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3

Ghostly Lie algebra cohomology Let $\mathfrak{g}$ be our Lie algebra and $V_\rho$ a representation space with representation map $\rho : \mathfrak{g} \to \mathrm{End}(V_\rho)$. $V_\rho$ is, by the action through the representation, naturally a $\mathfrak{g}$-module (people missing the ring structure in $\mathfrak{g}$ - just embed it into the universal ...

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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 ...

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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 ...

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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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