Quantum group, Bethe ansatz equations, and Bloch wave functions in magnetic fields

The wave functions for a two-dimensional Bloch electron in uniform magnetic fields at the mid-band points are studied by exploiting a connection to the quantum group ${\mathit{U}}_{\mathit{q}}$(${\mathit{sl}}_2$): A linear combination of its generators gives the Hamiltonian. We apply both analytic and numerical methods to obtain and analyze the wave functions, by solving the functional Bethe ansatz equations first proposed by Wiegmann and Zabrodin on the basis of the above observation. The semiclassical case with the flux per plaquette φ=1/Q is analyzed in detail, by exploring a structure of the Bethe ansatz equations. We also reveal the multifractal structure of the solutions to Bethe ansatz equations and corresponding wave functions when φ is irrational, such as the golden or silver mean. © 1996 The American Physical Society.