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

TL;DR

This work addresses the microscopic electronic origins of ferroaxial order in a nonmagnetic ferroaxial material by combining density-functional theory with a symmetry-adapted closest Wannier (SymCW) approach. By projecting the realistic electronic structure onto a complete SAMB multipole basis, the authors identify spinless electric toroidal dipole, octupole, and electric hexadecapole components as the key drivers, with two bond-cluster ETO terms (from P–O6i and Zr–O6i) dominating the transition. The analysis shows that these off-diagonal real hoppings and on-site hybridizations generate ferroaxiality, while relativistic spin–orbit coupling plays a negligible role. The results demonstrate that spin-independent orbital hybridization across different atoms and orbitals predominantly induces the ferroaxial transition, and they establish the SymCW framework as a powerful tool for disentangling electronic ferroaxial degrees of freedom in real materials.

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.

Figures (7)

• Figure 1: Schematic illustration of the intuitive images of (a) ETD and (b) ETQ moments under cluster-type orbital orderings. Red and blue indicate positive and negative phases, respectively, and green arrows denote the ETD.
• Figure 2: Crystal structure of ${\rm K_2Zr(PO_4)_2}$ in the nonferroaxial phase ($P\overline{3}m1$, top) and the ferroaxial phase ($P\overline{3}$, bottom). Atoms in blue, red, green, and orange correspond to Zr, P, ${\rm O_{6i}}$, and ${\rm O_{2d}}$ atoms, respectively, while K atoms are positioned underneath the P and ${\rm O_{2d}}$. Here, ${\rm O_{6i}}$ and ${\rm O_{2d}}$ denote the O atoms at the 6i and 2d Wyckoff positions in $P\overline{3}m1$. The displacement angle ${\varphi}$ of the ${\rm PO_4}$ tetrahedra characterizes the degree of ferroaxiality.
• Figure 3: Examples of SAMBs belonging to the ${\rm A_{2g}}$ irreducible representation (IR) of the $D_{\rm 3d}$ point group in a six-site cluster. The resulting ${\rm A_{2g}}$ multipoles are (a, d) the ETD, (b, e) the ETO, and (c,f) the EH. The atomic SAMBs are $\mathcal{P}$-odd in (a)--(c) and $\mathcal{P}$-even in (d)--(f). All the SAMBs retain the spatial inversion symmetry in terms of the inversion center denoted by the black dots.
• Figure 4: Band structures and densities of states (DOS) obtained from DFT calculations, together with the fitting results of the CW and SymCW models. Panels (a) and (b) correspond to displacement angles of $\varphi = \pm16.89^\circ$ and $\varphi = 0^\circ$, respectively. In the left panel, gray lines represent the DFT bands, while red and blue lines show the band structures obtained by the CW and SymCW models. The DOS per atom for orbitals of P, ${\rm O_{6i}}$, and Zr are shown in the right panel; the in-plane ($xy$) and $d_{u} \equiv d_{3z^2-r^2}$ components are represented in blue, the out-of-plane component in orange, the $s$ orbital in green, and the $4p$ orbital in yellow. The proposed models reproduce the DFT results well up to about 7 eV above the Fermi level, which is the energy range relevant to most physical properties.
• Figure 5: (a) Schematic representation of $\mathbb{Z}_{7}^{(G_{3b})}$. (b) Focusing on the region enclosed by the black rectangle in (a), $\mathbb{Z}_{7}^{(G_{3b})}$ corresponds to the spin-independent off-diagonal real hopping between the $p_{x}$ orbital at O$_{\rm 6i}$ site and the $p_{y}$ orbital at P site, (c) and this off-diagonal real hopping is allowed in the ferroaxial phase with $\varphi \neq 0$. In the same manner, panels (d), (e), and (f) illustrate the schematic representation of $\mathbb{Z}_{11}^{(G_{z\alpha})}$, corresponding to the spin-independent off-diagonal real hopping between the $p_y({\rm O_{6i}})$-$d_{zx}({\rm Zr})$ and $p_x({\rm O_{6i}})$-$d_v({\rm Zr})$ orbitals, its hopping prossess at $\varphi=0$, and that at $\varphi \neq 0$, respectively. Panels (g) and (h) display the representation and the hopping process of $\mathbb{Z}_{1}^{(G_{z})}$ and $\mathbb{Z}_{3}^{(Q_{4b})}$, respectively.