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arXiv:2504.20140 · Physical review. B./Physical review. B

Multiband exact diagonalization and an iteration approach to search for fractional Chern insulators in rhombohedral multilayer graphene

Heqiu Li, Bogdan Andrei Bernevig, Nicolas Regnault

Published Aug 1, 2025• 4 citations
arXivHTMLPDF

Abstract

We perform a multi-band exact diagonalization (ED) study of rhombohedral pentalayer graphene twisted on hexagonal boron nitride with a focus on fractional Chern insulators (FCI) in systems with weak moir\'e gaps, complementing the results of [Yu et al., arXiv:2407.13770]. We consider both the charge-neutrality (CN) and average (AVE) interaction schemes. Saliently and surprisingly, we now find using the particle entanglement spectrum that the FCI at filling factor 1/3 in the CN scheme predicted by single-(Hartree-Fock) band ED is unstable towards a transition to charge density wave once a small fraction of electrons is allowed to occupy the higher bands. Meanwhile, the FCI at filling 2/3 in the AVE scheme remains more robust under similar band mixing until being suppressed when increasing band mixing. To tackle truncation errors that arise from including multiple bands in larger system sizes, we propose an ED iteration method that iteratively optimizes the single-particle basis so that the particles in the ground state should reside mainly in the lowest band. Nevertheless, we find that the FCI gap remains absent after convergence when the mixing with higher bands is considered. These findings highlight the delicate sensitivity of FCIs to multi-band effects and the shortcoming of all of the current models to explain the experimental emergence of such phases.

DOI
10.1103/zlqg-sj86

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We perform a multi-band exact diagonalization (ED) study of rhombohedral pentalayer graphene twisted on hexagonal boron nitride with a focus on fractional Chern insulators (FCI) in systems with weak moir'e gaps, complementing the results of [Yu et al., arXiv:2407.13770]. We consider both the charge-neutrality (CN) and average (AVE) interaction schemes. Saliently and surprisingly, we now find using the particle entanglement spectrum that the FCI at filling factor 1/3 in the CN scheme predicted by single-(Hartree-Fock) band ED is unstable towards a transition to charge density wave once a small fraction of electrons is allowed to occupy the higher bands. Meanwhile, the FCI at filling 2/3 in the AVE scheme remains more robust under similar band mixing until being suppressed when increasing band mixing. To tackle truncation errors that arise from including multiple bands in larger system sizes, we propose an ED iteration method that iteratively optimizes the single-particle basis so that the particles in the ground state should reside mainly in the lowest band. Nevertheless, we find that the FCI gap remains absent after convergence when the mixing with higher bands is considered. These findings highlight the delicate sensitivity of FCIs to multi-band effects and the shortcoming of all of the current models to explain the experimental emergence of such phases.
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.
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Chiral graviton modes are hallmark collective excitations of Fractional Quantum Hall (FQH) liquids. However, their existence on the lattice, where continuum symmetries that protect them from decay are lost, is still an open and urgent question, especially considering the recent advances in the realization of Fractional Chern Insulators (FCI) in transition metal dichalcogenides and rhombohedral pentalayer graphene. Here we present a comprehensive theoretical and numerical study of graviton-modes in fermionic FCI, and thoroughly demonstrate their existence. We first derive a lattice stress tensor operator in the context of the fermionic Harper-Hofstadter(HH) model which captures the graviton in the flat band limit. Importantly, we discover that such lattice stress-tensor operators are deeply connected to lattice quadrupolar density correlators, readily generalizable to generic Chern bands. We then explicitly show the adiabatic connection between FQH and FCI chiral graviton modes by interpolating from a low flux HH model to a Checkerboard lattice model that hosts a topological flat band. In particular, using state-of-the-art matrix product state and exact diagonalization simulations, we provide strong evidence that chiral graviton modes are long-lived excitations in FCIs despite the lack of continuous symmetries and the scattering with a two-magnetoroton continuum. By means of a careful finite-size analysis, we show that the lattice generates a finite but small intrinsic decay rate for the graviton mode. We discuss the relevance of our results for the exploration of graviton modes in FCI phases realized in solid state settings, as well as cold atom experiments.
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We investigate the many-body ground states in a family of fractionally filled bands where the Berry curvature fluctuations can be tuned while maintaining ideal quantum geometry. We numerically find that the neutral gap of the fractional Chern insulator (FCI) ground state decreases as the Berry curvature becomes less homogeneous, ultimately driving an instability to a charge density wave. We further extend our analysis to bands perturbed away from the ideal limit and give examples where a less ideal band geometry results in a more stable FCI phase. To explain our findings, we apply the single-mode approximation to the ground-state wave functions of the ideal band, from which we obtain analytic expressions for the FCI many-body gap in the limit of weak Berry curvature variations. Finally, we make a connection between our results and experimentally relevant systems where FCIs have been observed.