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Analyticity breaking of wave functions and fractal phase diagram for simple incommensurate systems
Author(s) -
Schellnhuber H. J.,
Urbschat H.
Publication year - 1987
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2221400221
Subject(s) - quasiperiodic function , integrable system , delocalized electron , quasiperiodicity , phase diagram , physics , fractal , renormalization group , hamiltonian (control theory) , diamagnetism , mathematical physics , universality (dynamical systems) , critical exponent , quantum mechanics , phase transition , condensed matter physics , mathematics , phase (matter) , magnetic field , mathematical analysis , mathematical optimization
The Aubry‐André transition for certain one‐dimensional quasiperiodic Schrödinger operators is discussed in terms of analyticity breaking for the states of the underlying periodic diamagnetic Ersatz‐Hamiltonian. The general concept is compared with recent developments in the theory cf non‐integrable Hamiltonian dynamics and illustrated by explicit calculations for the Harper model. It is shown that the latter is not structurally stable within the family of incommensurate systems derived from Bloch dynamics in irrational magnetic fields: A very simple non‐self‐dual quasiperiodic model is presented, which lies in the Harper universality class regarding critical behaviour but exhibits cascades of localization–delocalization transitions. The system has a “devil's fork” phase diagram, where localized and extended regions are separated by a fractal curve.