Conservation laws, vertex corrections, and screening in Raman spectroscopy
AbstractWe present a microscopic theory for the Raman response of a clean multiband superconductor, with emphasis on the effects of vertex corrections and long-range Coulomb interaction. The measured Raman intensity, R(Ω), is proportional to the imaginary part of the fully renormalized particle-hole correlator with Raman form factors γ(k⃗). In a BCS superconductor, a bare Raman bubble is nonzero for any γ(k⃗) and diverges at Ω=2Δmax, where Δmax is the largest gap along the Fermi surface. However, for γ(k⃗) = constant, the full R(Ω) is expected to vanish due to particle number conservation. It was sometimes stated that this vanishing is due to the singular screening by long-range Coulomb interaction. In our general approach, we show diagrammatically that this vanishing actually holds due to vertex corrections from the same short-range interaction that gives rise to superconductivity. We further argue that long-range Coulomb interaction does not affect the Raman signal for any γ(k⃗). We argue that vertex corrections eliminate the divergence at 2Δmax. We also argue that vertex corrections give rise to sharp peaks in R(Ω) at Ω<2Δmin (the minimum gap along the Fermi surface), when Ω coincides with the frequency of one of the collective modes in a superconductor, e.g., Leggett and Bardasis-Schrieffer modes in the particle-particle channel, and an excitonic mode in the particle-hole channel.