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

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is $\frac{1}{Q}$ with $Q$ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\to \infty$, the wave function is ${\left|\psi \mathrm{}\right(x\left)\right|}^2=\left(\frac{2}{sin \pi x}\right)$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.