Duality in the Azbel-Hofstadter problem and two-dimensional d-wave superconductivity with a magnetic field

A single-parameter family of the lattice-fermion model is constructed. It is a deformation of the Azbel-Hofstadter problem by a parameter h = Delta /t (quantum parameter). A topological number is attached to each energy band. A duality between the classical limit (h = +0) and the quantum limit (h = 1) is revealed in the energy spectrum and the topological number. The model has a close relation to two-dimensional d-wave superconductivity with a magnetic field. Making use of the duality and a topological argument, we shed light on how quasiparticles with a magnetic field behave, especially in the quantum limit.