Model Fractional Chern Insulators

We devise local lattice models whose ground states are model fractional Chern insulators—Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the ${\mathcal{Z}}_k$ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions $\nu =k/(\mathcal{C}+1)$ and (ii) nematic states at $\nu =k/2$, where $\mathcal{C}$ is the Chern number of the lowest band. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local ($k+1$) body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well suited for the study of novel phenomena involving, e.g., twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.

Figure 1

Typical particle entanglement spectra (PES), here displayed for the $k=2$, $\nu =2/3$ non-Abelian state in a $\mathcal{C}=M=3$ band with $N=8$, $N_x\times N_y=4\times 4$, $\varphi =1/3$ and hoppings truncated at (a) $d=\sqrt{2}$, (b) $d=3$, (c) $d=\sqrt{10}$, and (d) $d=\infty$ lattice constants. Generally, a PES obtained by numerical diagonalization includes three parts: the low-lying levels with quasihole excitation information (blue), the high nonuniversal levels (red), and numerical noise (gray) set by the double precision above ${\xi }_c\approx \ln (2^{-53})\approx 36.7$. The high, nonuniversal levels merge into numerical noise for large $d$, preventing us from further numerically tracking the growth of the entanglement gap, which we argue to increase without bound with increasing $d$.

Figure 2

The entanglement gap ${\mathrm{\Delta }}_{\xi }$ in the $N_A=\lceil N/2\rfloor$ sector versus the inverse hopping distance $1/d$ in the (a) $\mathcal{C}=2$ and (b) $\mathcal{C}=3$ band. For each $\mathcal{C}$, we consider both Abelian states at $\nu =1/(\mathcal{C}+1)$ and non-Abelian states at $\nu =2/(\mathcal{C}+1)$ and $\nu =3/(\mathcal{C}+1)$, with lattice geometry of either $\gcd (N_x,\mathcal{C})=\mathcal{C}$ or $\gcd (N_x,\mathcal{C})=1$. $\varphi$ is chosen as $1/\mathcal{C}$. For $d$ longer than three or four lattice constants, the size of ${\mathrm{\Delta }}_{\xi }$ cannot be tracked further due to the limitation of machine precision.

Figure 3

The finite-size scaling of the energy gap for the Abelian $\nu =1/(\mathcal{C}+1)$ states and non-Abelian $\nu =2/(\mathcal{C}+1)$ states in $\mathcal{C}=2$ and $\mathcal{C}=3$ bands. Note that the spread of the ground state manifold is exactly zero here. $N_x$ and $N_y$ are appropriately chosen to make samples as isotropic as possible. $\varphi$ is chosen as $1/\mathcal{C}$.