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.

Fig. 1

Statics and dynamics of the d = 2 glass former. a The smooth evolution of the static structure factor from T_onset down to the lowest studied temperature T = 0.026 indicates that the system remains fully amorphous at all T. b Snapshot of an equilibrium configuration at T = 0.026. c Arrhenius representation of the structural relaxation time τ_α using SWAP and normal Monte Carlo dynamics, rescaled by the relaxation time at the onset temperature. The mode-coupling temperature, T_MCT (gray dashed line), and the estimated range of experimental glass temperature, T_g (navy strip), are indicated. The Arrhenius fit to the low-T data provides a lower bound for the growth of τ_α. SWAP can equilibrate systems down to T ≈ 0.3T_g, where the Arrhenius fit gives ${\tau }_{\alpha }^{\mathrm{normal}}∕{\tau }_0~10^{46}$

Fig. 2

Zero-temperature Kauzmann transition. a Decrease of the configurational entropy with temperature using the potential energy landscape (PEL), Frenkel–Ladd (FL), and point-to-set (PTS) length estimates. The error bars for FL correspond to the ambiguity of defining the plateau regime in the mean squared displacement of the FL construction. b Once rescaled by their value at T_g, all estimates evolve nearly identically, and the collapsed data are well fitted by a quadratic function of T for T < T_g (dashed blue line: s_conf(T)/s_conf(T_g) = 0.01 + 1.48(T/T_g) − 0.49(T/T_g)² indicates the quadratic fit for the point-to-set estimate). All results are consistent with a linearly vanishing s_conf at T_K = 0. c The specific heat, c_V, obtained from the derivative of the potential energy increases monotonically above the Dulong–Petit law for d = 2 (dashed horizontal line), which is also consistent with a thermodynamic transition at T_K = 0

Fig. 3

Approaching the random first-order transition. a Phase diagram showing the low-Q region for large cavities and high-Q region for small cavities, separated by the boundary determined by the point-to-set correlation length, ξ_PTS. The dashed blue line is the same quadratic fit (after unit conversion) as in Fig. 2b. Inset: Representative configurations with overlap field for T = 0.035 at R = 6.6 (low Q, white) and 4.8 (high Q, dark). b Evolution of the probability distribution function of overlap P(Q) at T = 0.035 from R = 4.8 to R = 6.6. Bimodality signals a first-order-like phase coexistence