Sum rules and density wave spectrum for non relativistic fermions

TL;DR

This paper derives frequency sum rules for non-relativistic fermion systems, revealing how density wave spectra behave at large and low wavelengths, with implications for plasmon energies and correlation functions.

Contribution

It introduces general sum rules for density waves in fermionic systems and analyzes their spectral behavior, including specific results for Coulomb models and decay potentials.

Findings

Density wave spectrum converges to zero frequency at large wavelengths.

Plasmon energy spectrum in jellium model matches zero momentum limit.

Low momentum behavior of <ω²(k)> depends on potential decay properties.

Abstract

Frequency sum rules are derived in extended quantum systems of non relativistic fermions from a minimal set of assumptions on dynamics in infinite volume, for ground and thermal states invariant under space translations or a lattice subgroup. For the jellium Coulomb model, they imply the one point result 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 large wavelenght, to a point measure at zero frequency, for any number of fermion fields and potentials with integrable second derivatives. For low momentum, <\omega^2(k)> ~ k^2 for potentials V with r^2 d_i d_j V integrable, <\omega^2(k)> ~ 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 range interactions, the fourth momentum of the frequency is expressed, at lowest order in k, purely in terms of the three point correlation function of the density.