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 crystalsand 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.Mermin-Wagner fluctuations | 2D ensembles | glass transition | phase transition | confined geometry F or 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 longrange order (including magnetic order) can exist in 1D or 2D at all (16)(17)(18)(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 n and particle n + 1 at finite temperature may be ξ. Similar is the fluctuation between particle n + 1 and n + 2. The relative fluctuation between second nearest neig...

Systems in which a short-ranged attraction and long-ranged repulsion compete are intrinsically frustrated, leading their structure and dynamics to be dominated either by mesoscopic order or by metastable disorder. Here we report the latter case in a colloidal system with long-ranged electrostatic repulsions and short-ranged depletion attractions. We find a variety of states exhibiting slow non-diffusive dynamics: a gel, a glassy state of clusters, and a state reminiscent of a Wigner glass. Varying the interactions, we find a continuous crossover between the Wigner and cluster glassy states, and a sharp discontinuous transition between the Wigner glassy state and gel. This difference reflects the fact that dynamic arrest is driven by repulsion for the two glassy states and attraction in the case of the gel.

Using positional data from video-microscopy of a two-dimensional colloidal system and from simulations of hard discs we determine the wave-vector-dependent elastic dispersion relations in glass. The emergence of rigidity based on the existence of a well defined displacement field in amorphous solids is demonstrated. Continuum elastic theory is recovered in the limit of long wavelengths which provides the glass elastic shear and bulk modulus as a function of temperature. The onset of a finite static shear modulus upon cooling marks the fluid/glass transition in an intuitive and unique way.PACS numbers: 82.70. Dd, 61.20.Ja While fluids flow with a finite viscosity, solids respond elastically to deformation. At the glass transition, a supercooled liquid transforms into a disordered solid which possesses mechanical rigidity to shear deformations. The corresponding elastic constant is the shear modulus µ [1]. In crystalline solids, shear rigidity results from the longranged correlations of displacements heralding the breaking of translational invariance. The transverse acoustical displacements are the Goldstone modes, whose (squared) amplitude scales with thermal energy in the equipartition theorem.Yet, the emergence of rigidity at the glass transition, when an amorphous solid forms, remains poorly understood. Obviously, the disorder makes it a subtle fundamental problem to apply the concepts of spontaneous breaking of translational symmetry and of Goldstone modes. Already on the macroscopic level, different predictions exist for the behaviour of the shear modulus µ when an amorphous solid forms. It has been predicted to jump discontinuously by mode coupling theory [2, 3], or to increase continuously from zero by replica theory [4,5]. Such a continuous power-law increase from zero holds at the formation of a random gel, where a microscopic theory has established the link between the modulus and the Goldstone modes [6]. Also in the theory of granular systems, critical fluctuations close to point J (which is the density where a random athermal system becomes jammed) cause a characteristic algebraic increase of the shear modulus µ from zero [7]. A more recent calculation in replica theory again finds a jump in shear modulus at the glass transition and identifies the displacement field of a disordered solid, necessary to discuss elastic acoustic modes. [8]. Definitions of displacement fields in disordered solids have already been given in [9], yet the equipartition theorem could not be established in this approach.Colloidal dispersions offer the unique possibility that the particle trajectories can be observed by video microscopy, and thus are ideally suited to study displacements microscopically. In recent work on colloidal glass [10][11][12], the covariance matrix of the particle displacements was obtained and the mechanical density of states and the associated local modes were studied. But the shear modulus and the elastic dispersion relations were not obtained. Up to now only in crystalline colloidal solids, the e...