Linear Response Theory for Shear Modulus $C_{66}$ and Raman Quadrupole Susceptibility: Significant Evidence for Orbital Nematic Fluctuations in Fe-Based Superconductors

The emergence of the nematic order and fluctuations has been discussed as a central issue in Fe-based superconductors. To clarify the origin of the nematicity, we focus on the shear modulus $C_{66}$ and the Raman quadrupole susceptibility ${\chi }_{x^2-y^2}^{Raman}$. Due to the Aslamazov-Larkin vertex correction, the nematic-type orbital fluctuations are induced, and they enhances both $1\mathrm{/}{C}_{66}$ and ${\chi }_{x^2-y^2}^{Raman}$ strongly. However, ${\chi }_{x^2-y^2}^{Raman}$ remains finite even at the structure transition temperature $T_S$, because of the absence of the band Jahn-Teller effect and the Pauli (=intra-band) contribution, as proved in terms of the linear response theory. The present study clarifies that origin of the nematicity in Fe-based superconductors is the nematic-orbital order/fluctuations.