# Critical Slowing Down of Quadrupole and Hexadecapole Orderings in Iron Pnictide Superconductor

## TL;DR

This study uses ultrasonic measurements to reveal critical slowing down of quadrupole and hexadecapole orderings in iron pnictide superconductors, linking these to structural and superconducting transitions and suggesting a spontaneous rotation of the superconducting state.

## Contribution

It demonstrates the critical slowing down of multipole orderings in iron pnictides and proposes a connection between hexadecapole ordering and the spontaneous rotation of the superconducting state.

## Key findings

- Critical slowing down observed at structural transition due to quadrupole ordering.
- Critical slowing down observed at superconducting transition due to hexadecapole ordering.
- Hexadecapole ordering may induce spontaneous rotation of the superconducting state.

## Abstract

Ultrasonic measurements have been carried out to investigate the critical dynamics of structural and superconducting transitions due to degenerate orbital bands in iron pnictide compounds with the formula Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$. The attenuation coefficient $\alpha_{\mathrm{L}[110]}$ of the longitudinal ultrasonic wave for $(C_{11}+C_{12}+2C_{66})/2$ for $x = 0.036$ reveals the critical slowing down of the relaxation time around the structural transition at $T_\mathrm{s} = 65$ K, which is caused by ferro-type ordering of the quadrupole $O_{x'^2-y'^2}$ coupled to the strain $\varepsilon_{xy}$. The attenuation coefficient $\alpha_{66}$ of the transverse ultrasonic wave for $C_{66}$ for $x = 0.071$ also exhibits the critical slowing down around the superconducting transition at $T_\mathrm{SC} = 23$ K, which is caused by ferro-type ordering of the hexadecapole $H_z^\alpha \bigl( \boldsymbol{r}_i, \boldsymbol{r}_j \bigr) = O_{x'y'}\bigl( \boldsymbol{r}_i \bigr) O_{x'^2 - y'^2}\bigl( \boldsymbol{r}_j \bigr) + O_{x'^2 - y'^2}\bigl( \boldsymbol{r}_i \bigr) O_{x'y'}\bigl( \boldsymbol{r}_j \bigr)$ of the bound two-electron state coupled to the rotation $\omega_{xy}$. It is proposed that the hexadecapole ordering associated with the superconductivity brings about spontaneous rotation of the macroscopic superconducting state with respect to the host tetragonal lattice.