Bosonic Laughlin and Moore-Read states from non-Chern flat bands

TL;DR

The paper addresses whether bosonic FCIs can emerge in non-Chern, flat-band lattices, challenging the view that nontrivial band topology is required. It employs exact diagonalization and iDMRG on a two-orbital honeycomb model engineered to realize either a singular flat band with a quadratic touching or a gapped trivial flat band with $C=0$, exploring both hard-core and soft-core onsite interactions and varying nearest-neighbor couplings. At filling $\nu=\tfrac{1}{2}$ in the singular flat band, a bosonic Laughlin FCI is stabilized for arbitrarily small to large onsite interactions, with ED/DMRG signatures including two-fold ground-state degeneracy, spectral flow under flux insertion, and edge-mode counting $\{1,1,2,3,5,...\}$ yielding $\sigma_H=\tfrac{1}{2}\frac{e^2}{h}$. The study also reports the first non-Abelian FCI in a non-Chern band: a Moore-Read state at $\nu=1$ with hard-core bosons, evidenced by Ising-anyon sector entanglement spectra and flux-induced edge-state evolution consistent with MR topological order. In the isolated $C=0$ flat band, soft-core bosons form a Laughlin FCI at $\nu=\tfrac{1}{2}$ with onsite interactions much smaller than the band gap, with momentum-space occupation $n(\mathbf{k})$ showing depletion near the peak of quantum metric and Berry curvature and entanglement spectra revealing the corresponding topological degeneracies, illustrating a geometry-driven mechanism beyond band topology that broadens the FCIs' landscape.

Abstract

The rapid advances in the study of fractional Chern insulators (FCIs) raise a fundamental question: while initially discovered in flat Chern bands motivated by their topological equivalence to Landau levels, is single- particle band topology actually a prerequisite for these many-body topological orders emergent at fractional fillings? Here, we numerically demonstrate bosonic FCIs in two types of non-Chern flat bands in honeycomb lattices, using exact diagonalization and density matrix renormalization group calculations. In a gapless flat band with a singular band touching, we observe a Laughlin state at half filling, stabilized by onsite interactions from the hard-core limit down to arbitrarily small strength. Furthermore, we report the first example of a non- Abelian FCI in a non-Chern band system: a Moore-Read state at $ν$ = 1 filling of the same singular flat band with hard-core bosons. Under lattice parameters that realize a gapped trivial band (C = 0) of exact flatness, we also find the Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much smaller than the band gap. In this case, the FCI forms as interacting bosons spontaneously avoid the peaks in quantum metric and Berry curvature, preferentially occupying Brillouin zone region with relatively uniform quantum geometry. Our work significantly expands the landscape for (non-)Abelian FCIs and broadens the understanding of their formation beyond the Chern band paradigm.

Figures (7)

• Figure 1: Non-interacting model. (a) Two orbital model on honeycomb lattice including up to third NN hoppings with $-t_1=t_2=t_3=1$. (b) When $\eta=1$, the lowest band has exact flatness and a singular quadratic touching point to a dispersive band (grey) at $\Gamma$. (c) When $\eta<1$, the flat band is isolated from the dispersive band by a band gap $\Delta$, while retaining the exact flatness, which necessarily means the band Chern number $C=0$. Both (b) the singular flat band and (c) the isolated trivial flat band are color-coded by their Berry curvature distribution $\Omega(\bf k)$.
• Figure 2: Laughlin FCI in the singular flat band with arbitrarily small and large onsite interactions. We consider $\nu=1/2$ filling of the singular flat band ($\eta=1$). ED spectra of soft-core bosons with a very small $U_2=0.001$ interaction from $N=4\times3\times2$ and $N=4\times4\times2$ tori are shown in panels (a,b), respectively. For each system size, the spectral flow of the well gapped two-fold ground states is shown in the inset. The iDMRG results of hard-core bosons ($U_{n\geq2}=\infty$) are shown in panel (c-e). In panel (c), the momentum-resolved entanglement spectrum (ES) of the charge sector ($N_\mathrm{L}=0$) with the largest eigenvalue of the reduced density matrix shows the characteristic counting $\{1,1,2,3,5,...\}$ of the edge conformal field theory. We further show (d) the evolution of ES of different charge sectors ($N_\mathrm{L}=-2,-1,0,1$) and (e) charge pumping results by adiabatically inserting $4\pi$ fluxes.
• Figure 3: Moore-Read state in the singular flat band. We show the iDMRG results of $\nu=1$ filling of the singular flat band ($\eta=1$) with hard-core bosons ($n_\mathrm{max}=2$). (a) The ES of $\Psi_\mathbf{I}$ ground state from the $N_1=6$ cylinder, showing the edge mode counting $\{1,2,4,7,... \}$ and $\{1,1,3,5,10... \}$ for the $N_\mathrm{L}=-1$ and $N_\mathrm{L}=0$ charge sectors, respectively. (b) The ES of the $\Psi_\sigma$ ground state from the $N_1=5$ cylinder, showing the same degeneracy pattern $\{1,2,4,8,... \}$ in all charge sectors ($N_\mathrm{L}=-1,0$ plotted for example). (c) Charge pumping results starting from the $\Psi_\mathbf{I}$ or the $\Psi_\sigma$ ground state. (d) The evolution of ES after flux insertion, starting from the $\Psi_\mathbf{I}$ ground state.
• Figure 4: Laughlin FCI in the isolated $C=0$ flat band with soft-core bosons. The iDMRG results at $\nu=1/2$ filling of the isolated flat band ($\eta=0.8$ with band gap $\Delta=0.08$) with only $U_2=0.02$. The inset of panel (a) shows the trace of quantum metric of the trivial flat band with a peak at $\Gamma$, and the blue dashed line shows the path along which we compute the momentum-space boson distribution $n(\mathbf{k})$ in (a). $n(\mathbf{k})$ is measured from the ground state in the identity sector for example and the bosons tend to avoid occupying around $\Gamma$. (b) The ES of the $N_\mathrm{L}=0$ charge sector of the ground state in the identity sector, showing the characteristic edge-mode counting $\{1,1,2,3,5,7,... \}$. (c) The ES of the ground state in the semion sector, where all charge sectors show the same degeneracy pattern and the ES is symmetric about $N_\mathrm{L}=0.5$.
• Figure S 1: Supplementary spectral flow of the Laughlin FCI in the singular flat band. In Figs. 2(a-b) of the main text, the spectral flow of the degenerate ground states are shown in the insets. Here, we zoom out and add the low-energy excited states to further support the robust gap. The parameters are the same as Fig. 2(a-b) of the main text with $\nu=1/2$ of the singular flat band ($\eta=1$), and only very small $U_2=0.001$ interaction. Panels (a) and (b) are from $N=3\times4\times2$ and $N=4\times4\times2$ tori, respectively.