Momentum Distribution and Contact of the Unitary Fermi Gas

We calculate the momentum distribution $n(k)$ of the unitary Fermi gas by using quantum Monte Carlo calculations at finite temperature $T/{ϵ}_F$ as well as in the ground state. At large momenta $k/k_F$, we find that $n(k)$ falls off as $C/k^4$, in agreement with the Tan relations. From the asymptotics of $n(k)$, we determine the contact $C$ as a function of $T/{ϵ}_F$ and present a comparison with theory. At low $T/{ϵ}_F$, we find that $C$ increases with temperature, and we tentatively identify a maximum around $T/{ϵ}_F\simeq 0.4$. Our calculations are performed on lattices of spatial extent up to $N_x=14$ with a particle number per unit volume of $\simeq 0.03–0.07$.

Figure 1

Momentum distribution $n(k)$ from QMC calculations for $N_x=10$ as a function of $k/k_F$, for various temperatures ranging from zero to $T/{ϵ}_F\simeq 0.5$. The curves are intended as a guide to the eye, and the statistical errors are the size of the symbols. Inset: $n(k)$ for $N_x=14$ in a log-log scale, showing the asymptotic $\sim k^{-4}$ behavior.

Figure 2

Plot of $3{\pi }^2(k/k_F{)}^{4}n(k)$ for $N_x=12$ as a function of $k/k_F$ at $T/{ϵ}_F=0.178$ and 0.404. The “plateaux” at large $k/k_F$ give the (intensive) dimensionless quantity $C/(N{k}_F)$. At low $T/{ϵ}_F$, the asymptotic region is reached at $k/k_F\simeq 2$. Inset: $N_x=10$ results at $T=0$ showing only slight dependence on the Suzuki-Trotter step $\tau$.

Figure 3

Summary of QMC results for $C/(N{k}_F)$ as a function of $T/{ϵ}_F$, as determined from the large $k/k_F$ behavior of $n(k)$. The error bars are dominated by systematics related to the residual fluctuations in the plateaux, as exhibited in Fig. 2. Also shown are the $t$-matrix calculations of Refs. 27, 28, the virial expansion of Ref. 29, and the diagrammatic Monte Carlo result of Ref. 34.