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Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions
Open accessAbstract
We describe a method which gives the weak disorder expansion (λ → 0) of the Lyapounov exponent γ(E) of a discretized one-dimensional Schrödinger equation ψn+1 + ψ n-1 + λVnψn = Eψn with a random potential Vn. Near the band edge of the pure system (E → 2), the weak disorder expansion of y(E) is non analytic and we show that γ(E) ∼ λ2/3 when λ → 0. At the band centre (E → 0), we recover the anomaly which has already been explained by Kappus and Wegner. We find another anomaly at the energy E = 2 cos (π/3) and we believe that similar anomalies should occur at all energies E = 2 cos (απ) with α rational.
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Cited by
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications
Bouchaud, J.-P., & Georges, A. (1990) · Physics Reports · doi:10.1016/0370-1573(90)90099-n · Cited by 4,162
Localization: theory and experiment
Kramer, B., & MacKinnon, A. (1993) · Reports on Progress in Physics · doi:10.1088/0034-4885/56/12/001 · Cited by 1,926
Anomalous localization in low-dimensional systems with correlated disorder
Izrailev, F. M., Krokhin, A. A., & Makarov, N. M. (2012) · Physics Reports · doi:10.1016/j.physrep.2011.11.002 · Cited by 274
Localization and its consequences for quantum walk algorithms and quantum communication
Keating, J. P., Linden, N., Matthews, J. C. F., & Winter, A. (2007) · Physical Review A · doi:10.1103/physreva.76.012315 · Cited by 123
Liu (2016) · doi:10.1088/1751-8113/49/11/115303 · Cited by 11
Metadata
- Authors
- Derrida, B., & Gardner, E.
- Venue
- Journal de Physique
- Publisher
- EDP Sciences
- Published
- 1984
- Volume
- 45
- Issue
- 8
- Pages
- 1283-1295
- ISSN
- 0302-0738 (ISSN-L)
- OpenAlex
- W2108156913
- Cited by
- 289
- References
- 0
- FWCI
- 10.08
- Type
- journal article
- Language
- en
- Open access
- green
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- Derrida, B., & Gardner, E. (1984). Lyapounov exponent of the one dimensional Anderson model : weak disorder expansions. Journal de Physique. https://doi.org/10.1051/jphys%3A019840045080128300
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