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

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

2000-11-22

arxiv.org/abs/cond-mat/0011396v2

Physics

Condensed Matter

Disordered Systems and Neural Networks

21 pages, 19 figures

Scientific paper

10.1103/PhysRevE.64.045204

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 $\epsilon$ versus Energy $E$ with irrational magnetic flux $\omega$. 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 $U_q(sl_2)$ with $q^2=e^{i2\pi\omega}$. The magnetic flux is taken to be the golden mean $\omega^*=(\sqrt{5}-1)/2$ and is obtained by successive rational approximants $\omega_m=F_{m-1}/F_m$ with $F_m$ given by the Fibonacci sequence $F_m$.[OUTP-00-08S, \texttt{cond-mat/0011396}]

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