Thread by @j_bertolotti

#PhysicsFactlet (331)
"Anderson localization" is a weird phenomenon that is not well known even among Physicists, but has the habit of popping up essentially everywhere.
An introductory thread 🧵
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The idea of "localization" originally came about as an explanation (by P.W. Anderson, hence the name) of why the spins in certain materials did not relax as fast as expected.
nobelprize.org/prizes/physics…
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What Anderson realized was that when you have a wave (in this case a quantum mechanical wavefunction) that propagates in a random system, interference can play a major role, and potentially impede propagation completely.
journals.aps.org/pr/abstract/10…
(Paywalled)
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The original paper (and, frankly, most of the literature on the subject) is pretty impenetrable, but thankfully Anderson localization can happen any time we have a wave and a random medium, doesn't matter what kind of wave, so we can try to look at a simple system.
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Let's start VERY simple with the simple pendulum, and let's make it even simpler by assuming the oscillations are small, and thus we only have to deal with an harmonic oscillator.
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The pendulum has a natural frequency (i.e. the frequency at which it will naturally oscillate if you just let it go), which will depend on its length and the gravity acceleration pulling it down: ω₀= √(g/L).
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If you take a bunch of such pendula, they will all oscillate with the same natural frequency.
Let's complicate the problem a bit and add a (elastic) coupling between the pendula.
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https://twitter.com/j_bertolotti/status/1518882862329651200

The system in its entirety now will have a number of natural frequencies equal to the number of pendula, resulting in a complex motion that is the superposition of the oscillations at all those different frequencies.
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Let's complicate the system again and make the mass of one of the pendula different. At first sight not much has changed: we still have the same number of natural frequencies (they just moved a bit), and the overall motion is the superposition of all those oscillations
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To get a better feeling at what the different mass did, we can look at the eigenmodes of the system, i.e. at those collective oscillations that don't change over time.
The different mass did change a bit the eigenmodes (shown, the lowest energy one)
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With just one mass changed just a little bit the difference is not enormous, but the heavier pendulum tends to oscillate a bit less, and the ones on the sides tend to oscillate a bit more.
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Ok, so what it is going to happen if we change all the masses at random?
At first sight the answer might look to be again "not much": we still have the same number of natural frequencies, and the overall motion is the superposition of all those oscillations.
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It is looking at the eigenmodes that the true difference becomes apparent (again, showing the lowest energy eigenmode here).
Most of the masses are essentially not moving, and all the oscillation is concentrated in a small area.
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This is the meaning of the term "localization" in "Anderson localization". The presence of randomness is not just scrambling the natural frequencies and distorting the eigenmodes a bit, it is changing their nature dramatically!
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What happens if the "disorder" is strong enough and your system is big enough, is that the eigenmodes of the system start shrinking, becoming more and more peaked around a particular position (a different one for each mode).
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If the disorder is not very strong, or the system is not very big, most of the modes will be quite wide, and they will overlap with each other. In this case any excitation will couple to a large number of modes, and you will see the excitation travel around as you expect.
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But if you have a lot of disorder and the system is big enough, the modes will be very narrow and won't overlap very much. As a result any excitation will couple to only a few, and the excitation won't be able to propagate very far.
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(Remember: how far you can propagate depends on how far the modes you excited extend.)
Therefore energy in a "Anderson localized" system can't easily propagate very far, which explains why the spins Anderson was studying had troubles relaxing.

https://twitter.com/j_bertolotti/status/1126488445101199360

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Anderson localization is still an active field of research, with still both theoretical and experimental open questions (and people arguing at conferences). And it is a LOT more complicated than this super-simplified description might make you think 😉
19/19 THE END