  Topological defects shape the material and transport properties of physical
systems. Examples range from vortex lines in quantum superfluids,
defect-mediated buckling of graphene, and grain boundaries in ferromagnets and
colloidal crystals, to domain structures formed in the early universe. The
Kibble-Zurek (KZ) mechanism describes the topological defect formation in
continuous non-equilibrium phase transitions with a constant finite quench
rate. Universal KZ scaling laws have been verified experimentally and
numerically for second-order transitions in planar Euclidean geometries, but
their validity for discontinuous first-order transitions in curved and
topologically nontrivial systems still poses an open question. Here, we use
recent experimentally confirmed theory to investigate topological defect
formation in curved elastic surface crystals formed by stress-quenching a
bilayer material. Studying both spherical and toroidal crystals, we find that
the defect densities follow KZ-type power laws independent of surface geometry
and topology. Moreover, the nucleation sequences agree with recent experimental
observations for spherical colloidal crystals. These results suggest that KZ
scaling laws hold for a much broader class of dynamical phase transitions than
previously thought, including non-thermal first-order transitions in non-planar
geometries.
