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We study one dimensional quasiperiodic system obtained from the tight-bindingmodel on the square lattice in a uniform magnetic field with the spin orbitinteraction. The phase diagram with respect to the Harper coupling and theRashba coupling are proposed from a number of numerical studies including amultifractal analysis. There are four phases, I, II, III, and IV in this orderfrom weak to strong Harper coupling. In the weak coupling phase I all the wavefunctions are extended, in the intermediate coupling phases II and III mobilityedges exist, and accordingly both localized and extended wave functions exist,and in the strong Harper coupling phase IV all the wave functions arelocalized. Phase I and Phase IV are related by the duality, and phases II andIII are related by the duality, as well. A localized wave function is relatedto an extended wave function by the duality, and vice versa. The boundarybetween phases II and III is the self-dual line on which all the wave functionsare critical. In the present model the duality does not lead to pure spectra incontrast to the case of Harper equation.

Localization problem of the quasiperiodic system with the spin orbit interaction