't Hooft–Polyakov monopole

Remarks

At infinity the Higgs field points radially, $\phi^a \approx v \, x^a / r$. This "hedgehog" map from a large sphere to the sphere $\phi^a \phi^a = v^2$ has degree $1$, a topological invariant that finite-energy deformations cannot change. Far away the symmetry is spontaneously broken from $SU(2)$ to $U(1)$, and the unbroken $U(1)$ field is that of a magnetic charge $4\pi/e$ in these units. So the monopole is a smooth, finite-energy soliton that looks from far away like a Dirac magnetic monopole.

The energy satisfies the Bogomolny bound $E \ge 4\pi v \abs{n} / e$, where $n$ is the degree of the Higgs field at infinity. In the limit $\lambda \to 0$, the bound is attained by solutions of the first-order equations $B_i^a = \pm (D_i\phi)^a$. For $n = 1$ the solution is explicit, the Prasad–Sommerfield solution: with $\xi = e v r$, $$K = \frac{\xi}{\sinh\xi}, \qquad h = v \left( \coth\xi - \frac{1}{\xi} \right).$$

Gerard 't Hooft and Alexander Polyakov found the solution independently in 1974. It shows that a gauge theory whose unbroken group is $U(1)$ inside a compact non-abelian group predicts magnetic monopoles.