Fig. 1: Elementary magnetic excitations in TbMn₆Sn₆.

a Ferrimagnetic structure of TbMn₆Sn₆ with key interlayer magnetic interactions indicated by black arrows and where the superscripts M and T stand for Mn and Tb, respectively. b 2D hexagonal Brillouin zone showing reciprocal lattice vectors a^* = (1, 0) and b^* = (0, 1) and representative high symmetry Γ (black circles), \({\Gamma }^{{\prime} }\) (empty circles), M (blue circles), and K (gold circles) points. INS data are described with the orthogonal vectors (1,0) and (-1, 2). Red arrows correspond to the three reciprocal space slices shown in panels (c–h). Panels c–e show the intensity of slices through the INS data along the (1 − K, 2K) (slice #1), (H, 0) (slice #2), and (H − 0.5, 1) (slice #3) directions, respectively, after averaging over L = − 7 to -7 rlu. High-symmetry points are indicated by vertical dashed white lines. In panels c and d, data with E < 150 meV ( > 150 meV) were collected with E_i = 250 meV (500 meV), respectively, as indicated by the black horizontal line. Data in (e) were collected with E_i = 500 meV. Panels f–h show identical slices as c–e calculated from linear spin wave theory using the parameters described in the main text and averaged over L = − 7 to -7 rlu. In panels c–g, the solid and dashed pink lines correspond to model dispersions with L = 0 and L = 1/2, respectively. In panel h, blue, pink, green, and white lines label the model dispersions of the Tb, AE/AO, OE/OO, and FE/FO modes, respectively, as described in the text. White (yellow) dashed rectangles outline overdamped excitations at the \({\Gamma }^{{\prime} }\) (K) point, respectively. i Energy cuts centered at the \({\Gamma }^{{\prime} }=(1,0)\) point (red), at the \({{{{{{{\rm{K}}}}}}}}=\left(\frac{2}{3},\frac{2}{3}\right)\) point (blue), and at the Γ = (2, 0) point (gray) with E_i = 500 meV. The data are scaled to the imaginary part of the dynamical susceptibility, χ^″(E), after correcting for the L-averaged magnetic form factor. Error bars from counting statistics are one standard deviation. Solid lines correspond to damped harmonic oscillator fits, as described in the Supplementary Information.