Dispersion of plasmons in three-dimensional superconductors

We study the plasma branch of an homogeneous three-dimensional electron gas in an $s$-wave superconducting state. We focus on the regime where the plasma frequency $ω_p$ is comparable to the gap $Δ$, which is experimentally realized in cuprates. Although a sum rule guarantees that the departure of the plasma branch always coincides with the plasma frequency, the dispersion and lifetime of the plasmons is strongly affected by the presence of the pair condensate, especially at energies close to the pair-breaking threshold $2Δ$. When $ω_p $ is above $1.7Δ$, the level repulsion is strong enough to give the plasma branch an anomalous, negative dispersion with a minimum at finite wavelength. At non-zero temperature and at $ω_p>2Δ$, we treat in a non-perturbative way the coupling of plasmons to the fermionic excitations, and show that a broadened plasma resonance inside the pair-breaking continuum coexists with an undamped solution in the band gap. This resonance splitting is associated with the presence of multiple poles in the analytic continuation of the propagator of the Cooper pairs.

💡 Research Summary

The paper investigates the collective charge‑density oscillations (plasmons) of a homogeneous three‑dimensional electron gas that has entered an s‑wave superconducting state. The authors focus on the experimentally relevant regime where the bare plasma frequency ωp is of the same order as the superconducting gap Δ, a situation realized in many cuprate compounds. Using a microscopic Random Phase Approximation (RPA) framework that includes both the long‑range Coulomb interaction V C (q)=m ωp²/(ρ q²) and a short‑range contact pairing interaction g δ(r₁−r₂), they derive a 3×3 response matrix χ(ω,q) coupling density fluctuations to phase (δθ) and amplitude (δ|Δ|) fluctuations of the order parameter. The collective mode spectrum is obtained from the zeros of the determinant det