Adiabatic continuity between Hofstadter and Chern insulator states

We show that the topologically nontrivial bands of Chern insulators are adiabatic cousins of the Landau bands of Hofstadter lattices. We demonstrate adiabatic connection also between several familiar fractional quantum Hall states on Hofstadter lattices and the fractional Chern insulator states in partially filled Chern bands, which implies that they are in fact different manifestations of the same phase. This adiabatic path provides a way of generating many more fractional Chern insulator states and helps clarify that nonuniformity in the distribution of the Berry curvature is responsible for weakening or altogether destroying fractional topological states.

Figure 1

Lattice model and band structure for the square-checkerboard lattice model. Panel (a) shows the lattice structure. The black dots show the lattice sites in a Hofstadter lattice with flux $2\pi /16$ through each plaquette. The dashed black square marks a magnetic unit cell containing 16 sites. The sites marked by the dashed rectangles form a checkerboard lattice where the two different orientations of the rectangles represent two sublattices. The Hamiltonian $H_{\text{sq-cb}}\left(R\right)$ in Eq. (9) interpolates between the Hofstadter and Chern insulator limits as $R$ is varied from 0 to 1. Panels (b)–(d) show the band structures at three values of $R$ (0, $0.5$, and 1, respectively) along the contour $\Gamma \to M\to X\to \Gamma$ in momentum space. In panel (b), flat Landau levels carry Chern number $C=1$ while the two nonflat bands at the middle have a total $C=-14$. In panel (d), the top and bottom bands of the checkerboard model have nontrivial Chern numbers $C=\mp 1$.

Figure 2

Phases of the square-lattice Hofstadter model. The numbers and arrows indicate the hopping phases along the bonds in units of $\pi$, and the star marks the plaquette where a $-2\pi$ flux is inserted.

Figure 3

Square-checkerboard hybrid lattice. (a) The single-particle band gap (top curve) and bandwidth (lower curve) as a function of $R$. The flatness ratio (band gap over bandwidth) is shown as the inset. (b) The Berry curvature at different k-points with $R=0.0$, $0.5$, and $1.0$ (dotted, dashed, and solid lines, respectively).

Figure 4

Lattice model and band structure for the triangular-kagome lattice model. Panel (a) shows the lattice structure. The black dots show the lattice sites in a triangular Hofstadter lattice with flux $\pi /16$ in each triangle. The dashed lines mark a magnetic unit cell, which contains 16 sites. The sites marked by the dashed circles form a kagome lattice where the three different colors represent the three sublattices. Panels (b)–(d) shows the band structures at different values of $R$ ($0.0$, $0.5$, and 1, respectively) along the contour $\Gamma \to K\to M\to \Gamma$ in momentum space.

Figure 5

Phases of triangular lattice Hofstadter model. With the exception of the outermost hopping bonds, the phase factor associated with a bond, in units of $\pi$, is indicated by the arrow on it and the number either below or to the left of it, and the star marks the triangle where a $-2\pi$ flux is inserted.

Figure 6

Triangular-kagome hybrid lattice (a) The single-particle band gap (top curve) and bandwidth (lower curve) as a function of $R$. The flatness ratio (band gap over bandwidth) is shown in the inset. (b) The Berry curvature at different k-points with $R=0.0$, $0.5$, and $1.0$ (dotted, dashed, and solid lines, respectively).

Figure 7

Energy spectra at filling $1/3$ ($N=8$, $N_x=4$, $N_y=6$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and $1.0$ (top to bottom). There are 3 quasidegenerate states at $(K_x,K_y)=(0,0)$, $(0,2)$, and $(0,4)$.

Figure 8

Energy spectra at filling $1/2$ ($N=10$, $N_x=4$, $N_y=5$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and $1.0$ (top to bottom). There are 6 quasidegenerate states: one each at $(K_x,K_y)=(0,0)$ or $(2,0)$, and two for $(1,0)$ or $(3,0)$.

Figure 9

Energy spectra at filling $2/5$ ($N=8$, $N_x=4$, $N_y=5$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and $1.0$. There are 5 quasidegenerate states in each panel at $(K_x,K_y)=(0,0),(0,1),(0,2),(0,3)$, and $(0,4)$.

Figure 10

Energy spectra at filling $2/5$ ($N=10$, $N_x=5$, $N_y=5$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, $0.6$, $0.8$, and $1.0$ (top to bottom). The 5 quasidegenerate states all appear in the $(K_x,K_y)=(0,0)$ sector and interaction causes splittings.

Figure 11

Evolution of the $1/3$ ground states for square-checkerboard (left panels) and triangular-kagome (right panels) models shown in Fig. 7 upon flux insertion in the $y$ direction. The quasidegenerate ground states are separated from the excited state at each point. Note that at $R=0.0$, the states are perfectly degenerate at each flux value and there is no obvious spectral flow.

Figure 12

Evolution of the $1/2$ ground states for square-checkerboard (left panels) and triangular-kagome (right panels) models shown in Fig. 8 upon flux insertion in the $x$ direction. The quasidegenerate ground states are separated from the excited state at each point. Note that at $R=0.0$, the states are perfectly degenerate at each flux value and there is no obvious spectral flow.

Figure 13

Quasihole spectra at $1/3$ filling ($N=7$, $N_x=4$, $N_y=6$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and $1.0$ (top to bottom). There are 12 states in the low-energy manifold (below the blue lines) in each momentum sector.

Figure 14

Quasihole spectra at filling $1/2$ ($N=10$, $N_x=3$, $N_y=7$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and $1.0$ (top to bottom). There are 6 states in the low-energy manifold (below the blue lines) in each momentum sector.

Figure 15

Quasiparticle spectra at filling $1/3$ ($N=9$, $N_x=4$, $N_y=6$) for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0.0$, $0.5$, and 1 (top to bottom). The number of states below the blue lines obey the FQH to FCI mapping in Eq. (14).

Figure 16

Quasiparticle spectra at filling $1/3$ (9 particle and 24 fluxes) on torus with Coulomb interaction.

Figure 17

Particle entanglement spectra at $1/3$ filling for the square-checkerboard (left panels) and triangular-kagome (right panels) models at $R=0$, $0.5$, and 1 (top to bottom). The number of states in the low-entanglement-energy manifold indicated by the blue lines are 46 states in the $K_y=0$, 3 momentum sectors, and 45 states in other sectors.

Figure 18

Particle entanglement spectra at $1/2$ filling for the square-checkerboard (left panels) and triangular-kagome (right panels) model at $R=0$, $0.5$, and 1 (top to bottom). The number of states in the low-entanglement energy manifold indicated by the blue lines are 200, 196, 201, and 196 states in the $K_x=0$, 1, 2, and 3 momentum sectors, respectively.

Figure 19

The plus signs, circles, squares, and asterisks show the gaps in the spectra of the $1/3$ ground states, $1/3$ quasihole states, $1/2$ ground states, and $1/2$ quasihole states at several $R$ for the square-checkerboard model (upper panel) and triangular-kagome model (lower panel). The continuous line shows the deviation of the Berry curvature $\delta \mathcal{F}$ (normalized by the average value $\overline{\mathcal{F}}$) as a function of $R$.