Sensitivity of quantum walks to a boundary of two-dimensional lattices: approaches based on the CGMV method and topological phases
Sensitivity of quantum walks to a boundary of two-dimensional lattices: approaches based on the CGMV method and topological phases
复制标题
量子游走对二维晶格边界的敏感性：基于 CGMV 方法和拓扑相的方法
DOI：
10.1088/1751-8121/aa8c5e
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Etsuo Segawa
中科院分区：
文献类型：
--
作者：
Takako Endo;Norio Konno;Hideaki Obuse;Etsuo Segawa
在本文中，我们处理二维晶格中的量子行走，沿着沿着Asboth和Edge（2015 Phys. Rev. A 91 022324）引入的直边界切割边缘，以研究源自物质拓扑相的一维边缘态，并获得量子步行者如何对边界作出反应的附带证据。首先，我们将该模型连接到CMV矩阵，该矩阵提供了与单位圆上的谱测度相关的洛朗多项式的5项递归关系。其次，我们利用CMV矩阵的谱分析，明确地导出了带边界量子行走的体态和边态的谱。第三，虽然迄今为止所研究的模型的拓扑数是很好地定义时，在体光谱中存在的间隙，我们发现一个新的拓扑数定义时，只有没有间隙的体光谱。我们确认，来自CMV矩阵的边缘状态的频谱的存在是一致的预测从散装边缘对应使用拓扑数计算的情况下，在散装频谱中的间隙不存在或不存在。最后，我们展示了边缘态如何对量子穿越发现概率极限定理的渐近行为做出贡献。相反，我们也提出了一个微分方程使用这个极限分布的解决方案是潜在的边缘状态。
In this paper, we treat quantum walks in a two-dimensional lattice with cutting edges along a straight boundary introduced by Asboth and Edge (2015 Phys. Rev. A 91 022324) in order to study one-dimensional edge states originating from topological phases of matter and to obtain collateral evidence of how a quantum walker reacts to the boundary. Firstly, we connect this model to the CMV matrix, which provides a 5-term recursion relation of the Laurent polynomial associated with spectral measure on the unit circle. Secondly, we explicitly derive the spectra of bulk and edge states of the quantum walk with the boundary using spectral analysis of the CMV matrix. Thirdly, while topological numbers of the model studied so far are well-defined only when gaps in the bulk spectrum exist, we find a new topological number defined only when there are no gaps in the bulk spectrum. We confirm that the existence of the spectrum for edge states derived from the CMV matrix is consistent with the prediction from a bulk-edge correspondence using topological numbers calculated in the cases where gaps in the bulk spectrum do or do not exist. Finally, we show how the edge states contribute to the asymptotic behavior of the quantum walk through limit theorems of the finding probability. Conversely, we also propose a differential equation using this limit distribution whose solution is the underlying edge state.