Figure 1

Panels (a) and (b) show the spin and magnetoelastic mechanisms that couple the emergent gauge electric field $\mathbf{e}$ to an applied external field $\mathbf{E}$ shown here along the vertical ($y$) direction. Panel (a) depicts a spin configuration that has a net polarization along the electric field due to virtual charge fluctuations. Dark shaded bonds labeled $+$ and light shaded bonds labeled $-$ respectively indicate strong and weak spin-singlet correlations. The arrows schematically indicate spin orientation; since the individual spin operators ${\mathbf{S}}_{\mathbf{i}}$ do not commute with the polarization operator the true quantum states associated with definite polarization will be in some entangled superposition of similar spin configurations. Panel (b) shows a distortion of the Cu${}^{2+}$ ions induced by the electric field. This distortion alters the bond distance, which increases the exchange coupling $J_{ij}$ for dark bonds labeled $+$ and decreases it for light bonds labeled $-$. Both mechanisms have the same symmetry.Reuse & Permissions

Figure 2

The kagome point group is generated by mirror reflections, $M_x$, about the $x$ axis passing through the center of the vertical hour-glasses, and $\pi /3$ rotations, $R_{\pi /3}$, about the center of the hexagons.Reuse & Permissions

Figure 3

Blue and green dots are Cu and O ions, respectively. (a) DM arises from the bent Cu-O-Cu bonds. The pattern of $\mathbf{D}$ vectors is shown by $\otimes$ and $\odot$ indicating $\mathbf{D}$ into and out of the plane, respectively, for the bond orientation shown with arrows. (b) In an electric field, the oppositely charged Cu${}^{2+}$ and O${}^{2-}$ ions undergo relative displacements. The O ion displacement changes the bond angle, and the Cu displacement changes the bond exchange strength as described in Fig. 1. As explained in the text, this alters the DM pattern as indicated by the $+$ and $-$ labels, which couples the electric field to the spin current.Reuse & Permissions

Figure 4

Space-time Wilson loop corresponding to the emergent electric field $e_{ij}$ on lattice bond $\langle ij\rangle$.Reuse & Permissions