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arXiv:2210.13477 · Physical Review Research

Many-body ground states from decomposition of ideal higher Chern bands: Applications to chirally twisted graphene multilayers

Junkai Dong, Patrick J. Ledwith, Eslam Khalaf, Jong Yeon Lee, Ashvin Vishwanath

Published Jun 16, 2023• 55 citations
arXivHTMLPDF

Abstract

Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with arbitrary Chern number C with ideal quantum geometry. While C > 1 bands differ from Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states for short-range interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit an SU ( C ) action that combines real space and momentum space translations. Remarkably, they also allow for analytic construction of exact many-body ground states, such as generalized quantum Hall ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU ( C ) action is promoted to a symmetry on the ground-state subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to translation broken states and admit charged skyrmion excitations corresponding to a spatially varying density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems such as monolayer graphene twisted relative to a Bernal bilayer. Published by the American Physical Society 2023

DOI
10.1103/physrevresearch.5.023166

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Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with arbitrary Chern number C with ideal quantum geometry. While C > 1 bands differ from Landau levels, we show that these bands host exact fractional Chern insulator (FCI) ground states for short-range interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit an SU ( C ) action that combines real space and momentum space translations. Remarkably, they also allow for analytic construction of exact many-body ground states, such as generalized quantum Hall ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit of short-range interactions. In this limit, the SU ( C ) action is promoted to a symmetry on the ground-state subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to translation broken states and admit charged skyrmion excitations corresponding to a spatially varying density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems such as monolayer graphene twisted relative to a Bernal bilayer. Published by the American Physical Society 2023
Motivated by recent experiments on correlated van der Waals materials, including twisted and rhombohedral graphene and twisted WSe 2 , we perform an analytical and numerical study of the effects of strong on-site and short-range interactions in fractionally filled ideal Chern bands. We uncover an extensive nontrivial ground state manifold within the band filling range 0 < ν < 1 and introduce a general principle, the “three-rule,” for combining flat band wave functions, which governs their zero-energy property on the torus geometry. Based on the structure of these wave functions, we develop a variational approach that reveals distinct phases under different perturbations: metallic behavior emerges from a finite dispersion, and superconductivity is induced by attractive Cooper channel interactions. Our approach, not reliant on the commonly applied mean-field approximations, provides an analytical expression for the macroscopic wave function of the off-diagonal long-range order correlator, attributing pairing susceptibility to the set of nontrivial zero-energy ground state wave functions. Extending to finite screening lengths and beyond the ideal limit using exact diagonalization simulations, we demonstrate the peculiar structure in the many-body wave function's coefficients to be imprinted in the low-energy spectrum of the topologically ordered Halperin spin-singlet state. Our findings also make connections to frustration-free models of noncommuting projector Hamiltonians, potentially aiding the future construction of exact ground states for various fractional fillings. Published by the American Physical Society 2024
Experimental realizations of Abelian fractional Chern insulators (FCIs) have demonstrated the potentials of moiré systems in synthesizing exotic quantum phases. Remarkably, twisted multilayer graphene system may also host non-Abelian states competing with charge density wave under Coulomb interaction. Here, through larger scale exact diagonalization simulations, we map out the quantum phase diagram for ν=1/2ν=1/2ν=1/2 system with electrons occupying the lowest moirè band of the double twisted bilayer graphene. By increasing the system size, we find the ground state has six-fold near degeneracy and with a finite spectral gap separating the ground states from excited states across a broad range of parameters. Further computation of many-body Chern number establish the topological order of the state, and we rule out possibility of charge density wave orders based on featureless density structure factor. Furthermore, we inspect the particle-cut entanglement spectrum to identify the topological state as a non-Abelian Moore-Read state. Combining all the above evidences we conclude that Moore-Read ground state dominates the quantum phase diagram for the double twisted bilayer graphene system for a broad range of coupling strength with realistic Coulomb interaction.
Abstract One of the most remarkable theoretical findings in magic angle twisted bilayer graphene (TBG) is the emergence of ferromagnetic Slater determinants as exact ground states for the interacting Hamiltonian at the chiral limit. This discovery provides an explanation for the correlated insulating phase which has been experimentally observed at half filling. This work is the first mathematical study of interacting models in magic angle graphene systems. These include not only TBG but also TBG-like systems featuring four flat bands per valley, and twisted trilayer graphene systems with equal twist angles. We identify symmetries of the chiral limit of the Bistritzer-MacDonald Hamiltonian that are responsible for characterizing the Hartree-Fock ground states as zero energy many-body ground states. Furthermore, for a general class of Hamiltonian, we establish criteria that the ferromagnetic Slater determinants are the unique ground states within the class of uniformly half-filled, translation invariant Slater determinants. We then demonstrate that these criteria can be explicitly verified for TBG and TBG-like systems at the chiral limit, using properties of Jacobi- θ{\theta }θ θ and Weierstrass- ℘{\wp }℘ ℘ functions.
DMRG study of strongly interacting Z2\mathbb{Z}_2Z2​ flatbands: a toy model inspired by twisted bilayer graphene
Strong interactions between electrons occupying bands of opposite (or like) topological quantum numbers (Chern =\pm1 = ± 1 ), and with flat dispersion, are studied by using lowest Landau level (LLL) wavefunctions. More precisely, we determine the ground states for two scenarios at half-filling: (i) LLL’s with opposite sign of magnetic field, and therefore opposite Chern number; and (ii) LLL’s with the same magnetic field. In the first scenario – which we argue to be a toy model inspired by the chirally symmetric continuum model for twisted bilayer graphene – the opposite Chern LLL’s are Kramer pairs, and thus there exists time-reversal symmetry ( \mathbb{Z}_2 ℤ 2 ). Turning on repulsive interactions drives the system to spontaneously break time-reversal symmetry – a quantum anomalous Hall state described by one particle per LLL orbital, either all positive Chern |{++\cdots+}\rangle | + + ⋯ + ⟩ or all negative |{--\cdots-}\rangle | − − ⋯ − ⟩ . If instead, interactions are taken between electrons of like-Chern number, the ground state is an SU(2) S U ( 2 ) ferromagnet, with total spin pointing along an arbitrary direction, as with the \nu=1 ν = 1 spin- \frac{1}{2} 1 2 quantum Hall ferromagnet. The ground states and some of their excitations for both of these scenarios are argued analytically, and further complimented by density matrix renormalization group (DMRG) and exact diagonalization.
Twisted bilayer materials provide a versatile platform for realizing correlated topological states. Motivated by the recent experimental realization of atomic Bose-Einstein condensates in twisted bilayer optical lattices, we investigate the emergence of fractional topological phases in such systems. For experimentally realistic parameters, the single-particle spectrum hosts a pair of quasi-degenerate nearly flat moiré bands. We show using self-consistent Hartree-Fock calculations that atomic interactions spontaneously break time-reversal symmetry, lift the quasi-degeneracy, and generate an isolated nearly flat band with a nonzero Chern number. At fractional filling, exact diagonalization of the Hamiltonian projected to this band reveals a threefold degenerate ground-state manifold and a characteristic particle entanglement spectrum, providing strong evidence for a fractional Chern insulator. The resulting many-body gap is of order 10−3ER10^{-3}E_R10−3ER​, corresponding to a nanokelvin energy scale accessible to state-of-the-art cold-atom experiments. Our scheme requires neither spin-orbit coupling, as in transition metal dichalcogenides, nor a finite magnetic field, as in existing twisted-bilayer-graphene experiments, providing a highly tunable route to strongly correlated topological phases in twisted bilayer optical lattices.
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.