In a nutshell explanation of topological defects and charge?

Question

I read this great answer:

What is a topological defect?

in which it seems that a topological defect is a region of the domine of a function in which the function is not defined. In fact, here it is written that

Topological defects are parts of the domain where the quantity of interest is not necessarily everywhere well defined and the configuration is unable to satisfy the usual physical criteria

Therefore, my question is: if we have a physical quantity defined by a function $f(R^n)$, so that $\exists x\in R^n/\nexists f(x)$, can we name $x$ as a topological defect of $f$? If so, can we say that the topological charge of $f$ is the number of such undefined points of its domine?

Answer 1

Defects : topological or not ? A topological defect is a region where a physical system has configuration which cannot be continuously deformed into one another (we say that they are homotopically distinct).

This happens for fields, which are functions $f:M\to T$ where $T$ is some target space. Broadly speaking, for homotopically distinct configurations to exist, there must be some holes in $M$ and $T$. This is the case for example when $f$ is a unit vector field (like a normalized velocity field) (then $T =\mathbb S^1$). The usual space for $M$ is usually the euclidean plane, which has no holes. If, however, $f$ is not defined at some points $p_1,\ldots,p_n$ in $E_2 = \mathbb R^2$, then we have $M = E_2 \backslash \{p_1,\ldots,p_n\}$ in which case we say that $p_1,\ldots,p_n$ are topological defect.

In general, we might call defects physical points which are not in the domain of $f$. Those defects are not necessarily topological though : we also need the condition on the target space $T$. If $f$ is a real scalar field (like temperature), there cannot be topological defects.

Topological charges However, the topological charge is not the number of topological defects. At each defect $p_1,\ldots,p_n$, a field configuration $f$ will have some quantity which is invariant when $f$ is continuously deformed (we say that they are topologically protected). Those invariants are called topological charges.

In our example ($ M = E_2 \backslash \{p_1,\ldots,p_n\}$ and $T = \mathbb S^1$), we can define $n_i$ to be the winding number of $f$ around the defect $p_i$. Each $n_i$ can take any integer value.