Colloidal test bed for universal dynamics of phase transitions

Early insight on the critical dynamics of phase transitions arose in a cosmological setting in an effort to understand the origin of structure formation in the early Universe. Kibble pointed out that in a spontaneous symmetry breaking scenario, when a system is driven across a phase transition from a high-symmetry phase to a topologically nontrivial vacuum manifold, causally disconnected regions of the system choose independently the broken symmetry (1, 2). These conflicting choices result in the formation of topological defects, such as domain walls in a ferromagnet and vortices in a superfluid, to name a couple of familiar examples. Soon after, Zurek indicated that signatures of universality in the dynamics of a phase transition could be tested in condensed matter systems, e.g., superfluid helium (3, 4). Further, he improved the estimate for the average size of the domains and predicted a universal power law for the density of topological defects as a function of the rate at which the phase transition is crossed. The combination of these ideas is known as the Kibble−Zurek mechanism (KZM) and has been a lively subject of both theoretical and experimental research during the last decades. The abundant attempts to verify the KZM in the laboratory have, however, faced a variety of shortcomings, and, while different aspects of the mechanism have been confirmed, a definite test is still missing (5). A remarkable step forward is reported by Deutschländer et al., who used colloidal monolayers as a test bed for universal critical dyamics with unprecedented accuracy (6). In a nutshell, the paradigmatic KZM provides a framework to describe the dynamics across a continuous phase transition. At equilibrium, the correlation length diverges as a universal power law in the thermodynamic limit when the critical point λc is approached by tuning a control parameter λ, i.e., ξ= ξ0=jej\ with e= ðλc − λÞ=λc. The equilibrium relaxation time exhibits a similar power-law behavior, τ= τ0=jej ∝ ξ, that is responsible for the critical slowing down and the nonadiabatic character of the phase transition dynamics. Above, ν and z are critical exponents associated with the universality class of the transition. An arbitrary time-dependent modulation of λ= λðtÞ leading to a spontaneous symmetry breaking scenario can be linearized around the critical point λc so that λðtÞ ’ λcð1− t=τQÞ and eðtÞ ’ t=τQ, where τQ is the quench time in which the transition is crossed. As illustrated in Fig. 1A, the KZM exploits the adiabatic impulse approximation to “chop” the evolution through the phase transition in three sequential stages, where the dynamics is quasi-adiabatic, frozen, and quasi-adiabatic again, as the control parameter takes values λ λc, λ≈ λc, and λ λc, respectively. The crossover between these stages occurs at the freeze-out time t̂, when the equilibrium relaxation time matches the time lagged after crossing the critical point. This characteristic time plays a key role in the KZM and already inherits an imprint of universality, scaling with the quench time as t̂ = ðτ0τQÞ. The beautiful insight behind the KZM is the estimate of the average size of the domains by the value of the equilibrium correlation length at the freeze-out time, i.e., ξ̂= ξ1⁄2eð̂tÞ . As a result, the distance between topological defects increases with the quench time following a universal power law of the form ξ̂= ξ0ðτQ=τ0Þ ν ð1+νzÞ, which is the main prediction of the KZM. Despite the symmetry ξ1⁄2eð−t̂Þ = ξ1⁄2eð̂tÞ within this simplified description, numerical simulations indicate that +t̂ plays the dominant role in determining ξ̂ (7). In addition, coarsening of domains can already occur in the effectively frozen-out stage for very slow ramps (8). The quest for a conclusive test verifying the KZM scaling faces the following major challenges. Given that the mechanism uses equilibrium properties of the system to account for the nonequilibrium dynamics, measurements of the equilibrium correlation length and relaxation time and the associated critical exponents (z and ν) are required beforehand. Important deviations from the power-law behavior are expected in finitesize systems. Both the time modulation of the control parameter and the system itself should be spatially homogeneous. In addition, the range of available quench rates in the laboratory should be wide enough B A