Bound States in a Quasi-Two-Dimensional Fermi Gas

We consider the problem of $N$ identical fermions of mass $m_{\uparrow }$ and one distinguishable particle of mass $m_{\downarrow }$ interacting via short-range interactions in a confined quasi-two-dimensional (quasi-2D) geometry. For $N=2$ and mass ratios $m_{\uparrow }/m_{\downarrow }<13.6$, we find non-Efimov trimers that smoothly evolve from 2D to 3D. In the limit of strong 2D confinement, we show that the energy of the $N+1$ system can be approximated by an effective two-channel model. We use this approximation to solve the $3+1$ problem and we find that a bound tetramer can exist for mass ratios $m_{\uparrow }/m_{\downarrow }$ as low as 5 for strong confinement, thus providing the first example of a universal, non-Efimov tetramer involving three identical fermions.

Figure 1

Critical mass ratio for the appearance of trimers and tetramers in quasi-2D, where the 2D limit corresponds to ${ϵ}_b/{\omega }_z\to 0$. The solid line follows from the solution of the full three-body quasi-2D problem, Eq. (7). Dashed lines follow from an effective two-channel model. The vertical dotted line marks unitarity, where the 3D scattering length diverges.

Figure 2

The diagrams which give the binding energy of the $N+1$ bound state in quasi-2D. Black dots indicate the initial interaction inside $f$.

Figure 3

Energy of the trimer in quasi-2D for mass ratios $m_{\uparrow }/m_{\downarrow }=3.5$, 4, 5, 6.64 (from top to bottom). The solid lines correspond to the full calculation, while the dashed lines are derived from the effective two-channel model, Eq. (6).

Figure 4

Energy of the dimer (solid line), trimer (dotted line), and tetramer (dashed line) in 2D as a function of mass ratio. The trimer binds when $m_{\uparrow }/m_{\downarrow }>3.33$, consistent with Ref. 26, while the trimer-tetramer transition occurs at $m_{\uparrow }/m_{\downarrow }=5.0$. Inset: The difference between trimer and tetramer energies, $E_3-E_4$.