Bethe ansatz for the Harper equation: Solution for a small commensurability parameter

Abstract

The Harper equation describes an electron on a 2D lattice in magnetic field and a particle on a 1D lattice in a periodic potential, in general, incommensurate with the lattice potential. We find the distribution of the roots of Bethe ansatz equations associated with the Harper equation in the limit as alpha=1/Q tends to 0, where alpha is the commensurability parameter (Q is integer). Using the knowledge of this distribution we calculate the higher and lower boundaries of the spectrum of the Harper equation for small alpha. The result is in agreement with the semiclassical argument, which can be used for small alpha.

17 pages including 5 postscript figures, Latex, minor changes, to appear in Phys.Rev.B

References in corpus (2)

• Hierarchical Structure of Azbel-Hofstader Problem: Strings and loose ends of Bethe Ansatz
• Asymptotically exact solutions of Harper equation

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