  The magnon-phonon coupling has received growing attention in recent years due
to its central role in spin caloritronics and the emerging field of acoustic
spintronics. At resonance, this magnetoelastic interaction drives the formation
of magnon polarons, which underpin exotic phenomena such as magnonic heat
currents and phononic spin, but has with a few recent exceptions only been
investigated using mesoscopic spin-lattice models. Motivated to integrate the
magnon-phonon coupling into first-principle many-body electronic structure
theory, we set up to derive the non-relativistic exchange-contribution, which
is more subtle than the spin-orbit contribution, using Schwinger's method of
functional derivatives. To avoid having to solve the famous Hedin-Baym
equations self-consistently, the phonons are treated as a perturbation to the
electronic structure. A formalism is developed around the idea of imposing
crossing symmetry on the interaction, in order to treat charge and spin on
equal footing. By an iterative scheme, we find that the spin-flip component of
the ${\mathit collective}$ four-point interaction, $\mathcal{V}$, which is used
to calculate the magnon spectrum, contains a first-order "screened T matrix"
part and an arguably more important second-order part, which in the limit of
local spins describes the same processes of phonon emission and absorption as
obtained from phenomenological magnetoelastic models. Here, the "order" refers
to the ${\mathit screened}$ ${\mathit collective}$ four-point interaction,
$\mathcal{W}$ - the crossing-symmetric analog of Hedin's $W$.
Proof-of-principle model calculations are performed at varying temperatures for
the isotropic magnon spectrum in three dimensions in the presence of a flat
optical phonon branch.
