Finite - size scaling at the dynamical transition of the mean - field 10 - state Potts glass

We use Monte Carlo simulations to study the static and dynamical properties of a Potts glass with infinite range Gaussian distributed exchange interactions for a broad range of temperature and system size up to N = 2560 spins. The results are compatible with a critical divergence of the relaxation time τ at the theoretically predicted dynamical transition temperature TD, τ ∝ (T−TD) −∆ with ∆ ≈ 2. For finite N a further power law at T = TD is found, τ(T = TD) ∝ N z with z⋆ ≈ 1.5 and for T > TD dynamical finite-size scaling seems to hold. The order parameter distribution P (q) is qualitatively compatible with the scenario of a first order glass transition as predicted from one-step replica symmetry breaking schemes. 29. September, 2000 PACS numbers: 64.70.pf, 75.10.Nr, 75.40.Gb Developing a theory of the glass transition of a fluid from its Hamiltonian within first principles statistical mechanics is still a formidable problem [1–4]. While some researchers attribute glassy freezing to the (hypothetical) vanishing of the configurational entropy [5] at the “Kauzmann temperature” TK [6] (which is lower than the experimental [1] glass transition temperature Tg), others emphasize the dynamical transition at the critical temperature Tc of mode coupling-theory [2] from the ergodic fluid to a non ergodic state. Since, for atomic systems, Tc > Tg, this frozen phase can have only a finite lifetime and is thought to decay by thermally activated (so-called “hopping”) processes. Recently evidence has been given [3,4] that these two seemingly different scenarios could both result as two complementary aspects of the same unifying theory [7]. In view of the questions that still exist on the various theoretical approaches, it is valuable to have exactly solvable models that exhibit a similar behavior: a dynamical transition at a temperature TD and a static first order glass transition at a temperature T0 < TD. One of these models is the p-state infinite range Potts glass with p > 4 [8–13], where at T0 a static (Edwards-Anderson type [14,15]) spin glass order parameter qEA appears discontinuously. However at T0 there is neither a latent heat nor a divergence of the static spin glass susceptibility χSG. The latter would diverge only at an extrapolated spinodal temperature Ts < T0, see Fig. 1. The dynamical behavior of the spin autocorrelation function C(t) for T & TD is described by the same type of equations [12,13] as found in mode-coupling theory [2]. Thus this model seems indeed to have many properties in common with structural glasses. Apart from being a possible prototype model for the structural glass transition, the Potts glass can also be