拓扑激发

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

若系统处于均匀温度，则大小 $|m|$ 与位置无关，而 $m$ 仅由其方向确定。在基态下，可以预期 $m$ 本身也与位置无关。为确定 $m$ 的方向，引入极坐标 $(\theta ,\varphi )$ 是方便的。$m$ 与球面 $S^2$ 上的点之间存在一一对应。假设 $m$ 随位置变化：$m=m(x)$。在空间的每一点 $x$ 处都指定了 $S^2$ 的一个点 $(\theta ,\varphi )$，于是我们得到一个从空间到 $S^2$ 的映射 $(\theta (x),\varphi (x))$。除基态（以及由基态附近的小振动（自旋波）所描述的激发态）之外，系统还可能具有各种无法通过小扰动从基态得到的激发态。可能存在何种激发取决于空间的维数和序参量。例如，若空间是二维的，海森堡铁磁体就可能容许一种称为 Belavin-Polyakov 单极子(Belavin-Polyakov monopole)的激发，如图 4.20所示（Belavin and Polyakov 1975）。注意此时 $m$ 趋于一个常矢量 $(\hat{z}$，因此能量并不发散。这一条件保证了该激发的稳定性；在远离原点处保持 $m$ 不变的前提下，不可能把这种位形形变为均匀位形。这类稳定性依赖于拓扑论证的激发称为拓扑激发(topological excitation)。注意，场 $m(x)$ 定义了一个映射 $m:S^2\to S^2$，因而由同伦群 ${\pi }_2(S^2)=\mathbb{Z}$ 来分类。

英文原文

If the system is at a uniform temperature, the magnitude $|m|$ is independent of the position and $m$ is specified by its direction only. In the ground state, $m$ itself is expected to be independent of position. It is convenient to introduce the polar coordinate $(\theta ,\varphi )$ to specify the direction of $m$ . There is a one-to-one correspondence between $m$ and a point on the sphere $S^2$ . Suppose $m$ varies as a function of position: $m=m(x)$ . At each point $x$ of the space, a point $(\theta ,\varphi )$ of $S^2$ is assigned and we have a map $(\theta (x),\varphi (x))$ from the space to $S^2$ . Besides the ground state (and excited states that are described by small oscillations (spin waves) around the ground state) the system may carry various excited states that cannot be obtained from the ground state by small perturbations. What kinds of excitation are possible depends on the dimension of the space and the order parameter. For example, if the space is two dimensional, the Heisenberg ferromagnet may admit an excitation called the Belavin-Polyakov monopole shown in figure 4.20 (Belavin and Polyakov 1975). Observe that $m$ approaches a constant vector $(\hat{z}$ in this case) so the energy does not diverge. This condition guarantees the stability of this excitation; it is impossible to deform this configuration into the uniform one with $m$ far from the origin kept fixed. These kinds of excitation whose stability depends on topological arguments are called topological excitations. Note that the field $m(x)$ defines a map $m:S^2\to S^2$ and, hence, are classified by the homotopy group ${\pi }_2(S^2)=\mathbb{Z}$ .

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