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 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 →x, the wave function is | ψ (x) l2 = (2/sin TTX), 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.