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KibbleZurek mechanism in colloidal monolayersSven Deutschlnder, Patrick Dillmann, Georg Maret, and Peter Keim1
Department of Physics, University of Konstanz, 78464 Konstanz, Germany
Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved March 27, 2015 (received for review January 13, 2015)
The KibbleZurek mechanism describes the evolution of topolog-ical defect structures like domain walls, strings, and monopoleswhen a system is driven through a second-order phase transition.The model is used on very different scales like the Higgs field inthe early universe or quantum fluids in condensed matter systems.A defect structure naturally arises during cooling if separated re-gions are too far apart to communicate (e.g., about their orienta-tion or phase) due to finite signal velocity. This lack of causalityresults in separated domains with different (degenerated) locallybroken symmetry. Within this picture, we investigate the nonequi-librium dynamics in a condensed matter analog, a 2D ensemble ofcolloidal particles. In equilibrium, it obeys the so-called KosterlitzThoulessHalperinNelsonYoung (KTHNY) melting scenario withcontinuous (second order-like) phase transitions. The ensemble isexposed to a set of finite cooling rates covering roughly threeorders of magnitude. Along this process, we analyze the defectand domain structure quantitatively via video microscopy and de-termine the scaling of the corresponding length scales as a func-tion of the cooling rate. We indeed observe the scaling predictedby the KibbleZurek mechanism for the KTHNY universality class.
KibbleZurek mechanism | nonequilibrium dynamics | spontaneoussymmetry breaking | KTHNY theory | colloids
In the formalism of gauge theory with spontaneously brokensymmetry, Zeldovich et al. and Kibble postulated a cosmo-logical phase transition during the cooling down of the earlyuniverse. This transition leads to degenerated states of vacuabelow a critical temperature, separated or dispersed by defectstructures as domain walls, strings, or monopoles (13). In thecourse of the transition, the vacuum can be described via anN-component, scalar order parameter (known as the Higgsfield) underlying an effective potential
V = a2 + b2 20
where a is temperature dependent, b is a constant, and 0 is themodulus of hi at T = 0. For high temperatures, V has a singleminimum at = 0 (high symmetry) but develops a minimumlandscape of degenerated vacua below a critical temperatureTc (e.g., the so-called sombrero shape for N = 2). Cooling downfrom the high symmetry phase, the system undergoes a phasetransition at Tc into an ordered (low symmetry) phase with non-zero hi. For T
hi= 0 within the path, and one remains with a monopole forD= 2 or a string for D= 3 (2, 3). A condensed matter analogwould be a vortex of normal fluid with quantized circulation insuperfluid helium. For N = 3 and D= 3, four domains can meet ata mutual point, and the degenerated solutions of the low tem-perature phase lie on a sphere: hi2 = hxi2 + hyi2 + hzi2. If thefield now again varies circularly on a spherical path (all fieldarrows point radially outward), a shrinking of this sphere leads toa monopole in three dimensions (2, 3).Zurek extended Kibbles predictions and transferred his con-
siderations to quantum condensed matter systems. He suggestedthat 4He should intrinsically develop a defect structure whenquenched from the normal to the superfluid phase (4, 5). Forsuperfluid 4He, the order parameter = j jexpi is complexwith two independent components: magnitude j j and phase (the superfluid density is given by j j2). A nontrivial, static so-lution of the equation of state with a GinzburgLandau potentialyields =0rexpin, where r and are cylindrical co-ordinates, nZ, and 00= 0. This solution is called a vortexline, which is topologically equivalent to a string for the caseN = 2 we discussed before. In the vicinity of the critical tem-perature during a quench from the normal fluid to the superfluidstate, will be chosen randomly in uncorrelated regions, leadingto a string network of normal fluid vortices. In condensed mattersystems, the role of the limiting speed of light is taken by thesound velocity (in 4He, the second sound). This upper boundaryleads to a finite speed of the propagation of order parameterfluctuations and sets a sonic horizon.Zurek argued that the correlation length is frozen-out close to
the transition point or even far before depending on the coolingrate (4, 5). Consider the divergence of the correlation length fora second-order transition, e.g., = 0jej, where e= T Tc=Tcis the reduced temperature. If the cooling is infinitely slow, thesystem behaves as in equilibrium: will diverge close to thetransition and the system is a monodomain. For an instantaneousquench, the system has minimal time to adapt to its surrounding: will be frozen-out at the beginning of the quench. For second-order phase transitions, the divergence of correlation lengths isaccompanied by the divergence of the correlation time = 0jej,which is due to the critical slowing down of order parameterfluctuations. If the time t it takes to reach Tc for a given coolingrate is larger than the correlation time, the system stays in equi-librium and the dynamic is adiabatic. Nonetheless, for every finitebut nonzero cooling rate, t eventually becomes smaller than , andthe system falls out of equilibrium before Tc is reached. Themoment when both the correlation time and the time it takes toreach Tc coincide defines the freeze-out time.
t= t. 
The frozen-out correlation length is then set at the temperaturee of the corresponding freeze-out time: = e= t. For a lin-ear temperature quench
e= T Tc=Tc = tq, 
with the quench time scale q, one observes t= 0q1=1+ and
= t= 0q0=1+. 
For the GinzburgLandau model (= 1=2, = 1), one finds thescaling 1=4q , whereas a renormalization group correction(= 2=3) leads to 1=3q (4, 5).A frequently used approximation is that when the adiabatic
regime ends at t before the transition, the correlation lengthcannot follow the critical behavior until again exceeds the timet when Tc is passed. Given a symmetric divergence of aroundTc, this is the time t after the transition. The period in between isknown as the impulse regime, in which the correlation length isassumed not to evolve further. A recent analytical investigation,however, suggests that in this period, the system falls into a re-gime of critical coarse graining (6). There, the typical lengthscale of correlated domains continues to grow because localfluctuations are still allowed, and the system is out of equilib-rium. On the other side, numerical studies in which dissipativecontributions and cooling rates were alternatively varied beforeand after the transition indicate that the final length scale of thedefect and domain network is entirely determined after thetransition (7). Several efforts have been made to provide ex-perimental verification of the KibbleZurek mechanism in avariety of systems, e.g., in liquid crystals (8) (the transition isweakly first order but the defect network can easily observedwith cross polarization microscopy), superfluid 3He (9), super-conducting systems (10), convective, intrinsically out of equilib-rium systems (11), multiferroics (12), quantum systems (13), ioncrystals (14, 15), and BoseEinstein condensates (16) (the lattertwo systems contain the effect of inhomogeneities due to, forexample, temperature gradients). A detailed review concerningthe significance and limitations of these experiments can befound in ref. 17.In this experimental study, we test the validity and applicability
of the KibbleZurek mechanism in a 2D colloidal model systemwhose equilibrium thermodynamics follow the microscopicallymotivated KosterlitzThoulessHalperinNelsonYoung (KTHNY)theory. This theory predicts a continuous, two-step melting be-havior whose dynamics, however, are quantitatively differentfrom phenomenological second-order phase transitions describedby the GinzburgLandau model. We applied cooling rates overroughly three orders of magnitude, for which we changed thecontrol parameter with high resolution and homogeneouslythroughout the sample without temperature gradients. Singleparticle resolution provides a quantitative determination of de-fect and domain structures during the entire quench procedure,and the precise knowledge of the equilibrium dynamics allowsdetermination of the scaling behavior of corresponding lengthscales at the freeze-out times. In the following, we validate thatthe KibbleZurek mechanism can be successfully applied to theKTHNY universality class.
Defects and Symmetry Breaking in 2D CrystallizationThe closed packed crystalline structure in two dimensions is ahexagonal crystal with sixfold symmetry. The thermodynamics ofsuch a crystal can analytically be described via the KTHNYtheory, a microscopic, two-step melting scenario (including twocontinuous transitions) that is based on elasticity theory and a
2+ 2 = 2path
Fig. 1. Emergence of defects in the Higgs field that is illustrated with redvectors (shown in 2D for simplicity). (A) For N= 1 and D= 3, domain walls canappear (strings for D= 2). (B) For N= 2 and D= 3, nontrivial topologies arestrings (monopoles for D= 2). The defects are regions where the order pa-rameter retains the high symmetry phase (hi= 0) to moderate betweendifferent degenerated orientations of the symmetry broken field.
6926 | www.pnas.org/cgi/doi/10.1073/pnas.1500763112 Deutschlnder et al.
renormalization group analysis of topological defects (1820). Inthe KTHNY formalism, the orientationally long range-orderedcrystalline phase melts at a temperature Tm via