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