Fig. 3. Stabilization mechanism and dynamical properties of the tetrahedral order in Co_1/3TaS₂.

a The in-plane crystal structure of Co_1/3TaS₂ demonstrates isolated CoS₆ octahedrons (purple-colored) and a long NN Co-Co distance. b The exchange interaction between Co local moments and conduction electrons from TaS₂ layers leads to an effective RKKY interaction between the local moments. c The Fermi surface of a 2D TLAF with 3/4 filling (shaded hexagons) and the Fermi surface of Co_1/3TaS₂ measured by ARPES. d The magnon spectra of Co_1/3TaS₂ at 5 K along the (00 L) direction. E_i = 7.9 and 14 meV data are plotted. e Antiferromagnetic NN interlayer coupling (J_c) of Co_1/3TaS₂, which is necessary for explaining the data in d and the refined spin configuration (Fig. 2g–i). f Const-E cuts of the INS data measured at 5 K (<T_N2). An energy integration range for each plot is $\pm$0.2 meV. The E = 1 and 1.5 meV (2.0~3.0 meV) plots are based on the E_i = 5 (7.9) meV data. In addition to bright circular spots centered at six M points (=linear modes), a weak, diffuse ring-like scattering which we interpret as the quadratic mode predicted by spin-wave theory, appears for E > 1.5 meV. g The calculated INS cross-section of the tetrahedral triple-Q ordering with J₁S² = 3.92 meV, J_cS² = 2.95 meV, J₂/J₁ = 0.19, and K_bq/J₁ = 0.02 (see Supplementary Materials). h The calculated INS cross-section of the single-Q ordering with three domains, using J₁S² = 3.92 meV, J_cS² = 2.95 meV, J₂/J₁ = 0.1, and K_bq/J₁ = 0. The line-shaped signal in h has a much higher intensity than in f or g. The simulations in g, h include resolution convolution (see Supplementary Fig. 9), and their momentum and energy integration range are the same as f. i Low-energy magnon spectra measured with E_i = 3.5 meV at 5 K, showing the energy gap of the linear magnon mode.