Linear Response Theory for Shear Modulus $C_{66}$ and Raman Quadrupole Susceptibility: Evidence for Nematic Orbital 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 ${\chi }_{x^2-y^2}^{\text{Raman}}$. Because of the Aslamazov-Larkin vertex correction, the nematic-type orbital fluctuations are induced, and they enhance both $1/C_{66}$ and ${\chi }_{x^2-y^2}^{\text{Raman}}$ strongly. However, ${\chi }_{x^2-y^2}^{\text{Raman}}$ remains finite even at the structure transition temperature $T_S$, because of the absence of the band Jahn-Teller effect and the Pauli (intraband) contribution, as proved in terms of the linear response theory. The present study clarifies that the origin of the nematicity in Fe-based superconductors is the nematic orbital order and fluctuations.

Figure 1

(a) Schematic phase diagram of Fe-based superconductors. (b) Fermi surfaces for $y=0$. The weight of the $d_{xz}$ orbital is stressed by green circles. (c) Relation ${\lambda }_{\text{photon}}\gg {\lambda }_{\text{ac}}$. (d) Particle-hole excitation continuum.

Figure 2

(a) $X_{k\text{−}\text{lim}}/T$ and (b) $X_{\omega \text{−}\text{lim}}/T$ as functions of $T$. Their $T$ dependences originate from $|{\mathrm{\Lambda }}_{\mathit{Q}}^{k(\omega )\text{−}\text{lim}}{|}^2$ since ${\xi }^2\propto 1/(1-{\alpha }_s)$ is fixed.

Figure 3

(a) Fittings of the data $C_{66}^{\text{exp}}$ normalized by the 33% Co-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ data in Ref. [5], shown by broad red lines. The dotted lines $C_{66}$ are the fitting result under the constraint $C_{66}=0$ at $T=T_S$, and the broken lines $C_{66}^{\prime }$ are the fitting without constraint. (b) The Weiss temperature $\theta$ given by the present fitting. ${\theta }_{\mathrm{NMR}}$ is the Weiss temperature of $1/T_{1}T$ [1], and ${\theta }_{\text{Raman}}$ is given by the Raman spectroscopy [20]. (c) $X_{k\text{−}\text{lim}}$ and $X_{\omega \text{−}\text{lim}}$ given by the fitting of $C_{66}$. Experimental data of ${\chi }_{{\mathrm{x}}^2-{\mathrm{y}}^2}^{\text{Raman}}$are shown by red circles [20].

Figure 4

Fittings of shear modulus for (a) underdoped and (b) overdoped $({\mathrm{Ba}}_{1-x}{\mathrm{K}}_x){\mathrm{Fe}}_2{\mathrm{As}}_2$. Experimental data $C_{66}^{\text{exp}}$ are shown by broad red lines [5].