Spontaneous Symmetry Breaking in 2D : Kibble-Zurek Mechanism In

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

1 1 1, Patrick Dillmann, Georg Maret, and Peter Keim ∗ 1University of Konstanz, D-78457 Konstanz, Germany (Dated: September 17, 2014) The Kibble-Zurek mechanism describes the formation of topological defects during spontaneous symmetry 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 of monopoles, strings, and membranes in the Higgs field since the phase of the symmetry broken field can not switch globally to gain the same value everywhere in space. Zurek pointed out that the same mechanism is relevant for second order phase transitions in condensed matter systems. Every finite cooling rate induces the system to fall out of equilibrium which is due to the critical slowing down of order parameter fluctuations: the correlation time diverges and the symmetry of the system can not change globally but incorporates defects between different domains. Depending on the cooling rate the heterogeneous order parameter pattern are a fingerprint of critical fluctuations. In the present manuscript we show that a monolayer of superparamagnetic colloidal particles is ideally suited to investigate such phenomena. In thermal equilibrium the system undergos continuous phase transitions according KTHNY-theory. If cooled rapidly across the melting temperature the final state is a polycrystal. We show, that the observations can not be explained with nucleation of a 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, separated with a finite barrier from the ground state, how a hypothesis is strengthened if the underlying con- compatible with a first order scenario. Hence defects cept is applicable at completely different scales. Based easily arise during the nucleation of grains of differently on spontaneous symmetry breaking, which is relevant at oriented phases being separated by causality. cosmic [1], atomistic (including suprafluids) [2–4], and Zurek pointed out that spontaneous symmetry braking elementary particle scale [5, 6], the Kibble-Zurek mecha- mediated by critical fluctuations of second order phase nism describes domain and defect formation in the sym- transitions will equally lead to topological defects in con- metry broken phase. densed matter physics [4]. While a scalar order parame- In big bang theory, the isotropy of a n-component ter (or one component Higgs field Φ1) can only serve grain scalar Higgs-field Φn is broken during the expansion and boundaries as defects (compare e.g. an Ising model), cooling of the universe. Regions in space which are sep- two component (or complex) order parameters Φ2 of- arated by causality can not gain the same phase of the fers the possibility to investigate membranes, strings and symmetry broken field a priory. The formation of mem- monopoles as topological defects depending on the ho- branes, separating the Higgs fields with different phases motopy groups [1, 18]. Examples are liquid crystals and [7], strings [8, 9] and monopoles [10, 11] is the conse- superfluid He4 and Zurek discussed finite cooling rates quence [1, 12]. Inflation [13–15], recently supported by [19] as well as instantaneous quenches [20]. For the latter b-mode polarization of the cosmic background ration (the the system has to fall out of equilibrium due to the criti- BICEP2 experiment) [16], explains why such defects are cal slowing down of order parameter fluctuations close to so dilute, that they are not observed within the visi- the transition temperature which is true for every finite ble horizon of the universe. A frequently used (but not but nonzero cooling rate. The symmetry of the order very precise) analogy is the formation of domains sep- parameter can not change globally since the correlation arated by grain boundaries in a ferromagnet below the time diverges. The role of the causality is incurred by the arXiv:1303.6821v2 [cond-mat.soft] 16 Sep 2014 Curie-temperature. The analogy is not precise since the maximum velocity for separated regions to communicate. ground state (or true vacuum) of the Higgs field would In condensed matter systems this is the sound velocity be a mono-domain whereas a ferromagnet has to be poly- (or second sound in suprafluid He defining a ’sonic hori- domain to reduce the macroscopic magnetic energy. The zon’. The order parameter pattern at the so called ’fall standard Higgs field order parameter expansion is visual- out time’ serves as a fingerprint of the frozen out critical ized with the ’Mexican-Sombrero’ such being compatible fluctuations with well defined correlation length. After with second order phase transition. To slow down the the fall out time, the broken symmetry phase will grow transition (to dilute monopoles and to solve the flatness- on expense of the residual high symmetry phase. The problem in cosmology [17]) a dip in the central peak (false domain size at the fall out time is predicted to grow al- vacuum) was postulated [1, 13] in the first models of in- gebraically as function of inverse cooling rate [19]. flation. Due to the dip the ’supercooled’ Higgs-field is Experiments have been done in He4 for the Lambda-

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-345263 2

The large density is due to the fact that the polystyrene beads are doped with iron oxide nano-particles leading to super-paramagnetic behavior of the colloids. After sedimentation particles are arranged in a monolayer at the planar and horizontal water-air interface of the droplet and hence form an ideal 2D system. Particles are small enough to perform Brownian motion but large enough to be monitored with video-microscopy. An external magnetic field H perpendicular to the water-air interface induces a magnetic moment in each bead (parallel to the applied field) leading to repulsive dipole-dipole interaction between all particles. We use the dimensionless control parameter Γ to characterize this interaction strength. Γ is given by the ratio of dipolar magnetic energy to thermal energy FIG. 1. Snapshot of the monolayer with symmetry broken domains, 120 s after a quench from Γ = 14 to a final coupling 2 3/2 µ0 (χH) (πρ) 1 strength of Γ = 140. The color code (from blue m = 0 Γ = T − (1) E 4π k T ∝ sys over yellow to red m = 1) is given by the magnitude m of the B local bond order parameter. Blue particles are disordered (low and thus can be regarded as an inverse system temper- order = high symmetry phase) and red particles have large ature. The state of the system in thermal equilibrium - sixfold symmetry indicating domains with broken symmetry. liquid, hexatic, or solid - is solely defined by the strength of the magnetic field H since the temperature T , the 2D particle density ρ and the magnetic susceptibility per transition, but the detection of defects (a network of bead χ are kept constant experimentally. In these units vortex lines with quantized rotation) can only be done the inverse melting temperature (crystalline - hexatic) indirectly [21]. Nematic liquid crystals offer the possi- is at about Γm = 60 and the transition from hexatic bility to visualize defects directly with cross polariza- to isotropic at about Γi = 57 [36]. Since the system tion microscopy [18, 22] but the underling phase tran- temperature is given by an outer field, enormous cooling sition is first order [23]. In two dimensions, quenched rates are accessible compared to atomic systems. Based dusty plasma have been investigated experimentally [24– on a well equilibrated liquid system [37] at Γ 15 we 26] and colloidal systems with diameter-tunable micro- initiate a temperature jump with cooling rates≈ up to 4 1 gel spheres [27] but were not interpreted with respect to dΓ/dt 10 s− into the crystalline region of the phase ≈ Kibble-Zurek mechanism. A careful theoretical study of diagram ΓM 60. This temperature quench triggers a quenched 2D XY-model is given in [28]. the solidification≥ within the whole monolayer (the heat In the present manuscript we show that a monolayer transfer is NOT done through the surface of the mate- of super-paramagnetic particles is ideally suited to inves- rial as usually in 3D condensed matter systems) and time tigate the Kibble-Zurek mechanism in two dimensions scale of cooling is 105 faster compared to fastest intrinsic for structural phase transitions. In thermal equilibrium scales, e.g. the Brownian time τB = 50 sec, particles need the system undergos two transitions with an intermediate to diffuse the distance of their own diameter. An elabo- hexatic phase according to the KTHNY-theory [29–33]. rate description of the experimental setup can be found Thus, this theory predicts two transition temperatures in [38]. The field of view is 1160 865 µm2 in size and × (for orientational and translational symmetry breaking) contains about 9000 colloids, the whole monolayer con- and calculates critical exponents for both diverging corre- tains of 300000 particles. Each temperature quench is ∼ lation lengths. The divergences are exponential (a unique repeated at least ten times to the same final value of the feature of 2D systems) as predicted by renormalization control parameter ΓF with sufficient equilibration times group theory [31, 33] and determined in experiment [34]. in between. Fig 1 shows the monolayer two minutes after The exponential divergence (one can take the derivatives the quench from deep in the fluid phase below the melting arbitrarily often) and the marginal ”hump” in the spe- temperature (above the control parameter Γm), (see also cific heat [35] is the reason for naming this transition movie bondordermagnitude.avi of the supplemental ma- continuous (and not second order). terial). The color code of particle l at position ~rl is given The 2D colloidal monolayer is realized out as follows. A by the magnitude ml = m(~rl) = ψl∗ψl of the local com- Nj i6θkl droplet of a suspension composed of polystyrene spheres plex (six-folded) bond order field ψl = 1/Nj k=1 e dispersed in water is suspended by surface tension in given by the Nj nearest neighbors. Blue particles have P a top sealed cylindrical hole (∅ = 6 mm) of a glass low bond orientational order m 0 (high isotropic sym- plate. The particles are 4.5 µm in diameter and have metry) and red particles indicate≈ domains of broken sym- a mass density of 1.5 g/cm3 leading to sedimentation. metry m 1 (high local sixfold order). Note that we in- ≈ 3

vestigate the orientational order and not the translational 1 0 τ ~ e t / f o r t ≤ 5 0 s one, since a) we can hardly distinguish between ’poly- Γ = α7 0 Γ E ≥ 17 = 6 3 hexalline’ and poly-crystalline on ﬁnite length scales, b) ~ t f o r t 0 s E a / 6

the hexatic phase is narrow compared to the quench ξ depth, and c) translational order parameters are based 1 ~ on reciprocal lattice vectors G being ill deﬁned in poly- 0 , 5 crystalline samples. The hexatic phase is not observable 1 1 0 t [ s ] 1 0 0 6 0 0 as function of time after a quench [39] and the bond-order 1 0 τ correlation function ~ e t / f o r t ≤ 4 5 s α Γ = 7 0 ~ t f o r t ≥ 120 s E

~ a g6(r) = ψ( ~rk)ψ∗(~rj) kj = ψ(~r)ψ∗(0) , (2) / 6

h| |i h| |i ξ always decays exponential g6(r, t) exp(r/ξ6(t)). The ∼ 1 orientational correlation length ξ6(t) grows monoton- 0 , 5 ically after the quench and the ﬁnal state is poly- 1 1 0 t [ s ] 1 0 0 6 0 0 crystalline. As a side remark; the Kibble-Zurek mech- 1 0 anism implies that poly-crystallinity can not solely be τ ~ e t / f o r t ≤ 3 5 s α Γ = 9 4 used as argument for ﬁrst order phase transitions since ~ t f o r t ≥ 85s E a

zero cooling rates can not be realized during preparation. / 6

To analyze the dynamics of locally broken symmetry with ξ respect to Kibble-Zureck mechanism, Figure 2 shows the 1 increase of the bond-order correlation length ξ6 for low 0 , 5 1 1 0 1 0 0 6 0 0 supercooling (ΓF = 63), intermediate (ΓF = 70and94), t [ s ] and deep supercooling (ΓF = 140). We use ξ6 as a mea- 1 0 τ ~ e t / f o r t ≤ 2 0 s sure for the average size of symmetry broken regions. Γ = 1 4 0 α E In the supplemental material we show, that the stan- ~ t f o r t ≥ 60 s a / 6

dard pair correlation length and the inverse defect den- ξ sity give qualitatively the same results [40]. In the early 1 state, we observe an exponential grow of the domain size 0 , 5 followed by an algebraic one for all magnitudes of inves- 1 1 0 t [ s ] 1 0 0 6 0 0 tigated supercoolings. The crossover from exponential to algebraic shifts so shorter times for deeper quenches. FIG. 2. Orientational correlation length as function of time 4 For the Lambda-transition of He Zurek proposed an al- in a log-log-plot for different quench depth. Shortly after gebraic decrease of the inverse defect density after the the quench, the growing behavior is exponential and switches quench [19]. For the XY-model a power-law increase of to algebraic later. The quench depth increases from top to the correlation length with a small logarithmic correc- bottom. Note the decreasing cross-over time (red arrows as tion is predicted [28] and in dusty plasma an algebraic guide for the eye) for deeper quenches. increase was found [24]. Whether or not the exponential growing of average domain sizes in the early state is a fingerprint of exponentially diverging correlation length variation in bond orientation ∆Θ = ψk ψl of neigh- | − | in KTHNY-melting can not be answered at present and boring particles k and l must be less than 2.3◦ in real should be topic of theoretical investigations. In Fig. 2 space (less than 14◦ in sixfolded space). An elaborate the cross-over appears for quite small values of the aver- discussion of criteria to define crystallinity in 2D on a lo- age correlation length (order of unity), even if individual cal scale can be found in [41]. Simply connected domains domain sizes e.g. 120 sec after a quench to ΓF = 140 are of particles which fulfill all three criteria are merged to a already extend over several particles (comparing Fig. 1 local symmetry broken domain. and Fig. 2 ξ6 is ’short eyed’ and does not measure the Figure 3 [left] shows the average number of local do- size of individual domains). Therefore we introduce a cri- mains as function of time up to 200 sec after the quench terium for locally broken symmetries what furthermore and the inset shows the mean size. As expected, the allows to follow and label individual domains in time: we mean size of the nuclei grows monotonically as function define a particle to be part of a symmetry broken do- of time for all investigated quench depths. The average main if the following three conditions are fulfilled for the number of domains (Fig. 3 [left]) first increases but finally particle itself and at least one nearest neighbor: i) The decreases in time. As can also been seen in the movies magnitude of the local bond order field m6k must exceed of the supplemental material, fewer but larger domains 0.6 for both neighboring particles. ii) The bond length cover the space as function of time as expected. The max- deviation ∆ l of neighboring particles k and l is less imum sifts to shorter times as function of quench depth - | kl| than 10% of the average particle distance la. iii) The deeper supercooling drives the system faster to the solid 4

800 800 700 700 600 600

500 500 D

15 D 15 B B S 400 S

400 N

N ] 2 10 ] 2 a 10 [ a Γ = 61 E [

300 > > 300 D

D Γ = 63 B B E S

5 S A 5 A 200 Γ = 70 <

E 200 < Γ E = 94 100 0 Γ 0 50 100 150 200 100 0 E = 140 0,0 0,2 0,4 0,6 0,8 1,0 t [s] X 0 0 0 50 100 150 200 0,0 0,2 0,4 0,6 0,8 1,0 t [s] 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 10 independent quenches with about 9000 particles in the field of view. Whereas the average size of the nuclei grows monotonically as expected, the number of symmetry broken domains first increases to a maximum but then decreases in favor of less but larger domains. The red arrows are a guide to the eye to identify the maxima. The right image shows the same data but plotted as fraction of symmetry broken area (crystallinity, which is implicity a function of time). The curves almost superimpose as function of crystallinity being independent of the quench depth. state. Note that the growing of domains starts immedi- coarse graining. In the supplemental material we show ately after the quench such that no lag time known from [40] that we can resolve the classical coarse graining af- classical nucleation theory (CNT) is detectable. (Further ter an instantaneous quench, too. Large domains grow discussion in the context of CNT can be found in [40].) on the expense of smaller ones with an algebraic increase Figure 3 [right] shows the same data but plotted as func- of the bond-order correlation length with reduced expo- tion of crystallinity X instead of time. Here, we define nent. The latter is due to grain boundary diffusion which crystallinity X (increasing monotonically in time) as the was recently described for colloidal mono-layers [43, 44]. fraction of particles belonging to a symmetry broken do- In the context of Kibble-Zurek-mechanism one would main identified with the three criteria mentioned above. expect all individual grains to grow after the quench Interestingly all curves almost superimpose as function of which is surprisingly only the case on average. In the crystallinity, re-scaling the time axis to an intrinsic sys- supplemental material we show that the shrinking prob- tem dependent parameter. Surprisingly, the position and ability p of individually labeled domains is always larger height of average number of symmetry broken domains s than the of growing probability p . This again rules (in the context nucleation this is called the mosaicity) g out a) critical nucleation with a finite nucleation bar- is independent of the quench depth. It appears when rier and b) allows for back and forth fluctuations of the roughly 37% belong to symmetry broken domains. Com- broken symmetry regions in 2D systems below the tran- paring the maximum on the real time axis (red arrows in sition temperatures after a quench. Zurek has argued Fig. 3 [left]) and the cross-over time in Fig. 2 for different that such fluctuations are suppressed in the vicinity of quench depth we propose the following scenario: After a the transition if the domain size ∆A times the energy- quench, locally symmetry broken domains start to grow density differences ∆ is less than k T . In equilibrium exponentially until about 37% of the space is covered. B this is nothing but another argument for critical slowing In our 2D system this marks a threshold where domains down. In 2D systems it is well known that fluctuations with different orientation (different phases of the symme- play a major role. This is obviously the case not only in try broken field) start to touch. The following dynamics equilibrium but also in non-equilibrium situations. is dominated by the conversion of the yet untransformed regions of the high symmetry phase marked by an alge- In conclusion, we have investigated the Kibble-Zureck braic increase of the bond order correlation length. For mechanism experimentally for a two-dimensional system finite cooling rates, G. Biroli et al. have argued [42] with complex order parameter given by the local bond that defect annihilation will alter the classical Kibble- order director field. After a sudden quench we resolve Zurek mechanism leading to two different time regimes two time regimes at an early state (followed by a third after the fall out time, separating critical and classical regime due to classical coarse graining, see [40]). First, the bond order correlation length grows exponentially un- 5 til roughly 40% of the area belongs to symmetry broken T.W. Hyde, A. Kovacs, Z. Donko: Phys. Rev. Lett., 105, domains with random orientation. While Zurek proposed 115004 (2010) an algebraic growing for 3D systems, we suggest the ex- [25] Yen-Shuo Su, Chi Yang, Meng Chun Chen, and Lin I: ponential behavior being caused by the underlying ex- Plas. Phys. Con. Fus., 54, 124010 (2012) [26] Chi Yang, Chong-Wai Io, and Lin I: Phys. Rev. Lett., ponential divergences in KTHNY-melting, being typical 109, 225003 (2012) for 2D systems. In a second regime an algebraic behav- [27] Z.R. Wang, A.M. Alsayed, A.G. Yodh, and Y.L. Han: J. ior is observed until no high symmetry phase is left, ex- Chem. Phys., 132 154501 (2010) cept grain boundaries separating domains with different [28] A. Jelic and L.F. Cugliandolo: J. of Stat. Mech., P02032 phase. The independence of the mosaicity strongly sup- (2011) ports the Kibble-Zureck mechanism. We hope our work [29] J.M. Kosterlitz and D.J. Thouless: J. Phys. C, 5, 124 will stimulate theory and simulations to investigate the (1972) [30] J.M. Kosterlitz and D.J. Thouless: J. Phys. C, 6, 1181 phenomena observed in phase transitions far from ther- (1973) mal equilibrium. Further studies in dusty plasma and [31] B.I. Halperin and D.R. Nelson: Phys. Rev. Lett. 41, 121 colloidal mono-layers might shed light on differences and (1978) similarities of the Kibble-Zurek mechanism in 2D and 3D [32] D.R. Nelson and B.I. Halperin: Phys. Rev. B., 19, 2457 systems at sudden quenches or finite cooling rates. (1979) P.K. acknowledges fruitful discussion with Sébastien [33] A.P. Young, Phys. Rev. B, 19, 1855 (1979) Balibar. Financial support from the German Re- [34] P. Keim, G. Maret, and H.H. von Grünberg: Phys. Rev. E, 75, 031402 (2007) search Foundation (DFG), SFB-TR6 project C2 and KE [35] S. Deutschländer, A.M. Puertas, G. Maret, and 1168/8-1 is further acknowledged. P. Keim: Phys. Rev. Lett., accepted (2014) arxiv.org/abs/1312.5511 [36] The transition temperatures correspond to those given in [34, 39, 41]. Recent publications have slightly different values based on new determinations of the magnetic ∗ [email protected] suszeptibility of a new batch of particles [35]. [37] The equilibration time to gain a completely flat water/air [1] T.W. Kibble: Phys. Rep., 67, 193 (1980) interface without any gradient of particle numberdensity [2] L.D. Landau: Zh. Eksp. Teor. Fiz. 7 19 (1937) within the plane is several weeks up to a few months. [3] V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, [38] F. Ebert, P. Dillmann, G. Maret, and P. Keim: Rev. Sci. 1064 (1950) Inst., 80, 083902 (2009) [4] W.H. Zurek: Nature, 317, 505 (1985) [39] P. Dillmann, G. Maret, and P. Keim: J. Phys. Cond. [5] Y. Nambu and G. Jona-Lasinio: Phys. Rev., 122, 345 Mat., 20, 404216 (2008) (1961) and 123, 246 (1961) [40] Further information can be found in the Supplemental [6] P.W. Higgs: Phys. Rev. Lett., 13, 508 (1964) Material [7] Y. Zeldovic, I.Y. Kobzarev, and L.B. Okun: Zh. Eksp. [41] P. Dillmann, G. Maret, P. and Keim: Euro. Phys. J. Teor. Fiz., 67, 3 (1974) Special Topic, 222, 2941 (2013) [8] A. Vilenkin: Phys. Rev. D, 24, 2082 (1981) [42] G. Biroli, L.F. Cugliandolo, and A. Sicilia: Phys. Rev. E, [9] A. Albrecht and N. Turok: Phys. Rev. Lett., 54, 1868 81, 050101 (2010) (1985) [43] T.O.E. Skinner, D.G.A.L. Aarts, and R.P.A. Dullens: [10] G tHooft: Nucl. Phys. B, 79, 276 (1974) Phys. Rev. Lett., 105, 168301 (2010) [11] A.M. Polyakov: JETP Lett., 20 194 (1974) [44] F.A. Lavergne, D.G.A.L. Aarts, and R.P.A. Dullens Oral [12] T.W. Kibble: J. Phys. A: Math. Gen., 9, 1387 (1976) contribution on the 9th liquid matter conference, Lisbon, [13] A.H. Guth: Phys. Rev. B, 23, 347 (1981) (2014) [14] A.D. Linde: Phys. Lett., 108 B, 389 (1982) [15] A. Albrecht and P.J. Steinhardt: Phys. Rev. Lett., 48, 1982 (1982) [16] BICEP2 Collaboration: Phys. Rev. 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Supplemental Material of Spontaneous symmetry breaking in 2D: Kibble-Zurek mechanism in temperature quenched colloidal monolayers (Dated: September 16, 2014) The supplemental material contains the crossover from algebraic to exponential for various measures, a discussion in the context of critical nucleation theory, and coarse graining due to grain boundary diﬀusion in a third time window.

CORRELATION LENGTH AND DEFECT DENSITY

t = - 9 , 7 s Two-dimensional dusty plasma and colloidal mono- 1 t = 4 9 , 8 s t = 2 2 8 , 4 s layers oﬀer the possibility to investigate structure for- t = 4 6 2 , 3 s t = 6 0 0 , 0 s

mation on single particle level. They are complemen-  1 -

tary in their single particle dynamics in that sense, that )

t , r

colloidal systems are over-damped due to the solvent ( 0 , 1 g whereas dusty plasma are not. Under equilibrium con- g  ditions this results in exponential relaxation after small local perturbations for colloidal systems whereas in dusty plasma oscillatory behaviour is possible. If the interac- 0 , 0 1 tion between particles is determined by an outer field, 0 5 both systems can be quenched almost instantaneously r [ a ] into different regions of the phase diagram. Instanta- neously means extremely fast compared to intrinsic time FIG. 1. Pair correlation function in a lin-log plot, before scales which is usually given by the Brownian time, the (black sysmbols) and after a quench from Γ = 13 to a final time a particle need to diffuse it’s own diameter. coupling strength of ΓE = 63. The pair correlation function For our colloidal monolayer with super-paramagnetic always decays exponential (this is same for quenches to ΓE = 70, 94, 140, not shown here). Solid lines are exponential fits interaction, the outer field is switched on timescales of at different times after the quench. about 5 ms whereas the Brownian time is of the order of 50 sec, four orders of magnitude larger. This way cooling rates are much larger in colloidal systems or dusty plasma tional correlation length ξ6. To complete the picture and compared to those being accessible in atomic systems. to follow Hartmann et al. [1] we analyzed the standard Structure formation and relaxation in dusty plasma has pair correlation function been studied in [1–3]. N Peter Hartmann et al. have carefully studied the crys- 1 1 g (r) = δ(r r ) , (1) tallization dynamics using the bond order orientational g 2πrρhN − ij i i=i correlation length ξ6, the standard pair correlation length X6 ξg and the inverse defect density 1/D as measures for where ρ is the 2D particle density. Fig. 1 shows the the size of crystallites. As in our colloidal system, both exponential decay of g (r, t) 1 for different times af- | g − | correlation functions always decay exponentially and the ter the quench. The correlation length ξg(t) (left column) correlation lengths grow monotonic in time. In [1] a and the inverse defect density (right column) is plotted in crossover from algebraic to algebraic with lower expo- Fig. 2 for different quench depth in our colloidal system. nent is observed for the correlation length but a single In all cases, the cross-over from exponential to algebraic algebraic behaviour for the inverse defect density. At is observed. The cross-over times do not match exactly the present stage we can not decide if the differences but are of the same order. They show qualitatively iden- are due to different kinds of kinetics (ballistic in plasma tical behaviour, e.g. the decreasing cross-over time with versus Brownian in colloids) or if the exponential in- increasing quench depth. Comparing Fig. 1 and Fig. crease in time of correlation length and inverse defect 2 of the main manuscript, the correlation length ξ6 at density is due to the exponential divergences in KTHNY- the cross-over time is still of the order of unity even if melting. Note, that the dusty plasma is reported not the symmetry broken domains are extended over several follow KTHNY-melting but seem to be closer to grain particles. The correlation length is ’short eyed’ since it is boundary induced melting [1]. The Kibble-Zurek mach- averaged over all particles and especially at the cross over anism on the other hand might explain the domain for- time 60% of the particles are still in the high symmetry mation for cooling rates different from zero. phase. Therefore we introduce a criterion for symmetry Fig. 2 from the main manuscript shows the orienta- broken domains as given in the main text and extensively 7

4 2 0 τ t / τ ~ e t / f o r t ≤ 5 0 s ~ e f o r t ≤ 5 0 s 3 α Γ = 6 3 α ≥ 17 E ~ t f o r t ≥ 170 s Γ a ~ t f o r t 0 s 1 0 = 6 3

/ E g

ξ 2

D / 1

1 2 1 1 0 t [ s ] 1 0 0 6 0 0 1 1 0 t [ s ] 1 0 0 6 0 0

4 2 0 τ t / τ ~ e t / f o r t ≤ 4 5 s ~ e f o r t ≤ 4 5 s 3 α α ≥ 12 Γ ~ t f o r t ≥ 120 s Γ a ~ t f o r t 0 s = 7 0 E 1 0 = 7 0

/ E g

ξ 2

D / 1

1 2 1 1 0 t [ s ] 1 0 0 6 0 0 1 1 0 t [ s ] 1 0 0 6 0 0

4 2 0 τ t / τ ~ e t / f o r t ≤ 3 5 s ~ e f o r t ≤ 3 5 s 3 α ≥ 85 Γ = 9 4 α ≥ 85 Γ ~ t f o r t s E ~ t f o r t s = 9 4 a 1 0 E / g

ξ 2

D / 1

1 2 1 1 0 t [ s ] 1 0 0 6 0 0 1 1 0 t [ s ] 1 0 0 6 0 0

4 2 0 τ τ ~ e t / f o r t ≤ 2 0 s ~ e t / f o r t ≤ 2 0 s 3 α ≥ 6 Γ = 1 4 0 α ≥ 6 Γ = 1 4 0 ~ t f o r t 0 s E ~ t f o r t 0 s E a 1 0 / g

ξ 2

D / 1

1 2 1 1 0 t [ s ] 1 0 0 6 0 0 1 1 0 t [ s ] 1 0 0 6 0 0

FIG. 2. Correlation length ξg from the pair correlation as function of time in a log-log-plot for different quench depth. Shortly after the quench, the growing behavior is exponential and switches to algebraic later. Note the decreasing cross-over time for deeper quenches. discussed in [4]. The threshold values are defined in comparison with mono-crystalline and isotropic fluid phases under equilibrium conditions and the qualitative results are robust LOCAL DOMAINS to (moderate) variations (about 20%) of the threshold values. Simply connected domains± of particles which ful- To repeat the criteria we note; individual particles are fill all three criteria are merged to a crystalline cluster. defined to be part of a symmetry broken domain if the Fig. 3 and the movie SBD.avi shows such domains. As following three conditions are fulfilled for the particle it- a side remark: comparing our results with colloidal sys- self and at least one nearest neighbour: tems, where an attractive interaction triggers a diffusion limited aggregation from a dilute ’gas’ [5, 6] and with The magnitude of the local bond order field m6 simulations and phase field calculations [7, 8], no amor- • k must exceed 0.6 for both neighboring particles. phous precursors (with increased local density) are found in our experiment, where purely repulsive interaction is The bond length deviation ∆ lkl of neighboring present. • particles k and l is less than 10%| | of the average particle distance la.

The variation in bond orientation ∆Θ = ψk ψl • of neighboring particles k and l must be| less− than| 2.3◦ in real space (less than 14◦ in sixfolded space). 8

Aj at ti with Bk at time ti+1. The index j ranges from 1

to Nti and k ranges from 1 to Nti+1 where Nti and Nti+1 are the numbers of cluster in the time steps ti and ti+1.

No passing of Cluster-label Aj to Bk if no overlap • in areas exists in the following time step. Cluster Aj has disappeared.

If cluster A has only overlap with one cluster B , • j k this cluster will become the label of cluster Aj in the previous time step.

If more than one cluster Bk has overlap with Aj, the • cluster with the largest overlap in area will become the label of Aj.

If there are cluster B which did not get a label • k FIG. 3. Snapshot of the monolayer with symmetry broken after all cluster Aj of time step ti has run through, domains (SBD), 64 s after a quench from Γ = 13 to a fi- they will get a new label. nal coupling strength of ΓE = 93. Small dots are fluid-like particles, whereas big ones are crystal-like. Different colors In Fig. 3 of the main manuscript, where the average indicate individual grains which are labeled in time. size and number of symmetry broken domains is plotted, one can see that the nucleation starts immediately after the quench. Therefore, no lack time of critical nucleation DISCUSSION OF CRITICAL NUCLEATION is detectable. Additionally, with the parameters given above about 10% of the particles are identified having low Crystal nucleation is one of the basic physical phenom- symmetry (large local order) already before the quench, ena that govern the growth and the structure of crys- deep in the fluid phase at Γ = 14. Such domains consists talline state in three dimensions [9–23]. The most widely of 2.7 particles on average and the lifetime of the domains used concept, the Classical Nucleation Theory (CNT), is a few seconds. In the context of nucleation such cluster attempts to describe this complex process by treating are usually interpreted as ’precritical nuclei’. In the con- the nuclei as compact spheres that grow if their size ex- text of second order phase transitions, those fluctuations ceeds a critical value [9–11]. This critical value is de- should be interpreted as local order parameter fluctua- termined by a maximum in the free energy, where the tions and are the analogue to quantum or vacuum fluc- energy gain in the volume r3 overcompensates the sur- tuations of the Higgs field in the Kibble mechanism. The face energy cost r2 of the∼ nucleus. However, experi- large amount (compared to 3D systems) of those ’precrit- ments on hard sphere∼ colloidal systems [15, 16, 19], where ical nuclei’ (or order parameter fluctuations) is due to the individual particle interactions are sufficiently simple to fact that the preferred local symmetry in 2D is sixfold, be modeled theoretically differ from the predictions of both in the crystalline as well as fluid phase. If the crite- critical nucleation theory. In more complex systems, the ria for symmetry broken domain particles are tightened, CNT calculation underestimates the nucleation rate by the amount of ’precritical nuclei’ decreases but then a many orders of magnitude [24]. It is a priory not clear monocrystal in equilibrium is not detected as such any if Classical Nucleation Theory and the concept of super- more. The latter is due to Mermin-Wagner fluctuations cooled fluids might be applicable for monolayers. Never- being typical for low dimensional systems [25]. theless, the energy gain in volume of the low symmetry The formation of a nucleus in 2D after a temperature nuclei can be translated into an energy gain of area in quench is not a rare event and the critical nucleation bar- two dimensions. Similarly the 3D surface tension might rier (if it exist at all) is very low. In this case one would be translated into a 2D line tension such that a critical expect all individual nuclei 2 particles to grow after nucleus size exist in 2D, too. Following Urs Gasser et the quench which is surprisingly≥ not the case. In Figure al. [16] in a 3D colloidal ensemble, we label individual 4 the difference of growing pg and shrinking probability grains in time and compare the size (area) in consecutive ps of nuclei is plotted for different time windows after time steps. This procedure is published in [? ] and only a quench to ΓF = 93.5 where M is the size of the nu- briefly summarized here. If the crystalline domains are clei given by the number of the particles which belong identified for the whole duration of the experiment one to the cluster. The probability for shrinking and grow- can label the domain of consecutive time steps to follow ing for individually labeled clusters of size M is given by their temporal behaviour. Cluster of time ti and ti+1 are pg/s = Ng/s/NM where NM is the number of clusters of compared by investigating the overlap in area of cluster size M and Ng or Ns are the numbers of those clusters 9

FIG. 4. The shrinking probability after all quenches (ΓF = 63, 94, 140 is shown, ΓF = 70 is not shown here) for a crystalline domain composed of M particles is always larger than the growing probability. Unlike in 3D systems one can not determine a critical nucleus size on single particle level and hence 2D systems out of equilibrium can not be described with critical nucleation theory.

which are larger Mτ > M or smaller Mτ < M after a is the reason why we do not use the terminus ’nuclei’ finite waiting time τ. but ’symmetry broken domain’ in the main manuscript. For all time windows and nucleus sizes the probabil- Furthermore the ’subcritical nuclei’ in the isotropic face ity of shrinking is larger than the probability of growing. should be interpreted as local order-parameter fluctua- This is true for all quench depths performed, close to tions. melting (ΓF = 63) as well as deep into the crystalline In the context of Kibble-Zurek mechanism the large phase (ΓF = 140). The waiting time was varied over al- shrinking probability for individual is still surprising: it most two decades from 0.1 10 sec and no qualitative indicates that fluctuations of high symmetry and symme- differences were found. Obviously− shrinking (often, but try broken phase is still allowed beyond the fall out time. small areas) and growing (less often, but larger areas) is This might be a special feature of 2D systems where the quite asymmetric and the solidification far from equilib- energy-density differences between high symmetry and rium is strongly dominated by fluctuations. Practically symmetry broken phase seem to be small. Furthermore all grains which are present shortly after the quench dis- fluctuations in 2D are known to have dramatic effects on appear again and those domains wich are present at late the phase behaviour already in equilibrium situations. times were generated in between. Two mechanisms appear to be responsible for this. First, small grains dis- solve completely into fluidlike particles and second, at a COARSE GRAINING later stage, grains of intermediate size fluctuate strongly in orientation such that they disrupt into smaller subdo- In Fig. 2 of the main manuscript and in Fig. 2 we mains (looking like Mermin-Wagner fluctuations on a lo- showed the cross-over from exponential to algebraic grow- cal scale). Such subdomains frequently merge again but ing of linear measures of the domain size averaged over then the net contribution to the growing probability is at least ten independent quenches for each quench depth. typically zero, since one grain has grown but the other(s) The longest investigated times were 10 min Fig. 5 we has/have ’disappeared’. Whereas the latter might be show the bond order correlation lenght ξ of a single an artefact based on difficulties to define symmetry bro- 6 quench to a final system temperature Γ = 153 for up to ken domains on a local scale in a (on average) sixfold F 2 h after the quench. On this time scale, a third regime background, the disappearance of local order in the early is observable which again shows algebraic growing with stage is completely unexpected. Interestingly the lifetime reduced exponent. In this regime grain boundaries (given of symmetry broken domains shortly after the quench as chains of dislocation pairs) as the only high symmetry is again a few seconds and comparable to the lifetime residuals separate symmetry broken domains with differ- of ’subcritical nuclei’ in the isotropic phase before the ent orientation. The temporal evolution of the system quench. Alltogether: the absence of a lack time, the large is given by classical coarse graining mediated by grain amount of ’precritical nuclei’ and the dominating shrink- boundary diffusion [26, 27] recently analyzed in colloidal ing probability for individual grains indicates that the monolayers. formation of a nucleus in 2D after a temperature quench is not a rare event and a critical nucleation barrier does not exist. Since the underlying equilibrium phase transition is continuous according KTHNY-theory, it is finally not to surprising that the concept of a supercooled fluid [1] P. Hartmann, A. Douglass, J.C. Reyes, L.S. Matthews, and classical nucleation does not fit to our system. This T.W. Hyde, A. Kovacs, Z. Donko: Phys. Rev. Lett., 105, 10

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