Self-consistent Vertex Correction Analysis for Iron-based Superconductors: Mechanism of Coulomb Interaction-Driven Orbital Fluctuations

We study the mechanism of orbital or spin fluctuations due to multiorbital Coulomb interaction in iron-based superconductors, going beyond the random-phase approximation. For this purpose, we develop a self-consistent vertex correction (VC) method, and find that multiple orbital fluctuations in addition to spin fluctuations are mutually emphasized by the “multimode interference effect” described by the VC. Then, both antiferro-orbital and ferro-orbital ($=\mathrm{\text{nematic}}$) fluctuations simultaneously develop for $J/U\sim 0.1$, both of which contribute to the $s$-wave superconductivity. Especially, the ferro-orbital fluctuations give the orthorhombic structure transition as well as the softening of shear modulus $C_{66}$.

Figure 1

(a) Fermi surfaces of iron pnictides. The colors correspond to $2=xz$ (green square), $3=yz$ (red circle), and $4=xy$ (blue diamond), respectively. (b) ${\alpha }_{\mathit{Q}}^c$, ${\alpha }_{\mathbf{0}}^c$, and $U$ as function of $J/U$ in RPA under the condition ${\alpha }_{\max }^s=0.97$. (c) ${\chi }_{xz}^c(\mathit{q})$ and ${\chi }_{x^2\mathrm{\text{−}}{y}^2}^c(\mathit{q})$ given by the RPA for $(J/U,U)=(0.088,1.53)$.

Figure 2

(a) The MT and AL terms: The wavy and solid lines are susceptibilities and electron Green functions, respectively. ${\Lambda }_{l{l}^{\prime },ab,ef}$ is the three-point vertex. (b) Dominant AL terms for ${\chi }_{x^2\mathrm{\text{−}}{y}^2}^c(\mathbf{0})$; the first (second) term represents the two-orbiton (two-magnon) process. (c) Dominant AL terms for ${\chi }_M^c(\mathit{Q})$ ($M=xz$, $yz$); higher-order terms with bubbles made of ${\chi }_M^c(\pm \mathit{Q})$ ($=\mathrm{\text{multifluctuation}}$ process) are relevant. (d) Enhancement of ${\Lambda }_{l{l}^{\prime },ab,ef}$ due to charge VCs.

Figure 3

${\chi }_{xz}^c(\mathit{q})$ and ${\chi }_{x^2\mathrm{\text{−}}{y}^2}^c(\mathit{q})$ given by the SC-VC method. The relation ${\alpha }_{\max }^s={\alpha }_0^c=0.97$ is satisfied in all cases: (a) $n=6.1$ and $J/U=0.88$ (${\alpha }_{\mathit{Q}}^c=0.86$), (b) $n=6.1$, $J/U=0.9$, and $g=0.05$ (${\alpha }_{\mathit{Q}}^c=0.96$), and (c) $n=6.05$, $J/U=0.11$, and $g=0.065$ (${\alpha }_{\mathit{Q}}^c=0.87$).