Superfluid phases of fermions with hybridized $s$ and $p$ orbitals

We explore the superfluid phases of a two-component Fermi mixture with hybridized orbitals in optical lattices. We show that there exists a general mapping of this system to the Lieb lattice. By using simple multiband models with hopping between $s$- and $p$-orbital states, we show that superfluid order parameters can have a $\pi$-phase difference between lattice sites, which is distinct from the case with hopping between $s$ orbitals. If the population imbalance between the two spin species is tuned, the superfluid phase may evolve through various phases due to the interplay between hopping, interactions, and imbalance. We show that the rich behavior is observable in experimentally realizable systems.

Figure 1

The lattice we study in one dimension (a), where on the $B$ lattice sites the $p$ orbitals are sketched. Hopping within the specified unit cell is denoted by a full line, whereas hopping to the neighboring unit cell is denoted by a dashed line. In (b) the mapping from the $s\text{−}p$ lattice to the lattice with $s$ orbitals only is shown.

Figure 2

(a) The $s\text{−}p$ lattice in two dimensions, where on the $B$ lattice sites the $p_x$ and $p_y$ orbitals are sketched. Hopping within the specified unit cell is denoted by a full line and to the neighboring unit cells by a dashed line. (b) Illustration of the two decoupled lattices. On the left the full lattice is sketched and the two different hopping routes for the particles are denoted by full and dashed lines, which are shown as separate lattices in the middle and right. (c) Mapping from the $s\text{−}p$ lattice to the lattice with $s$ orbitals only.

Figure 3

(a) Zero temperature phase diagram. Dashed lines denote continuous phase transitions; all other transitions are first order. (b) Local minima of ${\mathrm{\Omega }}^{1\mathrm{D}}$ as functions of $h$ at different values of $U$. The different curves represent ${\mathrm{\Omega }}_1-{\mathrm{\Omega }}_6$ in Table 1, where the color coding is the same as in (a). The dashed, dotted, and solid curves are for the cases of ${\mathrm{SF}}_0,{\mathrm{SF}}_{\pi }$, and normal phases, respectively.

Figure 4

LO order parameter and the ${\mathrm{SF}}_{\pi }$ phase order parameters. For ${\mathrm{\Omega }}_4$ the axes origin is at zero, whereas for ${\mathrm{\Omega }}_5$ it is at some nonzero value.

Figure 5

(a) Lattice potential in Eq. (11) with $V_0=10$ and $\theta \approx 0.556\pi$ in one unit cell, with the shallower $A$ site at the origin and the deeper $B$ sites at four corners. The bottom of the potential at $B$ sites is $-24.67$, while it is $-15.33$ for $A$ sites. The two arrows indicate nearest neighbor hopping. (b) Corresponding dispersions of single-particle states along $k_x=k_y$.

Figure 6

Dispersions of the three hybridized states along (a) $k_x=k_y$ and (b) along $k_x$ with $k_y=0$, for the lattice with $V_0=10$ and $\theta \approx 0.556\pi$. The solid curves are exact dispersions solved numerically, while the dashed curves are the fitted dispersions of the reduced Hamiltonian with only two parameters ${\epsilon }_0$ and $t$.

Figure 7

Dispersions ${\omega }_i$ for (a) the ${\mathrm{SF}}_0$ phase and for (b) the ${\mathrm{SF}}_{\pi }$ phase as functions of $k_x$ at fixed $k_y=0$, with the parameters $t\approx 0.075$ and ${\epsilon }_0=\mu \approx -11.42$, and lattice parameters $V_0=10$ and $\theta \approx 0.556\pi$. (a) ${\mathrm{\Delta }}_0\approx 0.187$ and ${\mathrm{\Delta }}_1\approx 0.185$, which minimize $\mathrm{\Omega }$ at $h=0,U=0.10$, and $T=0.01$, while in (b) the sign of ${\mathrm{\Delta }}_1$ is reversed to make a comparison between the ${\mathrm{SF}}_0$ and ${\mathrm{SF}}_{\pi }$ phases.

Figure 8

Phase diagrams as functions of $h$ and $U$ for lattice parameters $V_0=10$ and $\theta \approx 0.556\pi$ at different temperatures. In all phase diagrams the white region denotes the normal phase, red corresponds to the ${\mathrm{SF}}_0$, and blue to the ${\mathrm{SF}}_{\pi }$ phases. The crosses mark the values for which the momentum distributions are calculated in Figs. 10 and 11.

Figure 9

Phase diagrams as functions of $h$ and $U$ for fixed temperature $T=0.01$ at different lattice parameters. A shallower lattice is considered in (a) with $V_0=8,\theta \approx 0.564\pi ,t\approx 0.110,{\epsilon }_0\approx -8.44,U_0\approx 3.98U$, and $U_1\approx 3.65U$. A deeper lattice is used in (b) with $V_0=12$ and $\theta \approx 0.550\pi$, with $t\approx 0.0517,{\epsilon }_0\approx -14.5,U_0\approx 5U$, and $U_1\approx 4.39U$. As before, the white regions denote the normal phase (N), red corresponds to the ${\mathrm{SF}}_0$, and blue to the ${\mathrm{SF}}_{\pi }$ phases.

Figure 10

Momentum distributions of the spin-up (top) and spin-down (bottom) particles averaged over the three bands as functions of $k_x$ and $k_y$ in a lattice with parameters $V_0=10$ and $\theta \approx 0.556\pi$ for the points marked in Fig. 8. The temperature is $T=0.01$, the interaction is $U=0.10$, and the chemical potential differences (a) $h=0.10 \left({\mathrm{SF}}_0\right)$, (b) $h=0.14 \left({\mathrm{SF}}_{\pi }^1\right)$, (c) $h=0.17 \left({\mathrm{SF}}_{\pi }^2\right)$, and (d) $h=0.22$ (normal state).

Figure 11

The same as in Fig. 10 but at temperature $T=0.03$, corresponding to the points in Fig. 8. Again the interaction is $U=0.10$ and now the chemical potential differences are (a) $h=0.10 \left({\mathrm{SF}}_0\right)$, (b) $h=0.135 \left({\mathrm{SF}}_{\pi }^1\right)$, (c) $h=0.17 \left({\mathrm{SF}}_{\pi }^2\right)$, and (d) $h=0.22$ (normal state).