Abstract: Some boundary phenomena in free-fermion systems fall outside symmetry-based classifications—for example, zero modes in Hermitian models with broken sublattice symmetry and skin modes in non-Hermitian systems without symmetry. This talk presents a general organizing principle based on a “subspace property”: if a Bloch Hamiltonian H(k) maps a k-independent subspace M into M’ (H(k)M ⊆ M’), point-gap invariants can be defined from the restriction H|_M. These invariants imply a bulk–boundary correspondence: when M = M’ they yield zero-winding skin effects under open boundaries, while M ≠ M’ enforces boundary zero modes even without sublattice symmetry. I will give minimal 1D examples (a triangular Hatano–Nelson double chain and an extended SSH model) and show how additional BDI-type constraints on H|_M produce a \mathbb{Z}_2 subclass verified by stacking. I will conclude with remarks on possible realizations in coupled lattices, circuits, and open systems.
