Existence of a topological subspace in apparently non-topological systems

It is well known that inversion symmetry in one-dimensional periodic systems leads to the quantization of the Zak phase, which is either 0 or π. When the system has particle-hole symmetry, this topological property ensures the existence of zero-energy interface states at the interface of two bulk systems with different Zak phases. In the absence of inversion symmetry, the Zak phase can take any value and interface states are not ensured. Here we show that this is not true when the unit cell contains multiple degrees of freedom which can support hidden inversion symmetry in a subspace of the system. Specifically, we consider a model system of two SSH (Su-Schrieffer-Heeger) chains coupled by a coupler chain. Although the introduction of coupler chain breaks the inversion symmetry of the system, certain hidden symmetry ensures the existence of a decoupled 2 × 2 SSH Hamiltonian in the subspace of the entire system and the associated two SSH bands have quantized Zak phases. These “quantized” bands in turn can provide topological boundary or interface states in such systems. Since the entire system has no inversion symmetry, the bulk-boundary correspondence may not hold exactly. The above is also true when next-nearest-neighbor hoppings are included. We will also show that the quantization of Zak phase in the subspace still holds in the presence of gain and loss as long as the topological subspace is in the PT exact phase. Finally, we show the existence of Anderson localized states in the continuum of topological subspace when randomness is introduced to the on-site energies of the coupler sites. The above results are also true for multi-SSH chains. We will show that our systems can be realized by coupled acoustic cavities or photonic waveguides.