THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THEORY OF NON-LINEAR SUSCEPTIBILITY AND CORRELATION LENGTH IN GLASSES AND LIQUIDS

Within the framework of the effective potential theory of the structural glass transition, we calculate for the p-sp'm model and a hard sphere liquid in the hypernetted chain approximation a static nonlinear susceptibility related to a four-point density correlation function, and show that it grows and diverges in mean field with exponent 7 = 1/2 as the critical temperature Tc is approached from below. When Tc is approached from above, we calculate for the p-spin model a time dependent nonlinear susceptibility and show that there is a characteristic time where this susceptibility has a maximum, and that this time grows with decreasing T. We find that this susceptibility diverges as Tc is approached from above, and has key features in common with the "displacement-displacement susceptibility" recently introduced to measure correlated particle motion in simulations of glass-forming liquids. Tempted by the possibility of treating the glass transition within the framework of conventional critical phenomena, researchers have long searched for evidence of a static correlation length that becomes large as the glass transition is approached. However, no clear evidence for such a length has been reported [1]. Recent numerical studies of glass-forming liquids have identified a dynamical length associated with the range over which particle motions are correlated [2-7]. By introducing a "displacement-displacement" correlation function and generalized "displacement susceptibility" Xu{t), Refs. [4,5] showed in two different model liquids that this length grows with decreasing temperature T as the mode coupling temperature Tc is approached from above, despite the fact that density and composition correlations remain short-ranged. Evidence for a growing length associated with correlated particle motion at fixed T in simulations below Tc has also been reported [8]. In this Letter we calculate within the effective potential theory a new susceptibility \4 associated with a four-point density correlation function [9]. We show that below Tc, \4 diverges with exponent 1/2 as T —> T~. We show that above Tc, XA 1S time-dependent and resembles the susceptibility calculated in [4,5]. In particular, we show that Xi{t) has a maximum which diverges in mean field as T —> Tc . We argue that (1) the diverging correlation length implied by the diverging susceptibility is associated with incipient ergodicity breaking at Tc, and (2) this length underlies the growing range of correlated particle displacements measured in Refs. [4,5]. Finally, we test our theoretical predictions above Tc using data from molecular dynamics simulations of a model glass-forming liquid. We use two different approaches depending on T. In the low T regime (T < Tc), we calculate a static, nonlinear susceptibility using the effective potential theory [10], and in the high T regime (T > Tc), we calculate a dynamic, nonlinear susceptibility in a dynamical approach. The low T calculations are performed both for a hard-sphere liquid in the hypernetted chain (HNC) approximation [12] and for the spherical p-spin model [11]; the dynamical calculations are performed only for the spherical p-sp'm model. This is the simplest model that (i) allows both static and dynamic quantities to be calculated exactly, and (ii) exhibits several key features common to liquids in and close to their glassy regime [12-14]. For example, the high-T dynamics of the p-spin model is described exactly by the ideal mode coupling equations [15,16] and displays a sharp transition at Tc. In the effective potential theory, below Tc phase space is split into separate ergodic components which remain stable at all temperatures (this stability is due to the mean field nature of the theory). A key hypothesis of our approach is that in real systems the ergodic components are metastable states with a long but finite escape time. In this framework we can use a static approach to compute the properties of these metastable states below Tc. We can also obtain information on the dynamical properties above Tc, where mean field theory predicts that the time to escape from a metastable state diverges on approaching Tc. The divergence of a static susceptibility inside the metastable state when we approach Tc from below is directly related to the divergence of the corresponding dynamical susceptibility when we approach Tc from above. As is typical for mean field theories, which neglect spatial fluctuations, a diverging correlation length is deduced from the diverging susceptibility. We first describe the essential elements of the effective potential theory (a complete description can be found in [13]). The theory is formulated using a measure of the similarity or "overlap" q between two configurations X and Y as an order parameter to detect vitrification. Different definitions of q can be used in different systems and the main results of the theory do not depend on the particular definition adopted. In the case of simple liquids with N particles at fixed density [13,17], one can define q(X,Y) = ±Ylw(xi-yj) (1) 1 f = — j dxdyw{x-y)px(x)pY(y), where X = {x\, . ,XN}, Y = {y\, . ,yjv}, and pz(x) = S^=i $( ~ i) 1S the microscopic density corresponding to the configuration Z = X, Y. Here w{r) is chosen to be a smooth, continuous, short-range function close to one for r < aro and close to zero otherwise (ro is the radius of a particle). The value of a < 1 is arbitrary, and a = 0.3 is a good compromise for an overlap insensitive to small thermal fluctuations [13,17]. The effective potential V(q), which is a constrained free energy, is constructed by choosing a fixed reference equilibrium configuration Y at temperature T, and calculating the free energy of a configuration X that has an overlap q with Y: V(q) = -^log JdX exp(-pH(X))6(q(X, Y) q). (2) Here j3 = 1/ksT and H{X) is the potential energy of X. V{q) is self-averaging with respect to the choice of Y. The typical mean-field shape of V(q) for a system undergoing vitrification is shown in Fig. 1 for several values of T. The shape of V(q) allows one to distinguish the liquid from the glassy phase since the presence of a single or multiple minima indicates either ergodicity or broken ergodicity, respectively. At high T, the system is ergodic and V{q) is convex, with a single minimum at a small value of the overlap q between any two configurations chosen with the Boltzmann weight. Upon lowering T, the curvature changes sign, and at Tc, V(q) develops a secondary minimum at a higher value of q. This signals breaking of ergodicity: at Tc the configuration space become disconnected into an exponentially large number of "ergodic components" JV ~ exp(TVS), each carrying vanishing weight in the Boltzmann distribution [13]. A fundamental result of the theory, central in the following discussion, is that physical quantities calculated in the primary minimum represent averages computed with the Boltzmann weight, while the same quantities calculated in the secondary minimum represent averages computed only within a single ergodic component.