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Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem

P. B Wiegmann·A. V. Zabrodin
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Abstract

We present a new approach to the problem of Bloch electrons in magnetic ( sometimes called Azbel-Hofstadter problem) field, by making explicit a natural relation between the group of magnetic translations and the quantum group $U_{q}(sl_2)$. The approach allows us to express the "mid" band spectrum of the model and the Bloch wave function as solutions of the Bethe-Ansatz equations typical for completely integrable quantum systems. The zero mode wave functions are found explicitly in terms of $q$-deformed classical orthogonal polynomials.In this paper we present solution for the isotropic problem. We also present a class of solvable quasiperiodic equations related to $U_{q}(sl_2)$.

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P. B Wiegmann, A. V. Zabrodin. 1993-12-19. Quantum Group and Magnetic Translations. Bethe-Ansatz Solution for Azbel-Hofstadter Problem. https://doi.org/10.1016/0550-3213(94)90443-x

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