Fig. 1

Characteristics of mixed topological semimetals. a The magnetization direction \(\widehat {\mathbf{m}}\) = (sin θ, 0, cos θ) of a two-dimensional magnet encloses the angle θ with the z-axis perpendicular to the film plane. b Acting as sources or sinks of the Berry curvature, emergent band crossings in the mixed phase space of crystal momentum k = (k_x, k_y) and θ can be identified with jumps of the momentum Chern number \({\cal{C}}\) and the mixed Chern number \({\cal{Z}}\) upon passing through the nodal points. Alternatively, the topological nature of such a mixed Weyl point can be confirmed by calculating its charge as the flux of Berry curvature through the closed surface indicated by the grey box. c If the magnetic system is symmetric with respect to reflections at z = 0, nodal lines with the Berry phase γ = π may manifest in the corresponding (k_x, k_y)-plane of the mixed phase space. The inset illustrates the distribution of the generalized Berry curvature field Ω around the nodal line. d Mixed topological semimetals can host additionally a very distinct type of nodal lines that are one-dimensional manifolds evolving in θ as well as in k. Originating from the complex topology in the mixed phase space as revealed by a non-trivial Berry phase γ, these nodal lines give rise to a characteristic distribution of the Berry curvature as exemplified in the inset