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Weak disorder expansion for localization lengths of quasi-1D systems
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Roemer, Rudolf A. and Schulz-Baldes, H.. (2004) Weak disorder expansion for localization lengths of quasi-1D systems. Europhysics Letters, Vol.68 . pp. 247-253. ISSN 0295-5075
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Official URL: http://dx.doi.org/10.1209/epl/i2004-10190-9
Abstract
A perturbative formula for the lowest Lyapunov exponent of an Anderson model on a strip is presented. It is expressed in terms of an energy-dependent doubly stochastic matrix, the size of which is proportional to the strip width. This matrix and the resulting perturbative expression for the Lyapunov exponent are evaluated numerically. Dependence on energy, strip width and disorder strength are thoroughly compared with the results obtained by the standard transfer matrix method. Good agreement is found for all energies in the band of the free operator and this even for quite large values of the disorder strength.
| Item Type: | Journal Article |
|---|---|
| Subjects: | Q Science > QC Physics |
| Divisions: | Faculty of Science > Centre for Scientific Computing Faculty of Science > Physics |
| Library of Congress Subject Headings (LCSH): | Lyapunov exponents, Anderson model |
| Journal or Publication Title: | Europhysics Letters |
| Publisher: | EDP Sciences |
| ISSN: | 0295-5075 |
| Date: | October 2004 |
| Volume: | Vol.68 |
| Page Range: | pp. 247-253 |
| Identification Number: | 10.1209/epl/i2004-10190-9 |
| Status: | Peer Reviewed |
| Access rights to Published version: | Open Access |
| Funder: | Deutsche Forschungsgemeinschaft (DFG), Sonderforschungsbereich 288 -- Differentialgeometrie und Quantenphysik |
| References: | [1] I. Goldsheid, S. Molcanov, L. Pastur, Funct. Anal. Appl. 11, 1 (1977). [2] D. J. Thouless, in Ill-condensed Matter, edited by G. Toulouse and R. Balian (North-Holland, Amsterdam, 1979), p. 1. [3] L. Pastur, A. Figotin, Spectra of Random and Almost- Periodic Operators, (Springer, Berlin, 1992). [4] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979). [5] A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546 (1981); |, Z. Phys. B 53, 1 (1983). [6] J.-L. Pichard and G. Sarma, J. Phys. C 14, L127 and L617 (1981). [7] J. FrÄohlich, T. Spencer, Commun. Math. Phys. 88, 151 (1983). [8] M. Aizenman, S. Molchanov, Commun. Math. Phys. 157, 245-278 (1993). [9] D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977). [10] C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). [11] H. Schulz-Baldes, Perturbation theory for Lyapunov ex- ponents of an Anderson model on a strip, mp arc/03-369, submitted to GAFA, May 2003. [12] In [11], the distribution of the v(x)'s is merely supposed to be centered and of unit variance. The present normal- ization of the disorder strength is the standard choice in the physics literature. [13] P. Bougerol, J. Lacroix, Products of Random Matri- ces with Applications to SchrÄodinger Operators, (Birk- hÄauser, Boston, 1985). [14] M. Kappus and F. Wegner, Z. Phys. B 45, 15 (1981). [15] F. Milde, R. A. RÄomer, M. Schreiber, and V. Uski, Eur. Phys. J. B 15, 685 (2000), ArXiv: cond-mat/9911029. [16] B. Derrida, E. J. Gardner, J. Physique 45, 1283 (1984). [17] H. Schulz-Baldes, Lifshitz tails for the 1D Bernoulli- Anderson model, mp arc 03-368, to appear in Markow Processes and Related Fields, (2004). [18] B. Kramer and A. MacKinnon, Rep. Prog. Phys. 56, 1469 (1993). |
| URI: | http://wrap.warwick.ac.uk/id/eprint/347 |
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