Pairing states of a polarized Fermi gas trapped in a one-dimensional optical lattice

We study the properties of a one-dimensional (1D) gas of fermions trapped in a lattice by means of the density matrix renormalization group method, focusing on the case of unequal spin populations, and strong attractive interaction. In the low-density regime, the system phase separates into a well-defined superconducting core and a fully polarized metallic cloud surrounding it. We argue that the superconducting phase corresponds to a 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, with a quasicondensate of tightly bound bosonic pairs with a finite center-of-mass momentum that scales linearly with the magnetization. In the large density limit, the system allows for four phases: in the core, we either find a Fock state of localized pairs or a metallic shell with free spin-down fermions moving in a fully filled background of spin-up fermions. As the magnetization increases, the Fock state disappears to give room for a metallic phase, with a partially polarized superconducting FFLO shell and a fully polarized metallic cloud surrounding the core.

Figure 1

(Color online) Results for the 1D attractive Hubbard model ($U=-8t$, $L=80$), with $N=40$ fermions confined by a parabolic potential with strength $V=0.002t$, for different magnetizations: (a) density profile, (b) magnetization profile, (c) charge fluctuations, and (d) spin fluctuations.

Figure 2

(Color online) (a) Pair momentum distribution function for $N=40$ fermions with $N^{\mathrm{pair}}={\sum }_k{n}_k^{\mathrm{pair}}$. The dotted lines show results for the sum in Eq. (2) restricted to the center of the trap. (b) Spatial decay of pair correlations (squares). Dashed line: Fit to $n_{lm}^{\mathrm{pair}}\propto \cos (k\mid l-m\mid )∕\mid l-m{\mid }^{\alpha }$ (compare Ref. 13). Data shown for $L=100$, $N=80$, $V=0.0005t$, $S^z=8$, corresponding to the same effective density ${\rho }_{\mathrm{eff}}=N\sqrt{V}$. (c) Momentum $k_{\max }$, at which the distribution shown in (a) is peaked, vs magnetization of the core $S_{\mathrm{eff}}^z∕L_{\mathrm{eff}}$ ($L_{\mathrm{eff}}\approx 40$, see text).

Figure 3

(Color online) Natural orbital ${\psi }_0$ of the pair OPDM for $N=40$ confined fermions and different magnetizations. The inset shows the OPDM’s eigenvalues ${\lambda }_{\alpha }$.

Figure 4

(Color online) Same parameters as in Fig. 1, but $N=80$ fermions. (a) Density profile, (b) magnetization profile, (c) charge fluctuations, and (d) spin fluctuations.

Figure 5

(Color online) (a),(b) Particle density for $⟨n_{i\sigma }⟩$, $\sigma =\uparrow ,\downarrow$, for $N=80$ fermions, and $S^z=2,8$, respectively. The double occupancy $⟨d_i⟩=⟨n_{i\uparrow }{n}_{i\downarrow }⟩$ is also included. (c) Pair momentum distribution function measured in concentric shells of different radii, for $S^z=8$. (d) First three NOs of the pair OPDM that are not just localized states. The inset shows the eigenvalues of the pair OPDM.