The Zak Phase and Topological Phase Transitions in One- and Two-Dimensional Models

The Zak phase is an open-path, gauge invariant integral of the Berry connection over the first Brillouin Zone, whose ends are connected by a reciprocal lattice vector. If the Zak phase changes value when varying parameters of a model, then the system has undergone a topological phase transition. In one-dimensional systems we calculate the Zak phase of multiorbital systems as a function of the location of the orbitals in the unit cell. Specifically, following work by Fuchs and Piéchon we study the one-dimensional Shockley and Su-Schrieffer-Heeger models and discuss how the first (second) model undergoes (does not undergo) a topological phase transition. In two dimensions, we start with the 3-band model of the CuO2 plane in a high-temperature superconductor and modify parameters along the two directions of space. Changing orbital hopping and onsite energies we find a potential phase transition when the local energies for the p and d-orbitals are equal.