Exotic superconducting phases of ultracold atom mixtures on triangular lattices

We study the phase diagram of two-dimensional Bose-Fermi mixtures of ultracold atoms on a triangular optical lattice, in the limit when the velocity of bosonic condensate fluctuations is much larger than the Fermi velocity. We contrast this work with our previous results for a square lattice system in the work of Mathey et al. [Phys. Rev. Lett. 97, 030601 (2006)]. Using functional renormalization-group techniques we show that the phase diagrams for a triangular lattice contain exotic superconducting phases. For spin-$1∕2$ fermions on an isotropic lattice we find a competition of $s$-, $p$-, extended $d$-, and $f$-wave symmetries, as well as antiferromagnetic order. For an anisotropic lattice, we further find an extended $p$-wave phase. A Bose-Fermi mixture with spinless fermions on an isotropic lattice shows a competition between $p$- and $f$-wave symmetries. These phases can be traced back to the geometric shapes of the Fermi surfaces in various regimes, as well as the intrinsic frustration of a triangular lattice.

Figure 1

(a) Fermi surfaces for an isotropic lattice, for ${\mu }_f<2t_{f,1∕2}$, ${\mu }_f=2t_{f,1∕2}$ (hexagonal shape), and ${\mu }_f>2t_{f,1∕2}$ (six disjoint arcs). (b) Diagram of the different types of Fermi surfaces that can be created on an anisotropic lattice, by varying the ratio $t_{f,2}∕t_{f,1}$ and ${\mu }_f$. (c) Fermi surfaces for $t_{f,2}<t_{f,1}$, for ${\mu }_f<4t_{f,2}-2t_{f,1}$, $4t_{f,2}-2t_{f,1}<{\mu }_f<2t_{f,1}$, and ${\mu }_f>2t_{f,1}$, corresponding to regimes I–III, respectively. (d) Fermi surfaces for $t_{f,2}>t_{f,1}$, for ${\mu }_f<2t_{f,1}$, $2t_{f,1}<{\mu }_f<4t_{f,2}-2t_{f,1}$, and ${\mu }_f>4t_{f,2}-2t_{f,1}$, corresponding to regimes IV-VI, respectively.

Figure 2

(a) Lattice geometry of the system. The continuous (dashed) bonds correspond to the hopping amplitudes $t_{b∕f,1\left(2\right)}$. For $t_{b∕f,1}=t_{b∕f,2}$, the lattice is an isotropic triangular lattice. (b) Schematic representation of the AF order corresponding to nesting vector ${\mathbf{Q}}_1$. [(c) and (d)] Order parameters of the extended $d$-wave orders $D_1$ and $D_2$. (e) Order parameter of the $f$-wave phase. This order can also be interpreted as two $s$-wave paired hole states whose order parameters are out of phase by $\pi$. (f) Order parameter of the extended $p$-wave phase, which appears in anisotropic lattices.

Figure 3

Phase diagram of a Bose-Fermi mixture on a 2D isotropic triangular lattice. The vertical axis corresponds to the interaction strength, $U_{ff}∕t_f$, whereas the horizontal axis corresponds to the filling fraction of the fermions per site, $n$. The other parameters are given by $\tilde{V}∕t_f=3$, and ${\xi }_a=1$ with $a=1,2$.

Figure 4

Phase diagram of a Bose-Fermi mixture on an anisotropic triangular lattice. The vertical axis corresponds to the ratio $t_{f,2}∕t_{f,1}$, the horizontal axis corresponds to the chemical potential ${\mu }_f$. The other parameters are given by $\tilde{V}∕t_f=3$, $U_{ff}∕t_{f,1}=2$, and ${\xi }_a=1$ with $a=1,2$.