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# Kibble–Zurek mechanism in colloidal monolayers

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved March 27, 2015 (received for review January 13, 2015)

## Significance

Spontaneous symmetry breaking describes a variety of transformations from high- to low-temperature phases and applies to cosmological concepts as well as atomic systems. T. W. B. Kibble suggested defect structures (domain walls, strings, and monopoles) to appear during the expansion and cooling of the early universe. The lack of such defects within the visible horizon of the universe mainly motivated inflationary Big Bang theories. W. H. Zurek pointed out that the same principles are relevant within the laboratory when a system obeying a second-order phase transition is cooled at finite rates into the low symmetry phase. Using a colloidal system, we visualize the Kibble–Zurek mechanism on single particle level and clarify its nature in the background of an established microscopic melting formalism.

## Abstract

The Kibble–Zurek mechanism describes the evolution of topological defect structures like domain walls, strings, and monopoles when a system is driven through a second-order phase transition. The model is used on very different scales like the Higgs field in the early universe or quantum fluids in condensed matter systems. A defect structure naturally arises during cooling if separated regions are too far apart to communicate (e.g., about their orientation or phase) due to finite signal velocity. This lack of causality results in separated domains with different (degenerated) locally broken symmetry. Within this picture, we investigate the nonequilibrium dynamics in a condensed matter analog, a 2D ensemble of colloidal particles. In equilibrium, it obeys the so-called Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) melting scenario with continuous (second order-like) phase transitions. The ensemble is exposed to a set of finite cooling rates covering roughly three orders of magnitude. Along this process, we analyze the defect and domain structure quantitatively via video microscopy and determine the scaling of the corresponding length scales as a function of the cooling rate. We indeed observe the scaling predicted by the Kibble–Zurek mechanism for the KTHNY universality class.

In the formalism of gauge theory with spontaneously broken symmetry, Zel'dovich et al. and Kibble postulated a cosmological phase transition during the cooling down of the early universe. This transition leads to degenerated states of vacua below a critical temperature, separated or dispersed by defect structures as domain walls, strings, or monopoles (1⇓–3). In the course of the transition, the vacuum can be described via an *N*-component, scalar order parameter *ϕ* (known as the Higgs field) underlying an effective potential*a* is temperature dependent, *b* is a constant, and *V* has a single minimum at

The geometry of the defect network that separates the uncorrelated domains is given by the topology of the manifold of degenerated states that can exist in the low symmetry phase. Thus, it depends strongly on the dimensionality of the system *D* and on the dimension *N* of the order parameter itself. Regarding the square root of Eq. **2**, the expectation value of a one-component order parameter (*A*). Consider now *B*). In a network of symmetry broken domains in two dimensions, at least three domains (in Fig. 1*B* separated by dashed lines) meet at a mutual edge. On a closed path around the edge, the expectation value *ϕ*) but can also vary by a multiple of

Zurek extended Kibble’s predictions and transferred his considerations to quantum condensed matter systems. He suggested that ^{4}He should intrinsically develop a defect structure when quenched from the normal to the superfluid phase (4, 5). For superfluid ^{4}He, the order parameter *r* and *φ* are cylindrical coordinates, *ψ* will be chosen randomly in uncorrelated regions, leading to a string network of normal fluid vortices. In condensed matter systems, the role of the limiting speed of light is taken by the sound velocity (in ^{4}He, the second sound). This upper boundary leads to a finite speed of the propagation of order parameter fluctuations 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 cooling rate (4, 5). Consider the divergence of the correlation length *ξ* for a second-order transition, e.g., *ξ* will diverge close to the transition and the system is a monodomain. For an instantaneous quench, the system has minimal time to adapt to its surrounding: *ξ* will be frozen-out at the beginning of the quench. For second-order phase transitions, the divergence of correlation lengths is accompanied by the divergence of the correlation time *t* it takes to reach *t* eventually becomes smaller than *τ*, and the system falls out of equilibrium before

A frequently used approximation is that when the adiabatic regime ends at *τ* again exceeds the time *t* when *τ* around ^{3}He (9), superconducting systems (10), convective, intrinsically out of equilibrium systems (11), multiferroics (12), quantum systems (13), ion crystals (14, 15), and Bose–Einstein condensates (16) (the latter two systems contain the effect of inhomogeneities due to, for example, temperature gradients). A detailed review concerning the significance and limitations of these experiments can be found in ref. 17.

In this experimental study, we test the validity and applicability of the Kibble–Zurek mechanism in a 2D colloidal model system whose equilibrium thermodynamics follow the microscopically motivated Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) theory. This theory predicts a continuous, two-step melting behavior whose dynamics, however, are quantitatively different from phenomenological second-order phase transitions described by the Ginzburg–Landau model. We applied cooling rates over roughly three orders of magnitude, for which we changed the control parameter with high resolution and homogeneously throughout the sample without temperature gradients. Single particle resolution provides a quantitative determination of defect and domain structures during the entire quench procedure, and the precise knowledge of the equilibrium dynamics allows determination of the scaling behavior of corresponding length scales at the freeze-out times. In the following, we validate that the Kibble–Zurek mechanism can be successfully applied to the KTHNY universality class.

## Defects and Symmetry Breaking in 2D Crystallization

The closed packed crystalline structure in two dimensions is a hexagonal crystal with sixfold symmetry. The thermodynamics of such a crystal can analytically be described via the KTHNY theory, a microscopic, two-step melting scenario (including two continuous transitions) that is based on elasticity theory and a renormalization group analysis of topological defects (18⇓–20). In the KTHNY formalism, the orientationally long range-ordered crystalline phase melts at a temperature *k*-sum runs over all *j*, and *k*th bond with respect to a certain reference axis. If particle *j* is perfectly sixfold coordinated [e.g., all *r* and *t*, respectively, each correlation attains a finite value in the (mono)crystalline phase, decays algebraically in the hexatic fluid, and exponentially *a* and *b* are constants (20, 22). This peculiarity is the reason why KTHNY melting is named continuous instead of second order. In equilibrium, the KTHNY scenario has been verified successfully for our colloidal system in various experimental studies (23⇓–25).

To transfer this structural 2D phase behavior into the framework of the Kibble–Zurek mechanism, we start in the high temperature phase (isotropic fluid) and describe the symmetry breaking with the spatial distribution of the bond order parameter. Because in 2D the local symmetry is sixfold in the crystal and the fluid, the isotropic phase is a mixture of sixfold and equally numbered five- and sevenfold particles (other coordination numbers are extremely rare and can be neglected). During cooling, isolated disclinations combine to dislocations that, for infinite slow cooling rates, can annihilate into sixfold particles with a uniform director field. This uniformity is given by a global phase, characterizing the orientation of the crystal axis. Spontaneous symmetry breaking implies that all possible global crystal orientations are degenerated, and the Kibble–Zurek mechanism predicts that in the presence of critical fluctuations the system cannot gain a global phase at finite cooling rates: Locally, symmetry broken domains will emerge, which will have different orientations in causally separated regions. The final state is a polycrystalline network with frozen-in defects. As in the case of superfluid ^{4}He, *A*) and by *B*). [In principle, also larger changes in

## Colloidal Monolayer and Cooling Procedure

Our colloidal model system consists of polystyrene beads with diameter ^{3}. The colloidal suspension is sealed within a millimeter-sized glass cell where sedimentation leads to the formation of a monolayer of beads on the bottom glass plate. The whole layer consists of *H* applied perpendicular to the monolayer, which induces a repulsive dipole–dipole interaction between the particles. The ratio between potential energy

## Structure and Dynamics of Defects and Domains

The key element of the Kibble–Zurek mechanism is a frozen-out correlation length

The length scale of the defect network can be measured by the overall defect concentration *ρ* (counting all particles being not sixfold, normalized by the total number of particles) in the *Upper*) shows the evolution of *ρ* for the same cooling rates *ρ* deviates from the equilibrium case in advance of the isotropic/hexatic transition at

The domain structure, on the other hand, can be characterized quantitatively by analyzing symmetry broken domains with a similar phase of *i*) the magnitude *ii*) the bond length deviation of neighboring particles is less than *iii*) the variation in the average bond orientation *i* and *j* must be less than *Lower*) shows the evolution of the ensemble average domain size *ρ*: domain formation significantly deviates from the equilibrium case before

Colloidal ensembles offer the unique possibility to monitor the system and its domain and defect structure on single particle level. Fig. 5 illustrates both (Left column for defects and right column for domains) at the freeze-out temperature *A* and *B*) and the slowest (Fig. 5 *C* and *D*) cooling rate. For *A* and *B*) where *C* and *D*) where

To allow relaxation of the defect and domain structure after the freeze-out time (6), we keep the temperature constant after *A* and *B*) where the quench time was *C* and *D*) where the quench time was

## Scaling Behavior

The main prediction of the Kibble–Zurek mechanism is a power law dependence of the frozen-out correlation length **5**), which results from the algebraic divergence of the correlation, presuming Eqs. **3** and **4**. In KTHNY melting, **10** to find the implicit dependency *z* (22). Here, **8**, one finds with **9** the expression*κ* can be explained by the relatively large value of the dynamical exponent *z*, which regulates the slope of *z* is due to quite long correlation times in this colloidal system (Fig. 3), which are caused by its overdamped dynamics. Note that the sonic horizon is set by the sound velocity of the colloidal monolayer (and not the solvent) being approximately millimeters per second, which is six orders of magnitude slower compared with atomic systems.

## Conclusions

We presented a colloidal model system, where structure formation in spontaneously symmetry broken systems can be investigated with single particle resolution. The theoretical framework is given by the Kibble–Zurek mechanism, which describes domain formation on different scales like the Higgs field shortly after the Big Bang or the vortex network in ^{4}He quenched into the superfluid state. Along various cooling rates, we analyzed the development of defects and symmetry broken domains when the systems fall out of equilibrium and fluctuations of the order parameter cannot follow adiabatically due to critical slowing down. Although 2D melting in the colloidal monolayer is described by KTHNY theory where the divergence of the relevant correlation lengths in equilibrium is exponential (rather than algebraic as typically found in 3D systems), the central idea of the Kibble–Zurek mechanism still holds, and the scaling of the observed domain network is correctly described. Implicitly, this shows that existence of grain boundaries cannot solely be used as criterion for first-order phase transitions and nucleation or to falsify second/continuous-order transitions because they naturally arise for nonzero cooling rates. Those will always be present on finite time scales in experiments and computer simulations after preparation of the system.

## Acknowledgments

P.K. thanks Sébastien Balibar for fruitful discussion. P.K. received financial support from German Research Foundation Grant KE 1168/8-1.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: peter.keim{at}uni-konstanz.de.

Author contributions: P.K. designed research; S.D. and P.K. performed research; P.D. contributed new reagents/analytic tools; S.D. analyzed data; and S.D., G.M., and P.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 6780.

Freely available online through the PNAS open access option.

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