Mode-coupling as a Landau theory of the glass transition

We derive the Mode Coupling Theory (MCT) of the glass transition as a Landau theory, formulated as an expansion of the exact dynamical equations in the difference between the correlation function and its plateau value. This sheds light on the universality of MCT predictions. While our expansion generates higher order non-local corrections that modify the standard MCT equations, we find that the square root singularity of the order parameter, the scaling function in the β regime and the functional relation between the exponents defining the α and β timescales are universal and left intact by these corrections. The Mode-Coupling Theory of glasses (MCT), developed since the mid-eighties following the seminal work of Götze [1] and Leutheusser [2], has significantly contributed to our understanding of the slowing down of supercooled liquids. One of its cardinal predictions is the appearance of a non trivial β-relaxation regime where dynamical correlation functions pause around a plateau value before finally relaxing to zero. In the vicinity of this plateau value, the theory predicts two power-law regimes in time (or in frequency), and the divergence of two distinct relaxation times, τα and τβ , at the MCT critical temperature Td. Although this divergence is smeared out by activated events in real liquids, the two-step relaxation picture suggested by MCT seems to account quite well for experimental and numerical observations [3], at least in weakly supercooled liquids and for hard sphere colloidal systems. Originally, MCT was obtained as an uncontrolled selfconsistent approximation within the Mori-Zwanzig projection operator formalism for Newtonian particles. This scheme yields an integro-differential equation for the dynamic structure factor C(k, t) that captures mathematically the slowing down of the dynamics and the appearance of a two-step relaxation at equilibrium. It provides very detailed predictions for the scaling properties of C(k, t) in the vicinity of the plateau value fkSk, where Sk is the static structure factor and fk is called the nonergodicity parameter (akin to the Edwards-Anderson parameter in spin glasses). Alternative derivations of MCT based on field theory have been sought for [4] and research on this topic has continued until now [5–8]. It was also realized that the same integro-differential equations describe the exact evolution of the correlation function of meanfield p-spin glasses [9]. This is important for at least two reasons: • Technically, it shows that the MCT approximation is realizable: there is a well defined system for which it is exact; hence MCT does not violate basic physical constraints. • Physically, it brings in a very useful interpretation of the MCT freezing transition in the “energy landscape” parlance. Above the transition Td, the dynamics is dominated by unstable saddles that become progressively less and less unstable as one approaches the transition; below the transition, there are only local minima that are separated by infinite barriers in mean-field, so that the system is forever trapped in one of them [9]. For non mean-field systems, these barriers are finite and the transition is smoothed. MCT can be naturally embedded within the broader Random First Order Theory [10] of the glass transition. In this context it describes the high temperature region where metastable states are still in embryo. However, this analogy shows that MCT is (at best) an