Exact solutions of fractional Chern insulators: Interacting particles in the Hofstadter model at finite size

We show that all the bands of the Hofstadter model on the torus have an exactly flat dispersion and Berry curvature when a special system size is chosen. This result holds for any hopping and Chern number. Our analysis therefore provides a simple rule for choosing a particularly advantageous system size when designing a Hofstadter system whose size is controllable, like a qubit lattice or an optical cavity array. The density operators projected onto the flat bands obey exactly the Girvin-MacDonald-Platzman algebra, like for Landau levels in the continuum in the case of $C=1$, or obey its straightforward generalization in the case of $C>1$. This allows a mapping between density-density interaction Hamiltonians for particles in the Hofstatder model and in a continuum Landau level. By using the well-known pseudopotential construction in the latter case, we obtain fractional Chern insulator phases, the lattice counterpart of fractional quantum Hall phases, that are exact zero-energy ground states of the Hofstadter model with certain interactions. Finally, the addition of a harmonic trapping potential is shown to lead to an appealingly symmetric description in which a new Hofstadter model appears in momentum space.

Figure 1

(a) Illustration of the system discussed in this work: a square lattice of $Q$ by $Q$ plaquettes with a uniform flux density of $n_{\varphi }=2\pi P/Q$ and periodic boundary conditions. (b) Single-particle spectrum for $Q=21$. All the bands are exactly flat.

Figure 2

Form factors in the LLL and the lattice case. The form factor in the lattice case is given for nearest-neighbor hopping. The system size is given by $N_y=21$.

Figure 3

Two-particle spectrum $E_{\left(2\right)}$ of the interaction Hamiltonian vs momentum sector $K_{y1}+K_{y2}$ for a contact interaction projected to the continuum LLL (left) and to the lowest band of the Hofstadter model with our special system size (right). These spectra were obtained by exact diagonalization for $N_y=21$. As indicated in the top panels, there are two nearly degenerate levels close to $E=1$ corresponding to the 0th pseudopotential [57]. The bottom panels show the same spectrum on a logarithmic scale, revealing the differences between the LLL and the Hofstadter case (see main text).

Figure 4

Interaction potential for the Hofstadter model which leads to a Hamiltonian with an exact zero-energy ground state given by the $\nu =1/2$ continuum Laughlin ground state. The left panels show $V\left(\mathbf{r}\right)$, the interaction leading to the LLL form factor, while the right panels show $V_s\left(\mathbf{r}\right)$, the interaction leading to the form factor of the Kapit-Mueller model [46]. The nearest-neighbor hopping is 1 and the NNN hopping is 0.27. The plots are given for $N_y=21$. The on-site interaction is normalized to 1. The second largest interaction is on the NNN and is equal to 0.029 for $V\left(\mathbf{r}\right)$ and 0.027 for $V_s\left(\mathbf{r}\right)$.

Figure 5

Form factors for $C=3$ in the LLL and lattice cases. The form factor in the lattice case is given for nearest-neighbor hopping. The flux density is $n_{\varphi }=11/32$ and $N_y=32$. The peaks centered at $(\pm N_y/3,\pm N_y/3)$ are too small to be seen clearly in (b).

Figure 6

Lattice interaction $V^{\left(1\right)}\left(\mathbf{r}\right)$ reproducing the $m=1$ pseudopotential. The form factor $|d_0{\left(\mathbf{q}\right)|}^2$ was calculated for a nearest-neighbor hopping of 1 and a NNN hopping of 0.27. The plot is given for $N_y=21$.

Figure 7

Real-space wave function of the trapping potential Hamiltonian eigenstates: $|{\psi }_l{\left(\mathbf{r}\right)|}^2$ in the case of $C=1$ and $N_y=21$.

Figure 8

(a) Normalized spectrum ${\epsilon }_l^{\left(n\right)}/\tilde{W}$ of the trapping potential projected to one band with Chern number $C=3$. (b)–(d) Real-space wave function $|{\psi }_l{\left(\mathbf{r}\right)|}^2$ of the first three eigenstates, corresponding to $L=0$ and $\sigma =0,1,2$. The system size is given by $N_y=32$ and the flux density is $n_{\varphi }=11/32$.