Fractional topological phases in generalized Hofstadter bands with arbitrary Chern numbers

We construct generalized Hofstadter models that possess “color-entangled” flat bands and study interacting many-body states in such bands. For a system with periodic boundary conditions and appropriate interactions, there exist gapped states at certain filling factors for which the ground-state degeneracy depends on the number of unit cells along one particular direction. This puzzling observation can be understood intuitively by mapping our model to a single-layer or a multilayer system for a given lattice configuration. We discuss the relation between these results and the previously proposed “topological nematic states,” in which lattice dislocations have non-Abelian braiding statistics. Our study also provides a systematic way of stabilizing various fractional topological states in $C>1$ flat bands and provides some hints on how to realize such states in experiments.

Figure 1

In panel (a), we give an example of the Hofstadter lattice with $n_{\varphi }=4$ that is used in panels (b) and (c). The indices of orbitals in a magnetic unit cell are shown in parentheses and the numbers on the bonds indicate the phases of the complex hopping amplitudes along the $y$ direction in units of $\pi$. In panel (b), a bilayer Hofstadter model is obtained by stacking the two layers together. In panel (c), the two Hofstadter layers are shifted relative to each other and then stacked together. The method used in panel (c) gives a color-entangled Hofstadter model in which the size of the magnetic unit cell is reduced by a factor of 2 and the lowest band has $C=2$. There are two orbitals on each lattice site (colored in red and blue) for both models and their indices are given in parentheses.

Figure 2

Energy spectra of bosons on the $C=2$ model at filling factors $1/2$ [(a) and (b)] and $2/3$ [(c) and (d)]. The system parameters are given in square brackets as $[N,N_x,N_y,L_x]$. The numbers above some energy levels indicate degeneracies that may not be resolved by inspection.

Figure 3

Energy spectra of fermions on the $C=2$ model at filling factor $1/4$. The system parameters are given in square brackets as $[N,N_x,N_y,L_x]$. The numbers above some energy levels indicate degeneracies that may not be resolved by inspection.

Figure 4

This figure shows a slice of the $C=2$ model constructed in Fig. 1 but the two orbitals are plotted separately for clarity. In panels (a) and (b), the unit cells are labeled by Roman numbers and the black lines represent the hopping terms along the $x$ direction. When $N_x$ is odd in (a) [even in (b)], this model maps into a single-layer (bilayer) system. The hopping terms along the $y$ direction do not change this mapping. Panel (c) shows certain interaction terms: (1) intracolor on-site term; (2) intercolor on-site term; (3) intracolor NN term within one unit cell; (4) intracolor NN term across the boundary of a unit cell; (5) intercolor NN term within one unit cell; (6) intercolor NN term across the boundary of a unit cell.