Topological phase transition in the Hofstadter-Hubbard model

We study the interplay between topological and conventional long-range order of attractive fermions in a time-reversal-symmetric Hofstadter lattice using quantum Monte Carlo simulations, focusing on the case of one-third flux quantum per plaquette. At half filling, the system is unstable towards $s$-wave pairing and charge-density-wave order at infinitesimally small interactions. At one-third filling, the noninteracting system is a topological insulator, and a nonzero critical interaction strength is needed to drive a transition from the quantum spin Hall insulator to a superfluid. We probe the topological signature of the phase transition by threading a magnetic flux through a cylinder and observe quantized topological charge pumping.

Figure 1

(a) Band structure and (b) density of states of the noninteracting Hofstadter model with $\varphi =1/3$. (c) Mean-field result of the pairing-order parameter as a function of attractive interaction strength at $1/2$ and $1/3$ fillings.

Figure 2

Pair and density correlation functions at $U/t=4$. The $\times$ symbol denotes the site $\mathbf{r}$ relative to which correlations are shown. (a) and (b) show results for the half-filled system, while (c) and (d) for the one-third-filled system.

Figure 3

(a) Pair and CDW structure factors versus band filling in an $L=12$ lattice with $U/t=4$. (b) The pairing structure factor versus $1/L$ for various interaction strengths at half filling. The CDW structure factors are identical at half filling due to a symmetry that is discussed in the text.

Figure 4

(a) The pairing structure factor of a $1/3$-filled system. Solid lines are fits with cubic function of $1/L$. (b) The charge gap ${\Delta }_c$ versus the interaction strength $U$ for various system sizes. The red arrow indicates the band gap of the noninteracting system. The inset shows the superfluid order parameter and the charge gap extrapolated to the thermodynamic limit close to the transition point.

Figure 5

Total particle number in the upper-half of a cylinder [Eq. (9)] versus flux ${\Phi }_x$ for various interaction strengths. Inset (a) illustrates a flux ${\Phi }_x$ (opposite for spin $\uparrow$ and $\downarrow$) threaded through the cylinder, which pumps particles vertically in the QSH phase. Sites in the upper half of the cylinder are shown in red and their occupation numbers sum to $N_{\mathcal{U}}$. Inset (b) shows single-particle energy levels versus the threaded flux ${\Phi }_x$ in a cylindrical noninteracting system. Color indicates the center of mass of single-particle wave functions, with red (blue) being closer to the upper (lower) edge.