Ground-state densities of repulsive two-component Fermi gases

We investigate separations of trapped balanced two-component atomic Fermi gases with repulsive contact interaction. Candidates for ground-state densities are obtained from the imaginary-time evolution of a nonlinear pseudo-Schrödinger equation in three dimensions, rather than from the cumbersome variational equations of the underlying energy density functional. With the employed hydrodynamical approach, gradient corrections to the Thomas-Fermi approximation are conveniently included and are shown to be vital for reliable density profiles. We provide critical repulsion strengths that mark the onset of phase transitions in a harmonic trap. We present transitions from identical density profiles of the two fermion species towards isotropic and anisotropic separations for various confinements, including harmonic and double-well-type traps. Our proposed method is suited for arbitrary trap geometries and can be straightforwardly extended to study dynamics in the light of ongoing experiments on degenerate Fermi gases.

Figure 1

Illustration of the generic structure of radial densities $n_{\pm }\left(r\right)$, depicted along the $x$ axis $\left(y=z=0\right)$, for ${\mu }_{\pm }=\mu$ and $N_{\pm }=5000$ particles in three dimensions. All quantities are given in harmonic oscillator units (osc). We disregard continuity, use the radial densities for $x>0$, and swap the roles of $n_{\pm }$ for $x<0$. The displayed anisotropic separated density profile has the same energy as the isotropic separated density profile. But it has a marginally lower energy than the symmetric solution, $E_{0,\mathrm{sep}}=287915<E_{0,\mathrm{symm}}=288487$. The radius $r_0$ at which both densities $n_{\pm }$ vanish is given by $V\left(r_0\right)=\mu$. The radius $r_1$ where the separated solutions merge with the symmetric solutions is given in the Appendix.

Figure 2

Two phase transitions for the particle densities $n_{\pm }$ with $N_{\pm }=5000$ in a harmonic trap are revealed as the repulsion strength $g$ increases. (a) The transition from nonseparated densities towards isotropic separation occurs at $g_{\mathrm{is}}$, with $2.93<g_{\mathrm{is}}<2.94$. (b) The transition from isotropic towards anisotropic separation occurs at $g_{\mathrm{as}}$, with $3.03<g_{\mathrm{as}}<3.04$.

Figure 3

Densities for $N_{\pm }=160$ in a harmonic trap for various values of $g$, illustrating the two phase transitions (cf. Fig. 2). (a, b) A transition from isotropic to anisotropic separation into two “semispheres” is encountered between $g=5.85$ and $g=5.9$, indicating spontaneous symmetry breaking as soon as the depletion of one component in the center is total. (c) With $g$ well beyond the second phase transition, we gain a quasicomplete separation of the two Fermi gas clouds.

Figure 4

Contour plots of the normalized densities ${\widehat{n}}_{\pm }\left(\mathit{r}\right)={\widehat{n}}_{\pm }\left(\mathit{r}\right)/{\max }_{\mathit{r}}{n}_{\pm }\left(\mathit{r}\right)$ (red/blue) in the $(z=0)$ plane for $N_{\pm }=160$ and $g=5.9$, with the color gradient from white $\left[{\widehat{n}}_{\pm }\left(\mathit{r}\right)<0.01\right]$ to the darkest color $\left[{\widehat{n}}_{\pm }\left(\mathit{r}\right)>0.93\right]$. The density profiles along the $x$ axis are displayed in Fig. 3.

Figure 5

Phase diagram of critical repulsion strengths $g_{\mathrm{is}}$ and $g_{\mathrm{as}}$ as functions of the particle numbers $N_{+}=N_{-}$ for the harmonic oscillator potential $V=r^2/2$.

Figure 6

Densities for $N_{\pm }=160$ in the harmonic oscillator potential $V=r^2/2$ for $g=5.4$ near the phase transition from symmetric profiles to isotropic separation.

Figure 7

Densities for $N_{\pm }=5000$ fermions in the harmonic oscillator potential $V=r^2/2$. Both the TF solutions (with ${\mu }_{+}={\mu }_{-}$) and the densities obtained from ITE show an isotropic separation in the trap center for $g=2.94$ (top), whereas an anisotropic separation for $g=3.04$ is only obtained by ITE, unless we disregard continuity for the TF solutions (bottom).

Figure 8

Densities for $N_{\pm }=5000$ from ITE with $\left(\xi =1/9\right)$ and without $\left(\xi =0\right)$ explicit gradient corrections. We observe isotropic separation for $g=2.94$ (top) and anisotropic separation for $g=3.04$ (bottom).

Figure 9

Employing a barrier at $x=0$ that marginally exceeds the chemical potentials, we observe a direct transition from a symmetric state to an anisotropic separation—very similar to the case of a higher barrier—between $g_1=2.92$ and $g_1=2.93$.

Figure 10

Isotropic densities for $N_{+}+N_{-}=320$ in the harmonic oscillator potential $V\left(\mathit{r}\right)={\mathit{r}}^2/2$. ITE (without explicit gradient corrections) is employed with $N_{\pm }=160$ for $g=5.4$. Disregarding the deviating particle numbers $N_{\pm }^{\mathrm{TF}}=160\pm 7$, we find that the TF approximation works reasonably well at the qualitative level.

Figure 11

Normalized total TF energies ${\widehat{E}}_0\left[n_{\pm }\right]=E_0[n_{\pm };g]/E_0[n_{\pm };g=0]$ of symmetric solutions $n_{+}=n_{-}$ in three dimensions as a function of the interaction strength $g$ for several $N_{\pm }$ (colored solid lines). The intersection with the constant value ${\widehat{E}}_0\left[{\tilde{n}}_{\pm }\right]$ (horizontal dash-dotted black line), obtained from the completely separated densities ${\tilde{n}}_{\pm }=n_{\pm }\left(2N_{\pm }\right)\mathrm{\Theta }(\pm x)$, serves as an estimate of the critical strengths $g^{\mathrm{TF}}\left(N_{\pm }\right)$ above which symmetric solutions of the TF equations cannot represent the ground state.