  Effective field theory of low-energy exitations-magnons that describes
antiferromagnets is mapped into scalar electrodynamics of a charged scalar
field interacting with an external electromagnetic potential. In the presence
of a constant inhomogeneous external magnetic field the latter problem is
technically reduced to the problem of charged-particle creation from the vacuum
by an electric potential step (x step). Magnetic moment plays here the role of
the electric charge, and magnons and antimagnons differ from each other in the
sign of the magnetic moment. In the framework of such a consideration, it is
important to take into account the vacuum instability (the Schwinger effect)
under the magnon-antimagnon production on magnetic field inhomogeneities (an
analog of pair creation from the vacuum by electric-like fields). We
demonstrate how to use the strong field QED with x steps developed by the
authors (SPG and DMG) to study the magnon-antimagnon pair production on
magnetic field inhomogeneities. Characteristics of the vacuum instability
obtained for some magnetic steps that allows exact solving the Klein-Gordon
equation are presented. In particular, we consider examples of magnetic steps
with very sharp field derivatives that correspond to a regularization of the
Klein step. In the case of smooth-gradient steps, we describe an universal
behavior of the flux density of created magnon pairs. We also note that since
the low-energy magnons are bosons with small effective mass, then for the first
time maybe the opportunity will arise to observe the Schwinger effect in the
case of the Bose statistics, in particular, the bosonic Klein effect in
laboratory conditions. Moreover, it turns out that in the case of the Bose
statistics appears a new mechanism for amplifying the effect of the pair
creation, which we call statistically assisted Schwinger effect.
