Abstract
Liquids cooled towards the glass transition temperature transform into amorphous solids that have a wide range of applications. While the nature of this transformation is understood rigorously in the meanfield limit of infinite spatial dimensions, the problem remains wide open in physical dimensions. Nontrivial finitedimensional fluctuations are hard to control analytically, and experiments fail to provide conclusive evidence regarding the nature of the glass transition. Here, we develop Monte Carlo methods for twodimensional glassforming liquids that allow us to access equilibrium states at sufficiently low temperatures to directly probe the glass transition in a regime inaccessible to experiments. We find that the liquid state terminates at a thermodynamic glass transition which occurs at zero temperature and is associated with an entropy crisis and a diverging static correlation length. Our results thus demonstrate that a thermodynamic glass transition can occur in finite dimensional glassformers.
Introduction
Difficult scientific problems can drastically simplify in some unphysical limits. For instance, very large dimensions (d → ∞, where d is the spatial dimensions) give relevant fluctuations a simple meanfield character^{1}, and onedimensional (d = 1) models can often be treated exactly. Yet these two solvable limits are crude idealizations of our threedimensional reality. The rich theoretical arsenal developed to interpolate between them has revealed the highly nontrivial role of spatial fluctuations in all areas of science. In particular, as the number of spatial dimensions decreases, a phase transition may change nature or even disappear. Dimensionality thus provides a key tool for understanding the essence of many natural phenomena.
The glass transition from a viscous liquid to an amorphous solid is no exception^{2}. Its meanfield description, which becomes mathematically exact as d → ∞, explains the dramatic slowdown of glassforming liquids through the rarefaction of the number of glassy metastable states upon approaching a critical temperature, T_{K}^{3,4}. The configurational entropy, s_{conf}, which is the logarithm of the number of such states, becomes subextensive when T ≤ T_{K}. The equilibrium glass transition thus corresponds to an entropy crisis, a hypothesis first suggested by Kauzmann in his visionary analysis of experimental data^{5} and initially formalized by Gibbs and DiMarzio^{6} in the context of a lattice polymer model.
The broad discussion that has since ensued^{2} has notably tried to describe the role of finited fluctuations beyond the meanfield framework^{7,8,9,10,11,12}, relating in particular the vanishing of s_{conf} to a diverging pointtoset correlation length, the key quantity for characterizing nonperturbative fluctuations in glass formers^{13}. These fluctuations, however, make it difficult to examine finitedimensional glass formers analytically, even for simple models composed of pointparticles such as those we study here. Exploring a broader diversity of models, from polymer^{14} to anisotropic patchy^{15} models, may yet provide additional theoretical insight.
Meanwhile, Kauzmann’s intuition has been repeatedly validated by experiments^{16,17}, but the conceptual and technical limits of his results have not been lifted. Current experiments access essentially the same restricted temperature range as his 70year old work. Theory and experiments thus currently fail to assess the status of the Kauzmann transition in finite d, or whether new mechanisms qualitatively change the underlying physics^{18,19}. Experimentally, it thus remains controversial whether the trend discovered by Kauzmann survives at much lower temperatures; entropy could go smoothly to zero^{20,21}, or to a finite residual value as temperature vanishes^{15,22,23}.
In this context, computer simulations are especially valuable. They allow direct measurements of both the configurational entropy and the pointtoset correlation length for realistic models of finitedimensional glass formers^{2}. The recent development of the swap Monte Carlo algorithm (SWAP) further allows the exploration of a temperature regime that experiments cannot easily access^{24}, even using ultrastable glassy materials^{25}. This has consolidated and extended Kauzmann’s experimental findings for threedimensional glass formers^{26}. Here, we report that SWAP is so efficient in d = 2 that it provides access to a temperature regime equivalent to experimental timescales 10^{18} larger than the age of the universe. This remarkable advance gives very strong evidence of a thermodynamic glass transition at T_{K} = 0 for d = 2, accompanied by an entropy crisis and the divergence of the pointtoset correlation length. Our results thus illuminate the lowdimensional fate of the glass transition and shed light on the nature of glassy dynamics in d = 2^{27,28,29,30}.
Results
Model and macroscopic behavior
We study a twodimensional mixture of soft particles interacting with a 1/r^{12} purely repulsive powerlaw pair potential and a size polydispersity chosen to minimize demixing, fractionation, and crystallization (see Methods). The average particle diameter is used as unit length, and the strength of the interaction potential as unit temperature. SWAP is implemented following the methodology recently validated for d = 3^{24}. Systems ranging from N = 300 to N = 20,000 particles within a periodic box are used to carefully track finitesize effects in both dynamics and thermodynamics. We mainly present results of N = 1000. Whereas experimental systems are typically composed of more complex particles (such as large molecules or polymers), the exact meanfield theory has thus far only been developed for the same type of point particles as we simulate here. In addition, such models have become a standard to study fundamental aspects of the glass transition, and are good representations of colloidal glasses.
Figure 1a shows that the static structure factor S(k) evolves smoothly over a broad temperature range, from the onset temperature T_{onset} = 0.250 down to T = 0.026, which is the lowest temperature for which our strict equilibrium criteria are met. The typical lowtemperature configuration depicted in Fig. 1b shows that particles of different sizes are well mixed, and that local ordering is extremely weak. In fact, no crystallization event was ever observed in our simulations, and the correlation lengths extracted from the pair correlation function for translational and bondorientational orders evolve modestly with T (see Supplementary Note 1). In other words, the model is an excellent glass former.
The bulk dynamics and equilibration are captured by the bondorientational order time correlation, C_{ψ}(t), which is not affected by longtime tails observed in simple twodimensional fluids^{27,31}. The 1/e decay of C_{ψ}(t) robustly defines bulk relaxation timescales τ_{α} both for SWAP (\(\tau _\alpha ^{{\mathrm{SWAP}}}\)) and normal (\(\tau _\alpha ^{{\mathrm{normal}}}\)) Monte Carlo dynamics (Fig. 1c). We normalize these timescales by \(\tau _0 \equiv \tau _\alpha ^{{\mathrm{normal}}}(T_{{\mathrm{onset}}})\). In agreement with earlier works^{27}, we find that translational correlation functions suffer from large finitesize effects, but that subtracting longrange Mermin–Wagner translational fluctuations results in systemsize independent measurements^{28,29,30} consistent with bondorientational dynamics (see Supplementary Fig. 1). The normal dynamics exhibits a wellknown superArrhenius growth of τ_{α}. Fitting its temperature evolution to a powerlaw divergence situates the modecoupling crossover at T_{MCT} = 0.123, which is roughly the lowest temperature accessible with this dynamics. Earlier work showed that thermalization can be achieved below T_{MCT} using Monte Carlo simulations^{32}. Following ref. ^{24}, we estimate the narrow range within which the experimental glass temperature takes place as T_{g} ∈ [0.0738, 0.0907]. (Henceforth we set T_{g} = 0.082.) The lower end of this interval stems from an Arrhenius fit which provides a lower bound to the true τ_{α}. By all estimates, SWAP dynamics is clearly much faster than the normal one. The speedup is about 5 orders of magnitude at T_{MCT}, 10 at T_{g}, and the Arrhenius lower bound suggests a formidable 42 orderofmagnitude speedup at T = 0.026. Using an atomistic value, τ_{0} = 10^{−10} s, converts this estimate to τ_{α} = 10^{36} s, or approximately 10^{18} times the age of the universe. Such a “cosmological” speedup leaves no doubt that the SWAP equilibration algorithm largely bypasses the slowdown associated with the glass transition in d = 2.
Configurational entropy
This computational advance permits the study of the d = 2 configurational entropy and its relationship to the putative entropy crisis far beyond the previous work^{33}. Extending earlier work on d = 3 systems^{26}, we obtain independent estimates of s_{conf} using stateoftheart methodologies, see Fig. 2a. Technical details are described in Supplementary Note 2. The first estimate stems from subtracting the vibrational contribution, measured by minimizing the potential energy of the system to an inherent structure and obtaining its vibrational spectrum, from the total liquid entropy^{34}. This potential energy landscape (PEL) approach needs to be complemented, for polydisperse systems, with an independent measure of the mixing entropy^{35}. Because minor but systematic additional adjustments are then required, two sets of PEL estimates are reported in Fig. 2a. The two are quantitatively close and similarly decrease with T, which confirms that methodological details do not affect our results in any essential way. This approach extends s_{conf} measurements from 1.5T_{g} in earlier d = 2 simulations^{33} down to a temperature five times smaller, 0.3T_{g}.
Our second estimate directly measures the glass entropy by performing a thermodynamic integration from the wellcontrolled harmonic solid limit. This approach, which is inspired by the Frenkel–Ladd method for crystals^{36}, was recently adapted to polydisperse amorphous solids^{37}. Because it does not count the number of inherent structures but measures instead the entropy of constrained glassy states, it is also very close in spirit (although not equivalent^{37}) to the freeenergy measurement that makes use of the Franz–Parisi potential^{38}. The Frenkel–Ladd estimate is smaller than the PEL ones, as expected, but exhibits a similar temperature dependence.
From the data in Fig. 2a, s_{conf} seemingly vanishes close to T_{K} = 0. This behavior sharply contrasts with that of threedimensional glass formers, for which evidence suggests that T_{K} > 0^{5,16,17,26}. The impending entropy crisis is expected to give rise to largescale fluctuations with a growing pointtoset correlation length^{13}. We use the computational tools developed in refs. ^{26,39,40} to analyze the thermodynamic properties of liquids confined within spherical cavities of radius R drawn from a reference equilibrium configuration (see Supplementary Note 3). The distribution P(Q) of the core cavity overlap Q among the confined equilibrium glassy configurations is then analyzed. The pointtoset correlation length, ξ_{PTS}, is determined from the decay with R of the average overlap. This length is then transformed into a third estimate, \(s_{{\mathrm{conf}}} \propto \xi _{{\mathrm{PTS}}}^{  (d  \theta )}\) with θ = 1. In d = 2, this choice of θ is natural because it both saturates the bound θ ≤ d − 1^{13} and satisfies the wetting relation θ = d/2^{3}. The resulting s_{conf}(T) = ξ_{PTS}(T_{g})/ξ_{PTS}(T) in Fig. 2a again has a similar temperature evolution as other estimates.
Figure 2b shows that rescaling all configurational entropies by their value at T_{g} collapses the entire set of measurements. This robustness is nontrivial because all four estimates make different types of approximations. The agreement of their temperature dependence may thus resolve earlier discrepancies and debates regarding conflicting estimates of the configurational entropy^{41,42}.
One expects s_{conf} to vanish linearly, s_{conf} ∝ (T − T_{K}), but this scaling arguably has a quadratic correction at higher temperatures. We thus perform a quadratic fit to the lowtemperature regime, T < T_{g}. This fitting yields T_{K} ≤ 0.003 for all cases. These estimates of T_{K} are 10 times smaller than our lowest temperature, T = 0.026, and 30 times smaller than T_{g}. The scaling behavior implied by this observation is presented in Supplementary Note 4. Known alternatives to an entropy crisis invoke a change in the concavity of s_{conf}^{14,22,43} and should be accompanied by a maximum in the specific heat c_{V}^{18,20,21}; we observe neither the convexity (Fig. 2a) nor the specific heat maximum (Fig. 2c). As T → T_{K}, c_{V} instead monotonically increases towards a finite value that is larger than the Dulong–Petit law. These observations thus strongly support the occurrence of a nontrivial entropy crisis at T_{K} = 0. The only alternative left is a change of behavior occurring at temperatures even lower than those we can study directly.
Pointtoset length scale
The thermodynamic glass transition at T_{K} = 0 also coincides with a divergence of the pointtoset correlation length. We illustrate the physical meaning of this length scale in Fig. 3a in the form of a (T, 1/R) diagram reminiscent of both the Franz–Parisi thermodynamic construction^{38} and of the random pinning approach^{44,45}. Upon decreasing the cavity size at a given temperature, the system crosses over from a lowQ regime at large R to a highQ regime at small R, as illustrated by the snapshots in Fig. 3a. For any T > 0, this crossover around R ≈ ξ_{PTS} corresponds to a finitesize version of the random firstorder glass transition with a rarefaction of the number of locally available states as R decreases^{46}. The evolution of P(Q) in Fig. 3b indeed exhibits features reminiscent of phase coexistence near an incipient random firstorder transition. The crossover also sharpens as T decreases, suggesting that the growing correlation length transforms it into a genuine thermodynamic phase transition as T → T_{K} = 0. In absolute values, ξ_{PTS} ≈ 6.5 at T = 0.028, which represents a very large static correlation length for glassy models^{26,39,40}. It implies that large clusters comprising about 120 particles are statically correlated, and should thus move collectively to restructure the liquid. These results are consistent with the sharp decay of the configurational entropy in Fig. 2 and the expected dramatic increase of the relaxation time in Fig. 1.
Discussion
The problem of the glass transition has two fundamental facets: thermodynamics and dynamics. While the current study focused on the thermodynamics of d = 2 glass formers, its dynamical counterpart, which involves obtaining a detailed functional form of the structural relaxation time, remains for now out of reach of computational work. Our results nonetheless suggest that in d = 2 the divergence of the relaxation time must take place at zero (rather than at finite) temperature. By identifying the thermodynamic properties that underlie the nature of glassy dynamics in d = 2^{27,28,29,30}, our results provide additional evidence that a thermodynamic transition can occur in finitedimensional systems, and that the lower critical dimension for the longrange amorphous order is d_{L} = 2 (see Supplementary Note 5). This finding lends indirect support to previous observations in d = 3^{26}, and will surely guide future analytical work.
Methods
Model
The glassforming model we consider consists of particles with purely repulsive softsphere interactions, and a continuous size polydispersity. Particle diameters, σ_{i}, are randomly drawn from a distribution of the form: f(σ) = Aσ^{−3}, for σ ∈ [σ_{min}, σ_{max}], where A is a normalization constant. The size polydispersity is quantified by \(\delta = \sqrt {\overline {\sigma ^2}  \bar \sigma ^2} /\bar \sigma\), where \(\overline{\cdots} \equiv {\int} {\mathrm{d}} \sigma \,\, f(\sigma )( \cdots )\), and is here set to δ = 0.23 by imposing σ_{min}/σ_{max} = 0.45. The average diameter, \(\bar \sigma\), sets the unit of length. The softsphere interactions are pairwise and described by an inverse powerlaw potential
where v_{0} sets the unit of energy (and temperature with Boltzmann constant k_{B} = 1), and \(\varepsilon = 0.2\) quantifies the degree of nonadditivity of particle diameters. We introduce \(\epsilon \, > \, 0\) to the model in order to suppress fractionation and thus enhance glass form ability^{24,47}. The constants, c_{0}, c_{1}, and c_{2}, enforce a vanishing potential and the continuity of the first and second derivatives of the potential at the cutoff distance r_{cut} = 1.25σ_{ij}. We simulate a system with N particles within a square cell of area V under periodic boundary conditions, at number density ρ = N/V = 1.01. Most simulations have N = 1000, but systems with N = 300, 3000, 8000, and 20,000 are also studied.
Observables
We monitor the system structure with two common liquid state quantities: the pairdistribution function g(r), and the structure factor S(k) = 〈ρ_{−k}ρ_{k}〉/N, where \(\rho _{\mathbf{k}} = \mathop {\sum}\nolimits_i {e^{i{\mathbf{k}}\cdot {\mathbf{r}}_i}}\) is the Fourierspace density. Orientational correlations are also considered, and are quantified using the sixfold bondorientational order parameter [?]^{48}
where the sum is performed over the n_{j} first neighbors of the jparticle. These neighbors are defined as particles with r_{ij}/σ_{ij} < 1.33, which is the location of the distance of the first minimum in the rescaled radial distribution function g(r/σ_{ij}). The angle θ_{jk} then measures the orientation of the axis between the two particles with respect to the xaxis. Because these correlations are orientationally invariant the choice of xaxis is made without loss of generality. Orientational correlations are then monitored through the twopoint bondorientational correlation function
where \(\psi _6(r) = \mathop {\sum}\nolimits_{i = 1}^N \delta ({\mathbf{r}}  {\mathbf{r}}_i)\psi _6^i\). The radial decay of the hexatic order correlation function, g_{6}(r)/g(r)^{48}, provides an hexatic correlation length ξ_{6}, as presented in Supplementary Note 1.
Translational dynamics is characterized by first measuring the intermediate scattering function
at the wave number k corresponding to the first peak of S(k). The relaxation time of the density fluctuations, \(\tau _\alpha ^{{\mathrm{TR}}}\), is then extracted from the exponential decay of the scattering function, i.e., \(F_{\mathrm{s}}(k,\tau _\alpha ^{{\mathrm{TR}}}) = e^{  1}\). Orientational dynamics is characterized similarly, replacing the Fourierspace density by the bondorientational correlation function in Eq. (3) defined by
In order to extract the bondorientational relaxation time τ_{α}, we use \(C_{\psi _6}(\tau _\alpha ) = e^{  1}\).
Equilibration and the glass ceiling
Normal MonteCarlo (MC) simulations allow only local particle displacements, drawing a random displacement vector on the (x, y) axis in the interval [−Δr_{max}, Δr_{max}] with Δr_{max} = 0.6 and moving a randomly chosen particle following a Metropolis acceptance criterion. Compounding N such displacement attempts defines a MC step, which is used as unit of time in this work. To ensure equilibration, we monitor both static and dynamical observables. Starting from a hightemperature liquid configuration, we quench the system at the final temperature and wait for the potential energy of the system to stop aging on a time window of ~10^{6} MC steps. We first estimate τ_{α} on simulations long enough to allow few decorrelations of \(C_{\psi _6}(t)\), and then perform simulations for 220τ_{α}. The system is left to equilibrate during the first 20τ_{α}; static and dynamical observables are computed over the following 200τ_{α}. Swap MC simulations include attempts at exchanging random pairs of particle diameters, which replace particle displacements with probability p_{swap} = 0.2. This algorithm defines the SWAP dynamics. The same equilibration and measuring protocol as for normal MC is then followed. Static observables monitor ordering and phase separation in the system, as discussed in Supplementary Note 1, whereas dynamical observables quantify the relaxation and equilibration timescales.
In Supplementary Fig. 1, we report orientational τ_{α} and translational \(\tau _\alpha ^{{\mathrm{TR}}}\) relaxation times for both normal and SWAP dynamics. Because the relaxation of local orientational degrees of freedom is slower, the associated timescale is used as reference. We perform three different fits to the τ_{α} results for the physical dynamics, in order to extract the temperatures relevant to the dynamical slowing down. First, we fit τ_{α} to a powerlaw function, as is predicted in the context of the modecoupling theory^{49},
over the interval τ_{α} ∈ (τ_{0}, 10^{3}τ_{0}). The resulting T_{MCT} = 0.123 roughly corresponds to the lowest temperature at which normal dynamics can reach equilibrium in simulations of reasonable duration^{24}.
Next, we estimate the laboratory glass transition temperature, T_{g}, at which experiments with atomic and molecular glass formers cannot be equilibrated anymore. At T_{g}, relaxation times have increased by 12 orders of magnitude with respect to their value at the onset of the supercooled dynamics^{50}. We thus fit the relaxation times both to a Vogel–Fulcher–Tallman (VFT) law
and to an Arrhenius law
where A and B are fitting constants. These two expressions respectively overestimate and underestimate the increase of relaxation times in experimental glassformers^{51,52}. We fit Eq. (8) using the whole temperature range T < T_{onset}, whereas we fit Eq. (9) only to T < 0.16 to ensure that the result serves as a proper lower bound on the relaxation time. Extrapolating up to the temperature at which \(\log _{10}(\tau _\alpha /\tau _0) \simeq 12\) gives \(T_{\mathrm{g}}^{{\mathrm{VFT}}} = 0.0907\) and \(T_{\mathrm{g}}^{{\mathrm{Arr}}} = 0.0738\). These two temperatures are, by construction, upper and lower bounds for T_{g}, and thus define an experimental glassceiling regime (blue shaded region)^{26} in Figs. 1, 2 and 3 as well as Supplementary Fig. 1. In all cases, SWAP dynamics equilibrates well beyond this experimentally limited regime, reaching T = 0.026. Supplementary Fig. 1 also shows the fitting curves to the dynamics. The modecoupling powerlaw prediction describes the growth of the relaxation times only within the first three decades of the glassy regime, but at lower temperatures it overestimates the results by many orders of magnitude. Whereas Eq. (8) adequately describes these same results over more than four decades, an Arrhenius law captures barely two decades.
Data availability
The data necessary to reproduce the figures in this paper are publicly available through the Duke University Libraries Digital Repository (https://doi.org/10.7924/r46w9b248)^{53}.
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Acknowledgements
The authors thank G. Tarjus for stimulating discussions. This research was supported by a grant from the Simons Foundation (#454933, L. Berthier and #454937, P. Charbonneau). Part of the computations was carried out through the Duke Compute Cluster.
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L.B., P.C., A.N., M.O., and S.Y. designed research, performed research, analyzed data, and wrote the paper.
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Berthier, L., Charbonneau, P., Ninarello, A. et al. Zerotemperature glass transition in two dimensions. Nat Commun 10, 1508 (2019). https://doi.org/10.1038/s41467019095123
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DOI: https://doi.org/10.1038/s41467019095123
Further reading

Physics of Disordered Systems
Resonance (2022)
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