Momentum Distribution in the Unitary Bose Gas from First Principles

We consider a realistic bosonic $N$-particle model with unitary interactions relevant for Efimov physics. Using quantum Monte Carlo methods, we find that the critical temperature for Bose-Einstein condensation is decreased with respect to the ideal Bose gas. We also determine the full momentum distribution of the gas, including its universal asymptotic behavior, and compare this crucial observable to recent experimental data. Similar to the experiments with different atomic species, differentiated solely by a three-body length scale, our model only depends on a single parameter. We establish a weak influence of this parameter on physical observables. In current experiments, the thermodynamic instability of our model from the atomic gas towards an Efimov liquid could be masked by the dynamical instability due to three-body losses.

Figure 1

Correlation functions for two unitary bosons. (a) Open (left) and closed (right) co-cyclic configurations in the path-integral representation. Closed configurations yield $g^{(2)}(\mathbf{r})$. Open configurations yield $n(\mathbf{k})$ and its inverse Fourier transform $g^{(1)}(\mathbf{r})$. (b) Pair-correlation function $g^{(2)}(\mathbf{r})$ (distance distribution in closed configurations), featuring a $r^{-2}$ divergence at small $r$. (c) Cut of $g^{(1)}(\mathbf{r})$ (distribution of the distance between open ends), for $\mathbf{r}=(x,y,0)$, illustrating the cusp at $\mathbf{r}\simeq 0$. (d) Momentum distribution $n(\mathbf{k})$ with asymptotic decay, $\propto 1/k^4$, at large $\mathbf{k}$.

Figure 2

Correlation functions for three unitary bosons. (a) Hyperradial probability distribution for three co-cyclical bosons with hyperradial cutoff at low temperature (cyan dashed line) and for the universal Efimov trimer (black solid line, from Ref. [4]). (b) Momentum distribution for three co-cyclical bosons (cyan dashed line), and for the universal trimer (black solid line, from Ref. [25]), in units of the trimer binding momentum ${\kappa }_0$.

Figure 3

Equilibrium phase diagram of unitary bosons. (a) Contact density $c_2{\rho }^{-4/3}$, as a linear interpolation of numerical results [extracted from $g^{(2)}(\mathbf{r})$, for $N=64$]. White stars: transition between normal gas and superfluid (Bose-Einstein condensed) phase. Black crosses: Phase-separated points. Gray area: Phase-coexistence region [23]. (b) Stable Efimov-liquid droplet coexisting with a normal gas ($N=256$). (c) Excitation free energy for the Efimov-liquid nucleation, vs nucleus size $l$. ${\lambda }_{\mathrm{th}}{\rho }^{1/3}$ varies between lines (see labels), between 0.5 (monotonically increasing, red line) and 0.9 (barrier, blue line). The hyperradial cutoff is fixed ($R_0{\rho }^{1/3}=0.03$), and the phase-separation region sets in at ${\lambda }_{\mathrm{th}}{\rho }^{1/3}\simeq 0.66$. (d) Contact density $c_2{\rho }^{-4/3}$ vs ${\lambda }_{\mathrm{th}}{\rho }^{1/3}$, for $R_0{\rho }^{1/3}=0.052$: Virial expansion (black dashed line) and numerical results, via the $n(\mathbf{k})$ and $g^{(2)}(\mathbf{r})$ estimators (crosses, circles). In the phase-coexistence region, the liquid and gas phases have different contact densities (for the gas, the virial expansion is used). (e) Momentum distribution [in units of the Fermi momentum $k_F=(6{\pi }^2\rho {)}^{1/3}$] for parameters corresponding to points $A$, $B$, and $C$, in panel (a).

Figure 4

Full momentum distribution $n(\mathbf{k})$ in the Bose-Einstein-condensed gas phase. (a) $n(\mathbf{k})$ at ${\lambda }_{\mathrm{th}}{\rho }^{1/3}=1.545$, $R_0{\rho }^{1/3}=0.184$ [point $D$ in Fig. 3]. First-principles results for $N=64$, 128, and 256 (black circles, green squares, brown diamonds, respectively), and $\propto C_2/k^4$ asymptotic behavior for $k\to \infty$ (for $N=64$, black solid line). Dashed lines are experimental data of Ref. [13] for two different densities. The momentum distribution for $N=256$ ideal bosons is also shown (cyan solid line). (b) Scaling of the condensed fraction $N_0/N$ with the system size, in the normal and condensed phases. The upper curve (at $T<T_c$) corresponds to the parameters in panel (a), and the exact numerical data are fitted by $N^{1/3}(N_0/N)\simeq 1.06+0.14N^{1/3}$ [same symbols for $N$ as in panel (a)]. The lower curve is at ${\lambda }_{\mathrm{th}}{\rho }^{1/3}\simeq 1.373$ (corresponding to $T>T_c$), and is fitted by $N^{1/3}(N_0/N)\simeq 1.71N^{-1/6}$. (c) Rescaled superfluid fraction vs temperature, at $R_0{\rho }^{1/3}=0.184$. The crossing point at $T/T_c^0\simeq 0.9$ (corresponding to ${\lambda }_{\mathrm{th}}{\rho }^{1/3}\simeq 1.45$) shows that $T_c$ is lowered by 10% with respect to the ideal Bose gas, in the limit $N\to \infty$.