Ideal glass-glass transitions and logarithmic decay of correlations in a simple system

The crossover from a liquid to an amorphous solid, observed near the calorimetric glass transition temperature Tg , exhibits as a precursor phenomenon an anomalous dynamics, called glassy dynamics. Its evolution is connected with a critical temperature Tc above Tg . It has been studied extensively in the recent literature of the glass-transition problem, experimentally @1‐5#, numerically @6,7#, and theoretically @8,9#. Experiments around Tc have been interpreted in the frame of the mode-coupling theory ~MCT! for structural relaxation. MCT deals primarily with closed equations of motion for the normalized density-fluctuation-correlation functions F q(t) for wave-vector moduli q. The equilibrium structure enters as input in these equations via the static structure factor Sq . The theory explains Tc as a glasstransition singularity resulting as a bifurcation phenomenon for the self-trapping problem of density fluctuations. Below Tc the interaction of density fluctuations leads to arrest in a disordered array, characterized by a Debye-Waller factor f q .0. Near the transition, the MCT equations can be solved by asymptotic expansions. Beyond the initial transient dynamics, correlation functions are predicted to decay with a power law toward a plateau value f q , the critical Debye-Waller factor. Above Tc the correlations decay from f q to zero, and this is the MCT interpretation of the a-process of the classical literature of glassy dynamics @10#. The initial part of this decay is another power law, called von Schweidler’s law. The values of the power-law exponents are controlled by the so-called exponent parameter l<1, which depends solely on Sq . For details and citations of the original literature the reader is referred to the review in Ref. @9#. A number of tests of MCT results against data, among them the ones in Refs. @1‐7,11,12#, demonstrates that this theory treats reasonably the evolution of structural relaxation in some systems. The MCT bifurcations are caused by a nondegenerate eigenvalue of a certain stability matrix to approach unity from below @13#. Therefore the bifurcation scenario for f q is that known for the zeroes of a polynomial as induced by changes of the polynomial’s coefficients @14#. The generic case for a