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

Pedro Castelo Ferreira

doi:10.1103/PhysRevE.64.045204

paper

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

arXiv:cond-mat/0011396 · doi:10.1103/PhysRevE.64.045204

Abstract

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 $ε$ 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 $U_{q} (s l_{2})$ with $q^{2} = e^{i 2 π ω}$ . The magnetic flux is taken to be the golden mean $ω^{*} = (5 - 1) /2$ and is obtained by successive rational approximants $ω_{m} = F_{m - 1} / F_{m}$ with $F_{m}$ given by the Fibonacci sequence $F_{m}$ .[OUTP-00-08S, \texttt{cond-mat/0011396}]

21 pages, 19 figures

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Strange Nonchaotic Attractors

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

Abstract

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