Kibble-Zurek mechanism and infinitely slow annealing through critical points Giulio Biroli, 1 Leticia F. Cugliandolo, 2 and Alberto Sicilia 3 $^{1}$Institut de Physique Th´eorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France and CNRS, URA 2306 $^{2}$Universit´e Pierre et Marie Curie – Paris VI, LPTHE UMR 7589, 4 Place Jussieu, 75252 Paris Cedex 05, France $^{3}$Department of Chemistry, University of Cambridge, Lensfield Road, CB2 1EW, Cambridge, UK (Dated: November 17, 2018) We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase ordering systems during an infinitely slow annealing across a second order phase transition. We elucidate the time and cooling rate dependence of the typical growing length and we use it to predict the number of topological defects left over in the symmetry broken phase as a function of time, both close and far from the critical region. Our results extend the Kibble-Zurek mechanism and reveal its limitations. PACS numbers: The out of equilibrium dynamics induced by a quench are the focus of intense research [1, 2]. Interesting re- alizations are quenches through a second order phase transition, which take the system from the symmetric phase into the symmetry broken one. Below the tran- sition, times scaling with the system size are needed to reach equilibrium and to realize the spontaneous symme- try breaking process. Before this–typically unreachable– asymptotic limit the symmetry is broken only locally: the system is formed by ordered regions of size growing with time [3]. Only when this size reaches the order of the vol- ume of the sample the symmetry is broken globally and the spatial average of the order parameter deviates from zero. The majority of theoretical studies focused on the dynamics after infinitely rapid quenches although experi- mentally quenches are performed at finite speed. Indeed, since the typical time-scale on which the system evolves is its age, i.e. the time elapsed since crossing the critical point, finite quench time-scales ( τ$_{Q}$ ) eventually become short compared to the relaxation time. Thus, they al- ter the out of equilibrium dynamics at short times only. The opposite limit of an extremely slow annealing, cor- responding to very long τ$_{Q}$ , needs a separate treatment. Surprisingly, this has not been studied in detail in the sta- tistical physics literature with, however, some exceptions for disordered systems [4–6]. It has, instead, attracted a lot of attention within the cosmology and, more recently, the condensed matter communities. An explanation of the slow dynamics induced by this protocol was given by the so-called Kibble-Zurek (KZ) mechanism [7–10]. This is an equilibrium scaling argument that yields an estimate for the density of topological defects left over in the or- dered phase as a function of the quenching rate close to the critical point. The argument has been recently gen- eralized to study very slow ‘quantum annealing’ across a quantum phase transitions in isolated systems [11–13]. The aim of this work is to obtain a more complete pic- ture of the slow dynamics induced by an extremely slow annealing. With numerical and analytical arguments we unveil the limitations of the KZ approach and we obtain a full scaling description of the slow dynamics. Our main result is that the dynamic evolution is characterized by a first adiabatic regime in agreement with KZ, followed by critical coarsening and, finally, standard coarsening at very long times. We find a new universal scaling function that characterizes the growth of the correlation length out of equilibrium under slow cooling procedures and we relate it to the number of topological defects in cases in which these exist. We start our discussion by recalling the KZ mecha- nism [8–10]. We take a system in equilibrium at equi- librium at a value g$_{0}$ > g$_{c}$ of the control parameter in the symmetric phase and subsequently anneal it at finite rate (in typical situations g corresponds to temperature). As KZ, we focus on the protocol g ( t ) = g$_{c}$ (1 − t/τ$_{Q}$ ) starting from, say, g$_{0}$ = g ( − τ$_{Q}$ ) = 2 g$_{c}$ . Henceforth we use the standard notation of dynamical critical phenom- ena [14] and we set the microsocopic time and length scales to one. Far from the critical point the equilib- rium relaxation time, τ$_{eq}$ , is barely larger than the mi- croscopic time. Thus, for very small annealing rate, i.e. very long τ$_{Q}$ , the system evolves adiabatically and re- mains in equilibrium at the running g ( t ). However, this regime must inevitably break down since τ$_{eq}$ diverges at the critical point as | Δ g | − νz$_{eq}$ with Δ g ≡ g − g$_{c}$ . KZ argued that the end of the adiabatic regime occurs when the remaining time, ˆ t , needed to reach g$_{c}$ becomes smaller than τ$_{eq}$ . This is certainly a lower bound and yields ˆ t ∝ τ$_{eq}$ (ˆ g ) ∝ τ νz$_{eq}$/ (1+ νz$_{eq}$ ) Q with ˆ g = g ( − ˆ t ). The dis- tance from the critical point at − ˆ t is Δˆ g ∝ τ − 1 / (1+ νz$_{eq}$ ) Q . KZ assumed that after − ˆ t the topological defect con- figuration remains frozen, in the sense that the order parameter ceases to evolve. In this so called ‘impulse’ regime the effect of lowering g is to reduce fluctuations, which are in general of thermal origin since very often g is related to the temperature. The main prediction of KZ is the number of topological defects, N , at the symmetric instant ˆ t where the coupling constant equals g$_{c}$ − Δˆ g . Within their approach N is inherited from the configuration at − ˆ t , it is therefore equal to the number of defects in equilibrium at ˆ g , and it is estimated to be arXiv:1001.0693v1 [cond-mat.stat-mech] 5 Jan 2010