拓扑物态的量子模拟和物性研究

拓扑物态是指一类特殊的物质态，其性质（尤其是电学性质）不受物理条件扰 动的影响。拓扑物态的鲁棒性使其在拓扑量子计算和自旋电子学器件领域具有广 泛的潜在应用。因此，拓扑物态的测量与表征的研究已成为物理学领域的研究热 点。本论文旨在研究拓扑物态，主要包括具有长程纠缠的拓扑态（即拓扑序）和受 对称性保护的短程纠缠拓扑态。我们对这两种拓扑物态进行了深入研究，探讨了 如何区分不同拓扑序的方法，提出并实现了完整确定拓扑序的实验测量方案；同 时也研究了对称保护拓扑态受维度变化的影响，包括一维拓扑态向二维拓扑态的 推广；以及三维拓扑态在量子限制下的输运性质变化。

本文的第二、三章讨论了长程纠缠的拓扑序的量子模拟、分类区分和完备测 量的问题。在第二章中，我们考虑在环面上的弦网模型，其中包括阿贝尔和非阿 贝尔的拓扑序模型，用量子散射电路测量了模数据信息（𝑆 矩阵和 𝑇 矩阵），成功 区分了阿贝尔和非阿尔贝的拓扑序。由于模数据信息并不能完备地区分所有的拓 扑序，因此在第三章中，我们研究了完备描述拓扑序信息的测量方案。基于体边 对应关系和任意子凝聚性质，我们设计了测量 𝐹 矩阵和 𝑅 矩阵的测量方案，并通 过核磁共振系统模拟了 toric code 模型且成功地实现了两个矩阵的测量，并证明了 这两个矩阵可以完备地描述拓扑序。

本文的第四、五章研究了维度变化对于对称保护拓扑态的影响。在第四章中， 我们对最简单的拓扑模型 Su-Schrieffer-Heeger（SSH）模型进行了维度的扩展，详 细地计算和分析了二维 SSH 模型的电子结构和拓扑性质，得到了拓扑边缘态的平 带特征。通过分析这些边缘态的空间分布，我们解释了边缘平带的起源，并表明 键合方块在平带的形成中起着至关重要的作用。在第五章中，我们通过对三维拓 扑节点环超导体施加量子维度限制，研究了拓扑节点环超导体的多个马约拉纳表 面态，探讨了（1）金属/势垒/拓扑节点环超导体结和（2）拓扑节点环超导体上接 两个薄膜这两种构型下马约拉纳表面态辅助的安德烈夫反射的特性。这两种构型 分别可以测量动量空间和实空间中的多个马约拉纳表面态分布。利用垂直于结方 向的马约拉纳表面态的非局域性，这种空间移动的安德烈夫反射以及干涉图案可 能提供另一种途径来确认量子限制系统中的马约拉纳束缚态。

在本论文中，我们取得了以下研究成果。首先，我们针对具有长程纠缠的拓扑 序进行了量子模拟、分类区分和完备测量，特别是编织操作相关的 𝑅 矩阵的测量，为拓扑序在拓扑量子计算中的应用奠定了基础。另一方面，我们从对称性保护拓 扑态的性质研究入手，通过改变维度来观察拓扑物态的物性变化。发现了拓扑边 缘态的平带特征以及多个马约拉纳表面态的存在，这些新奇的拓扑物态性质为自 旋电子学和拓扑量子计算的应用提供了新的可能。我们相信，本论文的研究成果 为进一步研究拓扑物态的理论研究和实验实现产生了积极的影响。

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