# Kibble-Zurek mechanism in curved elastic surface crystals

## TL;DR

This paper demonstrates that the Kibble-Zurek mechanism, originally studied in planar systems, also applies to curved elastic surface crystals during stress-induced phase transitions, revealing universal defect formation laws.

## Contribution

It extends the validity of Kibble-Zurek scaling laws to non-thermal, first-order transitions in curved, topologically nontrivial geometries like spheres and tori.

## Key findings

- Defect densities follow KZ power laws regardless of surface topology.
- Nucleation sequences match experimental observations in spherical crystals.
- KZ scaling laws are applicable to a broader class of phase transitions.

## Abstract

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.