缠绕数

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 4.8 凝聚态系统中的序

更数学地说，设 $X$ 是充满所考察介质的空间。序参量是一个经典场 $\psi (x)$，它也可视为一个映射 $\psi :X\to M$。假定介质中存在一个缺陷。为具体起见，我们考虑超导体这一三维介质中的线缺陷。设想一个环绕该线缺陷的圆周 $S^1$。若 $S^1$ 的每一部分都远离线缺陷，即比相干长度 $\xi$ 远得多，则可假定沿 $S^1$ 的序参量取值于序参量空间 $M=\mathrm{U}(1)$ 中，如图 4.22所示。基本群正是由此进入问题的：我们谈论的是拓扑空间 $\mathrm{U}(1)$ 中的环路。映射 $S^1\to \mathrm{U}(1)$ 由同伦类来分类。取一点 $r_0\in S^1$，并要求 $r_0$ 被映到 $x_0\in M$。注意到 ${\pi }_1(\mathrm{U}(1),x_0)=\mathbb{Z}$，我们便可给该线缺陷指定一个整数。这个整数称为缠绕数(winding number)，因为它计算 $S^1$ 的像绕空间 $\mathrm{U}(1)$ 缠绕了多少圈。若两个线缺陷具有

英文原文

To be more mathematical, let $X$ be a space which is filled with the medium under consideration. The order parameter is a classical field $\psi (x)$ , which is also regarded as a map $\psi :X\to M$ . Suppose there is a defect in the medium. For concreteness, we consider a line defect in the three-dimensional medium of a superconductor. Imagine a circle $S^1$ which encircles the line defect. If each part of $S^1$ is far from the line defect, much further than the coherence length $\xi$ , we may assume the order parameter along $S^1$ takes its value in the order parameter space $M=\mathrm{U}(1)$ , see figure 4.22. This is how the fundamental group comes into the problem; we talk of loops in a topological space $\mathrm{U}(1)$ . The map $S^1\to \mathrm{U}(1)$ is classified by the homotopy classes. Take a point $r_0\in S^1$ and require that $r_0$ be mapped to $x_0\in M$ . By noting that ${\pi }_1(\mathrm{U}(1),x_0)=\mathbb{Z}$ , we may assign an integer to the line defect. This integer is called the winding number since it counts how many times the image of $S^1$ winds the space $\mathrm{U}(1)$ . If two line defects have the

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