Angular momentum invoked band inversions in mirror symmetry protected topological states

The notion of a band inversion provides an intuitive physical paradigm for understanding the electronic band topology. Here we study the general band inversion mechanism, which exploits local atomic orbitals and lattice symmetry, in mirror protected topological crystalline insulators (TCIs). Based on low-energy effective theory analysis, we find that for these mirror protected TCIs, the topological invariant (i.e., the mirror Chern number ${\mathcal{C}}_{M}$) is determined by the difference of total magnetic quantum numbers ${m}_{j}$ of orbitals involved in the band inversion: $|{\mathcal{C}}_{M}|=|\mathrm{\ensuremath{\Delta}}{m}_{j}|$. This angular-momentum criterion is further verified by the atomic tight-binding model calculations in two-dimensional (2D) crystalline, quasicrystalline, and disordered lattices. Moreover, such an angular momentum invoked band inversion (AMBI) is also extendable to 3D lattices and gives rise to topological semimetals with Dirac points in the bulk and double Fermi arcs on surfaces. As a concrete material example, we further predict that the Ba monolayer is an AMBI-induced TCI by first-principles calculations. In addition, we also show that a large number of previously proposed TCI materials satisfy the AMBI mechanism. Our findings not only provide an alternative understanding of mirror protected band topology but also offer useful guidance for designing or engineering mirror protected TCI materials.