Chern绝缘体和拓扑绝缘体等拓扑态中无序效应研究

Using non-commutative geometry theory, we attempt to study the disorder effect in the real space on Chern insulators, topological insulator, and one-dimensional topological superconductor. The main research contents and goals include as the follows: (1) Using non-commutative Chern number formula and non-commutative Kubo formula, we would study the disorder effect on the transport properties of Chern insulators with Chern number n=1,2 and 3, and also the related topological quantum phase transitions. It is expected to obtain a precise critial exponent and uncover the classification of these phase transitions; (2)Using the method and techniques in the studies of mesoscopic systems, we would study the disorder effect on the transport properties of topological insulators, which consists of Chern insulators with Chern number n=1, 2, and 3. In addition, we would study the transport properties of topological crystal insulators. Through these studies, we try to confirm the time reversal symmetry is not the key for topological insulator. Given the time reversal symmetry is replaced by other resonable symmetries, the system can similarly show a topology-protected bulk property, and also robust edge or surface states; (3)We would attempt to find a real-space winding number formula to character the topological state of one-dimensional Chiral DIII superconductor, and use this formula to study the disorder effect and the related phenomena between topological phase transition in this one-dimensional system. In particular, we would summarize these phenomena between phase transitions, find the physical reasons behind them, and then pave the way for understanding the odd dimensional or other higher dimensional phase transitions in theory.

利用非对易几何理论，我们拟在实空间对Chern绝缘体、拓扑绝缘体及一维拓扑超导体等拓扑体系的无序效应展开研究。研究内容和目标主要包括：（1）利用非对易Chern数公式及非对易Kubo公式，研究无序存在时Chern数n=1,2,3等Chern绝缘体的各种输运性质和拓扑量子相变现象，通过数值方法确定这些相变中的临界标度指标，阐释这些拓扑量子相变的普适类别；（2）利用介观输运等方法和手段，研究由n=1,2,3等Chern绝缘体构成拓扑绝缘体的边界态输运性质及无序存在时变化特征。研究拓扑晶态绝缘体的输运性质和无序效应。期望揭示时间反演对称性和其它对称性对拓扑绝缘体形成及其边界态输运性质的影响；（3）寻找表征一维手性对称DIII超导体系的实空间拓扑绕数公式，并利用此类公式研究无序效应及与无序相关的拓扑量子相变现象，总结这些相变现象规律，为理解和认识奇数维或其它高维拓扑量子相变提供理论基础。

利用非平衡格林函数方法和紧束缚理论，我们研究了拓扑绝缘体相变现象，拓扑绝缘体热、电输运性质，拓扑绝缘体约瑟夫森结电输运性质，石墨烯约瑟夫森结电输运性质。研究发现，无序作用下外尔半金属和高陈绝缘体会展现丰富相图，加深了人们以往研究认知；拓扑绝缘体边界态具有强抗干扰特性，是未来制备低能耗、高保真量子器件的潜在材料之一；拓扑绝缘体约瑟夫森结会形成马约拉纳费米子束缚态，是研究和开发马约拉纳费米子的重要平台；强磁场下石墨烯约瑟夫森结使得正常和镜面安德烈夫反射变得可实验区分。我们的研究方向属于基础理论研究，是拓扑绝缘体材料迈向实际应用的必要环节，研究结果具有较高的科学价值，对研究同行具有一定的参考意义。

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