2 1 0 2 8 tr 2 6 2 1 0 2 8 sq 2 6 x 1 / 2 t | Δ g | νz$_{eq}$ R | Δ g | ν 10 5 0 -5 -10 6 5 4 3 2 1 0 t | Δ g | νz$_{eq}$ R | Δ g | ν 0 .5 0 -0.5 2 1 0 τ$_{Q}$ $_{−}$1 S i m u l a t i o n s τ$_{Q}$ N ( t = τ Q ) 1000 100 1000 100 10 FIG. 2: (Colour online.) Top panel: Test of the dynamic scaling hypothesis (2) and the limits (3) in the 2 d IM on a triangular and a square lattice, annealed at different cooling rates given in the key. A zoom on the critical region | x | < ∼ 1 is shown in the inset. The exponents are ν = 1, z$_{eq}$ = 2 . 17 and z$_{d}$ = 2. Bottom panel: Number of deffects at t = t$_{Q}$ for different cooling rates. Points represent the numerical data while the line corresponds to the prediction N ( τ$_{Q}$ ) = τ$_{Q}$ − d/z$_{d}$ rate quench is a simple generalization of the treatment of infinite rate ones (see, e.g. [1]). We find that the scal- ing (3) holds with ν = 1 / 2 and z$_{d}$ = z$_{eq}$ = 2 in all d > 2. Due to the coincidence of the z exponents the prefactor in the bottom expression of eq. (3) equals one and the dependence on τ$_{Q}$ disappears. As a summary we analyzed the dynamic evolution in- duced by annealing with rate 1 /τ$_{Q}$ ( τ$_{Q}$ → ∞ ) in pure systems characterized by conventional dynamic scaling and standard low temperature coarsening. We obtained a complete picture of the dynamics which is character- ized by three regimes: adiabatic, critical coarsening and standard coarsening. Using scaling arguments we found the growth law of the correlation length during the an- nealing and its τ$_{Q}$ scaling dependence. The cross-over between adiabatic and coarsening regimes is governed by a universal scaling function. We tested our findings with numerical simulations of the 2 d Ising model and a large N analysis of the O ( N ) model in d > 2. Our results gen- We close with a note on an exact study of the cool- ing rate effects in the relaxation of the classical Ising chain with Glauber dynamics by P. Krapivsky that shows qualitative but not quantitative agreement with the KZ mechanism [30]. This work was financially supported by ANR-BLAN- 0346 (FAMOUS). We thank C. Kollath, C. Godreche, A. Gambassi, D. A. Huse, A. Jelic, P. Krapivsky, J. Kur- chan, G. Lozano and A. Silva for very useful discussions. Physical situations in which understanding the evolu- tion during a slow annealing is important and which we plan to study in the future are disordered and quantum systems. Several studies have dealt with the former, see e.g. [4–6]. The latter have only recently received atten- tion in connection with quantum quenches and annealing in cold atoms. In these cases the conditions are differ- ent from the ones analyzed in this paper since isolated systems in which a coupling is slowly changed through a quantum critical point are usually considered. The ab- sence of the thermal bath may change drastically the physics. The KZ mechanism has been argued to ap- ply mutatis mutandis to the isolated quantum case as well [11, 12]. This has been verified in some integrable cases [13]. eralize the KZ mechanism and, at the same time, show its limitation. In particular we find that the defect dynam- ics are not frozen in the so-called impulse regime, as it can be found by using more general annealing protocols than a linear ramp in temperature. 4 [1] A. J. Bray, Adv. Phys. 43 , 357 (1994). [2] J.-P. Bouchaud, L. F. Cugliandolo, J. Kurchan, M. M´ ezard in “Spin-glasses and random fields”, A. P. Young Ed. (World Scientific). [3] The picture for cases characterized by the existence of topological defects is slightly different, see e.g. [1]. [4] D.A. Huse and D.S. Fisher, Phys. Rev. Lett. 57 2203 (1986). [5] H. Yoshino, K. Hukushima, H. Takayama Phys. Rev. B 66, 064431 (2002). [6] F. Alberici-Kious, J.-P. Bouchaud, L. F. Cugliandolo, P. Doussineau, A. Levelut, Phys. Rev. B 62 , 14766 (2000). [7] T. W. B. Kibble, J. Phys. A 9 , 1387 (1976). [8] W. H. Zurek, Nature (London) 317 , 505 (1985). [9] W. H. Zurek, Phys. Rep. 276 177 (1996). [10] W. H. Zurek, L. M. A. Bettencourt, J. Dziarmaga, and N. D. Antunes, Shards of broken symmetry in Topological defects and the non-equilibrium dynamcis of symmetry breaking phase transitions, Vol 549, Y. M. Bunkov and H. Godfrin eds. (Kluwer Academic Publishers, 1999). [11] W. H. Zurek, U. Dorner, P. Zoller, Phys. Rev. Lett. 95 , 105701 (2005). [12] A. Polkovnikov, Phys. Rev. B 72 , 161201 (2005). [13] J. Dziarmaga, Phys. Rev. Lett. 95 , 245701 (2005). [14] P. C. Hohenberg and B. Halperin, Rev. Mod. Phys. 49 , 435 (1977).