ar X iv : c on d - m at / 9 81 22 29 v 3 1 4 M ay 1 99 9 Mean - field theory of temperature cycling experiments in spin - glasses

We study analytically the effect of temperature cyclings in mean-field spinglasses. In accordance with real experiments, we obtain a strong reinitialization of the dynamics on decreasing the temperature combined with memory effects when the original high temperature is restored. The same calculation applied to meanfield models of structural glasses shows no such reinitialization, again in accordance with experiments. In this context, we derive some relations between experimentally accessible quantities and propose new experimental protocols. Finally, we briefly discuss the effect of field cyclings during isothermal aging. LPT-ENS/9842; LPTHE/9853. Glasses are characterized by having extremely slow relaxations and by the strong dependence of their behaviour upon the (“waiting”) time elapsed since their preparation. The latter property is usually called physical aging. A means to study the dynamics in the glassy phase in more detail consists in following the evolution of the sample under a complicated temperature history. The protocols that have been more commonly used include temperature cyclings within the low temperature phase. The results for different types of glasses are quite different.1–9 Spin-glasses show the puzzling phenomenon of reinitialization of aging following a decrease in temperature, combined with the recall of the situation attained before the negative jump when the original high temperature is restored. Remarkably, when similar protocols were applied to structural glasses, e.g. in dielectric constant measurements of glycerol by Leheny and Nagel, no substantial reinitialization was observed. This difference in the effect of temperature changes on spin and structural glasses is a fact that any generic theory of glasses is expected to explain. Different groups interpreted the behaviour of spin-glasses under temperature cyclings during aging as evidence for both the droplet and the hierarchical pictures of the dynamics (the former with some extra refinement with respect to the original versions of the eighties). The hierarchical dynamic picture is a heuristic way to think about the results from positive and negative cyclings inspired in the organization of equilibrium states in the Parisi solution of mean-field spin-glasses, such as the Sherrington-Kirkpatrick model. It is assumed that spin-glasses have a large number of metastable states that are organized in a hierarchical fashion just like the equilibrium states. It is then proposed that the system is composed of (independent) subsystems whose dynamics is given by the wandering in such a landscape. An average over subsystems has to be invoked in order to obtain smooth results as observed in experiments. A concrete realisation of a hierarchical dynamic system can be made with the trap models.13–15 These models have been solved analytically in isothermal conditions. Though a full analytic description of their dynamics in the presence of temperature cyclings is not available yet, a careful discussion of their effects yields very encouraging results. Surprisingly enough, the main features of the cycling experiments have never been derived analytically from microscopic models, while the numerical evidence24–27 is inconclusive. In this paper we shall show analytically how these effects arise in mean-field models of spin glasses, and why they are absent in mean field models of structural glasses. One of the questions that will receive a clear answer is why the effects should be hardly observable at very short times, such as are inevitably involved in simulations. We shall consider in detail the particular class of temperature cycling experi-