Conventional phase transitions relax through the growth of ordered domains — nucleation and coarsening, governed by the Allen-Cahn or Cahn-Hilliard equations depending on whether the order parameter is conserved. The dynamics are universal: domain walls move, small domains shrink, large ones grow, and the system settles into equilibrium. The microscopic details don't matter much.
Powell and Pal (arXiv:2501.03341) show that the Kasteleyn transition in spin ice relaxes through an entirely different mechanism: the seeding and growth of string excitations. After a magnetic-field quench near the critical point, the system doesn't form domains. It grows strings — one-dimensional defects that thread through the three-dimensional ice lattice. The scaling theory involves time, reduced temperature, and monopole fugacity (the cost of creating magnetic monopole defects at string endpoints).
The structural point: the Kasteleyn transition is topological, not symmetry-breaking. The ordered and disordered phases differ not in which symmetry is broken but in the topology of the ice-rule-satisfying configurations. Relaxation after a quench doesn't proceed by domains of the new phase growing into the old phase — it proceeds by strings of the new topology threading through the old topology. The dynamics are one-dimensional objects growing in a three-dimensional space, not two-dimensional domain walls sweeping through it.
The conventional assumption that phase transition dynamics are domain dynamics fails when the transition is topological. The shapes that matter are strings, not surfaces. The objects that seed the new phase are one dimension lower than expected.