  Exploration of topology protected by a certain symmetry is central in
condensed matter physics. A recent idea of sub-symmetry-protected (SSP)
topology--remains of a broken symmetry can still protect specific topological
boundary states--has been developed and demonstrated in an optical system [Nat.
Phys. 19, 992-998 (2023)]. Here, we extend this idea further by applying
sub-symmetry-protecting perturbation (SSPP) to one-dimensional topological
insulating and superconducting systems using the Su-Schrieffer-Hegger (SSH) and
Kitaev models. Using the tight-binding and low-energy effective theory, we show
that the SSP boundary states retain topological properties while the SSPP
results in the asymmetry of boundary states. For the SSH model, an SSP
zero-energy edge state localized on one edge possesses quantized polarization.
In contrast, the other edge state is perturbed to have non-zero energy, and its
polarization is not quantized. For topological superconductors, zero-energy SSP
Majorana boundary states for spinful Kitaev models emerge on only one edge,
contrary to the conventional belief that Majorana fermions emerge at opposite
edges. Our findings can be used as a platform to expand our understanding of
topological materials as they broaden our understanding of the symmetry in a
topological system and a method to engineer Majorana fermions.
