Topological domain wall states in a nonsymmorphic chiral chain

The Su-Schrieffer-Heeger (SSH) model, containing dimerized hopping and a constant onsite energy, has become a paradigmatic model for one-dimensional topological phases, soliton excitations, and fractionalized charge in the presence of chiral symmetry. Motivated by the recent developments in engineering artificial lattices, we study an alternative model where hopping is constant but the onsite energy is dimerized. We find that it has a nonsymmorphic chiral symmetry and supports topologically distinct phases described by a ${\mathbb{Z}}_{2}$ invariant $\ensuremath{\nu}$. In the case of multimode ribbon we also find topological phases protected by hidden symmetries and we uncover the corresponding ${\mathbb{Z}}_{2}$ invariants ${\ensuremath{\nu}}_{n}$. We show that, in contrast to the SSH case, zero-energy states do not necessarily appear at the boundary between topologically distinct phases, but instead these systems support a new kind of bulk-boundary correspondence: The energy of the topological domain wall states typically scales to zero as $1/w$, where $w$ is the width of the domain wall separating phases with different topology. Moreover, under specific circumstances we also find a faster scaling ${e}^{\ensuremath{-}w/\ensuremath{\xi}}$, where $\ensuremath{\xi}$ is an intrinsic length scale. We show that the spectral flow of these states and the charge of the domain walls are different than in the case of the SSH model.