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Theory of Generalized Landau Levels and Its Implications for Non-Abelian States

Zhao Liu1,2,*, Bruno Mera3,4,*, Manato Fujimoto5,6, Tomoki Ozawa4, and Jie Wang7,†

  • 1Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310058, China
  • 2Zhejiang Key Laboratory of Micro-Nano Quantum Chips and Quantum Control, School of Physics, Zhejiang University, Hangzhou 310027, China
  • 3Instituto de Telecomunicações and Departmento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal
  • 4Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Sendai 980-8577, Japan
  • 5Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
  • 6Department of Applied Physics, The University of Tokyo, Hongo, Tokyo 113-8656, Japan
  • 7Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA

  • *These authors contributed equally to this work.
  • †Contact author: jiewang.phy@gmail.edu
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Phys. Rev. X 15, 031019 – Published 16 July, 2025

DOI: https://doi.org/10.1103/1zg9-qbd6

Abstract

Landau levels (LLs) are cornerstones of the fractional quantum Hall effect, which is a paradigm of topologically ordered phases of matter. Predicting and understanding the emergence of fractional Chern insulators in a Chern band, to a large extent, relies on judging the similarity between the Chern band and LLs. In this work, we generalize the notion of LLs to Chern bands to allow for nonuniform quantum geometries while preserving certain geometric invariants. Mathematically, generalized LLs can be understood in terms of holomorphic curves and associated moving frames in the sense of Cartan. This formalism enables a thorough analytical discussion of their associated quantum geometries. In practical terms, generalized LLs allow for a systematic discussion of the geometric stability of fractional Chern insulators and provide useful guidance in material design. Approximating Chern bands as superpositions of a few generalized LLs and employing exact diagonalization, we numerically identify a universal geometric criterion for the non-Abelian Moore-Read state to be stabilized in Chern bands, providing useful guidance for searching for such phase in moiré materials. Moreover, a double-twisted bilayer graphene model at the second magic angle is used to suggest a possible experimental realization of generalized LLs and zero-field non-Abelian phases. More broadly, our results point to generalized LLs being an interdisciplinary research direction connecting Kähler geometry, topologically ordered phases, and moiré materials, which we expect will benefit the study of the geometric aspect of correlated phases in a more general sense.

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Physics Subject Headings (PhySH)

Popular Summary

In some quantum materials, electrons move in periodic patterns influenced by the material’s crystal structure. These movements can be described using energy bands, some of which possess hidden mathematical properties—such as shape (geometry) and more abstract traits related to symmetry and structure (topology)—that significantly affect how electrons interact. Even when two energy bands appear similar in one aspect, such as having the same overall symmetry, they can behave very differently if their internal geometry differs. A well-known example is the set of Landau levels, which describe electron behavior in strong magnetic fields. Although these levels share the same topological property, they can lead to very different physical effects. In our study, we introduce a new concept that helps explain this behavior: generalized Landau levels, which can describe energy bands with varying internal geometry.

To explore this idea, we develop a mathematical framework that allows the geometry within a band to change gradually while preserving key topological features. We find that even as the internal structure changes, certain quantities remain constant—serving as markers of the band’s overall behavior. Using this approach, we investigate how variations in internal geometry affect the stability of complex quantum states, focusing on a special state called the Moore-Read phase.

By broadening our understanding of these energy bands, we provide a tool for identifying when unusual and potentially useful quantum states can emerge. This work bridges mathematical theory and real materials, potentially guiding future experiments in materials like twisted layers of graphene, where such exotic behaviors may be observed.

Article Text

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Phys. Rev. X 15, 031019– Published 16 July, 2025
  • Received 4 June 2024
  • Accepted 25 June 2025
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