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As a non-specialist, I asked the question "What are disorders in condensed matter parlance?" about the meaning of disorder in condensed matter physics. I also wrote a non-specialist answer after some research. Here is yet another question that has been bugging me.

Crystalline metals with perfect periodicity perfectly conduct electric current (Correct me if I'm wrong). What happens when one gradually introduces more and more disorder into a periodic structure? Does the conductivity necessarily decrease?

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    $\begingroup$ Crystalline metals with perfect periodicity, of course, do not exist - at thermodynamic equilibrium at any non-zero temperature there are populations of point defects that can cause scattering. These are in addition to phonon scattering (again, at any non-zero temperature). Then there is surface scattering for any finite volume... $\endgroup$ – Jon Custer Jul 20 '18 at 13:47
  • $\begingroup$ @JonCuster There is no phonon are $T=0$. Right? So any scattering at $T=0$ must be resulting from scattering from disorders. Am I right? $\endgroup$ – SRS Jul 20 '18 at 13:51
  • $\begingroup$ You can't get to T=0. Further, the electron can scatter to create a phonon - you don't need one already existing. And, don't forget electron-electron scattering mechanisms. $\endgroup$ – Jon Custer Jul 20 '18 at 14:05
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    $\begingroup$ Actually, for an absolutely perfect crystal the conductivity should be zero; electrons will oscillate back and forth because they have negative mass when they get to the top of their Bloch bands. $\endgroup$ – knzhou Jul 22 '18 at 22:47
  • $\begingroup$ @knzhou Interesting! Will you expand on that or give some reference? $\endgroup$ – SRS Jul 29 '18 at 16:03
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Short answer is the resistivity increases until it transitions to insulator.

A band description makes things more clear. Before the transition, the electronic states are pertubatively connected to to disorderless band conductor. After the transition to an insulator the electronic states all become localized and in momentum space this shows up as a band gap around the fermi energy and effectively looks like a band insulator. This is anderson localization: https://en.wikipedia.org/wiki/Anderson_localization

Long answer: Its complicated and not fully understood. In 1D there is no transition. The smallest amount of disorder will localize the system. But at a certain interaction strength the system will delocalize and start conducting again.

Out side of 1D, the non interacting case can be described with different effect models, where usually another or multiple effective fields are introduced which interact with the electron field to cause localization. Here the normal mean field description of a phase transition works and can give you universal scaling relations at the transition. For some discussion you can see this review for the super symmetric method: http://arxiv-export-lb.library.cornell.edu/abs/1002.2632

But things are more complicated and there might not be a simple transition. The griffiths effect describes the possibility of rare regions to localize or delocalize the system in the traditionally conducting or insulating limits. This then creates a middle zone between the transition from conductor to insulator where rare regions play a more important role and may smooth the transition to a cross over.

Finally there has been observations that disorder can actually delocalize a mott insulator(where a gap opens around the fermi surface due to interactions). Here disorder increases conductivity: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.206402

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Long range order is less important to metallic behaviour than often believed. When alkali metals melt there is only a small jump in resistivity. When silicon melts, it turns from a semiconductor into a liquid metal, as does germanium. Amorphous Si and Ge are, however, semiconductors. These examples show that long range disorder or order are not deciding whether a material conducts or not. Long range disorder does increase resistivity.

A more useful picture of conduction is obtained by using local orbitals, as in the Hubbard model. This model in its basic form has two parameters, on-site electron-electron repulsion $U$ and nearest neighbour hopping $t$. It shows that the ratio $t/J$ is decisive. If the repulsion is strong localised orbitals and insulating behaviour results. In the opposite case, delocalised electron orbitals and conduction result.

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  • $\begingroup$ What you described is the Mott insulator - metal crossover. You should add that you can have band insulator just from filling of the band, even when interaction is small. $\endgroup$ – wcc Jul 31 '18 at 1:18
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    $\begingroup$ @IamAStudent this is exactly my point. It is generally believed that long range order determines whether a material is a conductor or not. In my answer I give a number of examples that contradict this view. $\endgroup$ – my2cts Jul 31 '18 at 7:41

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