AI 中文总结
该综述为自旋等二分量自由度在动量和实空间中的缠绕发展通用几何语言，探讨拓扑对工程师的作用，如在斯格明子、石墨烯、拓扑绝缘体和外尔半金属中的应用，表明拓扑是一种可用于多种功能设计的语言。
AI 中文摘要
自旋、赝自旋、能谷、极化等二分量自由度具有SU(2)几何性质，但其拓扑表现常被视为不同现象。本综述从贝里相位和狄拉克哈密顿量出发，为其在动量和实空间中的缠绕发展出通用几何语言，并扩展至石墨烯、拓扑绝缘体、外尔半金属和磁斯格明子。我们认为关键不仅在于拓扑本身，还在于对二分量波函数的连续性约束。当相关对称性得以保留时，缠绕决定哪些状态能跨界面或变形连续连接，从而控制传输、扭矩产生、光学选择规则等物理响应。接着探讨拓扑对工程师有何作用。在斯格明子中，缠绕划分磁构型空间并稳定具有可调动力学的超小信息载体；在石墨烯中，赝自旋匹配控制克莱因隧穿，实现栅极控制的传输间隙且不牺牲无质量狄拉克色散；在拓扑绝缘体和外尔半金属中，自旋 - 动量锁定和贝里曲率工程产生电可控自旋电流，螺旋度相关的光学跃迁产生圆光电流响应。这些例子表明拓扑不仅是量子物质的分类，更是一种设计语言，可利用对称保护的波函数连续性进行记忆、开关、驱动和传感等方面的设计。
英文摘要
Spin, pseudospin, valley, polarization, and other two-component degrees of freedom share the geometry of SU(2), yet their topological manifestations are usually discussed as separate phenomena. This review develops a common geometric language for their winding in momentum and real space,beginning with Berry phase and the Dirac Hamiltonian and extending to graphene, topological insulators, Weyl semimetals, and magnetic skyrmions. We argue that the common thread is not merely topology itself, but the continuity constraints imposed on two-component wavefunctions. Whenever the relevant symmetry is preserved, winding determines which states can continuously connect across an interface or deformation, thereby governing transmission, torque generation, optical selection rules, and other physical responses. We then ask a practical question: what does topology buy an engineer? In skyrmions, winding partitions magnetic configuration space and stabilizes ultrasmall information carriers with tunable dynamics. In graphene, pseudospin matching governs Klein tunneling, enabling a gate-controlled transmission gap without sacrificing the massless Dirac dispersion. In topological insulators and Weyl semimetals, spin-momentum locking and Berry-curvature engineering generate electrically selectable spin currents, while helicity-dependent optical transitions produce circular photogalvanic responses. Together, these examples suggest that topology is not merely a classification of quantum matter, but a design language in which symmetry-protected wavefunction continuity can be engineered for memory, switching, actuation, and sensing.
