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

TL;DR

This paper introduces generalized Landau levels that extend the concept of Landau levels to arbitrary quantum states, providing a universal geometric framework to analyze topological bands and fractionalized phases, with implications for moiré materials.

Contribution

It systematically extends the concept of Landau levels to generalized Landau levels with quantized quantum metric properties, aiding the study of topological Chern bands and non-Abelian states.

Findings

Derived geometric quantities for generalized Landau levels.

Proposed a superposition model capturing Chern band properties.

Identified geometric criteria for non-Abelian Moore-Read phase.

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 moir\'e 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 moir\'e 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.