Non-Abelian fractional Chern insulators from an exactly solvable two-body model

Abstract

We construct a class of lattice Hamiltonians whose single-particle spectrum consists of an arbitrary number of exactly degenerate flat bands that reproduce the analytic structure of the first $p$ Landau levels restricted to the lattice. When combined with local bosonic contact interactions, these models become exactly solvable frustration-free parent Hamiltonians for FCIs that realize both Abelian and non-Abelian parton quantum Hall states. Using exact diagonalization, we confirm the expected zero-mode counting for variants of the model stabilizing the bosonic Jain-21 state as well as the non-Abelian 22- and 33-states, which are expected to support Ising- and Fibonacci-type anyons, respectively. Our construction provides an exactly solvable lattice realization of multi Landau-level physics and offers a new framework for studying FCIs with Chern number $C > 1$. More broadly, it supplies a family of idealized lattice models that capture the analytic structure of continuum Landau levels while remaining compatible with exponentially local hopping.

Figures (8)

• Figure 1: A plot of the hopping parameter $t_R = |\braket{R_0|H|R}|$ for the 21-, 22-, and 33-state parent Hamiltonians at two different system sizes. Plots in each column are computed with a four, four and 12-site magnetic unit-cell for $H_{21}$, $H_{22}$ and $H_{33}$ respectively. In particular, the small lattices in the top row are of sizes $14\times14$, $18 \times 18$ and $24 \times 24$ again for the 21-, 22-, and 33-state parent Hamiltonians respectively, while the large lattices in the lower row have dimensions $28\times 28$, $36 \times 36$ and $48 \times 48$. Unless explicitly stated otherwise, all Hamiltonians here and elsewhere are normalized so that $t_0 = 1$. Distances are given in terms of the lattice constants $a = a_x = a_y$.
• Figure 2: A one-dimensional slice of the hopping parameter plot in Fig. \ref{['hoppingParameter']} i.e. for $t_x = |\braket{R_0|H|x,0}|$ for the 21-, 22-, and 33-state parent Hamiltonians, clearly demonstrating exponential decay of the hopping strength.
• Figure 3: A three-dimensional plot of the exponentially-decaying hopping parameter $t_R = |\braket{R_0|H|R}|$ for the 21-, 22-, and 33-state parent Hamiltonians. Plots for $H_{21}$ and $H_{22}$ are generated for lattices of size $14 \times 14$ and $18 \times 18$ with four site magnetic unit-cells, while the plot for $H_{33}$ corresponds to a $28 \times 28$ site lattice and a 12-site magnetic unit-cell.
• Figure 4: Band structure plots with and without finite-distance cutoffs, see (\ref{['rcoeq']}), for the three parent Hamiltonians discussed in the main text. Plots are computed for lattices of size $60 \times 20$, $40 \times 40$ and $24 \times 24$ with three, four and 12-site magnetic unit-cells for $H_{21}$, $H_{22}$ and $H_{33}$ respectively.
• Figure 5: The ten smallest eigenvalues of the 21- and 22-state parent Hamiltonians. The eigenvalues of both $H_{21}$ and $H_{22}$ are computed for four particles on a $4 \times 6$-site lattice with unit cells of dimension $m = n = 2$. The three smallest eigenvalues in each case are zero up to finite machine precision, in agreement with the expected topological ground state degeneracy.