in terms of the order parameter only. $T$ denotes the temperature, so ${\cal F}$ is the free energy in mean-field approximation. Throughout this paper we will use units such that Boltzmann’s constant and Planck’s constant are equal to unity, $k_{\text{B}} = \hbar = 1$.

For the order-parameter part of $S$, we consider an $O(3)$-symmetric $\phi^4$-theory, \[eqs:2.2\] S\_M = \_V d[x]{} . \[eq:2.2a\] $S_M$ represents a Landau-Ginzburg-Wilson (LGW) theory of an isotropic ferromagnet with volume $V \to \infty$. The parameter $t$ contains the exchange interaction that leads to a magnetic ordering transition. In mean-field approximation this transition occurs at $t=0$, with $t>0$ describing the paramagnetic phase, and $t<0$ the ferromagnetically ordered one. $a>0$ and $u>0$ are two additional model parameters, and $(\nabla{\bm M})^2 =
\partial_i\,M_j\,\partial^i\,M^j$. Here, and throughout the paper, summation over repeated vector, tensor, and spinor indices is implied unless otherwise noted. Note that $S_M$ is separately invariant under rotations in ${\bm M}$ (spin) space and real space, respectively.

The magnetization[@magnetization_footnote] ${\bm M}$ couples linearly to the curl of the magnetic vector potential ${\bm A}$: S\_ = \_V d[x]{} [M]{}([x]{})(([x]{})), \[eq:2.2b\] with $\muB = e/2\me c$ the Bohr magneton in terms of the electron charge $e$, the electron mass $\me$, 