Crystalline finite-size topology

Topological phases stabilized by crystalline point group symmetry protection are a large class of symmetry-protected topological phases subjected to considerable experimental scrutiny. Here, we show that the canonical three-dimensional (3D) crystalline topological insulator protected by time-reversal symmetry T and fourfold rotation symmetry C4 individually or the product symmetry C4T, generically realizes finite-size crystalline topological phases in thin film geometry [a quasi-(3−1)-dimensional, or q(3−1)D, geometry]: response signatures of the 3D bulk topology coexist with topologically protected, quasi-(3−2)D, and quasi-(3−3)D boundary modes within the energy gap resulting from strong hybridization of the Dirac cone surface states of the underlying 3D crystalline topological phase. Importantly, we find qualitative distinctions between these gapless boundary modes and those of strictly 2D crystalline topological states with the same symmetry protection and develop a low-energy, analytical theory of the finite-size topological magnetoelectric response. Published by the American Physical Society 2024