Transition between one- and two-dimensional topology in a Chern insulator of finite width

Abstract

Topology in quantum systems is typically considered in infinite crystals in one, two, or higher integer dimensions. Here, we show that one can continuously transform a system between a topological phase associated with one dimension and a topological phase associated with two dimensions without closing the energy gap. In this process, the dimension of the system itself changes. Concretely, we investigate a modified version of the Qi-Wu-Zhang model and develop a procedure to smoothly shrink the width of the system in one direction. By tracking gaps which remain open throughout the modulation, we establish a smooth transition from a two-dimensional to a one-dimensional topological insulator. In between the system exhibits both one- and two-dimensional topology, and the way the system accomplishes the transition is by making the one-dimensional topology more robust as the width decreases, while the two-dimensional topology becomes less robust. Finally, we show how the gaps arise from hybridization of edge states due to the finite width.