Extended and localized states of generalized kicked Harper models

Basic properties of quantum states for generalized kicked Harper models are studied using the phase-space translational symmetry of the problem. Explicit expressions of the quasienergy (QE) states are derived for general rational values q/p of a dimensionless Latin small letter h with stroke. The quasienergies form p bands, and the QE states are q-fold degenerate. With each band one can associate a pair of integers, σ and μ, determined from the periodicity conditions of the Qe states in the band. For q=1, σ is exactly the Chern index introduced by Leboeuf et al. [Phys. Rev. Lett. 65, 3076 (1990)] for a characterization of the classical-quantum correspondence. It is shown, however, that σ is always different from zero for q>1. The Chern-index characterization is then generalized by introducing localized quantum states associated in a natural way with σ=0. These states are formed from q QE bands with a total σ=0, and define q equivalent new bands, each with σ=0. While these states are nonstationary, they become stationary in the semiclassical limit p→.