Haldane statistics for fractional Chern insulators with an arbitrary Chern number

In this paper, we provide analytical counting rules for the ground states and the quasiholes of fractional Chern insulators with an arbitrary Chern number. We first construct pseudopotential Hamiltonians for fractional Chern insulators. We achieve this by mapping the lattice problem to the lowest Landau level of a multicomponent continuum quantum Hall system with specially engineered boundary conditions. We then analyze the thin-torus limit of the pseudopotential Hamiltonians, and extract counting rules (generalized Pauli principles, or Haldane statistics) for the degeneracy of its zero modes in each Bloch momentum sector.

Figure 1

Relabeling of the Wannier states $|X,k_y\rangle \leftrightarrow |j,s\rangle$ for $(N_x,N_y)=(5,6)$ and $C=4$. We focus on the principal region with $X\in \left[0..N_x\right)$ and $k_y\in \left[0..N_y\right)$. Each solid ellipse represents a Wannier center. The horizontal axis gives the $k_y$ index while the vertical axis gives the position of the Wannier center in the $x$ direction (mod $L_x$). The ellipses are colored according to the $X$ index, and labeled by the $(j,s)$ indices. We have employed Eq. (58) to shift $j$ to $[0..M)$ and $s$ to $[0..C)$. Upon a color-independent flux insertion, each Wannier center flows along the solid lines of its color.

Figure 2

Terms in the expansion of the pseudopotential Hamiltonian. Here we illustrate the example of $d=3$. Each dot represents a term $(q_y,\Delta )$ in Eq. (69). The weight of each term decays exponentially in its distance from the origin. The dashed circle marks the empirical threshold for truncation ${\left(q_{y}d\right)}^2+{(\Delta -q_{y}d)}^2=d^2$. The solid black dots inside are the density-density terms in Eq. (72), while the four solid gray dots contain the pair hopping and the density-density terms in Eqs. (75) and (85).

Figure 3

Terms in the expansion of the pseudopotential Hamiltonian at $d=1$. The presentation follows the same format as Fig. 2.

Figure 4

Energy spectrum of the pseudopotential Hamiltonian of two bosons on a $N_x\times N_y=3\times 2$ lattice with Chern number $C=2$. The three degenerate ground states at zero energy are marked in red.

Figure 5

Energy spectrum of the pseudopotential Hamiltonian of three bosons on a $N_x\times N_y=5\times 2$ lattice with Chern number $C=2$. The ten degenerate ground states at zero energy are marked in red.