Hierarchy of Fractional Chern Insulators and Competing Compressible States

We study the phase diagram of interacting electrons in a dispersionless Chern band as a function of their filling. We find hierarchy multiplets of incompressible states at fillings $\nu =1/3$, $2/5$, $3/7$, $4/9$, $5/9$, $4/7$, $3/5$ as well as $\nu =1/5$, $2/7$. These are accounted for by an analogy to Haldane pseudopotentials extracted from an analysis of the two-particle problem. Important distinctions to standard fractional quantum Hall physics are striking: in the absence of particle-hole symmetry in a single band, an interaction-induced single-hole dispersion appears, which perturbs and eventually destabilizes incompressible states as $\nu$ increases. For this reason, the nature of the state at $\nu =2/3$ is hard to pin down, while $\nu =5/7$, $4/5$ do not seem to be incompressible in our system.

Figure 1

(a) Illustration of the checkerboard lattice model with the relevant hopping amplitudes [2, 3]. The interaction term is a nearest-neighbor density-density interaction (interaction between pairs of white and grey sites). (b) Berry curvature in the first Brillouin zone for the indicated set of parameters. Dark (light) intensity denotes small (large) Berry curvature.

Figure 2

(a) Energy of a single hole in the fully filled lower band $E_h(\mathbf{K})≔E(N_p=N_s-1,\mathbf{K})-E(N_p=N_s,\mathbf{0})$. (b) Nonzero eigenvalues $E(N_p=2,\mathbf{K})$ of the two-particle problem on a representative path through the Brillouin zone for $t_2=0.5$. In general, there are four nonzero energies except at $\mathbf{K}=(0,0)$, where there are only two finite energies. Circles are data for (b) $6\times 6$ (empty circles) and $8\times 8$ (filled circles) unit cells [(a) $10\times 10$ unit cells], and the lines are guides to the eye.

Figure 3

(a)–(c) Numerical evidence for a fractional Chern insulator state at $\nu =2/5$. (d)–(f) Numerical evidence for a fractional Chern insulator state at $\nu =1/5$. For details, see the main text. Here, we focus on $t_2=0.5$, which is close to the value where the variance of the Berry curvature is minimal. See Table I for the samples that we consider in (c) and (f).

Figure 4

Momentum space orbital occupation $n(\mathbf{k})=⟨c_{\mathbf{k}}^{†}{c}_{\mathbf{k}}⟩$ plotted as a function of the single-hole energy $E_h(\mathbf{k})$. Different colors denote different fractions, while the symbols at the same fraction label different clusters. The dashed horizontal lines serve as reference values for a constant $n(\mathbf{k})$ at the same filling fraction. As the filling $\nu$ is increased, $n(\mathbf{k})$ displays a more pronounced (roughly linear) correlation with $E_h(\mathbf{k})$. Dotted lines ($\nu =2/3$, $\nu =4/5$) denote fractions which are most likely compressible, non-FCI states.