Relaxation Spectra at the Glass Transition: Origin of Power Laws

We propose a simple dynamical model of the glass transition based on the results from a non-randomly frustrated spin model which is known to form a glassy state below a characteristic quench temperature. The model is characterized by a multi-valleyed free-energy surface which is modulated by an overall curvature. The transition associated with the vanishing of this overall curvature is reminiscent of the glass transition. In particular, the frequencydependent response evolves from a Debye relaxation peak to a function whose high-frequency behavior is characterized by a non-trivial power law. We present both an analytical form for the response function and numerical results from Langevin simulations. 64.70.Pf, 64.60.Ht, 05.40+j, 02.60.Cb Typeset using REVTEX 1 (November 9, 1998) In supercooled liquids, the glass transition is heralded by anomalously slow relaxations [1]. The phase below the glass transition temperature is non-ergodic and the dynamics is characterized by “aging” [1]. Recent experiments indicate that the the approach to the glass transition has some universal features [2] when viewed in terms of the frequency-dependent response of the system. Some of these features are also shared by spin glass transitions [3]. Theoretical research has focussed on spin-glass-like phenomena in nonrandom systems [4] and some of these models have been shown to be equivalent to mean-field spin-glass models [5,6]. The dynamics of certain frustrated XY models have also been shown to exhibit glassy behavior [7]. In this paper, we present a dynamical model of the glass transition and analyze the trends in its frequency dependent response. The model is characterized by a multi-valleyed freeenergy surface modulated by an overall curvature and the glass transition is associated with the vanishing of this overall curvature. A multi-valleyed free-energy surface has long been associated with glasses [1]. The features of the present model were deduced from simulations of a nonrandomly frustrated spin model [8]. The frequency-dependent response of the system can be calculated in closed form, and its low temperature approximation becomes exact in the high-frequency limit. The results show that, as the overall curvature approaches zero, the frequency-dependent susceptibility changes from a characteristic Debye relaxation peak to a power-law spectrum reflecting the distribution of curvatures of the valleys. The predicted changes agree qualitatively with the generic features identified in experiments [2]. We begin with a description of the results of simulations of the frustrated spin system. The model considered is an Ising antiferromagnet on a deformable triangular lattice. Detailed analysis of this model has shown that there is a first-order transition from the disordered paramagnetic state to an ordered “striped” phase [9,10]. Frustration plays a crucial role in this phase transition [10]. The model is characterized by two order parameters: a staggered magnetization with Ising symmetry and a shear distortion with three-state Potts symmetry. Monte Carlo simulations [8] have demonstrated that the time evolution of various phys2 (November 9, 1998) ical quantities, such as the average energy, undergoes a qualitative change as the system is quenched below a characteristic temperature T , below the ordering transition [8]. To investigate the nature of the transition at T , we computed the fluctuation metric, Ω(t) [11]. This function has been shown to be sensitive to ergodicity breaking and has been used to study the glassy phase in Lennard-Jones systems [11]. The metric is defined by: Ω(t) =< (ν(t)− ν̄) > ; ν(t) = 1 t ∫ t