Contact in the Unitary Fermi Gas across the Superfluid Phase Transition

A quantity known as the contact is a fundamental thermodynamic property of quantum many-body systems with short-range interactions. Determination of the temperature dependence of the contact for the unitary Fermi gas of infinite scattering length has been a major challenge, with different calculations yielding qualitatively different results. Here we use finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods on the lattice within the canonical ensemble to calculate the temperature dependence of the contact for the homogeneous spin-balanced unitary Fermi gas. We extrapolate to the continuum limit for 40, 66, and 114 particles, eliminating systematic errors due to finite-range effects. We observe a dramatic decrease in the contact as the superfluid critical temperature is approached from below, followed by a gradual weak decrease as the temperature increases in the normal phase. Our theoretical results are in excellent agreement with the most recent precision ultracold atomic gas experiments. We also present results for the energy as a function of temperature in the continuum limit.

Figure 1

The contact $C$ (in units of $N{k}_F$) of the UFG as a function of temperature $T$ (in units of $T_F$). Our AFMC results in the continuum limit for $N=40$ particles (solid blue squares) are compared with the recent experimental results of the Swinburne group [34] (solid purple diamonds) and the MIT group [35] (solid red up triangles). We also compare with other theoretical results: the lattice diagrammatic Monte Carlo (LDMC) results of Ref. [39] (open gray diamonds), the bold diagrammatic Monte Carlo (BDMC) results of Ref. [40] (open black circles), the Luttinger-Ward (LW) results of Ref. [12] (solid pink line), and the $t$-matrix result of Ref. [36] (dotted purple line). We also show the $T=0$ quantum Monte Carlo results of Ref. [58] (open purple square) and the low-temperature experimental result of Ref. [59] (solid black down triangle). The second-order and third-order virial expansions for the contact are shown, respectively, by the dashed-dotted red line and dashed blue line. Virial coefficients were calculated in Refs. [38, 60, 61, 62]. The inset shows our continuum limit AFMC results for several particle numbers: $N=40$ (solid blue squares), $N=66$ (solid orange circles), and $N=114$ (solid green down triangles).

Figure 2

AFMC thermal energy $E$ (in units of the Fermi gas ground-state energy $E_{\mathrm{FG}}$) as a function of temperature $T$ (in units of the Fermi temperature $T_F$) for the UFG obtained in the continuum limit for $N=40$ particles (solid blue squares) and $N=66$ particles (solid orange circles). We compare with the experimental results of Ref. [48] (open black circles), and with the AFMC results of Ref. [64] (open blue squares). Using our lowest temperature results, we estimate the Bertsch parameter to be $\xi =0.367(7)$ (solid red circle), in close agreement with the ground-state quantum Monte Carlo estimate $\xi =0.372(5)$ of Ref. [56] (open green triangle). The inset shows the low-temperature regime.