Topological Order Without Band Topology in Moiré Graphene

Abstract

The discovery of zero-field fractional Chern insulators (FCIs) in moiré materials has attracted intense interest in the interplay between topology and correlations. Here, we demonstrate that fractionalized topological order can emerge under realistic conditions even within a topologically trivial moiré band. By projecting long-range Coulomb interactions into a trivial band of twisted multilayer graphene, we identify a set of incompressible FCI ground states exhibiting fractional quantized Hall conductance. Their Laughlin-like behavior is further confirmed through the particle-cut entanglement spectrum. We trace the origin of this phase to the strongly inhomogeneous distribution of quantum geometry within the moiré Brillouin zone, which reshapes interaction effects independently of the band topology. Extending this heuristic quantum geometric mechanism, we demonstrate that similarly unexpected Laughlin-like FCIs can also be stabilized in higher-Chern-number moiré bands under experimentally accessible conditions. Our results establish realistic scenarios under which many-body topological order can emerge independently of single-particle band topology.

Figures (11)

• Figure 1: FCIs in a $C=0$ band. (a) Single-particle band structure displaying a topologically trivial conduction band (marked in red). (b) Low-lying many-body energy spectrum, (c) particle-cut entanglement spectrum, and (d) energy gap scaling demonstrating the stabilization of Laughlin-like FCI ground states at $1/3$ filling of the trivial band. The data shown in (b) and (c) correspond to a finite system defined by the spanning vectors $\mathbf{T}_1=(6,1)$ and $\mathbf{T}_2=(9, 6)$. The number of states below the solid line in (c) matches the quasihole counting of the $1/3$ filling Laughlin state. In panel (d), $N_s$ is the number of moiré lattice sites. Additional details, including the system parameters, can be found in the End Matter and SM SupMat.
• Figure 2: Quantum geometry perspective on trivial-band FCIs. (a) Quantum geometry and (b) Berry curvature of the trivial band displaying a clear concentration around the $\kappa$ point. (c) The electron occupation is uniformly distributed along the mBZ in finite systems excluding the high-symmetry point $\kappa$. (d) In finite systems containing the $\kappa$ point, the electron occupation remains uniform but exhibits a hole at the $\kappa$ point. The form of the finite-size systems is detailed in the SM SupMat.
• Figure 3: Laughlin-like states in a realistic $C=2$ band. (a) FS metric distribution in the mBZ displaying the characteristic concentration at the $\kappa'$ point. (b) Low-lying many-body energy spectrum, (c) particle-cut entanglement spectrum, and (d) many-body energy gap scaling demonstrating the Laughlin-like nature of the many-body ground states at $1/3$ filling of the $C=2$ band. The number of states below the solid line in (c) correspond to the quasihole counting of the $1/3$ Laughlin state. The system parameters are detailed in the End Matter.
• Figure 4: Phase transition in an ideal $C=2$ band. (a) Many-body energy gap (black solid line) and many-body Chern number (red dashed line) as a function of interlayer coupling, displaying a clear phase transition. (b) The low-lying many-body energy spectrum at $\beta/(\hbar v_Fk_\theta)=0.1$ shows a Laughlin-like threefold degenerate ground state with $\mathcal{C}_{\text{ave}}=1/3$. (c) Low-lying many-body energy spectrum at $\beta/(\hbar v_Fk_\theta)=1$ characterized by a threefold degenerate $\mathcal{C}_{\text{ave}}=2/3$ ground state. (d) The corresponding particle-cut entanglement spectrum for the $\mathcal{C}_{\text{ave}}=2/3$ state contains $5508$ below the solid red line, matching the quasihole state counting of the quantum Hall bilayer $(\bar{1}\bar{1}2)$ state. Here, we considered a finite system defined by the spanning vectors $\mathbf{T}_1=(5,1)$ and $\mathbf{T}_2=(1, 5)$.
• Figure 5: The targeted band of TDBG above charge neutrality with $C=2$ (a) and the corresponding Berry curvature distribution over mBZ.