Bloch Model Wave Functions and Pseudopotentials for All Fractional Chern Insulators

We introduce a Bloch-like basis in a $C$-component lowest Landau level fractional quantum Hall (FQH) effect, which entangles the real and internal degrees of freedom and preserves an $N_x\times N_y$ full lattice translational symmetry. We implement the Haldane pseudopotential Hamiltonians in this new basis. Their ground states are the model FQH wave functions, and our Bloch basis allows for a mutatis mutandis transcription of these model wave functions to the fractional Chern insulator of arbitrary Chern number $C$, obtaining wave functions different from all previous proposals. For $C>1$, our wave functions are related to color-dependent magnetic-flux inserted versions of Halperin and non-Abelian color-singlet states. We then provide large-size numerical results for both the $C=1$ and $C=3$ cases. This new approach leads to improved overlaps compared to previous proposals. We also discuss the adiabatic continuation from the fractional Chern insulator to the FQH in our Bloch basis, both from the energy and the entanglement spectrum perspectives.

Figure 1

(a) shows the overlaps $\mathcal{O}$ between our FCI model states and the ground states of the fermionic ruby and the bosonic triangular lattice models, as a function of the Hilbert space dimension $d$. (b)–(d) demonstrate the adiabatic continuity between the triangular lattice model and the color-entangled Halperin pseudopotential Hamiltonian on $6\times 4$, $7\times 4$, $8\times 4$, and $6\times 6$ lattices ($\nu =1/4$ filling). We set $U=7.4237$, 7.0003, 6.9677, and 5.0955, respectively, to equalize the energy gaps at $\lambda =0$, 1, (b) shows the overlaps $\mathcal{O}$ between our FCI model states and the ground states of the interpolation Hamiltonian $H_{\lambda }$. (c) shows the energy gap $\Delta E$ above the ground states of $H_{\lambda }$. (d) shows the entanglement gap $\Delta \xi$ of the ground states of $H_{\lambda }$. $\Delta \xi$ is defined as the gap between the low-lying structure identical to the full entanglement spectrum of the model states (at $\lambda =1$) and the higher levels. By this definition, $\Delta \xi$ is infinity at $\lambda =1$.