Fig. 1: Spin transport in frustrated hyperkagome magnet GGG.

a Sketches of spin wave in ordered magnet and fluctuating spins in frustrated magnet. The opaque arrows represent the localized magnetic moments, while the transparent arrows represent the fluctuations of the moments. b Schematic of the long-range multipole order formed from ten-spin loops of Gd³⁺ in GGG. Orange double-headed arrows indicate the ten-spin director that are ordered normal to the loop plane. Smaller gray arrows represent the localized magnetic moments. c Crystal structure of hyperkagome magnet GGG with only Gd atoms shown. The two inter-penetrating corner-sharing Gd³⁺ triangular sublattices are colored in gray and pink, respectively. d Schematics of the nonlocal spin transport measurement setup with electrical connections. Two Pt wires are patterned on a GGG slab, one to inject spin current by thermal excitation and the other to detect the spin current through inverse spin Hall effect. Specifically, \({I}_{{in}}\): AC injection current; \({V}_{2\omega }\): the second harmonic thermal magnon inverse spin Hall signal; θ: the angle of the in-plane magnetic field B with respect to the direction perpendicular to the Pt electrodes. The inset shows angle dependent \({V}_{2\omega }\) of GGG with \(B=3{{\rm{T}}}\) and \({I}_{{in}}=80 \mu A\) and T = 2 K, in which the black solid line is a cosine function fit.