Time-Reversal-Invariant Hofstadter-Hubbard Model with Ultracold Fermions

We consider the time-reversal-invariant Hofstadter-Hubbard model which can be realized in cold-atom experiments. In these experiments, an additional staggered potential and an artificial Rashba–type spin-orbit coupling are available. Without interactions, the system exhibits various phases such as topological and normal insulator, metal as well as semi–metal phases with two or even more Dirac cones. Using a combination of real-space dynamical mean-field theory and analytical techniques, we discuss the effect of on-site interactions and determine the corresponding phase diagram. In particular, we investigate the semi–metal to antiferromagnetic insulator transition and the stability of different topological insulator phases in the presence of strong interactions. We compute spectral functions which allow us to study the edge states of the strongly correlated topological phases.

Figure 1

Magnetization $m=n_{\uparrow }-n_{\downarrow }$ in the Néel state is plotted versus interaction strength $U$. We show results for $\alpha =1/2$ (blue), $1/4$ (red), $1/6$ (magenta), $1/8$ (black), and $1/10$ (green) (curves from right to left). Inset: Fermi velocity $2\pi v_F$ (red symbols) for different $\alpha =1/q$ is shown versus $q$. $U_c$ (blue symbols) obtained within RDMFT versus $q$ is also shown. Magenta (upper) line is a fit of $v_F$ to $\propto 1/q$ and cyan (lower) line of $U_c$ to $\propto 1/q^2$. Note that odd $q$ denominators exhibit $U_c=0^{+}$ due to a nested Fermi surface.

Figure 2

Real space magnetization profile $\mathit{m}(\mathit{x})$ in $S^y\mathrm{\text{−}}{S}^z$ plane for $\alpha =1/6$, $U=5$, ${\lambda }_x=0$, and $\gamma =0.125$ (a) and $\gamma =0.25$ (b), respectively.

Figure 3

(a) $E_F\mathrm{\text{−}}{\lambda }_x$-phase diagram at $\gamma =U=0$. (b) $U\mathrm{\text{−}}{\lambda }_x$-phase diagram at $n_F=2/3$ and $\gamma =0$. (c) ${\lambda }_x\mathrm{\text{−}}\gamma$-phase diagram at half filling $n_F=1$ and $U=0$. (d) $U\mathrm{\text{−}}{\lambda }_x$-phase diagrams at $E_F=0$ for various values of $\gamma$ [indicated by arrows in part (c)] and inverse temperature $\beta =20$. We find (semi-)metal (red), normal insulator (NI) (green, in upper left), topological insulator (QSH) (blue), and magnetically ordered phases (purple, in lower right).

Figure 4

Spectral function $A(k_y,\omega )$ of interacting system clearly distinguishing between (a) NI phase with no edge states traversing the bulk gap at $U=0.5$ and (b) QSH phase at $U=2$ with a single pair of edge modes (per edge) connecting the two bulk bands. Both plots are for $\alpha =1/6$, $\gamma =0.25$ and ${\lambda }_x=1.5$.