Functional renormalization group study of orbital fluctuation mediated superconductivity: Impact of the electron-boson coupling vertex corrections

In various multiorbital systems, the emergence of the orbital fluctuations and their role on the pairing mechanism attract increasing attention. To achieve deep understanding on these issues, we perform a functional renormalization group (fRG) study for the two-orbital Hubbard model. The vertex corrections for the electron-boson coupling $(U$-VC), which are dropped in the Migdal-Eliashberg gap equation, are obtained by solving the RG equation. We reveal that the dressed electron-boson coupling for the charge channel ${\widehat{U}}_{\mathrm{eff}}^c$ becomes much larger than the bare Coulomb interaction ${\widehat{U}}^0$ due to the $U$-VC in the presence of moderate spin fluctuations. For this reason, the attractive pairing interaction due to the charge or orbital fluctuations is enlarged by the factor ${({\widehat{U}}_{\mathrm{eff}}^c/{\widehat{U}}^0)}^2\gg 1$. In contrast, the spin fluctuation pairing interaction is suppressed by the spin-channel $U$-VC, because of the relation ${\widehat{U}}_{\mathrm{eff}}^s\ll {\widehat{U}}^0$. The present study demonstrates that the orbital or charge fluctuation pairing mechanism can be realized in various multiorbital systems thanks to the $U$-VC, such as in Fe-based superconductors.

Figure 1

(a) The effective interaction ${\widehat{U}}^x$ for $x=c$ $(+)$ and $x=s$ $(-)$, which we call the dressed EBC. The filled circle represents the Coulomb interaction ${\widehat{U}}^{0;x}$, and the rectangle $\left({\mathrm{\Gamma }}^{I\left(U\right),x}\right)$ gives the $U$-VC. ${\mathrm{\Gamma }}^{I\left(U\right),x}$ is irreducible with respect to ${\widehat{U}}^{0;x}$ to avoid the double counting of the RPA-type diagrams. (b) Beyond the RPA: the irreducible susceptibility with the VC, where ${\widehat{\mathrm{\Lambda }}}^x={\widehat{U}}^x{\left\{{\widehat{U}}^{0;x}\right\}}^{-1}$. (c) Beyond the ME approximation: the gap equation with the three-point VCs for the coupling constant $(U$-VC). Only the single fluctuation exchange term is shown.

Figure 2

(a) Band dispersion of the two-orbital Hubbard model and (b) FSs composed of the $d_{xz}$ orbital (green) and $d_{yz}$ orbital (red). (c) The center of patches $\left(1\sim 64\right)$ on the FSs. The arrows represent the nesting vector. The tip and the tail of each arrow correspond to $(i_{\alpha },i_{\beta })=(6,37),(8,38),(10,39)$. (d) Definition of the full four-point vertex ${\mathrm{\Gamma }}_{l{l}^{\prime }m{m}^{\prime }}^{\sigma {\sigma }^{\prime }\rho {\rho }^{\prime }}(\mathit{k}+\mathit{q},\mathit{k};{\mathit{k}}^{\prime }+\mathit{q},{\mathit{k}}^{\prime })$ in the microscopic Fermi liquid theory.

Figure 3

The one-loop RG equation for the four-point vertex. The crossed lines represent the electron Green function with cutoff $\mathrm{\Lambda }$. The slashed lines represent the electron propagations having the energy shell $\mathrm{\Lambda }$.

Figure 4

(a) $\mathit{q}$ dependence of obtained total spin susceptibility ${\chi }^s\left(\mathit{q}\right)$ enlarged at $\mathit{q}\approx (2\pi /3,2\pi /3)$. (b) Obtained quadrupole susceptibility ${\chi }_{x^2-y^2}^c\left(\mathit{q}\right)$. (c) SC phase diagram obtained by RG+cRPA method.

Figure 5

(a) The bare interaction, (b) single-fluctuation-exchange term, (c) crossing-fluctuation-exchange term, and (d) the lowest particle-particle term. Here, $V_y^{l{l}^{\prime },m{m}^{\prime }}(\mathit{k},{\mathit{k}}^{\prime })=\frac{a_y}{4}{\mathrm{\Gamma }}_{ml{l}^{\prime }{m}^{\prime }}^s({\mathit{k}}^{\prime },\mathit{k};-\mathit{k},-{\mathit{k}}^{\prime })-\frac{1}{4}{\mathrm{\Gamma }}_{ml{l}^{\prime }{m}^{\prime }}^c({\mathit{k}}^{\prime },\mathit{k};-\mathit{k},-{\mathit{k}}^{\prime })$ $\left(a_t=-1,a_s=3\right)$. (e) Typical diagrams for the $U$-VC. For the charge sector, the Maki-Thompson (MT) term is negligibly smaller than the AL term in the presence of moderate spin fluctuations. The $O\left({\left\{U^0\right\}}^3\right)$ terms in MT and AL terms are dropped to avoid the double counting. In (a)–(e), spin indices are not written explicitly.

Figure 6

(a) $E_{1u}$-type TSC gap function ${\mathrm{\Delta }}_{t,x}\left(\theta \right)$ on the $\mathrm{FS}\alpha$ and $\mathrm{FS}\beta$ as functions of $\theta$. (b) $A_{1g}$-type SSC gap function ${\mathrm{\Delta }}_s\left(\theta \right)$. (c) ${\overline{\lambda }}_{t\left(s\right)}$ for ${\widehat{V}}_{t\left(s\right),\mathrm{RG}}$ as functions of ${\omega }_c$. (d) ${\overline{\lambda }}_{t\left(s\right)}$ for ${\widehat{V}}_{t\left(s\right),\chi }$.

Figure 7

Spin- and charge-channel pairing interactions obtained by using the RG+cRPA method: (a) spin-channel interaction ${\tilde{\mathrm{\Gamma }}}_{\chi }^s(\mathit{k},{\mathit{k}}^{\prime })$ and (b) charge-channel one ${\tilde{\mathrm{\Gamma }}}_{\chi }^c(\mathit{k},{\mathit{k}}^{\prime })$ in the absence of the $U$-VC. (c) ${\tilde{\mathrm{\Gamma }}}_{\mathrm{RG}}^s(\mathit{k},{\mathit{k}}^{\prime })$ and (d) ${\tilde{\mathrm{\Gamma }}}_{\mathrm{RG}}^c(\mathit{k},{\mathit{k}}^{\prime })$ in the presence of the $U$-VC. Here, $\left(\mathit{k},{\mathit{k}}^{\prime }\right)$ is the pair of momenta for $\left(i_{\alpha },i_{\beta }\right)$. (e) The ratios ${\tilde{\mathrm{\Gamma }}}_{\chi }^c(\mathit{k},{\mathit{k}}^{\prime })/{\tilde{\mathrm{\Gamma }}}_{\chi }^s(\mathit{k},{\mathit{k}}^{\prime })$ and ${\tilde{\mathrm{\Gamma }}}_{\mathrm{RG}}^c(\mathit{k},{\mathit{k}}^{\prime })/{\tilde{\mathrm{\Gamma }}}_{\mathrm{RG}}^s(\mathit{k},{\mathit{k}}^{\prime })$ as functions of $U$. $\mathit{k}$ and ${\mathit{k}}^{\prime }$ are set as the start and end positions of the nesting vector shown in Fig. 2. We take the average over the ellipsoidal area.

Figure 8

(a) The ratios ${(U_{\mathrm{eff}}^x/U^0)}_{\mathrm{diagram}}^2\equiv {(U_{\mathrm{with}\text{−}\mathit{U}\mathrm{VC}}^x(\mathit{k},{\mathit{k}}^{\prime })/U_{\mathrm{no}\text{−}\mathit{U}\mathrm{VC}}^x(\mathit{k},{\mathit{k}}^{\prime }))}^2$ $\left(x=c,s\right)$ given by the diagrammatic calculation as functions of the spin Stoner factor ${\alpha }_S$. For $U$-VC, we perform the diagrammatic calculation for Fig. 5. (b) Third-order term with respect to $U$ for $U$-VC: we put $U=U^{\prime }$ and $J=0$ for simplicity. This term is scaled as $\sim (2N_{\mathrm{orb}}-1)$, where $N_{\mathrm{orb}}$ is the number of $d$-orbital. (c) ${(U_{\mathrm{eff}}^x/U^0)}_{\mathrm{RG}}^2\equiv {\tilde{\mathrm{\Gamma }}}_{\mathrm{RG}}^x/{\tilde{\mathrm{\Gamma }}}_{\chi }^x$ given by the RG+cRPA method for $2.0\le U\le 3.1$. Inset: ${(U_{\mathrm{eff}}^s/U^0)}_{\mathrm{RG}}^2$ for $0\le U\le 3.1$.

Figure 9

The gap equation due to the $e$-ph interaction, where the dotted line represents the phonon propagator and $g$ is the $e$-ph coupling constant. Due to the charge-channel $U$-VC caused by spin fluctuations, the phonon-mediated attractive interaction is enlarged by the factor ${(U_{\mathrm{eff}}^c/U^0)}^2\gg 1$.