Three-body correlations in a two-dimensional SU(3) Fermi gas

We consider a three-component Fermi gas that has SU(3) symmetry and is confined to two dimensions (2D). For realistic cold atomic gas experiments, we show that the phase diagram of the quasi-2D system can be characterized using two 2D scattering parameters: the scattering length and the effective range. Unlike the case in three dimensions (3D), we argue that three-body bound states (trimers) in the quasi-2D system can be stable against three-body losses. Using a low-density expansion coupled with a variational approach, we investigate the fate of such trimers in the many-body system as the attractive interactions are decreased (or, conversely, as the density of particles is increased). We find that remnants of trimers can persist in the form of strong three-body correlations in the weak-coupling (high-density) limit.

Figure 1

Binding energy per atom of different few-body states. This corresponds to $-\mu$ of the many-body system in the limit of vanishing density (see text). The solid (orange) line is the ground-state trimer of the three-body problem, the dashed (gray) line is the excited trimer state of the three-body problem, and the dotted (purple) line is the dimer state of the two-body problem. At zero effective range, the ground-state and excited-state trimer energies are $E_{3}m{a}_{2\mathrm{D}}^2/3\approx -5.5$ and $-0.42$, respectively. The inset shows the same curves with linear scales on the axes in order to expose the behavior as $R_{2\mathrm{D}}/a_{2\mathrm{D}}\to 0$.

Figure 2

The solid (orange) lines show the rescaled decay rate $\mathrm{\Gamma }/\mathrm{\Delta }$ (top) and size $\langle r\rangle$ (bottom) of the ground-state trimer. The dotted (purple) lines show the corresponding values at zero effective range while the dashed (gray) line shows the critical effective range beyond which a dilute gas of trimers is unstable toward forming dimers.

Figure 3

Schematic phase diagram of the 2D SU(3) Fermi gas as a function of the effective range and scattering length. The blue SF region designates the fully paired superfluid phase, the orange shaded region corresponds to trimers only, and the remaining area has a mixture of a superfluid and fermions (trimers or unbound atoms). In the low-density limit, where $ln(1/k_F{a}_{2\mathrm{D}})\gg 1$ and $k_F{R}_{2\mathrm{D}}\ll 1$, the system corresponds to a weakly interacting gas of dimers and/or trimers, as illustrated by the particle clusters linked by solid lines. Here, when $R_{2\mathrm{D}}\to 0$, the ground state is a trimer Fermi gas, while increasing $k_F{R}_{2\mathrm{D}}$ yields a transition to a trimer-dimer mixture and then finally a dimer-only superfluid. In the limit $k_F{R}_{2\mathrm{D}}\gg 1$, there is a transition from the fully paired superfluid to weak BCS pairing plus unbound atoms as we decrease $ln(1/k_F{a}_{2\mathrm{D}})$. In the opposite regime $k_F{R}_{2\mathrm{D}}<1$, there is a region defined by strong three-body correlations (illustrated by orange particles linked with dashed lines), where the size of the three-body clusters can be comparable to or larger than the interparticle spacing when $ln(1/k_F{a}_{2\mathrm{D}})\lesssim -1$. Eventually, for sufficiently weak attraction, these three-body clusters are replaced by Cooper pairs coexisting with an atomic Fermi sea, which smoothly connects with the high-density phase in the $k_F{R}_{2\mathrm{D}}\gg 1$ limit.

Figure 4

Phase boundaries for the dimer-only (a) and trimer-only (b) regions calculated within different approximations. The shaded regions are the same as in Fig. 3. (a) The solid (black) lines are defined by the low-density condition (25), where the thickest line corresponds to $a_{dd}=0$, while the thinner solid lines are for $a_{dd}=0.1a_{2\mathrm{D}}$ and $0.56a_{2\mathrm{D}}$, in order of decreasing thickness. The $a_{dd}=0$ phase boundary is displayed as dotted once the dimer size $\sim 1/\sqrt{m|E_2|}$ is larger than the interparticle spacing $\sim k_F^{-1}$. The thin dashed (blue) line corresponds to the mean-field phase boundary described by Eq. (29). (b) The solid (black) lines depict the phase boundaries calculated using Eq. (24), where the thickest line is the solution with $\mathcal{F}=0$, while the thinner solid lines correspond to $\mathcal{F}=0.5$ and $-0.1$, in order of decreasing thickness. The thick lines are displayed as dotted when the trimer size $\sim 1/\sqrt{m|E_3|}$ is larger than the interparticle spacing. The thin dashed (orange) line marks the region at high density where three-body correlations are relevant, as discussed in Sec. 4c. This line is approximated by taking $\mathcal{F}=0.14$, which corresponds to the boundary of the (orange) shaded region.