Theory of Generalized Landau Levels and Implication for non-Abelian States

Abstract

Quantum geometry is a fundamental concept to characterize the local properties of quantum states. It is recently demonstrated that saturating certain quantum geometric bounds allows a topological Chern band to share many essential features with the lowest Landau level, facilitating fractionalized phases in moiré flat bands. In this work, we systematically extend the consequence and universality of saturated geometric bounds to arbitrary Landau levels by introducing a set of single-particle states, which we term as ``generalized Landau levels''. These generalized Landau levels exhibit exactly quantized values of integrated trace of quantum metric determined by their corresponding Landau level indices, regardless of the nonuniformity of their quantum geometric quantities. We derive all geometric quantities for individual and multiple generalized Landau levels, discuss their relations, and understand them in light of the theory of holomorphic curves and moving frames. We further propose a model by superposing few generalized Landau levels which is supposed to capture a large portion of the single-particle Hilbert space of a generic Chern band analogous to the first Landau level. Using this model, we employ exact diagonalization to identify a single-particle geometric criterion for permitting the non-Abelian Moore-Read phase, which is potentially useful for future engineering of moiré materials and beyond. We use a double twisted bilayer graphene model with only adjacent layer hopping term to show the existence of first generalized Landau level type narrow band and zero-field Moore-Read state at the second magic angle which serves as a promising starting point for more detailed future studies. We expect that generalized Landau levels will serve as a systematic tool for analyzing topological Chern bands and fractionalized phases therein.

arXiv comment40 pages, 11 figures

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• Quantum anomalous Hall crystals in moiré bands with higher Chern number
• Universal Wilson Loop Bound of Quantum Geometry
• Characterization of fractional Chern insulator quasiparticles in twisted homobilayer MoTe${}_2$
• Geometric bound on structure factor
• Twistronics and moiré superlattice physics in 2D transition metal dichalcogenides
• Electronic Crystal Phases in the Presence of Non-Uniform Berry Curvature and Tunable Berry Flux: The ${\lambda }_N$-Jellium model
• Ideal topological flat bands in chiral symmetric moiré systems from non-holomorphic functions
• Quantum geometric tensors from sub-bundle geometry
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• Gradient flow and Bogomolny bounds for quantum metric actions
• Electronic bounds in magnetic crystals
• Orbital Description of Landau Levels
• Abelian and non-Abelian fractionalized states in twisted MoTe${}_2$: A generalized Landau-level theory
• Robustness of real-space topology in moiré systems
• Probing the Quantum Geometry of Correlated Metals using Optical Conductivity
• Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice
• Topological Order Without Band Topology in Moiré Graphene
• Relaxation toward an Ideal Chern Band through Coupling to a Markovian Bath