Self-duality of one-dimensional quasicrystals with spin-orbit interaction

Abstract

Noninteracting spinless electrons in one-dimensional quasicrystals, described by the Aubry-Andr\'e-Harper (AAH) Hamiltonian with nearest-neighbor hopping, undergo a metal-to-insulator transition at a critical strength of the quasiperiodic potential. The AAH Hamiltonian is also known to be self-dual. Interestingly, the critical point and the self-dual point are identical in this case. In this work, we have studied the one-dimensional quasiperiodic AAH Hamiltonian in the presence of spin-orbit coupling of Rashba-type, which introduces an additional spin-conserving complex hopping and a spin-flip hopping. We have found that the AAH Hamiltonian remains self-dual in the presence of Rashba spin-orbit coupling, and the self-dual point follows a simple rescaled relationship among the parameters of the Hamiltonian. This system also undergoes a metal-to-insulator transition, and the nature of this transition has been found to be identical with the original AAH Hamiltonian. However, the critical points do not follow the same relationship as the self-dual points in general. In fact, the metal-to-insulator transition happens earlier than the self-dual point, except in some special cases in which they are observed to coincide.