Fractional Chern Insulators Transition in Non-ideal Flat Bands of Twisted Mono-bilayer Graphene

Abstract

Fractional Chern insulators (FCIs) in ideal flat bands with Chern number $C$ are commonly understood as color-entangled states constructed from $C$ copies of the lowest Landau level. In realistic moiré systems, however, the band geometry is generally non-ideal, and the mechanism that stabilizes such FCIs remains unclear. Using twisted monolayer-bilayer graphene as a platform, we find two FCIs separated by a continuous transition driven by a geometric instability of the Bloch wave functions.Below the transition, the target $C=2$ conduction band is geometrically stable, and the resulting fractional phase is naturally described by the Halperin-$(112)$ state. Above the transition, the system is geometrically unstable, entering a Laughlin-$1/3$ phase that persists despite further degradation of standard quantum-geometry indicators. To account for this unconventional phenomenon, we propose a color-separation mechanism beyond global geometric indicators: in the geometrically unstable regime, interactions hybridize electronic states near the $K$ point and generate an emergent ideal color component that supports the FCI. We corroborate this picture by applying a weak perpendicular magnetic field that acts as a "color separator," directly visualizing the ideal subcomponent in the single-particle level. Together, these results establish mechanisms which non-ideal flat bands stabilize FCIs, substantially enlarging their viable parameter space and clarifying the role of quantum geometry in strongly correlated topological phases.

Figures (6)

• Figure 1: (a) Moiré Brillouin zone of tMBG. (b) Non-interacting bands (dashed) and HF level bands of $\xi=1$,$s=\uparrow$ based on CM model at $\kappa=0.62$, with target conduction bands shaded in gray and orange. (c) Variation of geometric criteria in terms of $\kappa$ for non-interacting (dashed) and HF cases (dotted). Different Chern number for HF results in certain region is shaded in colors. (d) Evolution of Wannier centers as a function of $k_y$ for noninteracting (dotted) and HF (dashed black) cases. Blue and red dots denote the even and odd sectors, respectively Qi2012NematicFCI. The $y$-axis spans four unit cells; $L_1$ denotes the length of the moiré unit-cell vector. Insert: berry curvature distribution of non-interacting case. (e) The spread of hwF $w(k_y)$ evolution with $k_y$ of $C=2$ non-interacted band. The shaded box represents region where the berry curvature concentrate Insert: Metric trace distribution of non-interacting case.
• Figure 2: ED calculation on $N_{orb}=4\times 6$ momentum mesh with $N_e=8$ electrons. FCI-1 and FCI-2 stand for FCIs with $C=2/3$ and $C=1/3$. (a) Variation of ED energy levels with $\kappa$, where $\Delta E$ measures relative ground states. Degenerated 3-fold ground states are labeled with red, green and blue. The lowest 3-excited states with same total momentum with respect to ground states are labeled in purple. (b) Variation of PES density of states with $\kappa$, where $e^{-\xi/2}$ is eigenvalue of reduced density matrix $\rho_A$, which is obtained by bipartiting system into $N_A=3,N_B=5$ electrons, $1520$ states below FCI-1 gap while $1088$ sates below FCI-2 gap (see SI for details). (c) Density matrix fidelity of ground states for $8$ and $9$ electrons SI.
• Figure 3: Density of states (DOS) of the target flat bands versus $\kappa$ under a week magnetic field with $p/q = 1/20$. In region a, the bands are nearly ideal and, for $\kappa > \kappa_c = 0.55$, split into two branches. Region b hosts an exactly ideal flat band, whereas region c is non-ideal and offset from b by $\sim$1 meV. Inset: magnified view of 23--26 meV, with white lines are guided for eyes.
• Figure 4: (a, b) The phase diagram at $\kappa=0.5$ and $\kappa=0.7$ in terms of external bias $V_\mathrm{pot}$ and relative dielectric constant $\varepsilon_r$.
• Figure 5: (a) Energy spectrum of Halperin-(112) FQH states under coulomb interaction with $N_A=8,N_\phi=12,S_z=0$. (b) Corresponding PES which matches (1,2)-counting with $N_A=3$ for spin up with $S_z=0,1,2,3$, where $1520$ states below the dashed line.