Uniqueness of Landau levels and their analogs with higher Chern numbers

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Bruno Mera, Tomoki Ozawa


Lowest Landau level wavefunctions are eigenstates of the Hamiltonian of a charged particle in two dimensions under a uniform magnetic field. They are known to be holomorphic both in real and momentum spaces, and also exhibit uniform, translationally invariant, geometrical properties in momentum space. In this paper, using the Stone-von Neumann theorem, we show that lowest Landau level wavefunctions are indeed the only possible states with unit Chern number satisfying these conditions. We also prove the uniqueness of their direct analogs with higher Chern numbers and provide their expressions.

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Dear Dr Mera -- Thank you for posting an interesting summary. Question: In the presence of the gap, the Chern number is a topological invariant, which means that a small perturbation  can not change its quantized value. Suppose we start with a perfect lowest Landau level in a homogenous field and introduce ANY small perturbation. The Chern number must remain the same, but the functions will change. Is there a statement that they become non-analytic?
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Bruno Mera

Dear colleague -- Thank you for your interest and question. In general, small perturbations will change the wavefunction and break the holomorphicity constraint. However, there are perturbations that change the energy dispersion but not the lowest Landau level wavefunction, in which case the holomorphicity of the lowest Landau level is intact. 

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