Do metals *really* conduct at zero temperature? The questions is mostly in the title, but might expose another of my misunderstanding of the band structure of solids and how that leads to metals and insulators.
If we have a solid, and the fermi energy lies at the top of one of the bands, it will be an insulator, because there are no thermal fluctuations large enough to allow the electrons to move up to the next available state.
On the other hand, if the fermi level is in the middle of one of the bands, the solid will be a metal. If we apply a field, the electron can easily move up to a higher state within the band, and we will have conduction.
Here's my question: doesn't it take thermal fluctuations to "smear" the energy levels within a band? A band isn't really a continuous $\epsilon(k)$ spectrum; it's a series of discrete $\epsilon_k$ values. Does that not mean that at truly zero temperature, we would have the same (insulating) situation as above?
Thanks
 A: You are right. Perfect metal without interaction and impurity will not conduct direct electric current at zero temperature. It will do Bloch oscillation. However impurity scattering or thermal relaxation will destroy the Bloch oscillation, and lead to finite conductivity. With interaction, the metal may go superconducting at low temperature which is beyond the simple band theory consideration.
A: I'm not sure if this is how experts think about the issue, but in general phases of matter are only rigorously defined in the so-called "thermodynamic limit"- that is, taking infinite volume while keeping the density fixed. When applied to a band structure, this results in the spectrum within a band being truly continuous, and there is no energy gap.
A: A metal can only conduct if it is part of a circuit. There need to be contacts with the medium that provides the carriers. This will add electrons or holes. These can propagate unhindered through the metal. So the metal is a conductor with zero resistance.
