Hartree shift in unitary Fermi gases

The Hartree energy shift is calculated for a unitary Fermi gas. By including the momentum dependence of the scattering amplitude explicitly, the Hartree energy shift remains finite even at unitarity. Extending the theory also for spin-imbalanced systems allows calculation of polaron properties. The results are in good agreement with more involved theories and experiments.

Figure 1

The real and imaginary parts of $\Gamma (s,d)$ as a function of the center-of-mass momentum $s$ and the relative momentum $d$ calculated for a zero-temperature Fermi sphere.

Figure 2

The real and imaginary parts of $\Gamma (s,d)$ as a function of the center-of-mass momentum $s$ and the relative momentum $d$ calculated for a finite temperature $T=0.5$ $T_F$ Fermi sphere.

Figure 3

The real and imaginary parts of $\Gamma (s,d)$ as a function of the center-of-mass momentum $s$ and relative momentum $d$ calculated for the BPC ansatz occupation probabilities $n\left(x\right)=\frac{1}{2}(1-\frac{x-\mu }{\sqrt{{(x-\mu )}^2+{\Delta }^2}})$ with $\mu =0.59$ $E_F$ and $\Delta =0.69$ $E_F$.

Figure 4

The real part of $\Gamma (k_z,k_r)$ as a function of the atom momentum $\mathbf{k}=(k_z,k_r)$ for an atom with momenta $\mathbf{p}=0$ (top left), $\mathbf{p}=0.50$ $k_F$ (top right), $\mathbf{p}=1.0$ $k_F$ (bottom left), and $\mathbf{p}=2.0$ $k_F$ (bottom right). Shown is the absolute value of the real part, calculated assuming the zero-temperature Fermi distribution for the occupation numbers.

Figure 5

The real part of $\Gamma (k_z,k_r)$ as a function of the atom momentum $\mathbf{k}=(k_z,k_r)$ for an atom with momenta $\mathbf{p}=0$ (top left), $\mathbf{p}=0.50$ $k_F$ (top right), $\mathbf{p}=1.0$ $k_F$ (bottom left), and $\mathbf{p}=2.0$ $k_F$ (bottom right). Shown is the absolute value of the real part, calculated assuming the BPC ansatz for the occupation numbers.

Figure 6

The real and imaginary parts of the Hartree self-energy ${\Sigma }_{\mathrm{Hartree}}(p,{\varepsilon }_p)$ as a function of the atom momentum $p$ for the ideal Fermi sphere and the BPC ansatz.

Figure 7

The real and imaginary parts of the Hartree self-energy ${\Sigma }_{\mathrm{Hartree}}(p,{\varepsilon }_p)$ as a function of the atom momentum $p$ for the ideal Fermi sphere and the BPC ansatz, calculated using the semi-self-consistent scheme in which the imaginary part of the self-energy is neglected.

Figure 8

The Hartree self-energy of a polarized ($P=0.5$) zero-temperature Fermi gas. The real part is shown in the left and the imaginary part on the right. The Hartree self-energy for the balanced gas is shown for reference.

Figure 9

The real and imaginary part of the Hartree self-energy of a single minority component atom immersed in a zero-temperature Fermi sea of majority component atoms. Shown are results from both the fully self-consistent scheme and from the semi-self-consistent scheme in which the imaginary parts of the self-energy are neglected in the iteration.