Quantum phases in degenerate-$p$-orbital attractive one-dimensional fermionic optical lattices

We examine quantum phases emerging from the double degeneracy of $p$-orbital bands in attractive atomic Fermi gases loaded on a one-dimensional (1D) optical lattice. Our numerical simulations via the density-matrix renormalization group predict the emergence of a state with a charge excitation gap, namely the Haldane insulator phase. A mapping onto an effective spin-1 model reveals its physical origin. Moreover, we show that population imbalance leads to richer diversity of the quantum phases, including a phase-separated polarized state. Finally, we study the effects of the harmonic trap potential in this 1D chain.

Figure 1

Schematic diagram of fermionic gases on an optical lattice, with multiple bands (multiband Hubbard chain). The intraorbital interaction ($U_{pp}$) forms fermion pairs. The fermion pairs hop between different orbitals by the pair-hopping interaction ($U_{p_x{p}_y}$) [see Eq. (3)].

Figure 2

(a) Charge gap $E_{\mathrm{c}}$ vs filling rate $\tilde{n}$. (b) Particle density $n\left(i\right)$, with $\tilde{n}=1.0,0.8,0.2$. (c) Absolute value of the Fourier-transformed density fluctuations, $|\overline{n}\left(k\right)|$, on the $k$-$\tilde{n}$ plane. In all the figures, the population imbalance is $P=0$, the coupling parameters are $U_{pp}=-10$ and $U_{p_x{p}_y}=(4/9)U_{pp}$, and the total lattice site number is $L=100$.

Figure 3

Spatial distributions of the particle density $n\left(i\right)$ (solid line) and the spin density $m\left(i\right)$ (dashed line), with filling rates (a) $\tilde{n}=0.99$, (b) $\tilde{n}=0.6$, (c) $\tilde{n}=0.4$, and (d) $\tilde{n}=0.2$. In all the figures, the population imbalance is fixed as $P=12$. Other physical parameters are the same as in Fig. 2.

Figure 4

Schematic diagram of hopping processes of unpaired fermions (single up-arrow) in fermion-pair particles (pair of up- and down-arrows).

Figure 5

Spatial distributions of the particle density $n\left(i\right)$ (solid line) and the spin density $m\left(i\right)$ (dashed line) in the harmonic trap potential, with different population imbalances $P=0$,20,40. The trap potential strength is $V/t=0.4$. The total particle number is 180. The upper panels (a1)–(c1) show the results for the present doubly degenerate $p$-orbital 1D chain. For comparison, we show in the lower panels (a2)–(c2) the results for zero intraorbital interaction ($U_{p_x{p}_y}=0$), i.e., a single-band attractive Hubbard chain.