Spontaneous Symmetry Breaking in 2D: Kibble-Zurek Mechanism in TemperatureQuenched Colloidal Monolayers

Patrick Dillmann,1 Georg Maret,1 and Peter Keim1, ∗

1University of Konstanz, D-78457 Konstanz, Germany(Dated: September 18, 2014)

The Kibble-Zurek mechanism describes the formation of topological defects during spontaneoussymmetry breaking for quite different systems. Shortly after the big bang, the isotropy of the Higgs-field is broken during the expansion and cooling of the universe. Kibble proposed the formation ofmonopoles, strings, and membranes in the Higgs field since the phase of the symmetry broken fieldcan not switch globally to gain the same value everywhere in space. Zurek pointed out that the samemechanism is relevant for second order phase transitions in condensed matter systems. Every finitecooling rate induces the system to fall out of equilibrium which is due to the critical slowing downof order parameter fluctuations: the correlation time diverges and the symmetry of the system cannot change globally but incorporates defects between different domains. Depending on the coolingrate the heterogeneous order parameter pattern are a fingerprint of critical fluctuations. In thepresent manuscript we show that a monolayer of superparamagnetic colloidal particles is ideallysuited to investigate such phenomena. In thermal equilibrium the system undergos continuousphase transitions according KTHNY-theory. If cooled rapidly across the melting temperature thefinal state is a polycrystal. We show, that the observations can not be explained with nucleation ofa supercooled fluid but is compatible with the Kibble-Zurek mechanism.

PACS numbers: 05.70.Fh, 05.70.Ln, 64.60.Q-, 64.70.pv, 82.70.Dd

The Kibble-Zurek mechanism is a beautiful example,how a hypothesis is strengthened if the underlying con-cept is applicable at completely different scales. Basedon spontaneous symmetry breaking, which is relevant atcosmic [1], atomistic (including suprafluids) [2–4], andelementary particle scale [5, 6], the Kibble-Zurek mecha-nism describes domain and defect formation in the sym-metry broken phase.

In big bang theory, the isotropy of a n-componentscalar Higgs-field Φn is broken during the expansion andcooling of the universe. Regions in space which are sep-arated by causality can not gain the same phase of thesymmetry broken field a priory. The formation of mem-branes, separating the Higgs fields with different phases[7], strings [8, 9] and monopoles [10, 11] is the conse-quence [1, 12]. Inflation [13–15], recently supported byb-mode polarization of the cosmic background ration (theBICEP2 experiment) [16], explains why such defects areso dilute, that they are not observed within the visi-ble horizon of the universe. A frequently used (but notvery precise) analogy is the formation of domains sep-arated by grain boundaries in a ferromagnet below theCurie-temperature. The analogy is not precise since theground state (or true vacuum) of the Higgs field wouldbe a mono-domain whereas a ferromagnet has to be poly-domain to reduce the macroscopic magnetic energy. Thestandard Higgs field order parameter expansion is visual-ized with the ’Mexican-Sombrero’ such being compatiblewith second order phase transition. To slow down thetransition (to dilute monopoles and to solve the flatness-problem in cosmology [17]) a dip in the central peak (falsevacuum) was postulated [1, 13] in the first models of in-flation. Due to the dip the ’supercooled’ Higgs-field is

separated with a finite barrier from the ground state,compatible with a first order scenario. Hence defectseasily arise during the nucleation of grains of differentlyoriented phases being separated by causality.

Zurek pointed out that spontaneous symmetry brakingmediated by critical fluctuations of second order phasetransitions will equally lead to topological defects in con-densed matter physics [4]. While a scalar order parame-ter (or one component Higgs field Φ1) can only serve grainboundaries as defects (compare e.g. an Ising model),two component (or complex) order parameters Φ2 of-fers the possibility to investigate membranes, strings andmonopoles as topological defects depending on the ho-motopy groups [1, 18]. Examples are liquid crystals andsuperfluid He4 and Zurek discussed finite cooling rates[19] as well as instantaneous quenches [20]. For the latterthe system has to fall out of equilibrium due to the criti-cal slowing down of order parameter fluctuations close tothe transition temperature which is true for every finitebut nonzero cooling rate. The symmetry of the orderparameter can not change globally since the correlationtime diverges. The role of the causality is incurred by themaximum velocity for separated regions to communicate.In condensed matter systems this is the sound velocity(or second sound in suprafluid He defining a ’sonic hori-zon’. The order parameter pattern at the so called ’fallout time’ serves as a fingerprint of the frozen out criticalfluctuations with well defined correlation length. Afterthe fall out time, the broken symmetry phase will growon expense of the residual high symmetry phase. Thedomain size at the fall out time is predicted to grow al-gebraically as function of inverse cooling rate [19].

Experiments have been done in He4 for the Lambda-

2

FIG. 1. Snapshot of the monolayer with symmetry brokendomains, 120 s after a quench from Γ = 14 to a final couplingstrength of ΓE = 140. The color code (from blue m = 0over yellow to red m = 1) is given by the magnitude m of thelocal bond order parameter. Blue particles are disordered (loworder = high symmetry phase) and red particles have largesixfold symmetry indicating domains with broken symmetry.

transition, but the detection of defects (a network ofvortex lines with quantized rotation) can only be doneindirectly [21]. Nematic liquid crystals offer the possi-bility to visualize defects directly with cross polariza-tion microscopy [18, 22] but the underling phase tran-sition is first order [23]. In two dimensions, quencheddusty plasma have been investigated experimentally [24–26] and colloidal systems with diameter-tunable micro-gel spheres [27] but were not interpreted with respect toKibble-Zurek mechanism. A careful theoretical study ofa quenched 2D XY-model is given in [28].

In the present manuscript we show that a monolayerof super-paramagnetic particles is ideally suited to inves-tigate the Kibble-Zurek mechanism in two dimensionsfor structural phase transitions. In thermal equilibriumthe system undergos two transitions with an intermediatehexatic phase according to the KTHNY-theory [29–33].Thus, this theory predicts two transition temperatures(for orientational and translational symmetry breaking)and calculates critical exponents for both diverging corre-lation lengths. The divergences are exponential (a uniquefeature of 2D systems) as predicted by renormalizationgroup theory [31, 33] and determined in experiment [34].The exponential divergence (one can take the derivativesarbitrarily often) and the marginal ”hump” in the spe-cific heat [35] is the reason for naming this transitioncontinuous (and not second order).

The 2D colloidal monolayer is realized out as follows. Adroplet of a suspension composed of polystyrene spheresdispersed in water is suspended by surface tension ina top sealed cylindrical hole (∅ = 6 mm) of a glassplate. The particles are 4.5 µm in diameter and havea mass density of 1.5 g/cm3 leading to sedimentation.

The large density is due to the fact that the polystyrenebeads are doped with iron oxide nano-particles leading tosuper-paramagnetic behavior of the colloids. After sed-imentation particles are arranged in a monolayer at theplanar and horizontal water-air interface of the dropletand hence form an ideal 2D system. Particles are smallenough to perform Brownian motion but large enough tobe monitored with video-microscopy. An external mag-netic field H perpendicular to the water-air interface in-duces a magnetic moment in each bead (parallel to theapplied field) leading to repulsive dipole-dipole interac-tion between all particles. We use the dimensionless con-trol parameter Γ to characterize this interaction strength.Γ is given by the ratio of dipolar magnetic energy to ther-mal energy

Γ =µ0

4π

(χH)2(πρ)3/2

kBT∝ T−1

sys (1)

and thus can be regarded as an inverse system temper-ature. The state of the system in thermal equilibrium -liquid, hexatic, or solid - is solely defined by the strengthof the magnetic field H since the temperature T , the2D particle density ρ and the magnetic susceptibility perbead χ are kept constant experimentally. In these unitsthe inverse melting temperature (crystalline - hexatic)is at about Γm = 60 and the transition from hexaticto isotropic at about Γi = 57 [36]. Since the systemtemperature is given by an outer field, enormous coolingrates are accessible compared to atomic systems. Basedon a well equilibrated liquid system [37] at Γ ≈ 15 weinitiate a temperature jump with cooling rates up todΓ/dt ≈ 104 s−1 into the crystalline region of the phasediagram ΓM ≥ 60. This temperature quench triggersthe solidification within the whole monolayer (the heattransfer is NOT done through the surface of the mate-rial as usually in 3D condensed matter systems) and timescale of cooling is 105 faster compared to fastest intrin-sic scales, e.g. the Brownian time τB = 50 sec, particlesneed to diffuse the distance of their own diameter. Anelaborate description of the experimental setup can befound in [38]. The field of view is 1160 × 865 µm2 insize and contains about 9000 colloids, the whole mono-layer contains of ∼ 300000 particles. Each temperaturequench is repeated at least ten times to the same finalvalue of the control parameter ΓF with sufficient equili-bration times in between. Fig 1 shows the monolayer twominutes after the quench from deep in the fluid phasebelow the melting temperature (above the control pa-rameter Γm), (see also movie bond-order-magnitude-Gf-140-10min.avi of the supplemental material). The colorcode of particle l at position ~rl is given by the magnitudeml = m(~rl) = ψ∗

l ψl of the local complex (six-folded)

bond order field ψl = 1/Nj∑Nj

k=1 ei6θkl given by the Nj

nearest neighbors. Blue particles have low bond orien-tational order m ≈ 0 (high isotropic symmetry) and redparticles indicate domains of broken symmetry m ≈ 1

3

(high local sixfold order). Note that we investigate theorientational order and not the translational one, sincea) we can hardly distinguish between ’poly-hexalline’ andpoly-crystalline on finite length scales, b) the hexaticphase is narrow compared to the quench depth, and c)translational order parameters are based on reciprocallattice vectors ~G being ill defined in poly-crystalline sam-ples. The hexatic phase is not observable as function oftime after a quench [39] and the bond-order correlationfunction

g6(r) = 〈|ψ(~rk)ψ∗(~rj)|〉kj = 〈|ψ(~r)ψ∗(~0)|〉 , (2)

always decays exponential g6(r, t) ∼ exp(r/ξ6(t)). Theorientational correlation length ξ6(t) grows monoton-ically after the quench and the final state is poly-crystalline. As a side remark; the Kibble-Zurek mech-anism implies that poly-crystallinity can not solely beused as argument for first order phase transitions sincezero cooling rates can not be realized during preparation.To analyze the dynamics of locally broken symmetry withrespect to Kibble-Zureck mechanism, Figure 2 shows theincrease of the bond-order correlation length ξ6 for lowsupercooling (ΓF = 63), intermediate (ΓF = 70and94),and deep supercooling (ΓF = 140). We use ξ6 as a mea-sure for the average size of symmetry broken regions.In the supplemental material we show, that the stan-dard pair correlation length and the inverse defect den-sity give qualitatively the same results [40]. In the earlystate, we observe an exponential grow of the domain sizefollowed by an algebraic one for all magnitudes of inves-tigated supercoolings. The crossover from exponentialto algebraic shifts so shorter times for deeper quenches.For the Lambda-transition of He4 Zurek proposed an al-gebraic decrease of the inverse defect density after thequench [19]. For the XY-model a power-law increase ofthe correlation length with a small logarithmic correc-tion is predicted [28] and in dusty plasma an algebraicincrease was found [24]. Whether or not the exponentialgrowing of average domain sizes in the early state is afingerprint of exponentially diverging correlation lengthin KTHNY-melting can not be answered at present andshould be topic of theoretical investigations. In Fig. 2the cross-over appears for quite small values of the aver-age correlation length (order of unity), even if individualdomain sizes e.g. 120 sec after a quench to ΓF = 140 arealready extend over several particles (comparing Fig. 1and Fig. 2 ξ6 is ’short eyed’ and does not measure thesize of individual domains). Therefore we introduce a cri-terium for locally broken symmetries what furthermoreallows to follow and label individual domains in time: wedefine a particle to be part of a symmetry broken do-main if the following three conditions are fulfilled for theparticle itself and at least one nearest neighbor: i) Themagnitude of the local bond order field m6k must exceed0.6 for both neighboring particles. ii) The bond lengthdeviation ∆|lkl| of neighboring particles k and l is less

1 1 0 1 0 0 6 0 00 , 51

1 0ΓE = 6 3ΓE = 7 0

ξ 6/a

t [ s ]

~ e t / τ f o r t ≤ 5 0 s ~ t α f o r t ≥ 170 s

1 1 0 1 0 0 6 0 00 , 51

1 0

ξ 6/a

t [ s ]

~ e t / τ f o r t ≤ 4 5 s ~ t α f o r t ≥ 120 s ΓE = 7 0

1 1 0 1 0 0 6 0 00 , 51

1 0ΓE = 9 4

ξ 6/a

t [ s ]

~ e t / τ f o r t ≤ 3 5 s ~ t α f o r t ≥ 85s

1 1 0 1 0 0 6 0 00 , 51

1 0ΓE = 1 4 0

ξ 6/a

t [ s ]

~ e t / τ f o r t ≤ 2 0 s ~ t α f o r t ≥ 60 s

FIG. 2. Orientational correlation length as function of timein a log-log-plot for different quench depth. Shortly afterthe quench, the growing behavior is exponential and switchesto algebraic later. The quench depth increases from top tobottom. Note the decreasing cross-over time (red arrows asguide for the eye) for deeper quenches.

than 10% of the average particle distance la. iii) Thevariation in bond orientation ∆Θ = |ψk − ψl| of neigh-boring particles k and l must be less than 2.3◦ in realspace (less than 14◦ in sixfolded space). An elaboratediscussion of criteria to define crystallinity in 2D on a lo-cal scale can be found in [41]. Simply connected domainsof particles which fulfill all three criteria are merged to alocal symmetry broken domain.

Figure 3 [left] shows the average number of local do-mains as function of time up to 200 sec after the quenchand the inset shows the mean size. As expected, themean size of the nuclei grows monotonically as functionof time for all investigated quench depths. The averagenumber of domains (Fig. 3 [left]) first increases but finallydecreases in time. As can also been seen in the moviesof the supplemental material, fewer but larger domainscover the space as function of time as expected. The max-imum sifts to shorter times as function of quench depth -

4

0 50 100 150 2000

100

200

300

400

500

600

700

800

t [s]

NSB

D

ΓE = 61

ΓE = 63

ΓE = 70

ΓE = 94

ΓE = 140 0 50 100 150 200

0

5

10

15

<ASB

D>

[a2 ]

t [s]

0,0 0,2 0,4 0,6 0,8 1,00

100

200

300

400

500

600

700

800

NSB

D

X

0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

<ASB

D>

[a2 ]

X

FIG. 3. The left image shows the number of symmetry broken domains (SBD) and the average size of the domains (inset)as function of time for different quench depth (the label is valid for all four figures). The error bars are averages about 10independent quenches with about 9000 particles in the field of view. Whereas the average size of the nuclei grows monotonicallyas expected, the number of symmetry broken domains first increases to a maximum but then decreases in favor of less butlarger domains. The red arrows are a guide to the eye to identify the maxima. The right image shows the same data but plottedas fraction of symmetry broken area (crystallinity, which is implicity a function of time). The curves almost superimpose asfunction of crystallinity being independent of the quench depth.

deeper supercooling drives the system faster to the solidstate. Note that the growing of domains starts immedi-ately after the quench such that no lag time known fromclassical nucleation theory (CNT) is detectable. (Furtherdiscussion in the context of CNT can be found in [40].)Figure 3 [right] shows the same data but plotted as func-tion of crystallinity X instead of time. Here, we definecrystallinity X (increasing monotonically in time) as thefraction of particles belonging to a symmetry broken do-main identified with the three criteria mentioned above.Interestingly all curves almost superimpose as function ofcrystallinity, re-scaling the time axis to an intrinsic sys-tem dependent parameter. Surprisingly, the position andheight of average number of symmetry broken domains(in the context nucleation this is called the mosaicity)is independent of the quench depth. It appears whenroughly 37% belong to symmetry broken domains. Com-paring the maximum on the real time axis (red arrows inFig. 3 [left]) and the cross-over time in Fig. 2 for differentquench depth we propose the following scenario: After aquench, locally symmetry broken domains start to growexponentially until about 37% of the space is covered.In our 2D system this marks a threshold where domainswith different orientation (different phases of the symme-try broken field) start to touch. The following dynamicsis dominated by the conversion of the yet untransformedregions of the high symmetry phase marked by an alge-braic increase of the bond order correlation length. Forfinite cooling rates, G. Biroli et al. have argued [42]that defect annihilation will alter the classical Kibble-Zurek mechanism leading to two different time regimes

after the fall out time, separating critical and classicalcoarse graining. In the supplemental material we show[40] that we can resolve the classical coarse graining af-ter an instantaneous quench, too. Large domains growon the expense of smaller ones with an algebraic increaseof the bond-order correlation length with reduced expo-nent. The latter is due to grain boundary diffusion whichwas recently described for colloidal mono-layers [43, 44].

In the context of Kibble-Zurek-mechanism one wouldexpect all individual grains to grow after the quenchwhich is surprisingly only the case on average. In thesupplemental material we show that the shrinking prob-ability ps of individually labeled domains is always largerthan the of growing probability pg. This again rulesout a) critical nucleation with a finite nucleation bar-rier and b) allows for back and forth fluctuations of thebroken symmetry regions in 2D systems below the tran-sition temperatures after a quench. Zurek has arguedthat such fluctuations are suppressed in the vicinity ofthe transition if the domain size ∆A times the energy-density differences ∆ε is less than kBT . In equilibriumthis is nothing but another argument for critical slowingdown. In 2D systems it is well known that fluctuationsplay a major role. This is obviously the case not only inequilibrium but also in non-equilibrium situations.

In conclusion, we have investigated the Kibble-Zureckmechanism experimentally for a two-dimensional systemwith complex order parameter given by the local bondorder director field. After a sudden quench we resolvetwo time regimes at an early state (followed by a thirdregime due to classical coarse graining, see [40]). First,

5

the bond order correlation length grows exponentially un-til roughly 40% of the area belongs to symmetry brokendomains with random orientation. While Zurek proposedan algebraic growing for 3D systems, we suggest the ex-ponential behavior being caused by the underlying ex-ponential divergences in KTHNY-melting, being typicalfor 2D systems. In a second regime an algebraic behav-ior is observed until no high symmetry phase is left, ex-cept grain boundaries separating domains with differentphase. The independence of the mosaicity strongly sup-ports the Kibble-Zureck mechanism. We hope our workwill stimulate theory and simulations to investigate thephenomena observed in phase transitions far from ther-mal equilibrium. Further studies in dusty plasma andcolloidal mono-layers might shed light on differences andsimilarities of the Kibble-Zurek mechanism in 2D and 3Dsystems at sudden quenches or finite cooling rates.

P.K. acknowledges fruitful discussion with SebastienBalibar. Financial support from the German Re-search Foundation (DFG), SFB-TR6 project C2 and KE1168/8-1 is further acknowledged.

∗ peter.keim@uni-konstanz.de

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