Physical understanding of Anderson (disorder) localisation

My current understanding is that waves in disordered potentials experience localisation due to interference effects. (eg an electron in a disordered medium tries to take different paths of effectively random lengths, and so on average it destructively interferes with itself).

I'm not sure if this 'average' destructive interference idea is correct. I can't really justify why it would happen or why it would lead to localisation. (Destructive inteference in every direction so the wavefunction decays away?)

I have also heard an analogy, sound being produced in a disordered forest and after some localisation length the sound would be inaudible.

Anderson localization (AL) is, indeed, just a wave interference phenomenon, and can exist in completely classical systems. For example, it has been seen in ultrasound waves. The fact that it was first studied in the context of electrons is more of a historical accident than anything else, as far as I can tell. The required coherence and isolation of the system, as well as the strength of scattering normally required, just turn out to make it hard to see classically unless you are really looking for it.

Even though it is classical, the easiest way to understand AL is in a path-integral picture, so I will discuss it in those terms. Consider a particle in a system of fixed scatterers, like so:

Orange is our particle, and blue are the fixed scatterers. The solid arrows denote one path through the system that returns to the original location, which in a path integral formalism should have some amplitude $\phi$. However, for every path that returns to the origin like this there is also a time-reversed path, shown in this case by the dashed arrows. Notice that such time-reversed pairs only exist for paths returning to the original point.

Now, in a path integral you would sum up all the amplitudes of the paths going anywhere, and see where they interfere constructively or destructively, and that determines where the particle is likely to go. But these time-reversed pairs will always have the same amplitude, so they will always add constructively with each other. Crudely, for a round-trip path $i$,

$P(return)_i=|\phi_{i,forward}+\phi_{i,backward}|^2=4|\phi|^2$.

This means the particle will always have a higher probability of staying at its initial location than classically expected. This isn't the same as saying it is fully localized- it could only result in weak localization- but under some circumstances this increased probability to stay near the origin can win out over the probability for it to wander indefinitely away.

This isn't enough for a quantitative analysis of AL, but it will get you a long ways. For example, it is clear from this example that localization will only occur when the two time-reversed paths have the same total phase. This implies that if inelastic collisions, which effectively scramble the particle's phase, are common enough, they will ruin localization. Specifically, inelastic collisions set a phase coherence length, $l_{\phi}$, and if this length is short enough such that a particle's phase is scrambled over a typical return path, localization will not occur. Something that breaks time-reversal invariance like a magnetic impurity will also ruin localization.

Finally, as this picture suggests, it is useful to compare Anderson localization to a (classical) random walk. Anderson localization can be thought of as a quantum random walk, which has the important features of interference between paths but is fundamentally similar. In unbiased classical random walks, it is well-known that the probability of return depends crucially on the spatial dimension of the problem. In one or two dimensions, the walker will return to its origin an infinite number of times as the number of steps goes to infinity, but in three or more dimensions this is no longer true. A similar dependence on dimensionality occurs in Anderson localization: a particle in a disordered one- or two-dimensional system turns out to always be localized (assuming perfect coherence and time-reversal symmetry), but in three or more dimensions only particles below a critical energy are localized. This is known as the mobility edge, and it is an important problem in AL to figure out exactly where it is for a given system.

• What about weak anti-localization? In the above picture, if the electron is in a topological surface state, as it goes around a closed path, its momentum changes along the way. The spin-momentum lock require its spin changes continuesly around the path (solid line, for example) and aquire a $\pi$ phase around this path. Thus the electron will interfere destructively with itself. It seems that there is no need to account for its time-reversal path to explain weak antilocalization, am I right? Jan 9, 2016 at 8:46
• I am not familiar with this effect, so I cannot say for sure how it fits into this picture. But a natural question presents itself from your description: it is the case in these topological systems that time-reversal symmetry is broken by, e.g., a uniform magnetic field, in such a way that the time-reversed paths interfere destructively instead of constructively? If this is the case, then it would be a straightforward extension of the usual weak localization as described above. Jan 9, 2016 at 19:23
• In this accepted answer it is claimed that the above is more or less true- anti-localization is just the opposite effect of conventional localization, due to time-reversal breaking effects: physics.stackexchange.com/questions/38510/… . Jan 9, 2016 at 19:25
• the time reversal symmetry should not break in order to have this effect, according to my understanding. Jan 10, 2016 at 0:47
• If that is the case, I think it would be an interesting subject for its own question. Jan 10, 2016 at 3:24