Development of orbital and spin fluctuations in Fe-based superconductors based on the self-consistent vertex correction (SC-VC) method

To achieve unified understanding of the whole phase diagram of Fe-based superconductors, we analyze the multiorbital Hubbard model going beyond the random phase approximation (RPA). The 2nd-order non-magnetic structure transition at $T_{\mbox{S}} (>T_{\mbox{N}} )$, nematic order as well as large softening of shear modulus $C_{66} $ indicate the strong orbital fluctuations in the normal state. However, only the spin fluctuations develop within the RPA. To resolve this discrepancy, we develop the self-consistent vertex correction (SC-VC) method beyond the RPA, and find the mutual development of orbital and spin fluctuations due to the Aslamazov-Larkin VC, which describes the Kugel-Khomskii type spin-orbital coupling [1]. We find that (i) both the antiferro-orbital and ferro-orbital ($=$nematic) fluctuations develop for $J/U > 0.17$ by including the self-energy correction ($=$SC-V$\Sigma $ method): Both fluctuations contribute to the s-wave superconductivity, and the nematic fluctuations are the origin of the structure transition and the softening of $C_{66} $. (ii) The coexistence of orbital and spin fluctuations can induce the loop-shape nodes on the electron-pockets in BaFe$_{2}$(As,P)$_{2}$, as well as (impurity-induced) smooth $s_{\pm } \to s_{++} $crossover with high $T_{\mathrm{c}}$ [2,3]. Also, the horizontal node on the $z^{2}$-orbitlal hole-pocket predicted by RPA is filled by the inter-orbital fluctuations due to the VC, consistently with laser ARPES and other bulk experiments of 122 compounds. (iii) The same orbital nematic fluctuations are obtained in a simple two-orbital model for Sr$_{3}$Ru$_{2}$O$_{7}$, not only by the SC-VC method [4] but also by the two-dimensional RG method [5]. Therefore, the VC is expected to be the origin of novel orbital physics in various multioritital $d$- and $f$-electron systems.\\[4pt] [1] S. Onari and H. Kontani, PRL \textbf{109}, 137001 (2012).\\[0pt] [2] H. Kontani and S. Onari, PRL \textbf{104}, 157001 (2010).\\[0pt] [3] S. Onari and H. Kontani, PRL \textbf{103}, 177001 (2010).\\[0pt] [4] Y. Ohno, M. Tsuchiizu, S. Onari, and H. Kontani, arXiv:1209.3629.\\[0pt] [5] M. Tsuchiizu, S. Onari, and H. Kontani, arXiv:1209.3664.