Pairing and Vortex Lattices for Interacting Fermions in Optical Lattices with a Large Magnetic Field

We study the structure of a pairing order parameter for spin-$1/2$ fermions with attractive interactions in a square lattice under a uniform magnetic field. Because the magnetic translation symmetry gives a unique degeneracy in the single-particle spectrum, the pair wave function has both zero and finite-momentum components coexisting, and their relative phases are determined by a self-consistent mean-field theory. We present a microscopic calculation that can determine the vortex lattice structure in the superfluid phase for different flux densities. Phase transition from a Hofstadter insulator to a superfluid phase is also discussed.

Figure 1

(a) Three magnetic bands for $p/q=1/3$. (b) The Fermi surface of a half-filled (black solid line) and slightly away from half-filled (red dashed line) system. (c) and (d) the band dispersion along $k_x$ direction with $k_y=0$ (c) and along $k_y$ direction with $k_x=0$ (d). Various possible pairings included in our BCS theory are also illustrated in (c) and (d), which include intra- and interband pairing (c), and pairing with nonzero center-of-mass momentum (d).

Figure 2

Structure of the vortex lattice found from self-consistently solving the BCS mean-field Hamiltonian, where $n_B=1/3$ for (a) and $n_B=1/4$ for (b). This is a contour plot of pairing order parameter $\Delta (\mathbf{r})$. The gray area means low superfluid density and locates the center of vortex cores. The intersection of vertical and horizontal straight lines indicates lattice sites. In both plots we use $U=-5.5t$ and $n_{\uparrow }=n_{\downarrow }=1/3$ for (a) and $n_{\uparrow }=n_{\downarrow }=1/2$ for (b).

Figure 3

(a) For $p/q=1/3$, $I_{02}$, and $I_{21}$ (see text for definition) equal to unity within numerical accuracy, which verifies the symmetry relations. (b) Average superfluid order parameter $\Delta ={\overline{\Delta }}_i$ as a function of $U$ for $p/q=1/3$, $n=1/3$ (red solid line) shows a phase transition and $n=1/2$ (blue dashed line) does not.