Fourth- and Fifth-Order Virial Coefficients from Weak Coupling to Unitarity

In the current era of precision quantum many-body physics, one of the most scrutinized systems is the unitary limit of the nonrelativistic spin-$1/2$ Fermi gas, due to its simplicity and relevance for atomic, condensed matter, and nuclear physics. The thermodynamics of this strongly correlated system is determined by universal functions which, at high temperatures, are governed by universal virial coefficients $b_n$ that capture the effects of the $n$-body system on the many-body dynamics. Currently, $b_2$ and $b_3$ are well understood, but the situation is less clear for $b_4$, and no predictions have been made for $b_5$. To answer these open questions, we implement a nonperturbative analytic approach based on the Trotter-Suzuki factorization of the imaginary-time evolution operator, using progressively finer temporal lattice spacings. By means of these factorizations and automated algebra codes, we obtain the interaction-induced change $\mathrm{\Delta }{b}_n$ from weak coupling to unitarity. At unitarity, we find that $\mathrm{\Delta }{b}_3=-0.356(4)$ in agreement with previous results, $\mathrm{\Delta }{b}_4=0.062(2)$, which is in agreement with all previous theoretical estimates but at odds with experimental determinations, and $\mathrm{\Delta }{b}_5=0.078(6)$, which is a prediction. We show the impact of those answers on the density equation of state and Tan contact, and trace their origin back to their polarized and unpolarized components.

Figure 1

Top: our results for $\mathrm{\Delta }{b}_3$ (blue), $\mathrm{\Delta }{b}_4$ (red), and $-\mathrm{\Delta }{b}_5$ (green) shown with error bands as functions of the coupling strength given by $\mathrm{\Delta }{b}_2/\mathrm{\Delta }{b}_2^{\mathrm{UFG}}$. (We plot $-\mathrm{\Delta }{b}_5$ to avoid display interference with $\mathrm{\Delta }{b}_4$ around unitarity.) The dashed line shows $\mathrm{\Delta }{b}_3$ from Ref. [24]. The dark red cross (with errorbar) shows the Monte Carlo results of Ref. [29]: $\mathrm{\Delta }{b}_4^{\mathrm{UFG}}=0.078(18)$; the dark red plus sign (with small error bar) indicates the conjecture of Ref. [30]: $\mathrm{\Delta }{b}_4^{\mathrm{UFG}}=0.0620(8)$; finally, the dark red dot shows the approximate results of Ref. [31]: $\mathrm{\Delta }{b}_4^{\mathrm{UFG}}=0.06$. Bottom: subspace contributions $\mathrm{\Delta }{b}_{mj}$ as functions of the coupling strength. Our results are shown as error bands, color coded as in the top plot by $n=m+j$: blue for $\mathrm{\Delta }{b}_{21}$, red for $\mathrm{\Delta }{b}_{31}$ and $\mathrm{\Delta }{b}_{22}$, and green for $\mathrm{\Delta }{b}_{41}$ and $\mathrm{\Delta }{b}_{32}$. The red cross (with errorbar) shows Ref. [29]: $\mathrm{\Delta }{b}_{31}^{\mathrm{UFG}}=0.0848(64)$ and $\mathrm{\Delta }{b}_{22}^{\mathrm{UFG}}=-0.0920(128)$; the red dot shows Ref. [31]: $\mathrm{\Delta }{b}_{31}^{\mathrm{UFG}}=0.100$ and $\mathrm{\Delta }{b}_{22}^{\mathrm{UFG}}=-0.144$; finally, the red plus sign (with small error bar) shows Ref. [30]: $\mathrm{\Delta }{b}_{31}^{\mathrm{UFG}}=0.09188(16)$ and $\mathrm{\Delta }{b}_{22}^{\mathrm{UFG}}=-0.1220(8)$. Our results are closest to the latter; we obtain $\mathrm{\Delta }{b}_{31}^{\mathrm{UFG}}=0.0931(8)$ and $\mathrm{\Delta }{b}_{22}^{\mathrm{UFG}}=-0.1244(7)$.

Figure 2

Top: density equation of state at unitarity as a function of the fugacity $z$ showing our virial expansion results (error bands) compared with the data of Ref. [33]. The fourth-order virial expansion is also shown using the central value for $\mathrm{\Delta }{b}_4$ of Ref. [29]. Bottom: tan contact at unitarity as a function of temperature $T$ in units of the Fermi temperature $T_F=(3{\pi }^{2}n{)}^{2/3}/2$, where $n$ is the density, compared with the experimental measurements of Refs. [49, 50]. The vertical dash-dotted line shows the critical temperature $T_c/T_F=0.167(13)$ of Ref. [33].