Detecting the Elusive Larkin-Ovchinnikov Modulated Superfluid Phases for Imbalanced Fermi Gases in Optical Lattices

A system with unequal populations of up and down fermions may exhibit a Larkin-Ovchinnikov (LO) phase consisting of a periodic arrangement of domain walls where the order parameter changes sign and the excess polarization is localized. We find that the LO phase has a much larger range of stability in a lattice compared to the continuum; in a harmonic trap, the LO phase may involve 80% of the atoms in the trap, and can exist up to an entropy $s\sim 0.5k_B$ per fermion. We discuss detection of the LO phase (i) in real space by phase-contrast imaging of the periodic excess polarization; (ii) in k space by time-of-flight imaging of the single-particle and pair-momentum distributions; (iii) in energy space from the excess density of states within the gap arising from Andreev bound states in the domain walls.

Figure 1

The left panel shows the mean-field phase diagram of the cubic lattice Hubbard model at $U=-6t$, $T=0$ in the ($\mu$, $h$) plane. The thick dashed line indicates a slice through the phase diagram corresponding to suitable parameters (polarization fraction $P=0.37$). The right panel is a schematic of the corresponding shell structure in a harmonic trap within the local density approximation, i.e., $\mu (r)=\mu (0)-A{r}^2$. About 80% of the atoms are in the LO phase.

Figure 2

Critical polarization of the cubic lattice Hubbard model as a function of (a) average density $n=\frac{1}{2}(n_{\uparrow }+n_{\downarrow })$, (b) attraction $U$, (c) temperature $T$, and (d) entropy $S$. Energy scales are measured in units of the hopping $t$. BCS and LO represent superfluid states, 2FL represents a two-component Fermi liquid, and 1FL represents a fully polarized Fermi liquid (half-metal).

Figure 3

Results of BdG on a $24\times 20\times 29$ cubic lattice with modulation along [100], with parameters $T=0.01t$, $U=-4t$, $\mu =-0.5t$. (a) Strong LO state at $h=0.85t$, just above the critical field for domain wall penetration. At each domain wall, the order parameter changes sign, and the polarization is finite due to occupation of Andreev bound states. (b) Weak LO state at $h=t$, with sinusoidal pairing density $F_{\mathbf{r}}\sim \cos \mathbf{q}·\mathbf{r}$ accompanied by density oscillations at wave vector $2\mathbf{q}$.

Figure 4

Momentum distributions of up and down fermions, and their difference $m(\mathbf{k})=n_{\uparrow }(\mathbf{k})-n_{\downarrow }(\mathbf{k})$, along the slice $k_z=\frac{\pi }{2}$, for a weak LO state with $q=(\frac{\pi }{3},0,0)$. The distributions break fourfold symmetry but preserve inversion symmetry, in contrast to the results for FF states in Ref. 9.

Figure 5

(a) Total densities of states $A_{\uparrow }(E)$ (top, blue line) and $A_{\downarrow }(E)$ (bottom, green line) for a fully paired BCS state in a small Zeeman field ($T=0.01t$, $U=-4t$, $\mu =-0.5t$, $h=0.25t$). The dashed lines indicate $E=\pm {\Delta }_0$. (b) DOS’s for the strong LO state depicted in Fig. 3a, showing midgap states arising from domain walls.