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# Mermin–Wagner fluctuations in 2D amorphous solids

Edited by Giorgio Parisi, University of Rome, Rome, Italy, and approved December 19, 2016 (received for review August 4, 2016)

## Significance

Reducing dimensionality often entails striking new physics, as, e.g., the Quantum Hall effect in 2D electron systems or the different recurrence probabilities of random walks in various dimensions. In addition, phase transitions in 2D and 3D differ significantly: Long before graphene came into focus, theories for melting of 2D crystals were developed and probed with colloidal monolayers. In 2D, the crystalline and fluid phase is separated by a so-called hexatic phase, unknown in 3D. Furthermore, 2D crystals exhibit instabilities on large length scales. Here, we show that those instabilities also exist in 2D amorphous solids, but do not melt them—a topic that emerged only recently, 50 y after the prediction in 2D crystals by D. Mermin and H. Wagner.

## Abstract

In a recent commentary, J. M. Kosterlitz described how D. Thouless and he got motivated to investigate melting and suprafluidity in two dimensions [Kosterlitz JM (2016) *J Phys Condens Matter* 28:481001]. It was due to the lack of broken translational symmetry in two dimensions—doubting the existence of 2D crystals—and the first computer simulations foretelling 2D crystals (at least in tiny systems). The lack of broken symmetries proposed by D. Mermin and H. Wagner is caused by long wavelength density fluctuations. Those fluctuations do not only have structural impact, but additionally a dynamical one: They cause the Lindemann criterion to fail in 2D in the sense that the mean squared displacement of atoms is not limited. Comparing experimental data from 3D and 2D amorphous solids with 2D crystals, we disentangle Mermin–Wagner fluctuations from glassy structural relaxations. Furthermore, we demonstrate with computer simulations the logarithmic increase of displacements with system size: Periodicity is not a requirement for Mermin–Wagner fluctuations, which conserve the homogeneity of space on long scales.

For structural phase transitions, it is well known that the microscopic mechanisms breaking symmetry are not the same in two and in three dimensions. Whereas 3D systems typically show first-order transitions with phase equilibrium and latent heat, 2D crystals melt via two steps with an intermediate hexatic phase. Unlike in 3D, translational and orientational symmetry are not broken at the same temperature in 2D. The scenario is described within the Kosterlitz, Thouless, Halperin, Nelson, Young (KTHNY) theory (1⇓⇓⇓–5), which was confirmed (e.g., in colloidal monolayers) (6, 7). However, for the glass transition, it is usually assumed that dimensionality does not play a role for the characteristics of the transition, and 2D and 3D systems are frequently used synonymously (8⇓⇓⇓–12), whereas differences between the 2D and 3D glass transition are reported in ref. 13.

In the present work, we compare data from colloidal crystals and glasses and show that Mermin–Wagner fluctuations, well known from 2D crystals, are also present in amorphous solids (14, 15). Mermin–Wagner fluctuations are usually discussed in the framework of long-range order (magnetic or structural). However, in the context of 2D crystals, they have also had an impact on dynamic quantities like mean squared displacements (MSDs). Long before 2D melting scenarios were discussed, there was an intense debate as to whether crystals and perfect long-range order (including magnetic order) can exist in 1D or 2D at all (16⇓⇓–19). A beautiful heuristic argument was given by Peierls (17): Consider a 1D chain of particles with nearest neighbor interaction. The relative distance fluctuation between particle

What is the impact of Mermin–Wagner fluctuations? They are long(est) wavelength density fluctuations, and, mapping locally a perfect mathematical 2D lattice with commensurable density and orientation, one finds the displacement of particles to diverge. It is shown analytically that this displacement from perfect lattice sites increases in two dimensions logarithmically with distance (15, 20). Having a closer look at the arguments given in ref. 17, one finds that periodicity is not a requirement for those fluctuations. They will also be present in other 2D (and 1D) systems like quasicrystals or amorphous structures, provided the fact that nearest-neighbor distances have low variance (unlike, e.g., in a gas). D. Cassi, F. Merkl, and H. Wagner (22⇓–24) mapped the absence of spontaneously broken symmetries to the recurrence probability of random walks. In this work, it is proven that spontaneous magnetization on amorphous or fractal networks cannot occur in

We do not intend to enter the discussion about an “ideal” vs. “quasi-ideal” glass transition in the sense of infinite or just extremely large viscosities. Because infinite timescales are required to measure infinite viscosities, this is a purely academic discussion, and no experiment (or simulation) will prove this strictly. With respect to Mermin–Wagner fluctuations, we can state that it will depend on the way we measure: As in crystals, the viscosity will always be finite on arbitrary large length scales. On a local scale, Mermin–Wagner fluctuations do not change the cage-escape process, and thus the microscopic mechanism of 2D and 3D glass transitions are not necessarily different. Recent work by Vivek et al. using cage-relative intermediate scattering functions support this idea (31), and computer simulations by Shiba et al. independently found similar results (32).

Fig. 2 shows MSDs, where the sum runs over all particles **2** is the center of mass of the cage given by the *Left* shows the standard MSD as a function of time in red for a fluid system (red triangles) and two crystalline samples (red squares and circles). In a 2D crystal, the MSD is not confined. This indicates the failure of the Lindemann criterion in 2D. By using cage-relative coordinates (blue curves), the fluid data still diverge (blue triangles), but the CR-MSD from solid samples are confined (blue squares and triangles). The dashed line shows the critical value given by the dynamic Lindemann criterion (which is ^{∗} This is done by analyzing only particles that have a crystalline environment (six nearest neighbors) for the time of investigation. Fig. 2, *Center* shows the same analysis for a glass-forming system. The MSD of the fluid sample is labeled with red triangles; the transparent squares label a sample that is glassy but very close to the transition temperature; and the circles and diamonds represent amorphous solids (35, 36). Focusing on the CR-MSDs (blue curves), one finds that the amplitude of the local displacements is lower, but even for the deepest supercooled amorphous solid (blue diamond), there is an upturn for long times. Whereas for the CR-MSD (blue curves), long wavelength phonons are shortcut and thus invisible, the structural relaxation, which typically appears for glasses, is still visible. The so-called *Right* shows a 3D glass that lacks per definition Mermin–Wagner fluctuations. The amplitude of the CR-MSD (blue) is only slightly smaller, compared with the standard MSD (red), and the upturn seem to happen simultaneously. Only structural relaxation is measured that is shifted beyond the accessible time window for the system deepest in the glass (diamonds). The corresponding Fig. 2, *Insets* show typical snapshots of the 2D systems (see experimental details below and in *SI Text*, whereas for the 3D system, a sketch is shown, reconstructed from structural data of the amorphous solid.

## 1. Colloidal Systems

This difference of global and local fluctuations in 2D is already a hallmark of Mermin–Wagner fluctuations, but before we focus in orientational and structural decay, the experimental realization of 2D and 3D systems and details about the simulations are briefly discussed. The 2D systems are well established, and we investigated crystallization, defects (37⇓⇓–40), and the glass transition (30, 35, 36) with this setup. They consist of colloidal monolayers where individual particles are sedimented by gravity to a flat a water/air interface in hanging-droplet geometry. The colloids are a few microns in size and perform Brownian motion within the plane. The control parameter of the system is *SI Text*,

The 3D colloidal systems consist of more than a billion particles, dissolved in an organic solvent with identical mass density; thus, particles do not sediment. The colloids are slightly charged and the interaction is given by Coulomb interaction screened by a small amount of counterions in the solvent (Yukawa potential). Additional details are given in Table S1 and *SI Text*. Monitoring is performed with confocal microscopy, providing 3D images with several thousand particles being tracked in the field of view. Finite size effects in 2D were additionally investigated with computer simulations, specifically Brownian dynamics simulation of hard disks (see below). To prevent crystallization, a binary mixture of different sizes of disks was used. The phase diagram was controlled by entropy (not temperature), and the control parameter in this systems was solely given by the (area) packing fraction

Colloidal systems are so-called soft-matter systems: The interaction energy between particles is of the order (tenth of) eV, comparable to atomic or molecular systems. However, because length scales (distances between particles) are approximately

## 2. Structural and Orientational Decay in 2D and 3D

To measure the structural decay and to investigate the

Eq. **3** is nothing but distribution of displacements in Fourier space *Upper Left*) and 2D glass (Fig. 3, *Upper Right*) of dipolar particles as in Fig. 2, but omitting the fluid curves. After an initial decay due to thermal vibrations (which is hardly seen on the log-lin scale), the red curves enter a plateau, indicating the dynamic arrest. Only the stiffest glass (diamonds) is stable on the accessible timescale. Fig. 3, *Lower Right* shows data from simulations of a 2D hard disk system for comparison. The qualitative behavior is the same as for the 2D dipolar glass. Fig. 3, *Lower Left* shows the 3D glass, again with a typical two-step decay, except for the strongest glass (red diamonds), where the decay is hardly visible on the experimental accessible timescale.

In analogy to the CR-MSD, one can define a cage-relative intermediate scattering function given in blue in Fig. 3, where the displacement is reduced by the center of mass motion of the nearest neighbors (31). In 2D, the nearest neighbors are defined by Voronoi–Tessellation, whereas in 3D a cutoff value of

We further introduce the bond order correlation function

In Fig. 3, we now compare the cage-relative intermediate scattering function

For the soft glasses *Upper Right*), both correlation function *Lower Left* shows the 3D glass. The stiffest glass 1 (diamonds) is almost stable. In glass 2 (blue, red, and green circles) and glass 3 (blue, red, and green squares), all correlation functions decay on the same timescale due to structural relaxations, but without Mermin–Wagner fluctuations. We conclude that 2D crystals are affected by Mermin–Wagner fluctuations, and 2D glasses are affected by Mermin–Wagner fluctuations and

## 3. Finite Size Effects

In Fig. 3, a separation of timescales between standard structural and orientational decay was shown for the 2D glasses. However, the

Plotting the square root of the amplitude of the inflection point as a function of the logarithm of the linear system size

## 4. Discussion

In ref. 13, E. Flenner and G. Szamel reported fundamental differences between glassy dynamics in two and three dimensions and detected strong finite size effects in 2D. The localization in a 2D glass (e.g., measured by the plateau of the intermediate scattering function or MSD) is not well pronounced and decays faster in large systems. Bond-order correlation functions (taking only sixfolded particles into account) decay later and show less size dependence. Additionally, particle trajectories show sudden jumps in 3D, but not in 2D, and dynamical heterogeneities are significantly pronounced in 3D. They conclude that vitrification in 2D and 3D is not the same, calling for a reexamination of the present glass transition paradigm in 2D. We perform this reexamination, taking Mermin–Wagner fluctuations into account. We cannot address differences between Newtonian and Brownian dynamics (13) in colloidal experiments, and thus they are not discussed in the present manuscript. All other differences observed in ref. 13 for 2D and 3D disappear in our system when using local coordinates (25). E. Flenner and G. Szamel investigated remarkably large systems to reduce finite size effects, but this even enhances Mermin–Wagner fluctuations: Those fluctuations affect translational degrees of freedom, but not orientational ones, and depend logarithmically on the system size. In ref. 29, we showed that non-Gaussian behavior of the self part of the van-Hove function (which measures the variance of displacements at a given time) in 2D systems is only visible in local, cage-relative coordinates. Equivalently, the dynamical heterogeneities show significantly more contrast in cage-relative coordinates. Mermin–Wagner fluctuations simply “smear out” local events like hopping and cage escape if particles are measured in a global coordinate frame (29).

In ref. 30, we assumed that the presence of collective motion might be due to long wavelength fluctuations. Now this is validated in direct comparison with 2D crystals. A recent manuscript by S. Vivek et al. (31) reports results of a soft sphere and a hard sphere glass in 2D compared with a 3D glass. Using similar correlation functions, their results are essentially the same, but the (almost) hard sphere glass showed less signature of Mermin–Wagner fluctuations. This can be explained by the work of Fröhlich and Pfister (20): They determined the following conditions for Mermin–Wagner fluctuations to appear: (*i*) The pair-potential of particles has to be integrable in the far field and (*ii*) analytically at the origin. The first condition rules out Coulomb interaction, because for this long-range potential, the second, third, and higher nearest-neighbors interaction is strong enough that particle displacements cannot add up statistically independently. The second condition questions hard sphere interaction. An easy argument in the limit of zero temperature might go as follows: When all particles are at contact and closed packed, no positional fluctuations can appear at all. At finite temperature, the Mermin–Wagner fluctuations are excited, as shown in Fig. 4, but the separation of timescales is less pronounced in Fig. 3 for the hard-disks simulation, consistent with the results of Vivek et al. (31). An alternative ansatz to investigate Mermin–Wagner fluctuations is reported by H. Shiba et al., showing analogue results in large-scale computer simulations. Shiba et al. analyzed bond-breakage correlations, four-point correlations (46), and intermediate scattering functions in 2D and in 3D. Being a local quantity, bond-breakage correlations do not differ in 2D and 3D; thus, the microscopic nature of the glass transition is similar. However, the density of vibrational states of a 2D system, computed from the velocity autocorrelation function, shows an infinite growth of acoustic vibrations, very similar to 2D crystals (32). Those beautiful results are completely in line with our arguments.

Connecting Mermin–Wagner fluctuations and glassy behavior in 2D points to another question that is yet not completely solved, namely, how (shear) solidity appears. Glass is a solid on intermediate timescales, but on infinite times it may flow. A 2D crystal is soft on infinite length scales (lacking globally broken symmetries) and a solid only at intermediate scales.

## 5. Conclusions

Comparing experimental and simulation data in 2D and 3D, we show that a 2D crystal is affected by Mermin–Wagner fluctuations in the long time limit; a 2D glass has Mermin–Wagner fluctuations as a “second channel of decay” beside the standard

## SI Text

The confinement of colloids to two dimensions is achieved by sedimentation to an interface, which in our case is the lower water/air surface in hanging droplet geometry. The volume is regulated actively with a microsyringe to get a flat interface. The fluid interface guarantees free diffusion of the colloids (without any pinning) within the horizontal monolayer, but the sedimentation height is about ^{3}. The whole monolayer consists of several hundred thousand particles, where

Because of the superparamagnetic nature of the particles, the potential energy can be tuned by means of an external magnetic field

The 3D glass consists of monodisperse, charged polymethylmetacrylate (PMMA) particles of _{2}O, HBr, and any other polar molecules and ions (48). Dispersed in CHB, PMMA colloids acquire a moderate positive charge. This seems to originate from the adsorption of protons and bromide ions stemming from the decomposition of CHB (with protons being adsorbed more likely) (49). Together with the small abundance of ions in the solvent, one obtains a screened Coulomb interaction with the dimensionless Yukawa potential:

In the following, we will also compare experimental results with simulations. Those are made for a 2D binary mixture of hard disks undergoing Brownian motion by using an event-driven simulation algorithm (51, 52). The system contains

We observed Mermin–Wagner fluctuations in 2D crystal and 2D glass. This raises the question of which low-dimensional systems will show Mermin–Wagner fluctuations. Because well-defined neighbor distances with low variance as in glasses, quasicrystals, and crystals are required, we suggest hyperuniformity, originally formulated to characterize structures with isotropic photonic bandgaps (54, 55), to be a necessary requirement: in hyperuniform structures the number variance *n*-dimensional sphere of radius *n* − 1 dimensional surface of the sphere.

## Acknowledgments

P.K. thanks Herbert Wagner for fruitful discussion. H.K. thanks M. K. Klein and A. Zumbusch for the synthesis of polymethylmetacrylate colloids. P.K. was supported by the Young Scholar Fund, University of Konstanz. S.F. was supported in part by German Research Foundation FOR 1394, Project P3.

## Footnotes

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

Author contributions: P.K. designed research; B.I., S.F., H.K., C.L.K., and P.K. performed research; G.M. monitored research; B.I., S.F., and H.K. analyzed data; and P.K. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵

^{∗}Because real 3D monocrystals incorporate vacancies and interstitials due to entropic reasons, the MSD can, strictly spoken, not be finite due to defect migration in 3D, too.This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1612964114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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