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### The random phase property and the Lyapunov Spectrum for disordered multi-channel systems

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Roemer, Rudolf A. and Schulz-Baldes, H..
(2010)
*The random phase property and the Lyapunov Spectrum for disordered multi-channel systems.*
Journal of Statistical Physics, Vol.140
(No.1).
pp. 122-153.
ISSN 0022-4715

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Official URL: http://dx.doi.org/10.1007/s10955-010-9986-8

## Abstract

A random phase property establishing in the weak coupling limit a link between quasi-one-dimensional random Schrödinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a Markov process with a unique invariant measure which is of geometric nature. On the elliptic part of the transfer matrices, this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study. While the random phase property can up to now only be proved in special models or in a restricted sense, we provide strong numerical evidence that it holds in the Anderson model of localization. A main outcome of the random phase property is a perturbative calculation of the Lyapunov exponents which shows that the Lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary, orthogonal and symplectic ensemble differ by a factor 2 each. In an Anderson-Ando model on a tubular geometry with magnetic field and spin-orbit coupling, the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the Lyapunov spectrum.

Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Centre for Scientific Computing Faculty of Science > Physics |

Library of Congress Subject Headings (LCSH): | Localization theory, Mesoscopic phenomena (Physics), Lyapunov exponents, Schrödinger operator, Markov processes, Stochastic processes |

Journal or Publication Title: | Journal of Statistical Physics |

Publisher: | Springer New York LLC |

ISSN: | 0022-4715 |

Official Date: | 14 May 2010 |

Volume: | Vol.140 |

Number: | No.1 |

Page Range: | pp. 122-153 |

Identification Number: | 10.1007/s10955-010-9986-8 |

Status: | Peer Reviewed |

Access rights to Published version: | Restricted or Subscription Access |

References: | [And] P. W. Anderson, Absence of diffusion in certain random lattices , Phys. Rev. 109, 1492- |

URI: | http://wrap.warwick.ac.uk/id/eprint/3228 |

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