Conservation laws, vertex corrections, and screening in Raman spectroscopy

We 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, $R\left(\mathrm{\Omega }\right)$, is proportional to the imaginary part of the fully renormalized particle-hole correlator with Raman form factors $\gamma \left(\vec{k}\right)$. In a BCS superconductor, a bare Raman bubble is nonzero for any $\gamma \left(\vec{k}\right)$ and diverges at $\mathrm{\Omega }=2{\mathrm{\Delta }}_{\max }$, where ${\mathrm{\Delta }}_{\max }$ is the largest gap along the Fermi surface. However, for $\gamma \left(\vec{k}\right)$ = constant, the full $R\left(\mathrm{\Omega }\right)$ 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 $\gamma \left(\vec{k}\right)$. We argue that vertex corrections eliminate the divergence at $2{\mathrm{\Delta }}_{\max }$. We also argue that vertex corrections give rise to sharp peaks in $R\left(\mathrm{\Omega }\right)$ at $\mathrm{\Omega }<2{\mathrm{\Delta }}_{\min }$ (the minimum gap along the Fermi surface), when $\mathrm{\Omega }$ 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.

Figure 1

The summation scheme to calculate the nonresonant Raman response. The top line is the RPA sum involving the long-range Coulomb interaction $V_C\left(\vec{q}\right)$. The bottom line denotes the ladder approximation to account for the vertex corrections to the bare bubble. Here $V$ denotes a generic short-range interaction.

Figure 2

Diagrams in the ladder series for the Raman response that we call in the text particle-hole and particle-particle type contributions.

Figure 3

All possible interactions between a pair of bands $a,b$ (solid and dashed, respectively). $V_1^{ab}$ is the density-density interaction between band $a$ and band $b$; $V_2^{ab}$ is an exchange between the two bands; $V_3$ is a fermion-pair hopping term that is made possible due to umklapp processes; $V_{4,5}^{ab}$ are the density-density interaction terms within each band. Every pair of bands has a $V_{1,2,3}$ interaction component.

Figure 4

Schematic form of the integral vertex equations for $\mathrm{\Gamma }\left(\vec{k}\right)$ and $\overline{\mathrm{\Gamma }}\left(\vec{k}\right)$ (band indices have been dropped). The “1” in the equation for $\overline{\mathrm{\Gamma }}$ denotes the total density vertex [both $\gamma \left(\vec{k}\right)$ and “1” are proportional to ${\sigma }_3]$. Here $V({\vec{k}}^{\prime }-\vec{k})$ represents all the relevant short-range interactions that enter the renormalization of the respective vertices. The solid lines with arrows denote the Green's functions in Nambu space. The Green's functions in the Nambu space are matrices constructed of normal and anomalous Green's functions.

Figure 5

(a) The Raman response, for a 1-band $s$-wave SC in the $A_{1g}$ channel, where it is approximated by the bare-bubble contribution (dashed green line), the vertex-corrected (VC) contribution with a constant ${\gamma }_{\vec{k}}$ (red line), and the VC contribution with a momentum-dependent ${\gamma }_{\vec{k}}$ (black line). (b) The removal of the $2\mathrm{\Delta }$ edge singularity by vertex corrections from the subleading channel interaction $v_4^{22}\equiv {\nu }_F{V}_4^{22}$. The peak shifts to larger $\mathrm{\Omega }$ as the interaction strength in increased. We have used $V_4^{11}=-0.3/{\nu }_F,c_1=0.3,c_2=0.2$. A fermion lifetime of $0.005\mathrm{\Delta }$ was added to obtain the broadening.

Figure 6

The Raman response of a 2-band $s$-wave SC. The light line is the result when one includes only the bare bubble, dark line is the full result, with vertex corrections (VC) included. These corrections remove the edge singularity and move the peak to a frequency larger than $2{\mathrm{\Delta }}_{\min }={\mathrm{\Delta }}_a$. The Raman vertex for each band is chosen to be a constant with $c_1^a=0.3$ and $c_1^b=0.2$. Here $V_a=-0.2/{\nu }_F,V_b=-0.3/{\nu }_F,V_3=0.1/{\nu }_F$, and ${\nu }_{F,a}={\nu }_{F,b}={\nu }_F$.

Figure 7

(a) Solutions to the transcendental Eq. (63) given by the intersection of $\tilde{F}=F/2\nu$ with the right-hand side of Eq. (63) for different $v_{ph}$. (b) The Raman spectrum for $v_{ph}=0,v_{ph}=-0.5\in \{-1/|\tilde{\kappa }|,2\}$, and $v_{ph}=3$ (not in that bound). This case corresponds to the renormalization of the Leggett mode. Here $\tilde{\kappa }=-0.83$. (c) Solutions to the transcendental Eq. (63) and (d) the Raman spectrum for $v_{ph}=0,v_{ph}=1\in \{1/\tilde{\kappa },2\}$, and $v_{ph}=3$ (not in that bound). This shows the emergence of a possible excitonic mode. Here $\tilde{\kappa }=4.2$. Note the removal of the $2\mathrm{\Delta }$ edge singularity in all cases.

Figure 8

The Raman response in the $A_{1g}$ channel for a 2-band $s$-wave SC. The dashed line is the contribution from the bare bubble. The light red line is the response after accounting for vertex corrections with constant ${\gamma }_{\vec{k}}$: $c_1^a=1,c_1^b=0.3$. The dark line shows the Raman response for a nonconstant ${\gamma }_{\vec{k}}$ where we added $c_2^a=0.2,c_2^b=0.3$ (corresponding to the $cos4\theta$ harmonic). (a) and (b) refer to the case of intraband-driven attractive SC ($V_a{V}_b-V_3^2>0$) and show how the Leggett mode can be pushed to the continuum if $V_3$, denoted by $v$ in the figure, is increased. (c) is the case of interband-driven attractive SC ($V_a{V}_b-V_3^2<0$) which has no Leggett mode. Note the removal of the $2\mathrm{\Delta }$ edge singularity and strong suppression of the spectral weight around $2\mathrm{\Delta }$ in all cases. Here $v\equiv V_3{\nu }_b$; ${\nu }_a/{\nu }_b=0.6$; and $V_a{\nu }_b=-0.5,V_b{\nu }_b=-0.35$ for (a) and (b); $V_a{\nu }_b=0.3,V_b{\nu }_b=0.6$ for (c). A fermionic lifetime of $0.05\mathrm{\Delta }$ is added for broadening.