Uniqueness of Landau levels and their analogs with higher Chern numbers

TL;DR

This paper proves that Landau levels and their higher Chern number analogs are uniquely characterized by their holomorphic wave functions and flat geometrical properties, which are crucial for stabilizing fractional quantum Hall phases.

Contribution

It establishes the uniqueness of Landau levels and their higher Chern number analogs as the only holomorphic eigenstates with flat geometry, enabling the construction of all such states.

Findings

Landau levels are uniquely characterized by holomorphic wave functions with flat geometry.

Higher Chern number analogs of Landau levels also share these unique properties.

This uniqueness facilitates the construction of holomorphic eigenstates for fractional quantum Hall phases.

Abstract

Landau levels are the eigenstates of a charged particle in two dimensions under a magnetic field, and are at the heart of the integer and fractional quantum Hall effects, which are two prototypical phenomena showing topological features. Following recent discoveries of fractional quantum Hall phases in van der Waals materials, there is a rapid progress in understanding of the precise condition under which the fractional quantum Hall phases can be stabilized. It is now understood that the key to obtaining the fractional quantum Hall phases is the energy band whose eigenstates are holomorphic functions in both real and momentum space coordinates. Landau levels are indeed examples of such energy bands with an additional special property of having flat geometrical features. In this paper, we prove that, in fact, the only energy eigenstates having holomorphic wave functions with a flat geometry are the Landau levels and their higher Chern number analogs. Since it has been known that any holomorphic eigenstates can be constructed from the ones with a flat geometry such as the Landau levels, our uniqueness proof of the Landau levels allows one to construct any possible holomorphic eigenstate with which the fractional quantum Hall phases can be stabilized.