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

TL;DR

The paper addresses how topology in finite-width systems can interpolate between one- and two-dimensional topological phases without closing the energy gap. It analyzes a finite-width Chern insulator based on a modified Qi-Wu-Zhang model and introduces a smooth width modulation via edge potentials to drive a continuous 2D→1D transition. The authors employ a Wilson loop to characterize one-dimensional topology and an adiabatic flux insertion (adiabatic charge pump) to characterize two-dimensional topology, showing that gaps arise from finite-width edge-state hybridization and can persist throughout the modulation. The results reveal an inverse relation between the robustness of 1D and 2D topological signatures and provide a framework for exploring finite-size topological phases and potential experimental realizations in layered materials.

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.

Figures (9)

• Figure 1: The spectrum of $H_{\textrm{pp}}$ (black), $H_{\textrm{op}}$ (dark blue) and $H_{\textrm{oo}}$ (light blue) for $L_x=6$. Six intervals of length $\Delta M = 4$ are shown in magenta with their edges coinciding with the gap closings. The spectrum of $H_{\textrm{op}}(k_y=0)$ is shown with positive (negative) parity states in light (dark) green. Likewise, the spectrum of $H_{\textrm{op}}(k_y=\pi)$ is shown with positive (negative) parity states in gold (orange). The spectra are symmetric around $E=0$ due to charge conjugation symmetry and around $M=0$ due to \ref{['uniequ']}.
• Figure 2: A visualization of the smooth width modulation of the spectrum for $L_x=6$. (a) A snapshot of the spectrum for added edge potential $H_{\text{edge}}(u)$ with $u=0.918$. Two of the $\Delta M = 4$ intervals merge while they are shifted to the left away from the rest of the spectrum. The remaining part forms the $L_x=4$ spectrum. We label the four rightmost gaps from one to four for later reference as they never close. (b) The gap closings of the spectrum throughout the modulation. Gaps closings for $k_y=0$ ($k_y=\pi$) are highlighted in blue (red). Here gabs one through four can be seen to remain open throughout the whole process.
• Figure 3: The spectrum of $H_{\textrm{op}}$ for $L_x=6$. The gaps are colored according to the Wilson topological signature $N_{(-)}$ with trivial $N_{(-)}=0$ gaps depicted in white and non-trivial $N_{(-)}=1$ gaps depicted in green.
• Figure 4: Visualization of gaps in the spectrum throughout the smooth modulation of the width. (a) The size of a topological (non-topological) energy gap shown in blue (purple) at the $M$ value shown in red (orange) as it is tracked throughout the smooth width modulation. This shows that for smaller values of $L_x$ gaps become more prominent. (b) Gaps are shown throughout the modulation with topological (non-topological) gaps highlighted in green (white). The black lines show the gap closings. It is seen that the top six gaps remain open throughout the whole modulation.
• Figure 5: Visual representation of how a flux-tube (red) is threaded though the periodic crystal (blue) in a way that preserves inversion symmetry.