Nontrivial Haldane Phase of an Atomic Two-Component Fermi Gas Trapped in a 1D Optical Lattice

We propose a method to create a nontrivial Haldane phase in an atomic two-component Fermi gas loaded on a one-dimensional optical lattice with trap potential. The Haldane phase is naturally formed on a $p$-band Mott core in a wide range of the strong on-site repulsive interaction. The present proposal is composed of two steps, one of which is theoretical derivation of an effective 1D $S=1$ interacting-chain model from the original tight-binding Hamiltonian handling the two $p$ orbitals, and the other of which is a numerical demonstration employing the density-matrix renormalization group for the formation of the Haldane phase on a $p$-band Mott core and its associated features in the original tight-binding model with the harmonic trap potential.

Figure 1

Schematic figures for a 1D Fermi gas. (a) An optical lattice (OL) potential along the $z$ direction and a cylindrically symmetric vertical trap potential inside the $xy$ plane. The latter causes formation of discrete levels as depicted in the right-hand panel, in which the lowest level is $s$ orbital while the higher two degenerate ones are $p_{x,y}$ orbitals. The Haldane phase is formed under fully filled $s$ and partially filled $p$ orbitals (see text). (b) $p_{x,y}$ orbitals formed inside the vertical trap and their $p$ bands creatd by the OL. When $p$ bands are active, this system is described by a multiband Hubbard chain with the on-site intraorbital interaction ($U_{pp}$) and the interorbital one ($U_{p_x{p}_y}$) [see Eq.(2) and text].

Figure 2

(a) A typical spatial distribution profile of the particle density $n(i)=\sum _{\alpha =p_x,p_y}\sum _{\sigma }{n}_{\alpha ,\sigma ,i}$ for balanced 160 fermions ($80\uparrow$, $80\downarrow$) with the interaction constants $U_{pp}/t=15$, $U_{p_x{p}_y}=\frac{4}{9}{U}_{pp}$, and the trap potential strength $V/t=10$. (b) The Mott-core polarization $M$ versus the population imbalance ratio $p$. The employed parameters are the same parameters as in (a) except for $p$. For comparison, the inset is the case of $U_{pp}/t=15$ and $U_{p_x{p}_y}/t=0$, which is equivalent with two independent $S=1/2$ interacting chains.

Figure 3

Distribution profiles of particle density (dashed blue line) $n(i)$ and spin density (solid red line) $m(i)=\sum _{\alpha =p_x,p_y}(n_{\alpha ,\uparrow ,i}-n_{\alpha ,\downarrow ,i})$ for spin imbalance (a) $p=0(S_{\mathrm{tot}}^{(z)}=0)$ and (b) $p=0.0125$ ($S_{\mathrm{tot}}^{(z)}=1$). The inset of (a) is a focus on the spin density around the edge of the left Mott core. (c) The polarization in the left (right) surrounding phase $m_{L(R)}\equiv \sum _{i\in \mathrm{left}(\mathrm{right})}m(i)$ versus the population imbalance ratio $p$. (d) The spin gap $\Delta E_p\equiv E_p-E_{p=0}$ versus the population imbalance ratio $p$ . All other physical parameters are the same as in Fig. 2a.

Figure 4

Particle density (dashed blue line) and spin density (solid red line) distribution profiles for population imbalance ratios (a) $p=0.1625$ ($93\uparrow$, $67\downarrow$), (b) $p=0.475$ ($118\uparrow$, $42\downarrow$), and (c) $p=0.9375$ ($155\uparrow$, $5\downarrow$). All other parameters are the same as in Fig. 2a.