AI 中文总结
本文提出一类可变步长的拓扑量子行走，通过转移矩阵方法研究其拓扑保护的边缘态，实现对边缘态数量、空间轮廓和自旋结构的控制。
AI 中文摘要
拓扑绝缘体拥有受保护的边界态，这些边界态对无序具有鲁棒性，使其在应用中具有吸引力，并推动了在工程化系统中的实现。离散时间量子行走已在多种实验平台上实现，能够表现出拓扑非平凡相及其相关的边界态。在此，我们引入一类具有可变步长的拓扑量子行走，为这类行走的拓扑分类提供了理论见解。例如，这些行走可以实现更高的绕数并支持多个边缘态。它们还为时间反演对称性破缺时拓扑保护从$\mathbb{Z}$值不变量降为$\mathbb{Z}_2$值不变量提供了背景。我们使用转移矩阵方法研究了拓扑保护的边缘态，该方法描述了它们的空间轮廓和自旋结构。这些预测与数值结果以及适用情况下的狄拉克方程的Jackiw--Rebbi零模解高度一致。综合来看，我们的分析为设计量子行走提供了路线图，可控制其拓扑边缘态的数量、空间轮廓和自旋结构。
英文摘要
Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a $\mathbb{Z}$- to $\mathbb{Z}_2$-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw--Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.
Comments9 pages, 5 figures
