A set of perturbations, which preserves the topological invariant (or invariants) and respects a particular symmetry (or symmetries), is encircled with a grey line. Every boundary state, when it exists, is protected with its pertinent SubSy. In the illustration, two sets of SubSy-preserving perturbations (encircled with red and green lines) are sketched, but their number depends on the actual system. At the overlap region, one has an SPT phase with a topological invariant (illustrated as a Mobius strip) characterizing the bulk (illustrated with a thick black line), and all associated boundary states (illustrated with red and green circles). For the SSH lattice, the SPT phase is protected by a chiral symmetry (with the implicit assumption of inversion symmetry). The topological invariant is protected by the inversion symmetry (even when the chiral symmetry is broken). The edge state on the A sublattice is protected with A-SubSy, which is defined by a chiral symmetry equation that holds solely on the A sublattice. The same holds for the edge state on the B sublattice with B-SubSy as the protecting sub-symmetry. For the BKL with negligible long-range hopping (see main text), the C 3 symmetry protects the topological invariant, whereas there are three SubSys corresponding to three BKL sublattices, which protect the pertinent higher-order topological corner states. See text for details.
