# Zero-temperature glass transition in two dimensions

## 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 mean-field limit of infinite spatial dimensions, the problem remains wide open in physical dimensions. Nontrivial finite-dimensional 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 two-dimensional glass-forming 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 glass-formers.

## 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 mean-field character1, and one-dimensional (*d* = 1) models can often be treated exactly. Yet these two solvable limits are crude idealizations of our three-dimensional 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 exception2. Its mean-field description, which becomes mathematically exact as *d* → ∞, explains the dramatic slowdown of glass-forming liquids through the rarefaction of the number of glassy metastable states upon approaching a critical temperature, *T*K3,4. 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 data5 and initially formalized by Gibbs and DiMarzio6 in the context of a lattice polymer model.

The broad discussion that has since ensued2 has notably tried to describe the role of finite-*d* fluctuations beyond the mean-field framework7,8,9,10,11,12, relating in particular the vanishing of *s*conf to a diverging point-to-set correlation length, the key quantity for characterizing nonperturbative fluctuations in glass formers13. These fluctuations, however, make it difficult to examine finite-dimensional glass formers analytically, even for simple models composed of point-particles such as those we study here. Exploring a broader diversity of models, from polymer14 to anisotropic patchy15 models, may yet provide additional theoretical insight.

Meanwhile, Kauzmann’s intuition has been repeatedly validated by experiments16,17, but the conceptual and technical limits of his results have not been lifted. Current experiments access essentially the same restricted temperature range as his 70-year 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 physics18,19. Experimentally, it thus remains controversial whether the trend discovered by Kauzmann survives at much lower temperatures; entropy could go smoothly to zero20,21, or to a finite residual value as temperature vanishes15,22,23.

In this context, computer simulations are especially valuable. They allow direct measurements of both the configurational entropy and the point-to-set correlation length for realistic models of finite-dimensional glass formers2. The recent development of the swap Monte Carlo algorithm (SWAP) further allows the exploration of a temperature regime that experiments cannot easily access24, even using ultrastable glassy materials25. This has consolidated and extended Kauzmann’s experimental findings for three-dimensional glass formers26. Here, we report that SWAP is so efficient in *d* = 2 that it provides access to a temperature regime equivalent to experimental timescales 1018 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 point-to-set correlation length. Our results thus illuminate the low-dimensional fate of the glass transition and shed light on the nature of glassy dynamics in *d* = 227,28,29,30.

## Results

### Model and macroscopic behavior

We study a two-dimensional mixture of soft particles interacting with a 1/*r*12 purely repulsive power-law 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* = 324. Systems ranging from *N* = 300 to *N* = 20,000 particles within a periodic box are used to carefully track finite-size 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 mean-field 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 low-temperature 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 bond-orientational orders evolve modestly with *T* (see Supplementary Note 1). In other words, the model is an excellent glass former.