Fractal Structure of the Harper Map Phase Diagram from Topological Hierarchical Classification

It is suggested a topological hierarchical classification of the infinite many Localized phases figuring in the phase diagram of the Harper equation for anisotropy parameter ǫ versus Energy E with irrational magnetic flux ω. It is also proposed a rule that explain the fractal structure of the phase diagram. Among many other applications, this system is equivalent to the Semi-classical problem of Bloch electrons in a uniform magnetic field, the Azbel-Hofstadter model, where the discrete magnetic translations operators constitute the quantum algebra Uq(sl2) with q 2 = e. The magnetic flux is taken to be the golden mean ω = ( √ 5− 1)/2 and is obtained by successive rational approximants ωm = Fm−1/Fm with Fm given by the Fibonacci sequence Fm.[OUTP-00-08S, cond-mat/0011396]