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In the tight binding model, it's said that in a certain limit, we can regard electrons in the solid as localized to individual atoms. This statement shows up in most introductory condensed matter textbooks, like Ashcroft and Mermin, Kittel, and Altland and Simons.

However, the statement is basis-dependent! If we expand in the Wannier basis, then it's true that our electron wavefunctions are each localized about a particular atom. However, we can also expand in the Bloch basis, in which case "the" wavefunction of each electron is delocalized. Moreover the Bloch basis feels more natural to me since it diagonalizes the Hamiltonian.

The equivalence of these two perspectives follows from second quantization, as you can write the second-quantized ground state of the system in terms of creation operators in either basis. Hence there's no basis-independent meaning of "the" state of a single electron. How can localization be defined in a basis-independent way?

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    $\begingroup$ Who said that in a simple tight binding model you could regard the electrons as localised? The Wannier basis is convenient for the purposes of approximating the potential experienced by the electrons, but after that point the physically significant states are the eigenstates of the Hamiltonian, that is the Bloch states $\endgroup$ – By Symmetry Dec 19 '16 at 20:40
  • $\begingroup$ @BySymmetry Both Ashcroft and Mermin and Altland and Simons make this statement. In both cases, they use vague phrases like "the state of the electrons", even though this is basis dependent. $\endgroup$ – knzhou Dec 19 '16 at 20:42
  • $\begingroup$ As for your comment, what do you mean by 'the physically significant states'? What makes one basis physically significant, when I can also formulate the problem in another? $\endgroup$ – knzhou Dec 19 '16 at 20:44
  • $\begingroup$ I don't have either of those books available right now, so I can't comment on what the authors may have said. If somebody else does they may be able to comment. I will write a full answer to expand on the other aspects of your question $\endgroup$ – By Symmetry Dec 19 '16 at 20:56
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The potential experienced by electrons in a crystal is exceedingly complex. It is, therefore, desirable to try to find simple approximations for this potential.

In the tight binding model we write down a set of trial wavefunctions (the Wannier basis) in the hope that the can write the action of the Hamiltonian on these states in a simple form. We have not said that the electron is in any of these states, we have only asked the question of what would happen if it was. The physical insight of the model is that if the potential well at each site in the lattice is fairly deep and narrow, then

  1. The probability of an electron being between sites is (probably) small (in a sense the electrons are localised "on the atoms" as opposed to "between" them)
  2. The eigenstates of an isolated deep, narrow well will be widely spaced, so the dynamics of an electron within a single well are probably negligible
  3. If an electron were to be in a state localised to a single atom, the probability of tunnelling from one site to another would probably drop off quickly with distance, so only a small number of tunnelling terms would be needed.

This allows use to write down a good approximation for the matrix elements of the Hamiltonian in this particular basis. It does not, however, tell us what the electrons are actually doing. To do that for a system at finite temperature we need the density matrix, which, in thermal equilibrium, has the same eigenstates as the Hamiltonian. As you know, for any lattice system, these are the Bloch states and so these are the states which will determine the expectation values of any observables. Consequently, it is the non-localised Bloch states which are important to the physics of the system and not the Wannier states, which are simply an intermediate convenience.

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Here is the basis-independent statement that the electrons are "spatially localized:" in certain limits (which physically correspond to a strong insulator), the many-body electron wavefunction is unentangled in the position basis - i.e. it is (approximately) a product state of single-particle Wannier wavefunctions, whose spatial wavefunctions have negligible overlap with each other. In this case we can think of an "individual electron" living on each lattice site, and ignore the subtleties that come up with quantum entanglement. However, for a conductor, there will be very high spatial entanglement between the different lattice sites, so there is no sense in which the many-body wavefunction is simply a product of spatially localized single-particle wavefunctions.

In both cases, if the Hamiltonian is translationally invariant and noninteracting, then the many-body wavefunction will be unentangled in the momentum basis. The difference lies in the position basis.

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  • $\begingroup$ That isn't what I mean. The point is that you can write the joint wavefunction of all of the electrons either in terms of a Slater determinant of spatially localized states (the Wannier basis), or as a Slater determinant of momentum-localized states (the Bloch basis). So it's fundamentally interdeterminate what "the" states of the electrons are, since you get the same many-body state either way. $\endgroup$ – knzhou Dec 19 '16 at 23:32
  • $\begingroup$ @knzhou Thanks for clarifying; I've changed my answer. $\endgroup$ – tparker Dec 19 '16 at 23:45

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