Universal energy functionals for trapped Fermi gases in low dimensions

We study a system of trapped two-component Fermi gases with a zero-range interaction in two dimensions or one dimension. We calculate the one-particle density matrices of these systems at small displacements, from which we show that the $N$-body energies are linear functionals of the occupation probabilities of single-particle energy eigenstates. A universal energy functional was first derived in 2011 [S. Tan, Phys. Rev. Lett. 107, 145302 (2011)] for trapped zero-range interacting two-component Fermi gases in three dimensions. We also calculate the asymptotic behaviors of the occupation probabilities of single-particle energy eigenstates at high energies.

Figure 1

The configuration of $N$ fermions. (a) shows the region ${\mathcal{D}}_{\eta }$, where no spin-down fermion comes inside the circle centered at $\mathbf{r}$ with radius $\eta$. (b) shows that a spin-down fermion at position $\mathbf{s}$ is within the circle of radius $\eta$ centered at $\mathbf{r}$. (c) shows that two or more spin-down fermions come inside the small circle, and the probability amplitudes of these situations are suppressed by Fermi statistics.