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 $\mathcal{T}$ and four-fold rotation symmetry $\mathcal{C}_4$ individually or the product symmetry $\mathcal{C}_4 \mathcal{T}$, 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 co-exist with topologically-protected, quasi-(3-2)D and quasi-(3-3)D boundary modes within the energy gap resulting from strong hybridisation 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.