A general route to form topologically-protected surface and bulk Dirac fermions along high-symmetry lines

The band inversions that generate the topologically non-trivial band gaps of topological insulators and the isolated Dirac touching points of three-dimensional Dirac semimetals generally arise from the crossings of electronic states derived from different orbital manifolds. Recently, the concept of single orbital-manifold band inversions occurring along high-symmetry lines has been demonstrated, stabilising multiple bulk and surface Dirac fermions. Here, we discuss the underlying ingredients necessary to achieve such phases, and discuss their existence within the family of transition metal dichalcogenides. We show how their three-dimensional band structures naturally produce only small $k_z$ projected band gaps, and demonstrate how these play a significant role in shaping the surface electronic structure of these materials. We demonstrate, through spin- and angle-resolved photoemission and density functional theory calculations, how the surface electronic structures of the group-X TMDs PtSe$_2$ and PdTe$_2$ are host to up to five distinct surface states, each with complex band dispersions and spin textures. Finally, we discuss how the origin of several recently-realised instances of topological phenomena in systems outside of the TMDs, including the iron-based superconductors, can be understood as a consequence of the same underlying mechanism driving $k_z$-mediated band inversions in the TMDs.