Topological invariants to characterize universality of boundary charge in one-dimensionalinsulators beyond symmetry constraints

In the absence of any symmetry constraints we address universal properties of the boundary charge $Q_B$ for a wide class of nearest-neighbor tight-binding models in one dimension with one orbital per site but generic modulations of on-site potentials and hoppings. We provide a precise formulation of the bulk-boundary correspondence relating the boundary charge of a single band uniquely to the Zak phase evaluated in a particular gauge. We reveal the topological nature of $Q_B$ by proving the quantization of a topological index $eI=\mathrm{\Delta }{Q}_B-\bar{\rho }$, where $\mathrm{\Delta }{Q}_B$ is the change of $Q_B$ when shifting the lattice by one site towards a boundary and $\bar{\rho }$ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between the two lattice sites defining the boundary of a half-infinite system. For a given chemical potential we establish a central topological constraint $I\in \{-1,0\}$ related only to charge conservation of particles and holes. Our results are shown to be stable against disorder and we propose generalizations to multi-channel and interacting systems.