# Electric Toroidal Quadrupoles in Spin-Orbit Coupled Metal Cd$_2$Re$_2$O$_7$

## TL;DR

This paper proposes that electric toroidal quadrupoles (ETQs) explain the inversion symmetry breaking in Cd$_2$Re$_2$O$_7$, linking unconventional multipoles to observed phenomena and resolving experimental contradictions.

## Contribution

It introduces the concept of ETQs as the order parameters in Cd$_2$Re$_2$O$_7$, providing a new theoretical framework for understanding its phase transitions.

## Key findings

- ETQs are active in the phase transitions of Cd$_2$Re$_2$O$_7$
- ETQs can be detected via specific phenomena like spin splittings and nonreciprocal transport
- The proposed ETQ scenario reconciles previous experimental contradictions.

## Abstract

We report our theoretical results on the order parameters for the pyrochlore metal Cd$_2$Re$_2$O$_7$, which undergoes enigmatic phase transitions with inversion symmetry breaking. By carefully examining active electronic degrees of freedom based on the lattice symmetry, we propose that two parity-breaking phases at ambient pressure are described by unconventional multipoles, electric toroidal quadrupoles (ETQs) with different components, $x^2-y^2$ and $3z^2-r^2$, in the pyrochlore tetrahedral unit. We elucidate that the ETQs are activated by bond or spin-current order on Re-Re bonds. Our ETQ scenario provides a key to reconcile the experimental contradictions, by measuring ETQ specific phenomena, such as peculiar spin splittings in the electronic band structure, magneto-current effect, and nonreciprocal transport under a magnetic field.