# Real part of the AC conductivity has a discrete spectrum => What physics?

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?

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.