Fig. 6

Microscopics and prospects of nodal points in mixed topological semimetals. a The p-dominated valence and conduction states in the functionalized bismuth film realize an orbital inversion close to the Fermi energy, leading to an emergent band crossing for the generic direction θ  = 43°. The z-component of the orbital angular momentum \({\mathbf{L}} = - \mu _{\mathrm{B}}\mathop {\sum}\nolimits_{{\mathbf{k}}n}^{{\mathrm{occ}}} {\mathop {\sum}\nolimits_\mu {\left\langle {\psi _{{\mathbf{k}}n}^\theta | {\mathbf{r}}^\mu \times {\mathbf{k}}|\psi _{{\mathbf{k}}n}^\theta } \right\rangle _\mu } }\) of all occupied Bloch states \(|\psi _{{\mathbf{k}}n}^\theta \rangle\) is represented by colors, r^μ is the position relative to the μth atom, and the real-space integration is restricted to spherical regions around the atoms. b, c A finite electric field E repopulates the electronic states at the Fermi level \({\cal{E}}_{\mathrm{F}}\), which can be used to promote the net effect of mixed Weyl points on orbital magnetism. d, e Evolution of the orbital magnetization m_z(k) in the complex phase space of the crystal momentum k and θ in d the functionalized bismuth bilayer, and e the ferromagnet VOI₂. In both cases, the topological phase transition, which is accompanied by an emergent monopole in momentum space, happens for the critical value of θ that is indicated by the red star. f One-dimensional Fermi arcs connect the projections of the nodal points with opposite charge (red and blue dots) in a zigzag ribbon of the functionalized bismuth bilayer. Red and blue colors denote the localization of the Fermi arcs at opposite edges, and bold numbers refer to the evolution of the bulk Chern number \({\cal{C}}\) with the magnetization direction θ