Theory of Generalized Landau Levels and Its Implications for Non-Abelian States
Zhao Liu, Bruno Mera, Manato Fujimoto, Tomoki Ozawa, Jie Wang
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.