The Library

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

Tools

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

PDF
WRAP_Roemer_random_phase.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (490Kb)

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

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

Dates:

Date	Event
14 May 2010	Published

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-
1505 (1958).
[ATAF] P. W. Anderson, D. J. Thouless, E. Abrahams, D. S. Fisher, New method for a scaling
theory of localization, Phys. Rev. B 22, 3519-3526 (1980).
[Ando] T. Ando, Numerical study of symmetry effects on localization in two dimensions, Phys.
Rev. B 40, 5325-5339 (1989).
[Ben] C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69,
731-808 (1997).
[BB] C. W. J. Beenakker, M. B¨uttiker, Suppression of shot noise in metallic diffusive conductors,
Phys. Rev. Lett. 46, 1889-1892 (1992).
[BL] P. Bougerol, J. Lacroix, Products of Random Matrices with Applications to Schr¨odinger
Operators, (Birkh¨auser, Boston, 1985).
[CB] J. T. Chalker, M. Bernhardt, Scattering theory, transfer matrices, and Anderson localization,
Phys. Rev. Lett. 70, 982-985 (1993)
[Do1] O. N. Dorokhov, Electron localization in a multichannel conductor, Sov. Phys. JETP
58, 606-615 (1983).
[Do2] O. N. Dorokhov, On the coexistence of localized and extended electronic states in the
metalic phase, Solid State Commun. 51, 381-384 (1984).
[Do3] O. N. Dorokhov, Solvable model of multichannel localization, Phys. Rev. B 37, 10526-
10541 (1988).
[Dys] F. Dyson, The dynamics of a disordered linear chain, Phys. Rev. 92, 1331-1338 (1953).
[EF] K. B. Efetov, A. I. Larkin, Kinetics of a quantum particle in long metallic wires, Sov.
Phys. JETP 58, 444-451 (1983).
[FYMS] L. S. Froufe-P´erez, M. Y´epez, P. A. Mello, J. . S´aenz, Statistical scattering of waves
in disordered waveguides: From microscopic potentials to limiting macroscopic statistics,
Phys. Rev. E 75, 031113-031141 (2007).
[HP] F. Hiai, D. Petz, The Semicircle Law, Free Random Variables and Entropy, (AMS, Providence,
2000).
[HM] J. E. Howard, R. S. MacKay, Linear stability of symplectic maps, J. Math. Phys. 28,
1036-1051 (1987).
[MK] A. MacKinnon, B. Kramer, One-parameter scaling of localization length and conductance
in disordered systems, Phys. Rev. Lett. 47, 1546-1549 (1981).
[MC] A. M. S. Macˆedo, J. T. Chalker, Effects of spin-orbit interactions in disordered conductors:
A random-matrix approach, Phys. Rev. B 46, 14985-14994 (1992).
[Meh] M. L. Mehta, Random Matrices, Second Edition, (Academic Press, San Diego, 1991).
[MPK] P. A. Mello, P. Pereyra, N. Kumar, Macroscopic approach to multichannel disordered
conductors, Ann. Phys. 181, 290-317 (1988).
[MS] P. A. Mello, B. Shapiro, Existence of a limiting distribution for disordered electronic
conductors, Phys. Rev. B 37, 5860-5863 (1988).
[MSt] P. A. Mello, A. D. Stone, Maximum-entropy model for quantum-mechanical interference
effects in metallic conductors, Phys. Rev. B 44, 3559-3576 (1991).
[MT] P. A. Mello, S. Tomsovic, Scattering approach to quantum electronic transport, Phys.
Rev. B 46, 15963-15981 (1992).
[PF] L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, (Springer,
Berlin, 1992).
[PS] J.-L. Pichard, G. Sarma, Finite-size scaling approach to Anderson localisation I and II, J.
Phys. C 14, L127-132 and L617-625 (1981).
[RS] R. R¨omer, H. Schulz-Baldes: Weak disorder expansion for localization lengths of quasi-1D
systems, Euro. Phys. Lett. 68, 247-250 (2004).
[SS1] C. Sadel, H. Schulz-Baldes, Scaling diagram for the localization length at a band edge,
Annales Henri Poincare 8, 1595-1621 (2007).
[SS2] Ch. Sadel, H. Schulz-Baldes, Random Lie group actions on compact manifolds: a perturbative
analysis, preprint 2008.
[SS3] C. Sadel, H. Schulz-Baldes, Random Dirac operators with time reversal symmetry,
preprint 2009, to appear in Commun. Math. Phys..
[Sch] H. Schmidt, Disordered one-dimensional crystals, Phys. Rev. 105, 425-441 (1957).
[SB1] H. Schulz-Baldes, Perturbation theory for an Anderson model on a strip, GAFA 14,
1089-1117 (2004).
[SB2] H. Schulz-Baldes, Rotation numbers for Jacobi matrices with matrix entries, Math. Phys.
Elect. J. 13, 40 pages (2007).
[Tho] D. J. Thouless, Maximum metallic resistance in thin wires, Phys. Rev. Lett. 39, 1167-
1170 (1977).