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In an answer to this question, @user566 mentioned that there is a qualitative difference between gapped and gapless systems; that gapless systems are conducting and gapped system are insulating. Is this a rule? If so why? If not, why not?

I'm asking the question because I want to know how the presence/absence of a gap relates to what we can observe in an experiment. Applying a voltage and measuring a current, i.e. measuring resistance, is one such observable. Are there other experimental observables that depend on the presence/absence of a gap?

Just to be clear, I understand that a gap means that in the infinite size limit, there is an energy difference between the energy of the ground state subspace and the energy of the first excited state(s).

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The statement is not true, because there are counter examples.

  • A U(1) spin liquid is gapless, but it is insulating.
  • An $s$-wave superconductor is fully gapped, but it is (super)conducting.
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It is not generally true that a gapped system is insulating. Or more precisely, this statement is not detailed enough to be said true or false generically.

One case where this is true is for non-interacting particles (say, free electron in a lattice).

For interacting particles, it is much more subtle. In particular, just stating "gapped system" is not precise enough, as some correlation function can be gapped and other not. For example, in a BCS superconductor, the 2-point function is gapped (it cost energy to create a pair of electron from a Cooper pair), but the 4-point function is not (the order parameter, in absence of magnetic field, has arbitrary low-energy excitations of its phase corresponding to a Goldstone boson).

Another example: in an electronic Mott insulator, the charge degrees of freedom (dof) are gapped (no electric conduction), but the spin dof are not, since the ground-state is an antiferromagnet, with once again a gapless (Goldstone) mode. In some sense, there is a spin conduction.

The only case where we can be sure that the system is "absolutely" insulating is when the spectrum is fully gapped, i.e. all correlation functions are gapped.

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    $\begingroup$ What does it mean for a correlation function to be gapped? $\endgroup$ – sebastianspiegel Oct 7 '14 at 16:07
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    $\begingroup$ It means that the correlation function goes to zero exponentially at long distance. In momentum/frequency, it means that as $\omega,q\to 0$, the correlation function is finite. $\endgroup$ – Adam Oct 7 '14 at 18:24
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Your understanding is basically correct. Some materials, intrinsic semiconductors, have small band gaps so that at room temperature (for example) there is enough thermal energy around to promote some electrons to the conduction band, and some holes to the valence band. These materials have the kind of gap that you describe, but are not insulators.

Another phenomenon that depends on the presence of the kind of gap that you describe is photoconductivity. Light can promote electrons across the gap into the conduction band. These materials are conductors when exposed to light having enough energy to bridge the gap. Most common photoconductors are semiconductors so that photoconductivity occurs at visible wavelengths. But insulators with larger gaps can be photoconductive to UV radiation.

More complex systems exhibit behavior that depends on the presence of a gap, notably semiconductor junction devices in which two different types of material are joined at an abrupt interface. We have diodes, Zener diodes, transistors, thermocouples, lasers, avalanche photodiodes, and many more phenomena ... all of which depend on the existence of a gap of the type that you describe.

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  • $\begingroup$ Is there some mathematical justification for why a continuous band implies conduction? $\endgroup$ – sebastianspiegel Oct 7 '14 at 6:15
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    $\begingroup$ A partially filled band implies conduction. That is, no gap between filled and unfilled states. To have conduction you have to speed up an electron by a small amount ... add energy to it. Partially filled bands have states available for this, filled bands do not. Mathematically, conduction is proportional to the density of unoccupied states at the Fermi energy. That density is non-zero for partially filled bands, and zero for filled bands. $\endgroup$ – garyp Oct 7 '14 at 13:54

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