Purely Electronic Chirality without Structural Chirality

TL;DR

Purely electronic chirality (PEC) is introduced as an ordering of electronic degrees of freedom that yields a finite chirality without any atomic displacement, via an electric toroidal monopole G0 generated by the cross-coupling of quadrupole moments and orbital/spin textures. In a distorted kagomé lattice, a uniform q=0 120° quadrupole order produces hedgehog-like spin-momentum locking and spin textures, with the chirality order parameter C_R = \epsilon_{\mu\nu z} Q_{\mu} Q_{\nu z}^{R} that couples to conduction electrons, enabling magnetic-field control of chirality domains and associated magnetoelectric and nonreciprocal effects. URhSn is discussed as a concrete candidate, where a minimal localized spin-1 model reproduces a two-stage transition at T_o ~ 54 K and T_c ~ 15 K, with a ferromagnetic phase accompanied by a magnetic toroidal moment due to PEC. Furthermore, PEC induces coupling to lattice dynamics, giving rise to truly chiral phonons in achiral crystals with estimated relative splittings on the order of 1%, suggesting a measurable phonon analogue of PEC. The work thus establishes PEC as a distinct origin of chirality, distinct from structural chirality, and highlights the interplay between electronic order and lattice responses with potential for fast, field-tunable chirality control.

Abstract

We introduce the concept of purely electronic chirality (PEC), which arises in the absence of structural chirality. In condensed matter physics and chemistry, chirality has conventionally been understood as a mirror-image asymmetry in crystal or molecular structures. We demonstrate that certain electronic orders exhibit chirality-related properties without atomic displacement. Specifically, we investigate quadrupole orders to realize such purely electronic chirality with handedness that can be tuned by magnetic fields. As a representative example, we analyze a model featuring $120^circ$ antiferro quadrupole orders on a distorted kagomé lattice, predicting various chirality-related responses in the nonmagnetic ordered phase of URhSn. Furthermore, as a phonon analog, chiral phonons can emerge in achiral crystals through coupling with the PEC order. Our results provide a distinct origin of chirality and a fundamental basis for exploring the interplay between electronic and structural chirality.

Figures (5)

• Figure 1: Examples of purely electronic chirality in achiral crystals due to quadrupole moments. (a) breathing kagomé and (b) distorted kagomé structures. The sublattice labels are indicated as $n=1,2,3$. Circles indicate three-fold rotation axes. The filled, open, and grey circles represent inequivalent axes. In (b), the position of the unit cell is taken at the center of the distorted hexagon and denoted as ${\bm{R}}$. $\theta$ is the distortion angle parameter; $\theta=60^\circ$ for regular kagomé structure.
• Figure 2: (a) Dispersion $\epsilon_{\bm{k}}$ along the path K$^\prime$-$\Gamma$-K-M-$\Gamma$-M-K$^\prime$ for $k_xk_y\ge 0$ indicated by the line in (b) for $g=1/\sqrt{2}$, $t_\sigma=0.85$, $t_\pi=0.15$, and $\theta=75$ deg. with the other parameters being indicated in (c). Color represents the direction of spin $\bm{\sigma}_{\bm{k}}$ for the Bloch state on the $xy$ plane as illustrated in (b). (b) Fermi surfaces for $\mu_c=-1$. $\bm{\sigma}_{\bm{k}}$'s are on the $xy$ plane and indicated by arrows. (c) $\alpha_{xx}(=\alpha_{yy})$ as a function of $g$ for $\mu_c=-1$, $T=1.0$ and for several values of distortion angle $\theta$. $\theta=60$ deg. corresponds to the regular kagomé. (d) A schematic configuration of quadrupole and magnetic moments under magnetic fields along the $z$ direction inside the chiral phase or the FM state with the chiral quadrupole order. Green arrows are the magnetic moments that tilt toward the quadrupole principal axis, and this induces a finite scalar spin chirality for the triangular plaque.
• Figure S1: Multipole basis set for the sublattice degrees of freedom in the distorted kagomé lattice. The red and blue spheres represent the positive and negative charge, respectively. The arrows represent the directions of the current. Numbers on the spheres and arrows denote the values of the charge and current, respectively, which correspond to the matrix element in Eq. (\ref{['eq:basis']}).
• Figure S2: (a) $T$ dependence of the order parameters for $\bm{h}=0$. The arrows indicate the transition temperatures $T_{o,c}$. (b) $T$--$h_z$ phase diagram for ${\bm h}=(0,0,h_z)$, $J_z$$=$$4.5$ K, $J_\perp$$=$$15$ K, $J_\perp^{\rm ani}$$=$$10$ K, $K$$=$$45$ K, $\delta$$=$$5$ K, and $\Delta$$=$$40$ K. Schematic order parameter configurations are depicted for chirality and the non-coplanar ferromagnetic orders. The value of $h_z$ includes the size of the dipole moment.
• Figure S3: Examples of PEC caused by quadrupole orders. The sites are drawn by orange spheres, and the quadrupole moments are colored green/yellow. $\{Q_{yz},Q_{xz}\}$ type quadrupole order in (a) breathing honeycomb lattice and (b) breathing square lattice. (c) Antiferroic $Q_{x^2-y^2}$ quadrupole order in the tetragonal diamond structure. (d) Quadrupole order of icosahedron unit cell, which preserves cubic symmetry. The icosahedra are assumed to form a cubic Bravais lattice. In (a), the filled (open) circle is the C$_6$ (C$_3$) rotation axis. In (b), the filled (open) circle is the C$_4$ rotation axis at the center of a larger (smaller) square. In (c), the filled (open) circles are $4_1$ screw ($4_3$ screw) axes.