${\bm n}_{\text{s}}(x) =
{\bar\psi}_{\sigma_1}(x)\,{\bm\sigma}_{\sigma_1 \sigma_2}\,\psi_{\sigma_2}(x)$ are the electronic number and spin density,[@magnetization_footnote] respectively, with ${\bm\sigma} = (\sigma^x,\sigma^y,\sigma^z)$ the Pauli matrices, and the third term in Eq. (\[eq:3.1\]) describes the coupling of the electrons to the electromagnetic field, with ${\bm B} =
{\bm\nabla}\times{\bm A}$ the magnetic induction.[@ionic_density_footnote] Finally, the last term in Eq. (\[eq:3.1\]) describes the spin-orbit interaction. Note that both terms coupling the scalar potential $\varphi$ to the fermions carry an extra factor of $i$ compared to what one might expect from the first quantized Hamiltonian. This has the same origin as the Euclidian metric mentioned above.[@Euclidian_footnote]

Equation (\[eq:3.1\]) describes a continuum model. Some of the effects we are interested in are present only in the presence of certain types of lattices, and we will comment later on the modifications that occur if the electrons are put on a lattice.

Integrating out the photons {#subsec:III.B}
---------------------------

The action, Eq. (\[eq:3.1\]), depends only bilinearly on the electromagnetic potential. The latter can therefore be integrated out exactly, albeit at the expense of creating four-fermion terms. The latter represent electron-electron interactions that are mediated by the exchange of virtual photons. Technically, we need the photon propagator, which we can read off the second 