In a class of systems, there are gapped boundary-localized states described by a boundary Hamiltonian. The topological classification of gapped boundary Hamiltonians, same as the standard tenfold way for gapped bulk states, can lead to the emergence of boundary topological insulators (TIs) and superconductors (TSCs). In this work, we present a theoretical study of boundary TIs and TSCs of the full Altland-Zirnbauer tenfold symmetry classes. Based on the boundary projection analyses for a $d$-dimensional Dirac continuum model, we demonstrate that nontrivial boundary topology can arise at a $(d-n)$-dimensional boundary if the Dirac model incorporates ($n+1$) mass terms with $0<n<d$ although its bulk and $(d-1)\mathrm{D}, \cdots , (d-n+1)\mathrm{D}$ boundaries are topologically trivial. Furthermore, we present a unified criterion for the emergence of nontrivial boundary topology by extending bulk classification within the context of the Dirac model, which provides a unified framework for nontrivial bulk and boundary topology. Inspired by the Dirac continuum model analysis, we further construct bulk lattice Hamiltonians for realizing boundary TIs and TSCs of the full Altland-Zirnbauer tenfold symmetry classes, which enables the realization of higher-order TIs and TSCs in arbitrary dimensions with arbitrary orders. We analyze some typical examples of the constructed boundary TIs and TSCs in physical dimensions.

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• Topological insulators
• Topological phases of matter
• Topological superconductors
• CPT symmetry
• Topology