  Chiral symmetry plays an indispensable role in topological classifications as
well as in the understanding of the origin of bulk or boundary flat bands. The
conventional definition of chiral symmetry refers to the existence of a
constant unitary matrix anticommuting with the Hamiltonian. As a constant
unitary matrix has constant eigenvectors, boundary flat bands enforced by
chiral symmetry, which share the same eigenvectors with the chiral symmetry
operator, are known to carry fixed (pseudo)spin polarizations and be
featureless in quantum geometry. In this work, we generalize the chiral
symmetry and introduce a concept termed sub-chiral symmetry. Unlike the
conventional chiral symmetry operator defined as constant, the sub-chiral
symmetry operator depends on partial components of the momentum vector, so as
its eigenvectors. We show that topological gapped or gapless systems without
the chiral symmetry but with the sub-chiral symmetry can support boundary flat
bands, which exhibit topological spin textures and quantized Berry phases. We
expect that such intriguing boundary flat bands could give rise to a variety of
exotic physics in the presence of interactions or disorders.
