  A single-parameter family of lattice-fermion model is constructed. It is a
deformation of the Azbel-Hofstadter problem by a parameter $h={$B%((BDelta}/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 the two-dimensional d-wave superconductivity with a magnetic
field. Making use of the duality and a topological argument, we shed light on
how the quasiparticles with a magnetic field behave especially in the quantum
limit.
