Harmonically trapped Fermi gas: Temperature dependence of the Tan contact

Ultracold atomic gases with short-range interactions are characterized by a number of universal species-independent relations. Many of these relations involve the two-body Tan contact. Employing the canonical ensemble, we determine the Tan contact for small harmonically trapped two-component Fermi gases at unitarity over a wide range of temperatures, including the zero and high-temperature regimes. A cluster expansion that describes the properties of the $N$-particle system in terms of those of smaller subsystems is introduced and shown to provide an accurate description of the contact in the high-temperature regime. Finite-range corrections are quantified and the role of the Fermi statistics is elucidated by comparing results for Fermi, Bose, and Boltzmann statistics.

Figure 1

(a) The dotted, dashed, dash-dotted, and dash-dot-dotted lines show the scaled pair distribution function $r^2{P}_{12}\left(r\right)$ for $k_{B}T/\left(\hslash \omega \right)=0,0.6,1.2$ and 2, respectively, for the $(3,1)$ system interacting through $V_{\text{G}}$ with $r_0=0.06a_{\text{ho}}$. The $T=0$ curve is determined using the basis set expansion approach while the finite $T$ curves are determined using the PIMC approach. The thick solid lines in the inset of (a), which is a blowup of the small $r$ region, show the extrapolation to $r=0$. (b) The solid and dotted lines show the relative energy of the ground state with $L^{\Pi }=1^{-}$ and first excited state with $L^{\Pi }=0^{+}$ of the $(2,1)$ system interacting through $V_{\text{G}}$ with $r_0=0.06a_{\text{ho}}$ as a function of $-1/a_s$.

Figure 2

Contact $C_{N_1,N_2}$ as a function of $k_{B}T/E_{\text{ho}}$ for the (a) $(1,1)$, (b) $(2,1)$, (c) $(3,1)$, and (d) $(2,2)$ systems. The circles, squares, and triangles show $C_{N_1,N_2}$ for $V_{\text{G}}$ with $r_0/a_{\text{ho}}=0.06$, $0.08$ and $0.1$ obtained using the PIMC approach. The solid lines show $C_{N_1,N_2}$ for $V_{\text{G}}$ with $r_0=0.06a_{\text{ho}}$ obtained using the basis set expansion approach. For comparison, the dotted lines in (a) and (b) show $C_{N_1,N_2}$ obtained using $V_{\text{F}}$. (a) The dashed line shows the first-order Taylor expansion at high temperature. (b) The dashed line shows the cluster expansion, i.e., the quantity $2C_{1,1}$. (c)/(d) The dashed and dash-dotted lines show the leading order term of the cluster expansion and the full cluster expansion, respectively. The insets of (a) and (b) show blowups of the low-temperature regions.

Figure 3

Circles, squares, crosses, and triangles show $C_{N_1,N_2}$ for the $(1,1)$, $(2,1)$, $(3,1),$ and $(4,1)$ systems, respectively, interacting through $V_{\text{G}}$ with $r_0=0.06a_{\text{ho}}$ as a function of $T$ [43]. Dotted lines serve as a guide to the eye. Inset: Crosses, squares, and circles show the contact of the $(3,1)$ system for Fermi, Boltzmann, and Bose statistics, respectively.

Figure 4

Range dependence of the contact $C_{1,1}$ for $r_0=0.06a_{\text{ho}}$ as a function of $k_{B}T/E_{\text{ho}}$. The circles show the difference between the finite-range contact calculated using approach (i) and the zero-range contact, normalized by the zero-range contact. The solid line shows the difference between the finite-range contact calculated using approach (ii) and the zero-range contact, normalized by the zero-range contact. In the large $T$ limit, the difference approaches zero from below. In approach (ii), $r_{\text{eff}}=0.12178a_{\text{ho}}$—corresponding to $V_{\text{G}}$ with $r_0=0.06a_{\text{ho}}$—has been used.

Figure 5

(a) The solid, dotted, dashed, and dash-dotted lines show the weight factor $\exp [-E_j^{\text{rel}}/\left(k_{B}T\right)]/Z^{\text{rel}}$ (see right axis) as a function of the relative energy for $k_{B}T/E_{\text{ho}}=0.2,0.4,0.6,$ and $0.8$, respectively. The circles show the zero-temperature contact ${\overline{C}}_{2,1}$ (see left axis) for states with $E_j^{\text{rel}}<5E_{\text{ho}}$. (b) The dots show the zero-temperature contact ${\overline{C}}_{2,1}$ as a function of the relative energy for $E_j^{\text{rel}}\lesssim 30E_{\text{ho}}$. The solid (dashed) line connects the contacts of $L=1$ ($L=0$) states, which are characterized by the same $s_{L,\nu }$ value but different $q$ values; the $s_{L,\nu }$ values chosen are the smallest ones for both $L=1$ and 0.