Wilson Line and Disorder Invariants of Topological One-Dimensional Multiband Models

TL;DR

This work addresses the limitation of traditional 1D topological invariants in multiband systems by introducing an unwrapped Wilson line that yields a robust $\mathbb{Z}$ invariant $\gamma$. The method provides a numerically efficient tool to classify multiband SSH-type models, including SSH$_4$ and extended SSH families, and reconciles bulk topology with edge-state spectra through the bulk-boundary correspondence. It demonstrates that $\gamma$ can capture edge-state information beyond the winding number and Zak phase, including nonzero-energy edge states, and shows how the sign of $\gamma$ can be inferred from a Hall-like experimental configuration. The results emphasize the role of symmetry in protecting edge states and reveal a rich phase structure that extends to higher-dimensional generalizations, offering a practical framework for probing 1D topological phases in real systems.

Abstract

Topological invariants, such as the winding number, the Chern number, and the Zak phase, characterize the topological phases of bulk materials. Through the bulk-boundary correspondence, these topological phases have a one-to-one correspondence to topological edge states, which are robust to certain classes of disorder. For simple models like the Su-Schrieffer-Heeger (SSH) model, the computation of the winding number and Zak phase are straightforward, however, in multiband systems, this is no longer the case. In this work, we introduce the unwrapped Wilson line across the Brillouin zone to compute the bulk topological invariant. This method can efficiently be implemented numerically to evaluate multiband SSH-type models, including models that have a large number of distinct topological phases. This approach accurately captures all topological edge states, including those overlooked by traditional invariants, such as the winding number and Zak phase. To make a connection to experiments, we determine the sign of the topological invariant by considering a Hall-like configuration. We further introduce different classes of disorder that leave certain edge states protected, while suppressing other edge states, depending on their symmetry properties. Our approach is illustrated using different one-dimensional models, providing a robust framework for understanding topological properties in one-dimensional systems.

Figures (8)

• Figure 1: Top drawing illustrates the E$_n$SSH, which is an extended SSH model with two sites, A and B, per unit cell. In this example, the E$_2$SSH is plotted. The bottom drawing illustrates the SSH$_n$ model, where the unit cell is composed of $n$ atoms linearly coupled by $t_i$. In this example, the SSH$_4$ has a unit cell composed of A, B, C, and D sites.
• Figure 2: Wilson line invariant $\gamma$ from eq. \ref{['eq:gamma']} as a function of $t_3$ and $t_4$ with $t_1,t_2 = 1,2$ for the E$_1$SSH model.
• Figure 3: The winding number, $\nu$ (eq. (12)) and the Wilson line invariant, $\gamma$ (eq. (17)) of the E$_9$SSH is shown as a function of $t_2$, where ; $t_1=-t_2-0.7$, $t_3=1-t_2$, and $t_4=t_1-1$. Both, $\nu$ and $\gamma$, are computed numerically and show no visible differences. Also shown, is the number of edge states for the finite E$_9$SSH model using N=4000 and N=16000.
• Figure 4: (a) the linear representation of the periodic E$_3$SSH model. (b) The Hall configuration of the finite E$_3$SSH model. (c) The Wilson line invariant $\gamma$ of the periodic E$_3$SSH model as a function of $t_2$ and the Hall resistance as a function of $t_2$ and energy of the finite E$_3$SSH model for $N=800$ and all couplings to the leads are set to $t_L=0.3$. The Hall voltages ($V_1$ and $V_2$) are 3 atoms away from $V_S$ (like in the illustration). The other hopping parameters are given by $t_1=-t_2-0.7$, $t_3=1-t_2$ and $t_4=t_1-1$.
• Figure 5: The top graph shows the original phase and the bottom shows the unwrapped phase for parameters $t_1=0.52,t_2=1.5,t_3=0.39, t_4=2$.