  The proliferation of topological defects is a common out-of-equilibrium
phenomenon when a system is driven into a phase of broken symmetry. The
Kibble-Zurek mechanism (KZM) provides a theoretical framework for the critical
dynamics and generation of topological defects in such scenarios. One of the
early applications of KZM is the estimation of heavy magnetic monopoles left
behind by the cosmological phase transitions in the early universe. The
scarcity of such relic monopoles, which contradicts the prediction of KZM, is
one of the main motivations for cosmological inflationary theories. On the
other hand, magnetic monopoles as emergent quasi-particles have been observed
in spin ices, a peculiar class of frustrated magnets that remain disordered at
temperatures well below the energy scale of exchange interaction. Here we study
the annihilation dynamics of magnetic monopoles when spin ice is cooled to zero
temperature in a finite time. Through extensive Glauber dynamics simulations,
we find that the density of residual monopole follows a power law dependence on
the annealing rate. A kinetic reaction theory that precisely captures the
annihilation process from Monte Carlo simulations is developed. We further show
that the KZM can be generalized to describe the critical dynamics of spin ice,
where the exponent of the power-law behavior is determined by the dynamic
critical exponent $z$ and the cooling protocol.
