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arXiv:2406.09505 · Physical Review Research

Ideal Chern bands with strong short-range repulsion: Applications to correlated metals, superconductivity, and topological order

Patrick H. Wilhelm, Andreas Martin Läuchli, Mathias S. Scheurer

Published Dec 5, 2024• 6 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevresearch.6.043240

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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
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
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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.
Fractionalization in ideal Chern bands and non-Hermitian topological physics are two active but so far separate research directions. Merging these, we generalize the notion of ideal Chern bands to the non-Hermitian realm and uncover several striking consequences both on the level of band theory and in the strongly interacting regime. Specifically, we show that the lowest band of a Kapit--Mueller lattice model with an imaginary gauge potential satisfies a generalized ideal condition with complex Berry curvature in sync with a complex quantum metric. The ideal band remains purely real and exactly flat on both the torus and cylinder: eigenstates are extended on the torus, while on the cylinder all right and left eigenstates localize at the boundaries, yielding a non-Hermitian skin effect without spectral winding. In the interacting regime, we find that the generalized ideal condition stabilizes an incompressible liquid at fractional fillings, retaining intrinsic non-Hermitian features on both cylinder and torus, while strikingly distinct on different manifolds. On the cylinder, the ground states are always skin-Laughlin states. In contrast, on the torus, we instead observe an unconventional competition between topologically ordered Laughlin-like states and negative collective modes, arising purely from non-Hermiticity.
We devise local lattice models whose ground states are model fractional Chern insulators-Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Z_{k} parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1) and (ii) nematic states at ν=k/2, where C is the Chern number of the lowest band. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1) body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well suited for the study of novel phenomena involving, e.g., twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
In a flat band superconductor, bosonic excitations disperse while unpaired electrons are immobile. To study this strongly interacting system, we construct a family of multi-band Hubbard models with exact eta-pairing ground states in all space groups. We analytically compute their many-body excitations and find that the Cooper pair bound states obey an effective single-boson hopping Hamiltonian written in terms of the non-interacting wavefunctions. These bound states possess a unique zero-energy excitation whose quadratic dispersion is determined by the minimal quantum metric. The rest of the bound state spectrum is classified by topological quantum chemistry, which we use to identify Cooper pairs with Weyl nodes, higher angular momentum pairing, and fragile topology. Broadening the band dispersion, we find the strongest pairing occurs at half filling and not at the highest density of states. Our analytical results prove that the elementary fermionic and bosonic excitations, corresponding respectively to pairing and coherence, are governed by distinct properties of the flat band, rigorously establishing its unique phenomenology. Superconductivity in flat bands is a central feature of quantum geometric effects, supported by mean-field theory and numerics. Here, the authors exhibit a large family of flat band Hubbard models whose ground states, Bogoliubov-de-Gennes particles, and Cooper pairs are exactly solvable with analytically calculable dispersions.