Hidden magnetic order on a kagome lattice for ${\mathrm{KV}}_3{\mathrm{Sb}}_5$

${\mathrm{KV}}_3{\mathrm{Sb}}_5$ has recently attracted considerable attention due to its low-temperature superconducting properties, which are heralded by a charge-density wave. The apparent presence of a very weak magnetism does not result in long-range ordering. An explanation of the properties we present invokes higher-order terms in the vanadium magnetization density and a “hidden order” of Dirac (polar) multipoles. The Dirac dipole, known as an anapole or toroidal dipole, is one of a family of electronic multipoles visible in x-ray and magnetic neutron diffractions while undetectable with standard laboratory-based techniques. Actually, two viable magnetic structures, direct descendants of the established chemical structure, are studied with a view to testing their suitability in future experiments. One model structure is magnetoelectric and restricted to the linear type, whereas a second model cannot show a magnetoelectric effect of any type. The latter hosts a strange vanadium entity that is a true scalar and magnetic (time-odd), and associated in our paper with a fictitious charge distribution that is purely imaginary. Calculated x-ray and neutron-scattering amplitudes are symmetry-informed expressions of vanadium Dirac multipoles. Bragg diffraction patterns for the two models are found to be distinctly different, fortunately. Likewise, magnetochiral signals derived from our x-ray scattering amplitudes.

Figure 1

Depiction of a toroidal dipole also known as an anapole.

Figure 2

Primary $\left(\sigma ,\pi \right)$ and secondary $\left({\sigma }^{\prime },{\pi }^{\prime }\right)$ states of polarization. Corresponding wave-vectors q and q' subtend an angle 2θ. The Bragg condition for diffraction is met when $\mathbf{q}-{\mathbf{q}}^{\prime }$ coincides with the reflection vector indexed $\left(h,k,l\right)$. Crystal vectors a, ${\mathbf{b}}^{*}$, and c that define $\left(\xi ,\eta ,\zeta \right)$ and depicted Cartesian $\left(x,y,z\right)$ coincide in the nominal setting of the crystal.

Figure 3

Anapole motif in $P{6}^{\prime }/mm{m}^{\prime }$ (No. 1911. 237) viewed in two perspectives. They are depicted in Fig. 1 and are represented here by dark blue arrows. Two anapoles are normal to the crystal a axis and parallel to hexagonal $\left[1,2,0\right]\propto {\mathbf{b}}^{*}$ in the bottom panel. Cell vectors a and b subtend an angle ${120}^{\circ }$ (Sec. 2). Light blue and gray spheres represent Sb and K ions, respectively.

Figure 4

Ferromotif of anapoles in $P6/m^{\prime }mm$ (No. 191.235). Cell vectors a and b subtend an angle ${120}^{\circ }$ (Sec. 2). Light blue and gray spheres represent antimony and potassium ions, respectively.