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The proliferation of topological defects is a common out-of-equilibriumphenomenon when a system is driven into a phase of broken symmetry. TheKibble-Zurek mechanism (KZM) provides a theoretical framework for the criticaldynamics and generation of topological defects in such scenarios. One of theearly applications of KZM is the estimation of heavy magnetic monopoles leftbehind by the cosmological phase transitions in the early universe. Thescarcity of such relic monopoles, which contradicts the prediction of KZM, isone of the main motivations for cosmological inflationary theories. On theother hand, magnetic monopoles as emergent quasi-particles have been observedin spin ices, a peculiar class of frustrated magnets that remain disordered attemperatures well below the energy scale of exchange interaction. Here we studythe annihilation dynamics of magnetic monopoles when spin ice is cooled to zerotemperature in a finite time. Through extensive Glauber dynamics simulations,we find that the density of residual monopole follows a power law dependence onthe annealing rate. A kinetic reaction theory that precisely captures theannihilation process from Monte Carlo simulations is developed. We further showthat 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 dynamiccritical exponent $z$ and the cooling protocol.

Kibble-Zurek Mechanism for Nonequilibrium Generation of Magnetic Monopoles in Spin Ices