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If the real part of the AC conductivity $\text{Re}[\sigma(\omega)]$ has a discrete spectrum only, i.e.,

$\text{Re}[\sigma(\omega)]=a_1\delta(\omega-\omega_1)+a_2\delta(\omega-\omega_2)+\cdots,$

what can we say about the microscopic properties of this matter/material? Does it imply that the charge carriers are spatially ordered?

Other ways to ask the question are: If the electrons are spatially ordered, will $\text{Re}[\sigma(\omega)]$ have a discrete spectrum? Does the $\text{Re}[\sigma(\omega)]$ of a Wigner crystal have a discrete spectrum? Does the $\text{Re}[\sigma(\omega)]$ of a supersolid have a discrete spectrum?

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Personally, I find it more intuitive to think in terms of the closely related quantity, the loss function, $-\text{Im}\frac{1}{\epsilon}$, rather than the optical conductivity.

If one were to tune across a phase transition from a electronic liquid to an electronic solid, I suspect one would expect to see the softening of the free-carrier plasmon. Once the electron crystallizes, then perhaps one would be able to see a discrete energy-loss spectrum like you outlined. However, it may be difficult to separate this out from the phonon spectrum unless the energy scale was very different, which it very well may be.

However, just from symmetry, one may make the following statement: As the electronic liquid, which is translationally invariant, crystallizes into an electronic solid, one would expect to observe a Goldstone mode, i.e. a phonon of the electronic solid. This is obviously an idealistic scenario, however, and it would be a very interesting experiment indeed to undertake what you have suggested and investigate the optical properties of a Wigner crystal. Unfortunately, Wigner crystallization has been reported in very few materials and ones that are not capable of being studied easily by optical spectroscopy.

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