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Why are so many condensed matter phenomena so sensitive to impurities? In fact, quite a number of them depend upon impurities for their very existence!

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A. You cannot find a condensed matter system without impurities. Even with the most stringent manufacturing processes you are always left with some (however small) fraction of defects and impurities in any material.

B. The most interesting physics (Kondo Effect, Anderson Localization, Quantum Hall Effect) occurs only in the presence of defects. If you read Jain's excellent book "Composite Fermions" (or any other review on the QHE) you will discover that in a perfect sample one cannot have the QHE because of Lorentz invariance of the sample. Defects break this symmetry and one can no longer boost to a frame where the induced magnetic field cancels the external one.

Defects and impurities allow us extremely fine control over the electronic properties of materials and we would not have things as beautiful as transistors or SQUID devices without them.

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Just a quick comment: the integer QHE depends on impurities as you point out, but the fractional one needs a clean sample, which is why it was discovered later. The physics there depends on unscreened Coulomb interaction between the electrons. –  user566 Feb 25 '11 at 5:32
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@Moshe This is not true, FQHE still needs disorder even though interactions are crucial to its existence. The reason why it was discovered later than the IQHE was that the temperatures which are necessary to see it are much lower $T < 1.0$K –  DaniH Mar 7 '12 at 9:35
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Let me try to identify a generic reason for why are impurity-free condensed matter systems likely to be boring. A condensed matter system without defects:

  1. has a perfect spatial symmetry, so its low energy excitations can be described by an effective field theory, regularized at atomic energies by the actual lattice.

  2. the correspondng vaccuum is "empty", so there is not too much space for macroscopically observable qualitatively different phenomena unless they involve excitation on the cut-off (lattice) level. But such excitaions are nothing but defects/microscopic disorder/inhomogeneity that we want to avoid.

It is similar to QFT in the usual (God-given) vacuum: continuous symmetries put a strong restriction on the type of possible particles & interactions.

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Primary reference: http://www.tcm.phy.cam.ac.uk/~bds10/publications/lesh.ps.gz

Disorder is very important in the so-called mesoscopic regime, where the sample size is larger than the coherence length of electrons, and the coherence length is much larger than the Compton wavelength. Roughly speaking, due to the multiple scattering which occurs across the entire sample, transport is a diffusive process, but the scattering centres (i.e. impurities) are dilute and/or weak so that electrons propagate coherently between close ones.

One can imagine trying to compute the probability of transmission as the square of a propagator/two-point correlation function. Using field integral methods, one finds that the probability corresponds to a Feynmann diagram which contains a path going forwards then backwards. In particular, for the phases to not strongly cancel, the paths need to remain within a Compton wavelength of each other. The introduction of scattering centres gives rise to a quantum phenomenon where one can get a constructive interference by making a loop somewhere in the middle and making the forward and return paths go in the same direction instead.

The introduction of an intermediate length scale also introduces new effective modes, the most important of which are the diffuson (the classical diffusive mode) and the cooperon which is purely quantum mechanical (and corresponds roughly to the description above of forward and backward paths going in the same direction). The field theory then allow you to join these together arbitrarily and resumming the entire series then predicts various interesting phenomenon.

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+1; a very nice explanation of what is mesoscopic physics within the broad realm of condensed matter theory –  Slaviks Aug 18 '11 at 11:13
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