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arXiv:2505.09009 · Newton

Fractional Chern insulator states in an isolated flat band of zero Chern number

Zuzhang Lin, Hongyu Lu, Wenqi Yang, Dawei Zhai, Wang Yao

Published Dec 19, 2025• 6 citations
arXivHTMLPDF

Abstract

A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.

DOI
10.1016/j.newton.2025.100339

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A flat band with Chern number C = 0 , well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is an unexpected context for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band using large-scale exact diagonalization and infinite density matrix renormalization group simulations. Hosted by an anisotropic lattice, this band has a sharp peak for the quantum geometry. At 1/3 hole filling, we find the unexpected FCI with 3-fold ground-state degeneracy and σ H = − 1 / 3 ( e 2 / h ) . The momentum-space carrier distribution shows that the quantum geometry peak tends to push the interacting holes away, toward other regions of the Brillouin zone with the nearly "ideal" quantum geometry, underlying the formation of the FCI in the C = 0 flat band. In comparison, we find only trivial ground states when the quantum geometry peak becomes less avoidable. Our work not only shows that FCl states could exist without the single-particle topology, relying on the more intrinsic quantum geometry, but also suggests a possibly generic scenario to explore FCIs in otherwise trivial flat bands.
We study two models for spinless fermions featuring topologically nontrivial bands characterized by Chern numbers C=±1 at fractional filling. Using exact diagonalization, we show that, even for infinitely strong nearest-neighbor repulsion, the ground states of these models belong to the recently discovered class of quantum liquids called fractional Chern insulators (FCI). Thus, we establish that FCI states can arise even if interaction strengths are arbitrarily larger than the noninteracting band gap, going beyond the limits in which FCI states have been previously studied. The strong-coupling FCI states, therefore, depart from the usual isolated-band picture that parallels the fractional quantum Hall effect in Landau levels and demonstrate how a topologically ordered state can arise in a truly multiband system.
The integer quantum Hall state occurs when Landau levels are fully occupied by fermions, while the fractional quantum Hall state usually emerges when the Landau level is partially filled by strongly correlated fermions or bosons. Here, we report two fractional Chern insulating states of hard-core bosons in a multiband lattice model hosting topological flat bands with a high Chern number. The previously proposed $\ensuremath{\nu}=1/3$ fractional Chern insulating state inherited from the high Chern number C=2C=2C=2 of the lowest topological flat band is revisited by the infinite density matrix renormalization group algorithm. In particular, we numerically identify a bosonic 1/21/21/2-Laughlin-like fractional Chern insulating state at the integer fillings. We show two lower topological flat bands jointly generate an effective C=1C=1C=1 Chern band with half filling. Furthermore, we find a strictly particle-hole-like symmetry between the $\ensuremath{\nu}$ and $3\ensuremath{-}\ensuremath{\nu}$ filling in our model. These findings extend our understanding of quantum Hall states and offer a route to realize fractional states in the system with multibands and high Chern numbers.
We investigate a fractional Chern insulator (FCI) candidate arising from moir\'e bands with higher Chern number C=2C=2C=2 on a magic-angle twisted bilayer checkerboard lattice (MATBCB). There are two nearly flat low lying bands in the single-particle energy spectrum under the first magic angle $\ensuremath{\phi}\ensuremath{\approx}1.{608}^{\ensuremath{\circ}}$ and chiral limit. We find that MATBCB hosts a nearly uniform Berry curvature distribution and exhibits tiny violation of quantum geometric trace condition in the first moir\'e Brillouin zone (mBZ), indicating that there is a nearly ideal quantum geometry in MATBCB in single-particle level. Turning on projected Coulomb interactions, we perform exact diagonalization and find a 10-fold ground-state quasidegeneracy in many-body energy spectrum with filling fraction $\ensuremath{\nu}=1/5$. The 10-fold quasidegenerate ground states further show spectra flow under flux pumping. By diagnosing the particle entanglement spectrum (PES) of the ground states, we obtain a clear PES gap and quasihole state counting consistent with Halperin spin singlet generalized Pauli principle, suggesting that a C=2C=2C=2 Abelian fractional Chern insulator is realized in this system.
Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at 1/3 filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground state density is constant, and equal to 1/3 in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator - both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.
Fractional quantum anomalous Hall (FQAH) effect, a lattice analogue of fractional quantum Hall effect, offers a unique pathway toward fault-tolerant quantum computation and deep insights into the interplay of topology and strong correlations. The exploration has been successfully guided by the paradigm of ideal flat Chern bands, which mimic Landau levels in both band topology and local quantum geometry. Yet, given the boundless potential for Bloch bands in lattice systems, it remains a significant open question whether FQAH states can arise in scenarios fundamentally distinct from this paradigm. Here we turn to a class of gapless flat bands, featuring (i) ill-defined band topology, (ii) non-quantized Berry flux, (iii) divergent quantum geometry at singular band touchings, (iv) highly fluctuating and far-from-ideal quantum geometry across the Brillouin zone (BZ). Our exact diagonalization and density matrix renormalization group calculations unambiguously demonstrate FQAH phase that is virtually independent of the interaction strength, persisting from the weak-interaction to the strong-interaction limit. We find the stability of the FQAH states does not uniquely correlate with the singularity strength or the BZ-averaged quantum geometric fluctuations. Instead, the many-body topological order can adapt to the singular and fluctuating quantum geometric landscape by spontaneously developing an inhomogeneous carrier distribution, while its quenching accompanies the drop in the occupation-weighted Berry flux. Our work reveals a profound interplay between local quantum geometry and many-body correlation, and significantly expands the exploration space for FQAH effect and correlated phenomena in general.
Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3 × 10 6 . We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles.