Surfing on a critical line: Rejuvenation without chaos, Memory without a hierarchical phase space

– The dynamic behaviour of glassy materials displays strong nonequilibrium effects, such as ageing in simple protocols, memory, rejuvenation and Kovacs effects in more elaborated experiments. We show that this phenomenology may be easily understood in the context of the nonequilibrium critical dynamics of non-disordered systems, the main ingredient being the existence of an infinite equilibrium correlation length. As an example, we analytically investigate the behaviour of the 2D XY model submitted to temperature protocols similar to experiments. This shows that typical glassy effects may be obtained by ‘surfing on a critical line’ without invoking the concept of temperature chaos nor the existence of a hierarchical phase space, as opposed to previous theoretical approaches. The relevance of this phenomenological approach to glassy dynamics is finally discussed. ...comme des amateurs de surf qui glissaient sur la vague. Vassili Axionov, Une brûlure. Experiments on a large class of glassy materials demonstrate intriguing similarities in their nonequilibrium dynamic properties. Structural —in particular polymeric— glasses are the paradigm, and have been studied in a systematic way over a long period time [1]. Interest in ageing experiments grew with the realization that very similar effects are observed in spin glasses [2]. More recently, the family of glassy systems has been much enlarged to include systems as different as dirty type-II superconductors [3], several complex fluids [4], disordered dielectrics [5, 6] and ferromagnets [7], electron glasses [8], and granular materials [9]. These systems have in common that, in a part of their phase space, they cannot be equilibrated within the experimental time window: they undergo a ‘glass’ or ‘jamming’ transition. As a result, they exhibit nonequilibrium dynamics, where the whole history of the sample becomes relevant [10]. The conceptually simplest experiment performed in glassy materials consists of a sudden quench of the system into its glassy phase at the initial time tw = 0. The resulting ageing behaviour is best observed through the measurement of a two-time quantity, C(t, tw), typically a correlation function or a linear susceptibility. Experiments show that time translation invariance is absent, i.e. C(t, tw) 6= C(t − tw). Remarkably, they also show that the generic scaling behaviour C(t, tw) ∼ C(t/tw) is very often observed, implying that the relaxation time trel of the system is of the order of its age, trel ∼ tw. This ageing phenomenology can be explained by models of coarsening of domains of equivalent thermodynamic phases [10, 11]. The canonical example is a ferromagnet quenched into