Topological defects occur in many fields of physics, from magnetic vortices in solid-state materials to cosmic strings that are proposed to have formed in the early Universe just after the Big Bang. What is the mechanism for their formation and distribution? Two theoretical physicists, Kibble and Zurek, proposed several decades ago a unifying mechanism in which topological defects emerge from certain (mathematically specific) types of transitions that take systems in question from their high-symmetry states to certain lower-symmetry ones. An important consequence follows from this so-called Kibble-Zurek mechanism: The density of the topological defects has a power-law dependence on the rate at which the system in question passes through the transition. Demonstrating this mechanism experimentally on cosmological scales is out of reach, and demonstrating it in condensed-matter systems has proved elusive. In this paper, we report the first unambiguous experimental demonstration of the Kibble-Zurek mechanism in the solid-state hexagonal manganite materials and propose this class of materials as a laboratory platform where the analogue of cosmic-string formation in the early Universe can be studied.

Hexagonal manganites have earned their own place of distinction among condensed-matter systems for their multiferroism—the simultaneous occurrence of ferroelectricity and magnetic ordering. Upon cooling, a ferroelectricity transition takes place between structures with zero and nonzero macroscopic electric polarization. In this work, we have shown that the lattice symmetries associated with the transition set the right symmetry conditions for the Kibble-Zurek mechanism to operate. In addition, the resulting topological defects—vortices of electric polarization—that form at the transition can be readily detected because they are associated with the formation of easily observable ferroelectric domains. By controlling and varying the rate of cooling across the transition temperature of the material and by monitoring with polarization force microscopy the density of polarization vortices and its dependence on the cooling rate, we have established that the density of vortex formation is consistent with the Kibble-Zurek scaling. Adding to the richness of physics in these materials, a crossover is also found of the defect-formation physics from the Kibble-Zurek scaling to a new regime at very fast cooling.

Our work represents a significant advance in the fundamental understanding of topological-defect generation in solid-state materials. We hope that it will also stimulate studies of the beyond–Kibble-Zurek scaling in a cosmological context.