Phase Diagrams of the Harper Map and the Golden Staircase

We present phase diagrams of the Harper map, which is equivalent to the problem of Bloch electrons in a uniform magnetic field (Azbel-Hofstadter model). We consider the cases where the magnetic flux $\omega$ assumes either the continued fraction approximations towards the golden mean or the golden mean itself. The phase diagrams for rational values of $\omega$ show a finite number of Arnol'd tongues of localized electronic states with rational winding numbers and regions of extended phases in between them. For the particular case of $\omega = \frac{\sqrt{5}-1}{2}$, we find an infinite number of Arnol'd tongues of localized phases with extended phases in between. In this case, the study of the winding number gives rise to a Golden Staircase, where the plateaux represent localized phases with winding numbers equal to sums of powers of the golden mean. We also present evidence of the existence of an infinite number of strange nonchaotic attractors for $\epsilon=1$ in points analogous to critical points in the pressure-temperature phase diagram of the water.