comparison between static MF and DMFT (marked by open dots). Blue dashed line is for the cubic lattice, solid red line for the semicircular DOS, and dot-dashed green line for the lattice gas.

Figure 7

(Color online) Dimensionless renormalized superfluid order parameter as a function of density. Comparison between static MF and DMFT (marked by open dots). Blue dashed line is for the cubic lattice, solid red line for the semicircular DOS, and dot-dashed green line for the lattice gas.

Figure 8

(Color online) Dependence on the density of the chemical potential divided by the lattice free Fermi energy for the three lattices under study (dimensionless quantity).