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).