Spinless electric toroidal multipoles in ferroaxial ${\rm K_2Zr(PO_4)_2}$ revealed by symmetry-adapted closest Wannier analysis

Abstract: From a symmetry perspective, ferroaxial order belongs to the same symmetry as time-reversal-even pseudovectors. Experimentally, ${\rm K_2Zr(PO_4)_2}$ is known to undergo a displacive-type phase transition from a non-ferroaxial to a ferroaxial phase. To identify the key microscopic ingredients driving this transition, we carry out a quantitative analysis combining density-functional theory calculations and symmetry-adapted closest Wannier analysis. As a result, we show that electric toroidal dipole, electric toroidal octupole, and electric hexadecapole, which belong to the same irreducible representation, make dominant contributions to the ferroaxial transition. In particular, we find that spinless electric toroidal octupoles, which originate from spin-independent off-diagonal real hopping between the $p$ orbitals on P and O atoms and between the $d$ orbitals on Zr atoms and $p$ orbitals on O atoms, provide the most significant contributions. Moreover, we explicitly analyze the orbital characters involved in the relevant hybridizations associated with these multipoles. We further show that the relativistic spin--orbit coupling has a negligible influence on the ferroaxial transition. These results demonstrate that spin-independent orbital hybridization between different orbitals on different atoms plays a crucial role in inducing the ferroaxial transition.