The Kibble-Zurek Mechanism - 2D Quench Space-Time Map and Defect Scaling

Spontaneous symmetry breaking

A real order parameter ψ(x,t) obeys the time-dependent Ginzburg-Landau equation ∂ψ/∂t = -r(t)ψ - ψ³ + ∂²ψ/∂x² + η, where the noise η has strength set by the temperature. For r > 0 the field fluctuates around ψ = 0 (disordered); for r < 0 it must pick ψ = +√|r| or -√|r| (two equal minima, a Z₂ symmetry). The control parameter is ramped linearly, r(t) = -t/τ_Q, so a larger τ_Q is a slower cooling.

Freeze-out and the scaling law

Near the transition the correlation length diverges as ξ = |r|^-ν and the relaxation time as τ_rel = |r|^-zν. The field can follow the ramp only while τ_rel is shorter than the time left to the transition. It freezes at t̂ where τ_rel(t̂) = t̂, giving t̂ = τ_Q^zν/(1+zν) and a frozen-in domain size ξ̂ = τ_Q^ν/(1+zν). The mean-field exponents of this model are ν = 1/2 and z = 2, so t̂ = τ_Q^1/2, ξ̂ = τ_Q^1/4 and the defect (domain-wall) density scales as n ∝ τ_Q^-1/4. This is the Kibble-Zurek mechanism, proposed for cosmic defects by Kibble and for condensed matter by Zurek, and tested in superfluids, liquid crystals and cold atoms.

Honest simplifications: a one-dimensional lattice (spacing 1) with domain walls as the only defects; dimensionless units; the prefactor of the scaling law depends on the noise level and the counting rule, so only the exponent is compared; thermal kink-antikink pairs appear at high temperature and add a constant background.

• Press the Fast quench preset, then Slow quench: the slow ramp leaves far fewer domain walls.
• Raise the temperature: thermal walls appear that do not depend on the quench rate.
• Watch the dashed line on the map: the field is frozen above it and orders below it.
• Compare the red measured points with the Kibble-Zurek line of slope -1/4 on the log-log graph.