Fermion pairing in mixed-dimensional atomic mixtures

We investigate the quantum phases of mixed-dimensional cold atom mixtures. In particular, we consider a mixture of a Fermi gas in a two-dimensional lattice, interacting with a bulk Fermi gas or a Bose-Einstein condensate in a three-dimensional lattice. The effective interaction of the two-dimensional system mediated by the bulk system is determined. We perform a functional renormalization group analysis, and demonstrate that by tuning the properties of the bulk system, a subtle competition of several superconducting orders can be controlled among $s-\mathrm{wave}, p-\mathrm{wave}, d_{x^2-y^2}-\mathrm{wave}$, and $g_{xy(x^2-y^2)}-\mathrm{wave}$ pairing symmetries. Other instabilities such as a charge-density-wave order are also demonstrated to occur. In particular, we find that the critical temperature of the $d-\mathrm{wave}$ pairing induced by the next-nearest-neighbor interactions can be an order of magnitude larger than that of the same pairing induced by doping in the simple Hubbard model. We expect that by combining the nearest-neighbor interaction with the next-nearest-neighbor hopping (known to enhance $d-\mathrm{wave}$ pairing), an even higher critical temperature may be achieved.

Figure 1

Schematics of the system that we consider. A 2D Fermi gas (red dots) interacts with 3D particles (blue dots) via a local density-density interaction $V$ at $z=N^{\prime }$. We use an open boundary condition for the $z$ direction of the 3D system with $2N^{\prime }-1$ sites.

Figure 2

(a) The effective interaction given by Eq. (15) in momentum space in units of $4V^2/t^{\prime }$. We use $N=50$ and $N^{\prime }=11$. (a1)–(a4) represent the dependence of $U_{\text{eff}}^{\text{F}}\left(\mathbf{q}\right)$ on ${\mu }^{\prime }$. We observe a peak at $\mathbf{q}=(\pi ,\pi )$ for ${\mu }^{\prime }=0$ shifting to $(0,0)$ as ${\mu }^{\prime }$ increases. (b) The effective interactions in real space in units of $4V^2/t^{\prime }$. The NN and next-nearest-neighbor (NNN) interactions change signs as ${\mu }^{\prime }$ changes.

Figure 3

The total “bare” on-site and nearest-neighbor interaction for 2D fermions with the original on-site interaction $U=1$. Dashed lines show the line where the interaction is zero.

Figure 4

Schematics of the patching scheme we use and the Fermi surface. Thick lines represent Fermi surfaces for various chemical potentials in each panel. [In the right panel, the inner (outer) path is for $\mu =-1(-0.1)$.] Thin lines in the left panel show how to dissect the first Brillouin zone into narrow 28 patches.

Figure 5

Phase diagrams of a 2D Fermi gas in contact with a noninteracting 3D Fermi gas. We use $N=50, N^{\prime }=11$, and $t^{\prime }=8t$. Blank regions correspond to Fermi liquid; no instability is found.

Figure 6

The spatial profile of the condensation obtained for the parameters we used for the calculations.

Figure 7

(a) The effective interaction $U_{\text{eff}}^{\text{B}}\left(\mathbf{q}\right)$ in Eq. (32) in units of $V^2$. We take $N^{\prime }=11, t_{\text{B}}=10t, {\mu }_B=-5.5t_{\text{B}}$, and $U_{\text{B}}=0.2t$. (b) A phase diagram obtained by fRG with the same parameters as (a) and $V=0.8t$. Blank regions correspond to Fermi liquid; no instability is found.

Figure 8

(a) The critical temperatures of phases found in the Fermi-Fermi mixture at half-filling and $V^{\prime }/t^{\prime }=1$ as in Fig. 5. (b) The critical temperatures of phases found in the Fermi-BEC mixture at $\mu =-0.016$ as in Fig. 7. The vertical dotted lines are phase boundaries.

Figure 9

(a) The effective interaction in momentum space in units of $4V^2/t^{\prime }$. We use $N=50$ and $N^{\prime }=1$. (a1)–(a4) represent the dependence of $U_{\text{eff}}^{\text{F}}\left(\mathbf{q}\right)$ on ${\mu }^{\prime }$. (b) The effective interactions in real space in units of $4V^2/t^{\prime }$.

Figure 10

Phase diagrams of a 2D Fermi gas in contact with a noninteracting 2D Fermi gas. We use $N=50, N^{\prime }=1$, and $t^{\prime }=8t$. Blank regions correspond to Fermi liquid; no instability is found.