Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

TL;DR

This paper derives explicit solutions for Bethe ansatz equations in Bloch electrons under magnetic fields, revealing unique root distributions and wavefunction behaviors, including self-similarity and criticality, at specific flux values.

Contribution

It provides explicit solutions at the spectrum center for Bethe ansatz equations in magnetic fields, highlighting root distributions and wavefunction structures for special flux cases.

Findings

Root distribution is uniform on the unit circle for odd Q.

Wavefunction obeys power law and is unnormalizable at large Q.

Self-similar root and wavefunction structures for golden mean flux.

Abstract

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations recently proposed by Wiegmann and Zabrodin. When the magnetic flux per plaquette is $1/Q$ where $Q$ is an odd integer, distribution of the roots is uniform on the unit circle in the complex plane. For the semi-classical limit, $ Q\rightarrow\infty$, the wavefunction obeys the power low and is given by $|\psi(x)|^2=(2/ \sin \pi x)$ which is critical and unnormalizable. For the golden mean flux, the distribution of roots has the exact self-similarity and the distribution function is nowhere differentiable. The corresponding wavefunction also shows a clear self-similar structure.