Fig. 4: Spin-orbit excitons: dispersions and Dirac node.

a Schematic level splitting for a Co²⁺ ion in an octahedral crystal field of trigonal symmetry including spin-orbit coupling. b INS energy scan observing transitions to the first two excited crystal field levels (the blue/red arrows above the peaks show the transitions indicated by matching colour vertical arrows in a, the solid line is a guide to the eye. c INS data probing the dispersions of the first crystal field level along high-symmetry directions, compared in d with a tight-binding model (thick solid/dashed lines through both graphs show best fit dispersions). e Angular intensity dependence around the nodal point (2/3,5/3) for the top/bottom exciton bands fitted to an \({{\mathcal{A}}}_{\pm }\pm {{\mathcal{B}}}_{\pm }\cos (\alpha -{\tilde{\alpha }}_{0})\) form (solid lines, \({\tilde{\alpha }}_{0}=155{(3)}^{\circ }\), calculated 153(1)^∘, in-plane radial wavevector range [0.075, 0.3] Å⁻¹). The black squares/white circles denote the inelastic neutron scattering intensity for the top/bottom exciton bands, respectively, with error bars representing one standard deviation. Note the analogous behaviour to the intensity dependence in azimuthal scans for Dirac magnons in Fig. 2c. f Exciton bands crossing at the two labelled nodal Dirac points, analogous to the magnon bands crossing in Fig. 2f). In e, f intensities are averaged for L = [0, 3.5], in c for a transverse wavevector range ±0.1 Å⁻¹, and in f for a transverse in-plane wavevector range ±0.025 Å⁻¹. Data were collected at 8 K with E_i = 83 meV in b and 45 meV in c, e, f. The colour bar in f also applies to c and d, and indicate scattering intensity in arbitrary units on a linear scale.