Fig. 3: Two types of surface states with double mirror symmetry.

Schematic band structures for Type I surface state with a single Dirac cone at \(\widetilde{\Gamma }\) (a) for unperturbed, under Zeeman perturbation with (b) \({\bf{B}}\parallel \hat{{\bf{x}}}\), (c) \({\bf{B}}\parallel \hat{{\bf{y}}}\), and (d) \({\bf{B}}\parallel \hat{{\bf{z}}}\). Insets show the surface Brillouin zones, with blue and magenta lines indicating k-planes invariant under \({M}_{x}\) and \({M}_{z}\), respectively. When mirror symmetry remains intact, the Dirac cone persists and shifts along the corresponding k-line associated with the mirror symmetry. Schematic band structures for Type II surface state with triple Dirac cones (e) for unperturbed, under Zeeman perturbation with (f) \({\bf{B}}\parallel \hat{{\bf{x}}}\), (g) \({\bf{B}}\parallel \hat{{\bf{y}}}\), and (h) \({\bf{B}}{||}\hat{{\bf{z}}}\). It consists of one TRS-protected Dirac cone at \(\widetilde{{\rm{U}}}\) and two (TRS + mirror)-protected ones at \(\widetilde{{\rm{X}}}\) and \(\widetilde{{\rm{Z}}}\). i–l Energy spectra for the zigzag surface of ε-phase terminated with In atoms, exhibiting a Type I surface state under magnetic field in the same directions as in (a-d). (m–p) Those terminated with a mixed atomic column of In and Te, exhibiting a Type II surface state under magnetic fields in the same directions as in (e–h).