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Effects of the Rashba spin–orbit coupling on Hofstadter’s butterfly

We study the effect of Rashba spin–orbit coupling on the Hofstadter spectrum of a two-dimensional tight-binding electron system in a perpendicular magnetic field. We obtain the generalized coupled Harper spin-dependent equations which include the Rashba spin–orbit interaction and solve for the energy spectrum and spin polarization. We investigate the effect of spin–orbit coupling on the fractal energy spectrum and the spin polarization for some characteristic states as a function of the magnetic flux α and the spin–orbit coupling parameter. We characterize the complexity of the fractal geometry of the spin-dependent Hofstadter butterfly with the correlation dimension and show that it grows quadratically with the amplitude of the spin–orbit coupling. We study some ground state properties and the spin polarization shows a fractal-like behavior as a function of α, which is demonstrated with the exponent close to unity of the decaying power spectrum of the spin polarization. Some degree of spin localization or distribution around + 1 or − 1, for small spin–orbit coupling, is found with the determination of the entropy function as a function of the spin–orbit coupling. The excited states show a more extended (uniform) distribution of spin states.