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。

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.