Ground-state energy of spin-$\frac{1}{2}$ fermions in the unitary limit

We present lattice results for the ground-state energy of a spin-$\frac{1}{2}$ fermion system in the unitary limit, where the effective range of the interaction is zero and the scattering length is infinite. We compute the ground-state energy for a system of 6, 10, 14, 18, and 22 particles, with equal numbers of up and down spins in a periodic cube. We estimate that in the limit of large number of particles, the ground-state energy is 0.25(3) times the ground-state energy of the free Fermi system.

Figure 1

${\xi }_{3,3}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$.

Figure 2

${\xi }_{5,5}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$.

Figure 3

${\xi }_{7,7}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$.

Figure 4

${\xi }_{9,9}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$.

Figure 5

${\xi }_{11,11}\left(t\right)$ versus $E_{F}t$ for $L=5,6$.

Figure 6

${\xi }_{N,N}\left(t\right)$ versus $E_{F}t$ for $N=3,5,7,9,11$. We have drawn lines showing the central value and error bounds for the estimate $\xi =0.25\left(3\right)$.

Figure 7

$O\left(a^2\right)$-improved and standard lattice action results for ${\xi }_{3,3}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$.

Figure 8

${\xi }_{5,5}\left(t\right)$ versus $E_{F}t$ for $L=4,5,6$ and $C^{\mathrm{phys}}=-1.08\times {10}^{-4}{\mathrm{MeV}}^{-2}$.

Figure 9

Comparison of ${\xi }_{7,7}\left(t\right)$, ${\xi }_{7,7}^1\left(t\right)$, and ${\xi }_{7,7}^2\left(t\right)$.