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What is Anderson localization, for someone with no previous knowledge on the subject?

I tried to read Anderson's original paper, but it was too terse for me. I have seen a couple of intuitive explanations, e.g. 50 years of Anderson localization on Physics Today. I also read "Localization of waves" by van Tiggelen, but it is more of a review with a lot of formulas and no deductions.

What I need is an introduction to the subject through one example of Anderson localization, worked out in detail.

You can skim over the tedious math. If you point out what needs to be done, I'll work it out (and if I get stuck I'll ask!).

(Related Phys.SE question: Introduction to Anderson localization)

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The toy example is localization on the Bethe lattice (AKA the regular Cayley tree). There is a paper by Abou-Chacra, Thouless and Anderson that discusses this. Or you can just google around.

R. Abou-Chacra et al 1973 J. Phys. C: Solid State Phys. 6 1734

or

R. Abou-Chacra and D.J. Thouless, J. Phys. C 7 (1974), 65.

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A nice example due to Michael Berry is to look at a stack of transparency sheets. Collectively the stack makes an excellent reflector precisely because of the randomness of the gaps between the individual sheets. If the gaps were constant you would just have a photonic band gap. Finite 1D systems require finite randomness for all the modes to be localized. But in infinite random 1D systems all modes are localized no matter how small the randomness. Then localization length is the inverse of the lyapunov exponent which you can show is bounded away from zero in this case. Purely 1D systems are somewhat pathological, but can be realized in single-mode waveguides. The key point is that you can have localized modes (e.g., defect states in condensed matter) without Anderson localization. The latter is a coherent multiple scattering effect. So, go find a box of transparency film :-)

Sorry I missed that you wanted the gory details. Here's an experimental/computational/theoretical example for millimeter waves: http://mesoscopic.mines.edu/preprints/PhysRevB_76_085118.pdf

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  • $\begingroup$ could you elaborate on the inverse relation between the localization length and the Lyapunov exponent? $\endgroup$
    – M. Zeng
    Commented Apr 27, 2015 at 3:11

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