Realizing strictly one-dimensional properties in atomic wire arrays on a semiconducting substrate through a dispersive two-dimensional system.

We consider models of one and two Su-Schrieffer-Heeger wires coupled to simple cubic semiconducting substrate and investigate the topological phases of these models with respect to the model parameters, eg. dimerization, wire-wire coupling, wire-substrate hybridization.The wire-wire coupling is either perpendicular or diagonal hopping respecting the particle-hole and time-reversal symmetries. The hybridization to the substrate renormalizes the model parameters of the wires towards the hopping parameter of the substrate without altering the basic nature of perpendicular or diagonal coupling and it can mediate effective perpendicular hopping but not diagonal hopping in the absence of direct wire-wire coupling. This motivates the investigation of multi uniform tight binding wires with perpendicular or diagonal hopping parameters while excluding the substrate. Perpendicularly coupled uniform wires reveal anisotropic two dimensional band dispersion. Diagonally coupled uniform wires reveal strictly one dimensional bands parallel to the wires direction under certain conditions. We demonstrated the possibIlity of realizing strictly one dimensional properties emerging from dispersive two dimensional systems if time-reversal and particle-hole symmetries are preserved. This supports discussion on one dimensional behavior of the Au/Ge(001) surace reconstructions.