(a) Δ g ( t ) τ$_{Q}$$_{2}$ τ$_{Q}$$_{1}$ 0 − ˆ t 1 − ˆ t 2 − ˆ t 3 R$_{τ}$$_{Q}$$_{4}$ R$_{τ}$$_{Q}$$_{3}$ R$_{τ}$$_{Q}$$_{2}$ R$_{τ}$$_{Q}$$_{1}$ ξ$_{eq}$ (b) ξ eq , R ˆ g 4 ˆ g 1 g$_{c}$ FIG. 1: (Colour online.) (a) The control parameter, Δ g ( t ) = [ g ( t ) − g$_{c}$ ] /g$_{c}$ , for different cooling rates. The crossover be- tween adiabatic and out of equilibrium dynamics are signaled as − ˆ t$_{i}$ for τ$_{Q}$$_{i}$ < τ$_{Q}$$_{i}$$_{+1}$ . (b) Sketch of the control parameter dependence of the equilibrium correlation length (thick red line) and the dynamic growing length R for four linear cool- ing rate proceedures with τ$_{Q}$$_{i}$ < τ$_{Q}$$_{i}$$_{+1}$ . Values of the control parameter at which the dependence changes from adiabatic to critical are shown as ˆ g$_{i}$ (for simplicity we plot them as singular points in the evolution of R . In reality they just correspond to cross-overs). For comparison the assumption of constancy during the impulse and subcritical regimes are shown with thin horizontal lines. topological defects. In the former case, the decaying typ- ical number of topological defects , N ( t ) ≃ R ( t ) − $^{d}$, reads for t ≳ τ$_{Q}$ Note that a qualitatively similar dependence on t and τ$_{Q}$ was found numerically in [29] in a system with vortex- antivortex pairs. Evaluating the above expression at t = ˆ t = τ νz$_{eq}$/ (1+ νz$_{eq}$ ) Q we recover KZ’s result, eq (1). Ours, however, is more general since it applies to any time t and it allows one to describe all the slow annealing evolution. For example, N ( t ) ∝ τ − d/z$_{d}$ Q on times of the order of the inverse annealing rate, t ≃ τ$_{Q}$ , thus show- ing that a substantial decrease takes place after ˆ t . In comparison with the KZ mechanism, our arguments al- low one to understand why R ( ˆ t ) ≃ fR ( − ˆ t ) with a factor A further test is provided by the analytic solution to the evolution of the O ( N ) model in the large N limit for a very slow annealing. This is a λφ 4 field theory in which the order parameter is upgraded to an N -dimensional vector and the fourth order term in the double-well po- tential is conveniently normalized to allow for an N → ∞ limit in which the model becomes solvable but still non- trivial [1]. Note that although there are no topological defects, since the large N limit is taken at fixed dimen- sion, N ≫ d , the dynamics are still characterized by a growing correlation length R ( t ). The analysis of a finite In the following we provide numerical evidence for the conclusions outlined above by presenting results of a Monte Carlo simulation (using the heat bath algorithm with random sequential updates) of the 2 d Ising model (IM) on square and triangular lattices. In particular, we check eqs. (2)-(4) and the universality of the scaling function f ( x ). We equilibrate the system at T$_{0}$ = 2 T$_{c}$ and we use a linear cooling rate that takes the temper- ature of the bath from T$_{0}$ at t = − τ$_{Q}$ to T = 0 at t = τ$_{Q}$ . We find that the system undergoes an adia- batic evolution until it falls out of equilibrium close to T$_{c}$ . The typical correlation length, R ( t ), is extracted from the analysis of the space-time correlation C ( r, t ) ≡ 〈 s ( ⃗ x ) s ( ⃗ x + ⃗ r ) 〉 ≃ g ( r/R ( t )) with the average taken over 100 initial conditions and noise realizations. We use vari- ous ways to determine R and verify that they yield equiv- alent results. Two of them are C ( R ( t ) , t ) = 1 / 2 and R ( t ) = ∫ d $^{2}$r r $^{ζ}$C ( r, t ) / ∫ d $^{2}$r r ζ − $^{1}$C ( r, t ) with ζ a pa- rameter that is chosen for convenience, namely to weigth differently shorter or longer distances. In the top panel of Fig. 2 we test the scaling hypothesis, eq. (2), and the limits of the scaling function f ( x ), eq. (3). For both the square and triangular lattices we find very good agree- ment between numerical data and theoretical expecta- tion. The square root growth at positive times demon- strates that standard coarsening cannot be ignored. The scaling collapse improves, as expected, restricting the range of | x | . A zoom on the small | x | region is shown in the inset. Moreover, we find that the scaling functions for square and triangular lattices coincide within numer- ical accuracy, confirming the universality of f ( x ). The bottom panel of Fig 2 displays N ( τ$_{Q}$ ) and confirms the τ − d/z$_{d}$ Q decay - the power is shown as a guide-to-the-eye next to the data. f that can be as large as 10 in some cases [15]. This was somewhat mysterious in the KZ scenario where defects are frozen out in the impulse regime. We understand the reduction in the number of topological defects as due to critical coarsening. Taking this phenomenon into ac- count is crucial for more general annealing protocols, e.g. T ( t ) = θ ( − t ) T$_{c}$ (1 − t/τ (1) Q )+ θ ( t ) T$_{c}$ (1 − t/τ (2) Q ). For a large ratio τ (2) Q /τ (1) Q the system spends a long time in the criti- cal region and R ( t ) evolves during the critical coarsening from [ τ (1) Q ] ν/ (1+ νz$_{eq}$ ) to [ τ (2) Q ] ν/ (1+ νz$_{eq}$ $^{)}$. 3 N ( t ) ≃ τ dν z $_{d}$( z$_{eq}$ − z$_{d}$ ) Q t − d z $_{d}$[1+ ν ( z$_{eq}$ − z$_{d}$ )] . (4)