Outside a defect core write
$$
\mathbf p=p_0(\cos\theta,\sin\theta).
$$
Around any closed curve on which $\mathbf p\ne0$, its <topological charge> is the <winding number>
$$
\boxed{
q=\frac1{2\pi}\oint d\theta
}.
$$
Single-valued polar order requires $q\in\mathbb Z$. A nonzero value prevents the field inside the loop from being continuously deformed to a uniform nonvanishing field, so the core must contain a zero or singularity.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 344 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/2/b/solution