Critical dynamics in glassy systems

Abstract

Critical dynamics in various glass models including those described by mode coupling theory is described by scale-invariant dynamical equations with a single non-universal quantity, i.e. the socalled parameter exponent that determines all the dynamical critical exponents. We show that these equations follow from the structure of the static replicated Gibbs free energy near the critical point. In particular the exponent parameter is given by the ratio between two cubic proper vertexes that can be expressed as six-point cumulants measured in a purely static framework.

Long-wavelength fluctuations lead to a model of the glass crossover

“…I have done this in such a way to keep the computation as close as possible to the static replica computation by means of a a superfield description of the dynamics. In that context it has been argued [25] that the dynamical field theory of the super-field correlator has the same structure of the replicated field theory (5) with the same coupling constants. Most importantly when the equations of state for this critical super-field correlator are translated into the those of the standard correlator one finds [25] that they have precisely the structure of the MCT critical equation (3):…”

The glass crossover from mean-field Spin-Glasses to supercooled liquids

“…As a consequence if we remove them we are left with a theory where formally all instants in time are interchangeable, in the sense that the theory is invariant under a permutation of the time indexes. Then we understand why such a theory is formally identical to a replica-symmetric theory as discussed in [9]. In spite of the fact that the equations are the same the key difference is that in the dynamical theory we can look for time-dependent casual solutions thus restoring a posteriori a time-ordering that is totally absent in the replica approach.…”

“…When performing the computation one immediately realizes that i) the corrections are singular near the critical point ,ii) the leading divergence depends only on the first terms beyond the quadratic ones in the expansion of H[Q] near Q * (in our case the cubic terms) iii) the fact that these terms have the symmetry of the original action H[Q]. The above statements amount to say that all critical behavior is determined by the fact that independently of higher order corrections H[Q] can be replaced by the following effective theory (referred as glassy critical theory (GCT) in the following ) that we identify the Landau theory of the Glass transition [2,9]:…”

“…In order to discuss them let us start with the leading order term g ± 0 (t). It was discovered long ago [6,7], (see also [10] and [9], for a recent discussion) that it obeys the very same equation of the so-called critical correlator of MCT [11] …”

“…The above expression is formal, the order parameter Q is not just a number but a complicated object that depends on two indexes (For details see [2,9]). The key point is the presence of a factor N in front of H[Q] that implies that we can take as the leading order the value of Q * that extremizes H[Q], dH[Q * ]/dQ = 0 and build a loop expansion around this value in order to obtain the 1/N corrections.…”