Fig. 4

Electronic properties of mixed topological semimetals. a Spin-resolved band structure and (b) energy dispersion of a finite ribbon, where the localization at the edge is indicated by colors ranging from blue (weak) to red (strong), of the TlSe monolayer with an in-plane exchange field of magnitude B = 0.5 eV. c Around the X point, for example, the emergence of nodal points with opposite topological charge (red and blue balls) for reversed in-plane directions θ of the magnetization imprints characteristic features on the phase-space distribution of the Berry curvature field \({\mathrm{\Omega }} = ( - {\mathrm{\Omega }}_{yy}^{\widehat {\mathbf{m}}{\mathbf{k}}},{\mathrm{\Omega }}_{yx}^{\widehat {\mathbf{m}}{\mathbf{k}}},{\mathrm{\Omega }}_{xy}^{{\mathbf{kk}}})\) as indicated by the arrows. A logarithmic color scale from blue (negative) to red (positive) is used to illustrate the mixed Berry curvature \({\mathrm{\Omega }}_{yx}^{\widehat {\mathbf{m}}{\mathbf{k}}}\) in the complex phase space of k and θ. d Spin-resolved band structure of one-layer Na₃Bi with an exchange field of magnitude B = 0.5 eV applied perpendicular to the film. Owing to the mirror symmetry of the system, the band crossings around Γ form a mixed nodal line in momentum space, which disperses as illustrated in the inset, where colors indicate energy differences between the crossing points and the Fermi level in meV