Dynamic spin and charge susceptibilities of a Fermi liquid

Motivated by recent observation of the charge susceptibility of superconducting cuprates [1], we study the dynamic spin and charge susceptibilities of a Fermi liquid beyond the random phase approximation (RPA) for a screened Coulomb interaction. Taking into account the self-energy (SE) and vertex (V) diagrams, which describe an excitation of two particle-hole pairs, gives a non-zero contribution to the imaginary part of the spin susceptibility, χ_s''(q,ω), beyond the continuum boundary. The resulting profile of χ_s''(q,ω) has two peaks, one is due to the damped spin mode below the continuum and another one is due to the non-RPA corrections above the continuum boundary. The tail of χ_s''(q,ω) falls off as q²/ω for frequencies far above the continuum boundary. In a Galilean-invariant system, the SE and V contributions to the charge susceptibility, χ_c(q,ω), are canceled by the Aslamazov-Larkin diagrams, and the tail of resulting χ_c''(q,ω) falls off as q⁴/ω. We also study an interplay between the plasmon peak and non-RPA corrections in χ_c''(q,ω).
[1] M. Mitrano et al. arXiv:1708.01929