# Sum rules and density wave spectrum for non relativistic fermions - Condensed Matter > Other Condensed Matter

Abstract: Frequency sum rules are derived in extended quantum systems of nonrelativistic fermions from a minimal set of assumptions on dynamics in infinitevolume, for ground and thermal states invariant under space translations or alattice subgroup. For the jellium Coulomb model, they imply the one pointresult for the plasmon energy spectrum in the zero momentum limit. In general,the density waves energy spectrum is shown to converge, in the limit of largewavelenght, to a point measure at zero frequency, for any number of fermionfields and potentials with integrable second derivatives. For low momentum,<\omega^2k> ~ k^2 for potentials V with r^2 d i d j V integrable,<\omega^2k> ~ k^{a-d+2} for potentials decaying at infinity as 1-r^a, d-2 < a< d, d the space dimensions. For one component models with short rangeinteractions, the fourth momentum of the frequency is expressed, at lowestorder in k, purely in terms of the three point correlation function of thedensity.

Author: A. Cintio, G. Morchio

Source: https://arxiv.org/