Stripe-ordered superfluid and supersolid phases in the attractive Hofstadter-Hubbard model

We use microscopic Bogoliubov–de Gennes formalism to explore the ground-state phase diagram of the single-band attractive Hofstadter-Hubbard model on a square lattice. We show that the interplay between the Hofstadter butterfly and superfluidity breaks spatial symmetry, and gives rise to stripe-ordered superfluid and supersolid phases in large parameter spaces. We also discuss the effects of a trapping potential and comment on the viability of observing stripe-ordered phases with cold Fermi gases.

Figure 1

(a) Typical $|{\Delta }_i|/t$ profiles are shown for the U-SF (left) and S-SF phases (right), where ${\varphi }_{\uparrow }={\varphi }_{\downarrow }=0$ and ${\varphi }_{\uparrow }={\varphi }_{\downarrow }=1/4$, respectively, and $\mu =0$ corresponding to a uniformly half-filled lattice. (b) Typical $|{\Delta }_i|/t$ (left) and $n_{i\uparrow }+n_{i\downarrow }$ (right) profiles are shown for the S-SS phase, where ${\varphi }_{\uparrow }={\varphi }_{\downarrow }=1/4$ and $\mu =-t$. Here, $(x,y)$ are in units of $\ell$, and we set $h=0$ and $g=7t$ in all figures.

Figure 2

The ground-state phase diagrams are shown for $\mu =0$ in (a) and $\mu =-t$ in (b), when $\varphi =p/q=1/4$, and the solid lines are guides to the eye.

Figure 3

The trap profiles are shown for a cut along the $y$ direction (in units of $\ell$) when $x=0,p/q=1/4,\mu =0$, and $h=t$.