Spin-glass dynamics in the presence of a magnetic field: exploration of microscopic properties

The main quantity used in our experiments of [5] is the relaxation function S(t, t_w; H):

$$\begin{equation}S\left(t,{t}_{\text{w}};H\right)=\frac{\mathrm{d}{M}_{\text{ZFC}}\left(t,{t}_{\text{w}};H\right)}{\mathrm{d}\mathrm{ln}\enspace t},\end{equation} \tag{ 11 }$$

which exhibits a local maximum at time ${t}_{H}^{\text{eff}}$. Experimentally, measurements of M_ZFC(t, t_w; H) enable the evaluation of the relaxation function S(t, t_w; H) directly. A representative set of data for T_m = 28.5 K and t_w = 10⁴ s is displayed in figure 2. Numerically, the calculation of S(t, t_w; H) is sensitive to the relative errors of the magnetisation density, which increase as

$$\begin{equation}\frac{\delta {M}_{\text{ZFC}}\left(t,{t}_{\text{w}};H\right)}{{M}_{\text{ZFC}}}\propto \frac{1}{H}.\end{equation} \tag{ 12 }$$

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We employ two tricks to extract the relaxation function S(t, t_w; H) from simulations. On the one hand, we perform a de-noising method to regularise the magnetisation density M_ZFC(t, t_w; H), exploiting the fluctuation–dissipation relations (FDRs) [32–36]

$$\begin{equation}\frac{T}{H}{M}_{\text{ZFC}}\left(t,{t}_{\text{w}};H\right)=\mathcal{F}\left(C;H\right),\end{equation} \tag{ 13 }$$

where $\mathcal{F}\left(C;H\right)$ behaves at large C(t, t_w; H) as $\mathcal{F}\left(C;H\right)=1-C\left(t,{t}_{\text{w}};H\right)$. We report the details in appendix A.1. On the other hand, we define S(t, t_w; H) as a finite-time difference

$$\begin{equation}S\left(t,{t}_{\text{w}},{t}^{\prime };H\right)=\frac{{M}_{\text{ZFC}}\left({t}^{\prime },{t}_{\text{w}};H\right)-{M}_{\text{ZFC}}\left(t,{t}_{\text{w}};H\right)}{\mathrm{ln}\left({t}^{\prime }/t\right)}.\end{equation} \tag{ 14 }$$

In simulations, time is discrete and we store configurations at times t_n = integer-part-of 2^n/4, with n an integer. Let us write explicitly the integer dependence of times t and t' as:

$$\begin{equation}t\equiv {t}_{n},\qquad {t}^{\prime }\equiv {t}_{n+k},\end{equation} \tag{ 15 }$$

where k is an integer time parameter. The reader will note that there is a trade-off in the choice of k. On the one hand, the smaller k is the better the finite difference in equation (14) represents the derivative. On the other hand, when k grows the statistical error in the evaluation of equation (14) decreases significantly. In this section we report only the case for k = 8 (more details about time discretisation are provided in appendix A.3). The numerical S(t, t_w, t'; H) are exhibited in figure 3, where a local maximum in the long-time region can be seen.

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As explained in the introduction, we are interested in the evaluation of the time when the relaxation function S(t, t_w, t'; H) peaks, namely ${t}_{H}^{\text{eff}}$. Two problems arise:

• (a)  
  The reader will note two separate peaks in the S(t, t_w, t'; H) curves of figure 3: namely the peak at microscopic times t ∼ 4, and the peak we are interested in at t ∼ t_w. Unfortunately, the distinction between the two is only clear at small H. Previous numerical work [23] did not face this problem, probably because of their smaller correlation length, ξ ≈ 8 (non-linear susceptibilities grow very fast with ξ, see the next section).
• (b)  
  We are most interested in the limit H → 0, which is extremely noisy, as we have explained above.

An interesting possibility emerges when plotting the relaxation function S(t, t_w, t'; H) in terms of the correlation function C(t, t_w; H), rather than as a function of time (see figure 4). C(t, t_w; H) is a decreasing function of time, the long-time region corresponding to small C(t, t_w; H), and vice versa. Hence, the physical peak in which we are interested is the one that appears at small C(t, t_w; H) (see figure 4). Analogously to figure 3, we report only the case for k = 8 in figure 4.

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The simulation data strongly suggest that, when H → 0, the correlation function C(t, t_w; H) approaches a constant value C_peak(t_w) at the maximum of the relaxation function. Hence, our proposal is to define ${t}_{H}^{\text{eff}}$ in simulations as the time when C(t, t_w; H) reaches the value C_peak(t_w):

$$\begin{equation}C\left({{t}^{\text{eff}}}_{H},{t}_{\text{w}};H\right)={C}_{\text{peak}}\left({t}_{\text{w}}\right).\end{equation} \tag{ 16 }$$

As the reader can see, equation (16) is applicable also at H = 0, solving the problem of the vanishing magnetisation in this limit. The crucial point for our new ${t}_{H}^{\text{eff}}$ definition, see equation (16), is, hence, the computation of C_peak(t_w). Two problems arise:

• (a)  
  The constant-value C_peak(t_w) is well defined only for small magnetic field H.
• (b)  
  The relaxation function as a function of the correlation, $\mathcal{S}\left(C;H\right)$, is an implicit function of a reparametrised time,

  $$\begin{equation}{t}_{\text{new}}=\frac{1}{2}\enspace \mathrm{ln}\left(\frac{{t}_{n+k}}{{t}_{n}}\right)\end{equation} \tag{ 17 }$$

  (see appendix A.3 for details).

Our strategy has been to study, for each run, the behaviour of $\mathcal{S}\left(C;H\right)$ for the two smallest magnetic fields H and for three different time parameters k. We report in figure 5 a closeup of the peak region for $\mathcal{S}\left(C;H\right)$, used for the evaluation of C_peak(t_w). We report our estimates for C_peak(t_w) in table 2.

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The relaxation function S(t_new, t_w; H) depends on the correlation length ξ(t_w), and on the applied magnetic field H, equation (2). We observe, however, that S(t_new, t_w; H) has a temperature dependence, which we extract by comparing runs 4 and 2 in figure 5. These two cases are characterised by

• (a)  
  A similar starting correlation length ξ(t_w; H = 0) ≈ 11.7 (see table 2),
• (b)  
  The same set of applied magnetic fields, namely H = 0.005 and H = 0.01.

Yet, there appear to be two different scenarios in the data plotted in figure 5. In run 2, the peak of $\mathcal{S}\left(C;H\right)$ is almost the same for all the rescaled time curves. In run 4, however, the peaks separate for different k. As will be explained in section 5.5, this difference in behaviour is caused by increasing non-linear effects in the magnetisation, M_ZFC(t, t_w; H).

In conclusion, equation (16) solves our two problems at once. We no longer need to resolve the short-time and the long-time peaks in figure 3 and it bypasses the problem of the vanishing magnetisation as H goes to zero.

Scaling laws for the spin-glass susceptibility in the vicinity of the condensation temperature have been proposed and analysed for decades. We first recall an important early approach, and then develop the scaling law that we have employed to analyse our experiments and simulations.

Non-linear magnetisation effects, and their scaling properties in spin glasses, were first introduced by Malozemoff, Barbara, and Imry [37–39], who introduced the relation for the singular part of the magnetic susceptibility,

$$\begin{equation}{\chi }_{s}={H}^{2/\delta }f\hspace{-3pt}\left({t}_{\mathrm{r}}/{H}^{2/\phi }\right),\end{equation} \tag{ 18 }$$

where f(x) is a constant for x → 0; f(x) = x^−γ for x → ∞; ϕ = γδ/(δ − 1) ≡ βδ; and t_r is the reduced temperature T/T_g. This form was used by Lévy and Ogielski [40] and by Lévy [41], who measured the AC non-linear susceptibilities of very dilute AgMn alloys above and below T_g as a function of frequency, temperature, and magnetic field. The critical exponents of equation (18) were evaluated as β = 0.9, γ = 2.3, δ = 3.3, ν = 1.3, and z = 5.4. They differ substantially from Monte Carlo simulations for short-range Ising systems: β = 0.782(10), γ = 6.13(11), ν = 2.562(42) from [27]. The discrepancy in the value of γ is very large, and most probably arises from the lack of an exact value for T_g in the experiments. This illustrates the value of and need for a different approach for scaling the non-linear magnetisation of spin glasses in the vicinity of T_g.

Our approach is to express the non-linear components of the magnetic susceptibility in terms of ξ(t, t_w), the spin-glass correlation length in a magnetic field H. ²¹ This approach gives the non-linear magnetisation a direct connection to a measurable quantity and obviates the need for an accurate value of T_g.

The argument goes as follows. Let M(t, t_w; H) be the magnetisation per spin, where explicit attention is paid to the waiting (ageing) time t_w in the preparation of the spin-glass state. The generalised susceptibilities χ₁, χ₃, χ₅, ... are defined through the Taylor expansion

$$\begin{equation}M\left(H\right)={\chi }_{1}H+\frac{{\chi }_{3}}{3!}{H}^{3}+\frac{{\chi }_{5}}{5!}{H}^{5}+\mathcal{O}\left({H}^{7}\right).\end{equation} \tag{ 19 }$$

where, for brevity's sake, we omit arguments t and t_w.

Under equilibrium conditions, and for a large-enough correlation length ξ_eq, there is a scaling theory for the magnetic response to an external field H [42, 43]. Our main hypothesis in this work is that this scaling theory holds not only at equilibrium, but also in the non-equilibrium regime for a spin glass close to T_g and in the presence of a small external magnetic field H:

$$\begin{equation}M\left(t,{t}_{\text{w}};H\right)={\left[\xi \left(t+{t}_{\text{w}}\right)\right]}^{{y}_{H}-D}\mathcal{F}\left(H{\left[\xi \left(t+{t}_{\text{w}}\right)\right]}^{{y}_{H}},\frac{\xi \left(t+{t}_{\text{w}}\right)}{\xi \left({t}_{\text{w}}\right)}\right).\end{equation} \tag{ 20 }$$

According to full-aging spin-glass dynamics (see, e.g. [44]), equation (20) tells us that ξ(t + t_w)/ξ(t_w) will be approximately constant close to the maximum of the relaxation rate (see figure 2), so we shall omit this dependence. Hence, combining equations (19) and (20), one can express the generalised susceptibilities χ₁, χ₃, χ₅, ... in terms of the spin-glass correlation length ξ(t, t_w; H):

$$\begin{equation}{\chi }_{2n-1}\propto \vert \xi \left({t}_{\text{w}}\right){\vert }^{2n\enspace {y}_{H}-D},\end{equation} \tag{ 21 }$$

where we have omitted the arguments t, H for convenience, and

$$\begin{equation}2{y}_{H}=D-\frac{\theta \left(\tilde {x}\right)}{2},\end{equation} \tag{ 22 }$$

with $\theta \left(\tilde {x}\right)$ the replicon exponent [23].

The first term of M(H) in equation (19) is χ₁, which contains the linear term as well as the first non-linear scaling term [4], so we write

$$\begin{equation}{\chi }_{1}=\frac{\hat{S}}{T}+\frac{{a}_{1}\left(T\right)}{{\xi }^{\theta \left(\tilde {x}\right)/2}}.\end{equation} \tag{ 23 }$$

where $\hat{S}$ is the function appearing in the FDRs [45] and a₁(T) is some unknown constant (hopefully smoothly varying with temperature).

The free-energy variation per spin in presence of a magnetic field can be obtained by integrating the magnetic density, equation (19), with respect to the magnetic field:

$$\begin{equation}{\Delta}F=-\left[\frac{{\chi }_{1}}{2}{H}^{2}+\frac{{\chi }_{3}}{4!}{H}^{4}+\frac{{\chi }_{5}}{6!}{H}^{6}+\mathcal{O}\left({H}^{8}\right)\right].\end{equation} \tag{ 24 }$$

Substituting the scaling behaviour from equations (21) and (23), the free energy ΔF can be written as (we drop the $\tilde {x}$ dependence of θ for brevity)

$$\begin{equation}{\Delta}F=-\left[\frac{\hat{S}}{2T}{H}^{2}+\frac{{a}_{1}\left(T\right)}{{\xi }^{\theta /2}}{H}^{2}+{a}_{3}\left(T\right){\xi }^{D-\theta }{H}^{4}+{a}_{5}\left(T\right){\xi }^{2D-\left(3\theta /2\right)}{H}^{6}+\mathcal{O}\left({H}^{8}\right)\right],\end{equation} \tag{ 25 }$$

where again the a_n (T) are unknowns and (again, hopefully) smoothly varying functions of temperature. We use the effective response time, ${t}_{H}^{\text{eff}}$, to reflect the total free-energy change at magnetic field H and H = 0⁺:

$$\begin{equation}\mathrm{ln}\left[\frac{{t}_{H}^{\text{eff}}}{{t}_{H\to {0}^{+}}^{\text{eff}}}\right]={N}_{\text{corr}}{\Delta}F,\end{equation} \tag{ 26 }$$

where N_corr is the number of correlated spins, ${N}_{\text{corr}}={V}_{\text{corr}}/{a}_{0}^{3}$, with V_corr the correlated spins volume and a₀ the lattice constant or average distance between magnetic moments. Combining equation (26) with equations (6) and (24) leads to

$$\begin{align}\hfill \mathrm{ln}\left[\frac{{t}_{H}^{\text{eff}}}{{t}_{H\to {0}^{+}}^{\text{eff}}}\right]& =-b\left[\left(\frac{\hat{S}}{2T}+\frac{{a}_{1}\left(T\right)}{{\xi }^{\theta /2}}\right){\xi }^{D-\left(\theta /2\right)}{H}^{2}\right.\hfill \\ \hfill & \quad +\left.{a}_{3}\left(T\right){\xi }^{2D-\left(3\theta /2\right)}{H}^{4}+{a}_{5}\left(T\right){\xi }^{3D-2\theta }{H}^{6}+\mathcal{O}\left({H}^{8}\right)\right],\hfill \end{align} \tag{ 27 }$$

where coefficient b is a geometrical factor, see equation (6), and we have absorbed the k_B T term in the a_n (T) coefficients. The correction term ${a}_{1}\left(T\right)/{\xi }^{\theta \left(\tilde {x}\right)/2}$ is small compared to $\hat{S}/T$, so it will be dropped in subsequent expressions. Equation (27) shows that the higher-order terms have the functional form

$$\begin{equation}{\chi }_{2n-1}\frac{{H}^{2n}}{\left(2n\right)!}={a}_{2n-1}\left(T\right){\xi }^{-\theta \left(\tilde {x}\right)/2}{\left[{\xi }^{2{y}_{H}}{H}^{2}\right]}^{n},\end{equation} \tag{ 28 }$$

where

$$\begin{equation}2{y}_{H}=D-\frac{\theta \left(\tilde {x}\right)}{2}.\end{equation} \tag{ 29 }$$

This leads to the new scaling relation,

$$\begin{equation}\mathrm{ln}\left[\frac{{t}_{H}^{\text{eff}}}{{t}_{H\to {0}^{+}}^{\text{eff}}}\right]=\frac{\hat{S}}{2T}\enspace {\xi }^{D-\theta \left(\tilde {x}\right)/2}{H}^{2}+\enspace {\xi }^{-\theta \left(\tilde {x}\right)/2}\mathcal{G}\left(T,{\xi }^{D-\theta \left(\tilde {x}\right)/2}{H}^{2}\right),\end{equation} \tag{ 30 }$$

where the geometrical factor b has been absorbed into the scaling function $\mathcal{G}$. Comparison with the previous, more classical, relation, equation (18), evinces the simplicity and power of our approach to scale the non-linear magnetisation in the vicinity of T_g.

We extract the effective waiting time ${t}_{H}^{\text{eff}}$ in equation (3) from the time at which S(t, t_w; H) is a maximum, as before. Our results for all four conditions in table 1 are exhibited as a function of H² in figure 6. The slope of the data in figure 6 at small values of the magnetic field H generates the spin-glass correlation length ξ(t_w; T_m) from equations (4) and (5), see table 1, which also lists the employed values for the replicon exponent $\theta \left(\tilde {x}\right)$. These results will allow us to express the non-linear susceptibility in terms of ξ(t_w).

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An example of the measured relaxation function S(t, t_w; H) is plotted for T_m = 28.5 K and t_w = 10 000 s in figure 2 for five different magnetic fields, while the effective response times, $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$, are plotted in figure 6 for all four experiments listed in table 1. Note the remarkable similarity in shape of the original experimental results for $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$ in figure 1 with our results in figure 6. Also, note the fits of all four of our results for $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$ in figure 6 to the scaling relationship for the non-linear magnetisation, equation (27), which will be described in more detail below.

Because the scaling relationship, equation (30), depends upon the magnitude of the waiting time in ξ(t, t_w; T_m), two different values of t_w were used at the same intermediate temperature T_m = 28.75 K, among the three temperatures (28.5 K, 28.75 K, and 29.0 K) listed in section 2 and in table 1, to test equation (30) at a given temperature. This allows us to discriminate between the influence of temperature and of waiting time on ξ(t, t_w; T_m). In this way, we are able to demonstrate explicitly that ξ(t, t_w; T_m) is the control parameter.

It is useful to display ${t}_{H}^{\text{eff}}$ against H² individually for each of the four values of T_m and t_w. They are exhibited above in figure 7. The data for $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$ is fitted to the function

$$\begin{equation}f\left(x\right)={c}_{0}+{c}_{2}x+{c}_{4}{x}^{2}+{c}_{6}{x}^{3}+\mathcal{O}\left({x}^{4}\right),\end{equation} \tag{ 31 }$$

where x ≡ H² and the c_n coefficients, according to equation (27), correspond to:

$$\begin{equation}{c}_{0}=\mathrm{ln}\left({t}_{H\to {0}^{+}}^{\text{eff}}\right),\end{equation} \tag{ 32 }$$

$$\begin{equation}{c}_{2}=\left[\frac{\hat{S}}{2{T}_{\mathrm{m}}}\right]{\xi }^{D-\theta \left(\tilde {x}\right)/2},\end{equation} \tag{ 33 }$$

$$\begin{equation}{c}_{4}={a}_{3}\left({T}_{\mathrm{m}}\right){\xi }^{2D-3\theta \left(\tilde {x}\right)/2},\end{equation} \tag{ 34 }$$

$$\begin{equation}{c}_{6}={a}_{5}\left({T}_{\mathrm{m}}\right){\xi }^{3D-2\theta \left(\tilde {x}\right)}.\end{equation} \tag{ 35 }$$

Notice that we have absorbed the geometrical prefactor b of equation (27) in the non-linear coefficients a_n (T_m) and in the linear coefficient $\hat{S}$, and we neglect the sub-leading coefficient ${a}_{1}\left({T}_{\mathrm{m}}\right)/{\xi }^{\theta \left(\tilde {x}\right)/2}$.

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The effect of increasing temperature with waiting time held constant can be seen in the difference between the measured ${t}_{H}^{\text{eff}}$ and the extrapolated value of the linear magnetisation term (quadratic in H²) for the largest magnetic field (H = 59 Oe) in exps. 1, 2 and 4 in figure 7. Non-linear effects grow for larger t_w, hence larger ξ(t, t_w; T_m) at the same temperature, which can be seen by comparing experiments 2 and 3. The linear and non-linear coefficients of equation (27) can be extracted from fits of the data in figure 7 (dashed lines), whose resulting coefficients are listed in table 3.

To test the scaling relationship of equation (30) we first consider the fits of the data at T_m = 28.75 K for the two waiting times, t_w = 2 × 10⁴ s and t_w = 10⁴ s. The linear term is proportional to ${\xi }^{D-\theta \left(\tilde {x}\right)/2}$. The ratio of the two correlation lengths from table 3 is, hence,

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{2}\left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{{c}_{2}\left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[D-\theta \left(\tilde {x}\right)/2\right]}\approx 1.0535.\end{equation} \tag{ 36 }$$

Should scaling hold according to equation (30), then consistency requires that the ratios of the correlation lengths from the non-linear terms be the same as that for the linear term. They are:

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{4}\left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{{c}_{4}\left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[2D-3\theta \left(\tilde {x}\right)/2\right]}\approx 1.0476,\end{equation} \tag{ 37 }$$

and

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{6}\left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{{c}_{6}\left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[3D-2\theta \left(\tilde {x}\right)\right]}\approx 1.0526.\end{equation} \tag{ 38 }$$

The equality (within experimental error) of equations (36)–(38) is an impressive experimental verification of scaling relationship (30). Another check is the growth of the correlation length itself. At temperature T_m = 28.75 K and for the two waiting times, it is possible to calculate the ratio of the two values of the correlation length directly, using the expression for power-law growth [5, 46],

$$\begin{equation}\xi \left({t}_{\text{w}};{T}_{\mathrm{m}}\right)={a}_{0}{\hat{C}}_{1}{\left(\frac{{t}_{\text{w}}}{{\tau }_{0}}\right)}^{{\hat{C}}_{2}\left({T}_{\mathrm{m}}/{T}_{\text{g}}\right)}\equiv {a}_{0}{\hat{C}}_{1}{\left(\frac{{t}_{\text{w}}}{{\tau }_{0}}\right)}^{{T}_{\mathrm{m}}/\left({z}_{\text{c}}{T}_{\text{g}}\right)},\end{equation} \tag{ 39 }$$

where ${\hat{C}}_{1}$ and ${\hat{C}}_{2}$ are constants, by definition ${\hat{C}}_{2}\equiv 1/{z}_{\text{c}}$, and τ₀ is a characteristic exchange time, here taken as ℏ/k_B T_g.

Using the growth-rate parameter z_c = 12.37(107) [2, 3] one finds

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=20\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}\approx {\left(\frac{2{\times}1{0}^{4}}{1{0}^{4}}\right)}^{{T}_{\mathrm{m}}/\left(12.37{T}_{\text{g}}\right)}={2}^{28.75/\left(12.37{\times}31.5\right)}\approx 1.0525.\end{equation} \tag{ 40 }$$

Comparing the ratio of ξ(t_w; T_m) for the two different waiting times, equation (40), from the growth law, equation (39), with the ratios from fitting to the scaling relationship, equations (36)–(38), is remarkable evidence for the consistency of our physical picture. It explicitly demonstrates the power of using the spin-glass correlation length as the primary factor for evaluating the spin-glass non-linear magnetisation in the vicinity of the transition temperature T_g.

The lingering issue from equations (6) and (30), according to Zhai et al [3] 'is that the replicon exponent [$\theta \left(\tilde {x}\right)$] (...) depends upon both the temperature and ξ through the crossover variable [$\tilde {x}$]', with

$$\begin{equation}\tilde {x}=\frac{{\ell }_{\text{J}}}{\xi \left({t}_{\text{w}};{T}_{\mathrm{m}}\right)}.\end{equation} \tag{ 41 }$$

From the notation of table 3 and equation (6) the number of correlated spins is

$$\begin{equation}{N}_{\text{corr}}=\frac{{k}_{\text{B}}{T}_{\mathrm{m}}\enspace {c}_{2}}{{\chi }_{\text{FC}}}={\xi }^{D-\theta \left(\tilde {x}\right)/2}={\xi }^{D-\left[\theta \left({\ell }_{\text{J}}/\xi \right)/2\right]}.\end{equation} \tag{ 42 }$$

The left-hand side is a number, the right-hand side is an implicit function of ξ and θ. Using the definition of the Josephson length ℓ_J and of the replicon $\theta \left(\tilde {x}\right)$ ²² , one can solve for ξ and θ at each of the four values of T_m and t_w explored experimentally. The results are displayed in table 1.

The ageing rate z_c varies as a function of ξ. Using the data at T_m = 28.75 K, the approximate aging-rate factor is z_c = 12.37(107) at ξ ∼ 200 lattice spacings a₀ [3]. Although the correlation length extracted at 28.75 K is larger than that at 28.5 K, the higher temperature sets the crossover variable x = ℓ_J(T_m)/ξ = 0.11 for t_w = 20 000 s and x = 0.12 for t_w = 10 000 s, in the range of the crossover variable obtained by us previously at T_m = 28.5 K [47]. Both measurements have exhibited slowing down of the spin-glass correlation growth rate near the critical temperature at large correlation lengths.

Using the average value of θ from table 1, θ = 0.343, and setting z_c = 12.37, the values exhibited in equations (36)–(40) are altered to 1.053, 1.048, 1.052, and 1.051, respectively. Using the values from table 1, the temperature-dependent coefficients a₃(T_m) and a₅(T_m) of equation (27) can be calculated for each of the four values of T_m and t_w. They are displayed in figure 8.

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From figure 8, one sees that the 'hope' expressed after equation (25), i.e. that the temperature dependence of the coefficients a_n (T_m) appearing in equation (25) be weak, is realised in this set of experiments. For a₃(T_m), within the experimental error bars, there is little or no change with temperature. The situation for a₅(T_m) is not as nice, but there appears to be little change with temperature at the two highest temperatures.

It is interesting to test the scaling relationship [4]

$$\begin{equation}{\chi }_{2n-1}\left({t}_{\text{w}};{T}_{\mathrm{m}}\right)\propto {a}_{2n-1}{\left[\xi \left({t}_{\text{w}};{T}_{\mathrm{m}}\right)\right]}^{\left(n-1\right)D-n\theta \left(\tilde {x}\right)/2}.\end{equation} \tag{ 43 }$$

Thus,

$$\begin{equation}{\chi }_{3}\propto {\xi }^{D-\theta \left(\tilde {x}\right)}\enspace {a}_{3},\qquad {\chi }_{5}\propto {\xi }^{2D-\frac{3\theta \left(\tilde {x}\right)}{2}}\enspace {a}_{5}.\end{equation} \tag{ 44 }$$

The measured non-linear susceptibilities are exhibited below for the three temperatures 28.5 K, 28.75 K and 29.0 K.

One can test the scaling relationships (43) and (44) by using the measured values for the spin-glass correlation length ξ, the replicon exponent $\theta \left(\tilde {x}\right)$ from table 1, and the values of c₂ and c₄ from temperatures 28.5 K and 29 K and t_w = 10 000 s. For T_m = 28.5 K, we have ξ = 320.36a₀, $\theta \left(\tilde {x}\right)=0.337$ (from table 1), and c₄ = 4.0 × 10⁻⁷ (from table 3, note that we have ignored the error bars). Similarly, for T_m = 29.0 K and t_w = 10 000 s we have ξ = 391.27a₀, $\theta \left(\tilde {x}\right)=0.349$, and c₄ = 10.2 × 10⁻⁷. Using equations (31) and (34) and the just quoted values of c₄(t_w; T_m) one finds

$$\begin{equation}\frac{{\chi }_{3}\left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s};{T}_{\mathrm{m}}=28.5\enspace \mathrm{K}\right)}{{\chi }_{3}\left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s};{T}_{\mathrm{m}}=29.0\enspace \mathrm{K}\right)}\approx 0.666.\end{equation} \tag{ 45 }$$

This ratio is well within the error bars of the measured non-linear susceptibilities in figure 9. A similar result is also found for χ₅.

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With these scaling observations in hand, it is interesting to wonder about using them to estimate the condensation temperature T_g. In principle, determination of T_g would require an infinite t_w, because ξ(T_m) → ∞ when T_m → T_g. One expects that any experiment at finite t_w would yield a maximum for the non-linear susceptibility at a temperature we shall call T_g(t_w) because t_w is finite.

In principle, then, by measuring T_g(t_w) for ever larger t_w, one could extrapolate to the true t_w → ∞ condensation temperature T_g. If nothing else, measurements at large values of t_w on laboratory time scales could establish a lower bound for T_g.

The non-linear susceptibility χ₃ diverges as

$$\begin{equation}{\chi }_{3}\left({t}_{\text{w}}\to \infty ;{T}_{\mathrm{m}}\right)={\chi }_{0}\enspace \frac{{T}_{\text{g}}\left({t}_{\text{w}}\to \infty \right)}{\vert {T}_{\text{g}}\left({t}_{\text{w}}\to \infty \right)-{T}_{\mathrm{m}}{\vert }^{\gamma }},\end{equation} \tag{ 46 }$$

where χ₀ is a constant independent of temperature, and γ = 6.13(11) [27]. For finite t_w, χ₃(t_w; T_m) only has a maximum as a function of temperature. A way of arriving at this maximum would be to fit the data to the function

$$\begin{equation}{\chi }_{3}\left({t}_{\text{w}};{T}_{\mathrm{m}}\right)={\chi }_{0}\enspace \frac{{T}_{\text{g}}\left({t}_{\text{w}}\right)}{\vert {T}_{\text{g}}\left({t}_{\text{w}}\right)-{T}_{\mathrm{m}}{\vert }^{\gamma }},\end{equation} \tag{ 47 }$$

and then use the data points from just two or three temperatures to extract T_g(t_w). For larger and larger t_w, one could in principle extrapolate to the true T_g. We emphasise that, though equation (47) suggests χ₃(t_w; T_m) diverges at T_m = T_g(t_w), it does not, arriving only at a maximum value for finite t_w. Nevertheless, equation (47) is a way of estimating T_g(t_w) for use in an extrapolation procedure.

To test whether this trick has any validity, consider the data exhibited in figure 9. Here, t_w = 10 000 s and, taking χ₃(t_w; T_m) at the centre of the error bars for the two temperatures 28.5 K and 29 K, one finds T_g(t_w = 10 000 s) = 32 K. This value is too high, as magnetisation measurements suggest T_g(t_w → ∞) = 31.5 K. More accurate determination of the parameters in table 3 would diminish the error in T_g(t_w), but it does suggest a feasible process for taking laboratory data for finite t_w and extrapolating to find T_g(t_w → ∞).

Given the above analysis of our recent data, it is convenient to revisit the work of Joh et al [5] and of Bert et al [9], to examine whether the Zeeman energy is proportional to H² or to H (alternatively, to the number of correlated spins or to the square root of the total number of spins, respectively). We have already alluded to the results of these works as displaying the effect of magnetisation non-linearity. We now explore this assertion in detail using the analysis of subsection 5.2.

Figure 1 of Joh et al and figure 3 of Bert et al are reproduced in figures 1 and 10 in this paper. Both exhibit significant deviations from an H² dependence of the $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$ with increasing values of H. Bert et al [9] go on to assert a linear dependence, as exhibited in their figure 3, reproduced here in figure 10. The magnetic fields in [9] are quite large, and the scale of their plot does not cover the dependence on H² for small H. Nevertheless, they claim their data fits a linear dependence of $\mathrm{ln}\enspace {t}_{H}^{\text{eff}}$ on H. A glance at the left panel of figure 10 suggests how they could rationalise their conclusion.

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Yet, as noted by the authors of the experiments in figure 1 [5], a linear dependence on H is a poor fit to the data at small H. Further, the argument for the magnetisation's growth with $\sqrt{{N}_{\text{corr}}}$ is supposedly valid at small H [48], while the dependence on the number of correlated spins is argued to be proportional to $\sqrt{{N}_{\text{corr}}}$, rather than linear in N_corr, as from equation (4). On the other hand, the data exhibited in figure 10 uses magnetic fields that are substantially larger than those considered in our experiments.

We assert that the departure from linearity in H² as H increases observed in [9] is simply the effect of non-linearity. To prove this, we apply the scaling relation, equation (30), to their data, doing our best to extract their measured values from their figure. Our fit to equation (31) is shown in the right panel of figure 10 and the resulting c_n are listed in table 4.

Although only 1–2 digits are significant in table 4, we write more digits, for the sake of reproducibility. The fitting quality for t_w = 10 000 s and t_w = 30 000 s is better than for the other two waiting times. Note that the coefficients c_n listed in table 4 are considerably smaller than in our table 3 for our current experiments on a CuMn 6 at. % single crystal. We believe this is because our measurements are for T_m ≈ 0.9T_g whereas Bert et al [9] worked at 0.72T_g, where non-linear terms are expected to be much smaller.

Using the fitting coefficients from table 4 and $\theta \left(\tilde {x}\right)=0.3$, we obtain

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=30\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{2}\left(30\enspace 000\enspace \mathrm{s}\right)}{{c}_{2}\left(10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[D-\theta \left(\tilde {x}\right)/2\right]}\approx 1.094,\end{equation} \tag{ 48 }$$

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=30\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{4}\left(30\enspace 000\enspace \mathrm{s}\right)}{{c}_{4}\left(10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[2D-3\theta \left(\tilde {x}\right)/2\right]}\approx 1.054,\end{equation} \tag{ 49 }$$

$$\begin{equation}\frac{\xi \left({t}_{\text{w}}=30\enspace 000\enspace \mathrm{s}\right)}{\xi \left({t}_{\text{w}}=10\enspace 000\enspace \mathrm{s}\right)}={\left[\frac{{c}_{6}\left(30\enspace 000\enspace \mathrm{s}\right)}{{c}_{6}\left(10\enspace 000\enspace \mathrm{s}\right)}\right]}^{1/\left[3D-2\theta \left(\tilde {x}\right)\right]}\approx 1.045.\end{equation} \tag{ 50 }$$

The three ratios, equations (48)–(50) do not agree with one another perfectly, but again, considering the rawness of the analysis, they are certainly suggestive. In summary, we believe that the assessment of [9] that their data is evidence for E_Z ∝ H is in error. Rather, we believe the departure they observe from linearity in H² arises from non-linear terms in the magnetisation as a result of the large magnetic fields utilised in their study.

We have exploited our proposed relation (16) to extract effective times ${t}_{H}^{\text{eff}}$, as explained in appendix A.3. Our results are displayed in figure 11. In the subsequent analysis in section 5.5, we shall need the derivative of $\mathrm{ln}\left({t}_{H}^{\text{eff}}/{t}_{H\to 0+}^{\text{eff}}\right)$ with respect to H², evaluated numerically at H² = 0. Our main scope here will be evaluating this derivative.

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An obvious strategy would be to fit the numerical data for $\mathrm{ln}\left({t}_{H}^{\text{eff}}/{t}_{H\to 0+}^{\text{eff}}\right)$ as we did for the experimental data in equation (31). Note that our sought derivative at H² = 0 is just the c₂ coefficient in the fit. A welcome simplification in the analysis of the numerical data is that we can explicitly put c₀ = 0 in the fit to equation (31) (indeed, we are able to carry out the fit for $\mathrm{ln}\left({t}_{H}^{\text{eff}}/{t}_{H\to 0+}^{\text{eff}}\right)$ thanks to equation (16)). Our fitting parameters are reported in table 5. Unfortunately, as the reader will note from figure 11, these fits predict unphysically wild oscillations that are not observed in the numerical data. A plausible explanation for these oscillations relies on the very large magnetic fields used (recall that H = 1 for the IEA model roughly corresponds to 5 × 10⁴ Oe in physical units). These huge magnetic fields probably exceed the radius of convergence of the Taylor expansion of equation (30). At any rate, the oscillations cast some doubts on the determination of the derivative at H² = 0. This is why we have turned to a different strategy in order to validate our computation.

Table 5. Results of the fits to equation (31) of the numerical data for the time ratio $\mathrm{ln}\left({t}_{H}^{\text{eff}}/{t}_{H\to {0}^{+}}^{\text{eff}}\right)$. Note that, in order to stabilise the fits, we needed to include an extra terms in equation (31) for two cases. In the table, ξ stands for ξ(t = 0, t_w; H = 0) and the fitting range is $0{\leqslant}{H}^{2}{\leqslant}{H}_{\mathrm{max}}^{2}$.

T	tw	ξ	c2(tw; T)	c4(tw; T)	c6(tw; T)	c8(tw; T)	${H}_{\mathrm{max}}^{2}$
0.9	222	8.294(7)	−6.01(6) × 102	3.45(13) × 104	−1.31(9) × 106	1.998(19) × 107	0.025
0.9	226.5	11.72(2)	−1.589(19) × 103	1.98(6) × 105	−1.41(6) × 107	3.86(18) × 108	0.015
0.9	231.25	16.63(5)	−4.380(15) × 103	1.34(8) × 106	−1.33(10) × 108	0	0.010
1.0	223.75	11.79(2)	−1.39(6) × 103	2.2(4) × 105	−1.8(5) × 107	0	0.008
1.0	227.625	16.56(5)	−3.36(9) × 103	9.4(6) × 105	−9.1(8) × 107	0	0.008
1.0	231.75	23.63(14)	−7.91(15) × 103	3.29(11) × 106	−3.56(13) × 108	0	0.008

Our starting point, recall [2] and equation (27), is the expected scaling behaviour for the coefficient c₂(t_w; T) listed in table 5. The non-linear coefficient c₂(t_w; T), reported in table 5, behaves as [4]

$$\begin{equation}{c}_{2}\left({t}_{\text{w}};T\right)={\xi }^{D-\theta \left(\tilde {x}\right)/2}\left(\frac{\hat{S}}{2T}+\frac{{a}_{1}\left(T\right)}{{\xi }^{\theta \left(\tilde {x}\right)/2}}\right),\end{equation} \tag{ 51 }$$

using the scaling of the susceptibility χ₁ from equation (23). Here, $\hat{S}$ is the function appearing in the FDR [45] and a₁(T) is a smooth function of temperature, and we have absorbed the geometrical prefactor b of equation (27) in a₁(T) and $\hat{S}\left(T\right)$. Notice that the ${a}_{1}\left(T\right){\xi }^{-\theta \left(\tilde {x}\right)/2}$ term is sub-leading compared to $\hat{S}/\left(2T\right)$ and it was neglected in the previous analysis.

We rewrite equation (51) as:

$$\begin{equation}\frac{{c}_{2}\left({t}_{\text{w}};T\right)}{{\xi }^{D-\theta \left(\tilde {x}\right)/2}}=\frac{\hat{S}}{2}+T{a}_{1}\left(T\right){\xi }^{-\theta \left(\tilde {x}\right)/2},\end{equation} \tag{ 52 }$$

and we study this quantity as a function of ${\left[\xi \left({t}_{\text{w}}\right)\right]}^{-\theta \left(\tilde {x}\right)/2}$ in figure 12. Note that in the above expression ξ was not obtained from the response to the magnetic field. Instead, we computed ξ from the correlation functions at H = 0 (see appendix A.5 and [7]). The data exhibit a constant value, except for the point correspondent to t_w = 2^31.75 at T = 1.0 (run 6). Therefore, we shall accept the numerical estimation of the derivative at H² = 0 through the coefficient c₂ for all cases but for our run 6. In order to clarify what is going on with run 6, we report in the right panel of figure 12 an enlargement of figure 11 in the small-magnetic-field regime for this case. As it could be guessed from the left panel, the fitting procedure clearly underestimates the slope of the curve at H² = 0. Therefore, in order to estimate the derivative for run 6, we have instead relied on equation (52) by averaging the constant value found in figure 12 over all other runs and by multiplying this averaged constant value by ${\left[\xi \left({t}_{\text{w}}\right)\right]}^{D-\theta \left(\tilde {x}\right)/2}$ (see equation (52)).

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In order to test the scaling form, equation (30), the data for all of the non-linear contributions to the magnetisation, experimental and numerical, are plotted according to the functional form

$$\begin{equation}{\xi }^{-\frac{\theta \left(\tilde {x}\right)}{2}}\mathcal{G}\left(T,{\xi }^{D-\frac{\theta \left(\tilde {x}\right)}{2}}{H}^{2}\right)\end{equation} \tag{ 53 }$$

in figures 13 and 14. The fit to scaling relationship (30) is remarkable and testimony to the agreement for both the experimental and numerical data.

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We first address the dynamical scaling law for a system in presence of a magnetic field at temperatures close to T_g in section 5.6.1. Then, in section 5.6.2, we analyse the dynamical scaling for ferromagnetic systems, either Ising or Heisenberg, in the presence of an external magnetic field.

5.6.1. Dynamical scaling close to T_g

We evaluate the growth of the correlation length ξ(t, t_w; H) in simulations that mimic the experimental field-cooling protocol (FC), where the temperature is lowered from above to below T_g in the presence of a constant magnetic field H.

We performed two independent simulations on Janus II at the critical temperature T_g = 1.102(3) [27] (in IEA units) and at T = 1.05 for several external magnetic fields and 16 samples. A protocol equivalent to FC, but convenient for simulations, is to place a random spin configuration instantaneously at the working temperature T, and turning on the external magnetic field at the same instant, so that t_w = 0.

According to equation (20), at the critical temperature T_g, and for small external magnetic fields H, there exists a scaling behaviour that connects ξ(t, t_w; H) with the external magnetic field H:

$$\begin{equation}\left[\xi \left(t,{t}_{\text{w}};H\right)\enspace {H}^{2/{y}_{H}}\right]\propto \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 54 }$$

The correlation length ξ(t, t_w; H) grows as

$$\begin{equation}\xi \left({t}_{\text{w}}\right)\propto {{t}_{\text{w}}}^{1/z\left(T\right)},\end{equation} \tag{ 55 }$$

with an exponent that, in a first approximation, is expected to behave near the critical temperature as [2]:

$$\begin{equation}z\left(T\right)\simeq {z}_{\text{c}}\enspace \frac{{T}_{\text{g}}}{T}\enspace ,\quad \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\enspace {z}_{\text{c}}=z\left({T}_{\text{g}}\right)=6.69\left(6\right).\end{equation} \tag{ 56 }$$

Hence, using equations (55) and (56) in the scaling argument of equation (54), we have equivalently,

$$\begin{equation}\left[t\enspace {\times}{H}^{2z\left(T\right)/{y}_{H}}\right]\propto \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 57 }$$

In table 6 we list the aging-rate factors z(T) used in our analysis. We plot our rescaled data in figure 15.

Table 6. The aging-rate factors z(T) used in figure 15.

T	z(T)
1.1	6.60
1.05	7.00
1.0	7.30
0.9	8.12

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The agreement with the scaling prediction, evinced by the data collapse, is striking. Our plots also exhibit overshooting, as evidence for the paramagnetic phase when the magnetic field is turned on.

The reader could wonder why we have used equation (54) for the scaling analysis at the temperature T = 1.05 < T_g, and whether this implies evidence of the absence of the dAT line in finite dimension. We shall address these questions in section 6.

5.6.2. Overshooting in a ferromagnetic system

By studying two ordered systems we can show that the overshooting phenomenon is, in fact, general. To demonstrate generality, we have simulated the three-dimensional Ising and Heisenberg models in a cubic lattice with periodic boundary conditions and size L at the critical point T_c. The N = L^D (D = 3) Heisenberg spins interact with their lattice nearest neighbours through the Hamiltonian

$$\begin{equation}\mathcal{H}=-\sum\limits _{\langle \boldsymbol{r},{\boldsymbol{r}}^{\prime }\rangle }{\boldsymbol{S}}_{\boldsymbol{r}}\cdot {\boldsymbol{S}}_{{\boldsymbol{r}}^{\hspace{1pt}\prime }}+\boldsymbol{H}\cdot \sum\limits _{\boldsymbol{r}}{\boldsymbol{S}}_{\boldsymbol{r}},\end{equation} \tag{ 58 }$$

where S _r are unit vector spins and H is an external magnetic field. The connected correlation function is

$$\begin{equation}C\left(r,t\right)=\frac{1}{{L}^{3}}\sum\limits _{\boldsymbol{x}}{\boldsymbol{S}}_{\boldsymbol{x}}\left(t\right)\cdot {\boldsymbol{S}}_{\boldsymbol{r}+\boldsymbol{x}}\left(t\right)-{\left[\boldsymbol{m}\left(t\right)\right]}^{2},\end{equation} \tag{ 59 }$$

with

$$\begin{equation}\boldsymbol{m}\left(t\right)=\frac{1}{{L}^{3}}\sum\limits _{\boldsymbol{x}}{\boldsymbol{S}}_{\boldsymbol{x}}\left(t\right).\end{equation} \tag{ 60 }$$

We only write equations for the Heisenberg model, equations (58)–(60), but the Ising analogues can be obtained trivially by just dropping the vector symbol in the spins. Furthermore, the correlation function C(r,t) will be averaged over different initial conditions (runs). We report the simulation details in table 7.

Table 7. Parameters of our ferromagnetic simulations for the Ising and Heisenberg models.

System	L	H	Number of runs
Ising	256	0	2200
			0.001	2600
			0.003	1700
			0.005	1200
Heisenberg	200	0	2660
			0.003	1000
			0.004	1600
			0.005	2400

The Ising and Heisenberg model have different symmetry properties, so they belong to two distinct universal classes. In other words, each model has a distinct value for the critical temperature and exponents. The Ising model has η = 0.036 297 8(20) [49], z = 2.0245(15) [50] and β_c = 0.221 654 626(5) [51]. Instead, for the Heisenberg ferromagnet η = 0.378(3) [52, 53], z = 2.033(5) [54] and β_c = 0.693 001(10) [55] (β_c ≡ 1/T_c).

As explained in appendix A.5, the correlation length ξ(t, t_w; H) can be calculated with integral estimators [7, 8],

$$\begin{equation}{I}_{k}\left(T,{t}_{\text{w}}\right)={\int }_{0}^{\infty }\mathrm{d}r\enspace {r}^{k}C\left(r,{t}_{\text{w}};T\right),\qquad {\xi }_{k,k+1}\left({t}_{\text{w}};T\right)=\frac{{I}_{k+1}\left({t}_{\text{w}};T\right)}{{I}_{k}\left({t}_{\text{w}};T\right)}.\end{equation} \tag{ 61 }$$

In this section, we evaluate the correlation length ξ₂₃(t, t_w; H). As the reader can notice, the growth of ξ₂₃(t) overshoots before reaching equilibrium for any external magnetic field for both ferromagnetic models, see figure 16.

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According to equation (20), at the critical temperature T_c, and for small external magnetic fields H, there exists a scaling law that connects ξ(t, t_w; H) with the external magnetic field H in ferromagnetic system:

$$\begin{equation}\left[\xi \left(t,{t}_{\text{w}};H\right)\enspace {H}^{1/{y}_{H}}\right]\propto \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 62 }$$

As the reader can notice, equation (62) differs from equation (54) in the power of the magnetic field. In the ferromagnetic system, the relevant external variable is H and not H² as it would be for spin glasses [42, 43]. Analogously to equation (57), we can rescale time as

$$\begin{equation}\left[t\enspace {\times}{H}^{z\left(T\right)/{y}_{H}}\right]\propto \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.\end{equation} \tag{ 63 }$$

We plot our rescaled data in figure 17. The agreement with the scaling prediction, both for the Heisenberg and for the Ising model, is remarkable.

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In conclusion, the overshooting phenomenon is general and we have observed it both in ferromagnetic systems, figures 16 and 17, and in disordered ones, figure 15.

Our numerical data for the magnetisation at small magnetic fields are rather noisy, which complicates the process of taking its derivative with respect to ln t. This derivative is the response function S(t, t_w; H) (recall equation (11)). This is why, before differentiating, we have followed a de-noising method first proposed in [45]. Because we work at larger correlation lengths (and closer to T_g) than in that work, however, we have found it preferable to change some technical details. We explain below the precise de-noising method that we have followed in this work.

Our starting observation is that the derivative of both M_ZFC(t, t_w; H) and TM_ZFC(t, t_w; H)/H peak at exactly the same time ${t}_{H}^{\text{eff}}$. However, TM_ZFC(t, t_w; H)/H enjoys the advantage of being a very smooth function of the correlation C(t, t_w; H). This smooth function is named the FDR [32–36]. The key point is that, at variance with the magnetisation, C(t, t_w; H) can be computed with high accuracy for any value of the field H, including H = 0. Thus, we follow a simple two-step de-noising algorithm:

• (a)  
  We fit our data for TM_ZFC(t, t_w; H)/H as a function of C(t, t_w; H), see equation (A.1).
• (b)  
  We replace our data for TM_ZFC(t, t_w; H)/H by the just-mentioned fitted function evaluated at C(t, t_w; H).

Our chosen functional form is as follows. Let the quantity TM_ZFC(t, t_w; H)/H be approximated by $f\left(\hat{x}\right)$, where for notational simplicity we do not write the explicit dependence on t, t_w and H,

$$\begin{equation}f\left(\hat{x}\right)={f}_{\text{L}}\left(\hat{x}\right)\frac{1+\mathrm{tanh}\left[\mathcal{Q}\left(\hat{x}\right)\right]}{2}+{f}_{\text{s}}\left(\hat{x}\right)\frac{1-\mathrm{tanh}\left[\mathcal{Q}\left(\hat{x}\right)\right]}{2}\end{equation} \tag{ A.1 }$$

with $\mathcal{Q}\left(\hat{x}\right)=\left(\hat{x}-{x}^{{\ast}}\right)/w$. The function $f\left(\hat{x}\right)$ has distinct behaviour for large and small $\hat{x}$. The crossover between the two functional forms is smoothed by the $\mathrm{tanh}\left[\mathcal{Q}\left(\hat{x}\right)\right]$ functional term, where x* is the crossover point and w is the crossover rate. The functional form for small $\hat{x}$ is

$$\begin{equation}{f}_{\text{s}}\left(\hat{x}\right)={a}_{0}+\sum\limits _{k=1}^{N}{a}_{k}\enspace \frac{{\left(\hat{x}-{\hat{x}}_{\mathrm{min}}\right)}^{k}}{k!}.\end{equation} \tag{ A.2 }$$

For the large-$\hat{x}$ region, we choose a polynomial expansion in terms of $\left(1-\hat{x}\right)$:

$$\begin{equation}{f}_{\text{L}}\left(\hat{x}\right)=\left(1-\hat{x}\right)+\sum\limits _{k=2}^{{N}^{\prime }+1}{b}_{k}\enspace \frac{{\left(1-\hat{x}\right)}^{k}}{k!}.\end{equation} \tag{ A.3 }$$

The polynomial expansion in $1-\hat{x}$ is quite natural in the large-$\hat{x}$ region [45], as a deviation from the fluctuation–dissipation theorem. This theorem, which holds only under equilibrium conditions, predicts N' = 0 for equation (A.3) and x* = w = 0 for equation (A.1) (so that, in equilibrium, one would have $f\left(\hat{x}\right)=\left(1-\hat{x}\right)$ in equation (A.1)). In the small-$\hat{x}$ region, there is not a strong justification (other than convenience) for our choice of ${f}_{\text{s}}\left(\hat{x}\right)$. In fact, the choice of [45] for ${f}_{\text{s}}\left(\hat{x}\right)$ was a Padé approximant. The quantity TM_ZFC(t, t_w; H)/H turns out to be affected by strong non-linear effects that increase with increasing external magnetic field H and upon approaching the glass temperature T_g (see figure 21).

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As we discuss in appendix A.2, it is necessary to select the appropriate order for the polynomials in equations (A.2) and (A.3). Our preferred choices are given in table 8.

Errors are computed using a jackknife procedure. We perform an independent fit for each jackknife block, and compute errors from the jackknife fluctuations of the fitted $f\left(\hat{x}\right)$. In figure 22 we show a comparison between the original and smoothed data for the case T = 1.0 and t_w = 2^31.75. As expected, the de-noising technique is most important for the smallest magnetic fields.

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A difficulty in our fits to equation (A.1) is that we can use only the diagonal part of the covariance matrix in the computation of the goodness-of-fit indicator χ². This is the reason underlying the very small values for χ² that we show in table 8. As a consequence, we cannot trust the χ² test for selecting the appropriate order for the polynomial expansions in equations (A.2) and (A.3). Hence, we follow a different strategy.

Fortunately, we can also compare the statistical errors that we find for the de-noised TM_ZFC(t, t_w; H)/H with different choices of the polynomial expansion (remember that these errors are not computed from χ², but from the jackknife fluctuations). As an example, consider the case at T = 0.9 for a waiting time t_w = 2^31.75 and H = 0.002, which is exhibited in figure 23. The figure compares the statistical errors of the original, non-de-noised data with the errors found with two possible choices for the polynomial fits in equations (A.2) and (A.3). Although both fits are indistinguishable from the point of view of the χ² test, see table 9, the resulting errors are very different. In one case, we find statistical errors that evolve rather smoothly with t. For the second choice, we find wild oscillations in the size of the errors as t varies. When in doubt, we have always taken the choice that provides the smoother t evolution for the errors. As we said above, our final choices are reported in table 8.

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As explained in the main text, the quantity used in experiment [5] to extract t^eff(H) is the relaxation function S(t, t_w; H) of equation (11). We calculate S(t, t_w; H) as a finite-time difference:

$$\begin{equation}S\left(t,{t}_{\text{w}},{t}^{\prime };H\right)=\frac{{M}_{\text{ZFC}}\left({t}^{\prime },{t}_{\text{w}};H\right)-{M}_{\text{ZFC}}\left(t,{t}_{\text{w}};H\right)}{\mathrm{ln}\left(\frac{{t}^{\prime }}{t}\right)}.\end{equation} \tag{ A.4 }$$

In simulations, time is discrete and we have stored configurations at t_n = integer-part-of 2^n/4, with n an integer. Let us write the integer dependence of times t and t' explicitly as:

$$\begin{equation}t\equiv {t}_{n},\qquad {t}^{\prime }\equiv {t}_{n+k},\end{equation} \tag{ A.5 }$$

where k is an integer time parameter. Hence, time is rescaled as

$$\begin{equation}{t}_{\text{new}}=\frac{1}{2}\enspace \mathrm{ln}\left(\frac{{t}_{n+k}}{{t}_{n}}\right).\end{equation} \tag{ A.6 }$$

We expressed our observables as functions of t_new:

$$\begin{equation}S\left(t,{t}_{\text{w}},{t}^{\prime };H\right)\quad \to \quad S\left({t}_{\text{new}},{t}_{\text{w}};H\right),\end{equation} \tag{ A.7 }$$

$$\begin{equation}C\left(t,{t}_{\text{w}};H\right)\quad \to \quad C\left({t}_{\text{new}},{t}_{\text{w}};H\right).\end{equation} \tag{ A.8 }$$

The relaxation function S(t_new, t_w; H) is trivial to construct, see equation (A.4). However, the correlation function C(t_new, t_w; H) needs to be calculated using a linear interpolation. For any given value of t_new, we looked for our original discrete time t_n such that

$$\begin{equation}\mathrm{ln}\left({t}_{n}\right){< }\mathrm{ln}\left({t}_{\text{new}}\right){\leqslant}\mathrm{ln}\left({t}_{n+1}\right).\end{equation} \tag{ A.9 }$$

Using a linear interpolation, we obtain

$$\begin{equation}C\left({t}_{\text{new}}\right)=\frac{\mathrm{ln}\left({t}_{\text{new}}\right)-\mathrm{ln}\left({t}_{n+1}\right)}{\mathrm{ln}\left({t}_{n}\right)-\mathrm{ln}\left({t}_{n+1}\right)}C\left({t}_{n}\right)-\frac{\mathrm{ln}\left({t}_{\text{new}}\right)-\mathrm{ln}\left({t}_{n}\right)}{\mathrm{ln}\left({t}_{n}\right)-\mathrm{ln}\left({t}_{n+1}\right)}C\left({t}_{n+1}\right).\end{equation} \tag{ A.10 }$$

Finally, one can express the relaxation function, S(t_new, t_w; H), as a function of the correlation function, $\mathcal{S}\left(C;H\right)$, in much the same manner as equation (A.10).

As explained in the main text, the extraction of ${t}_{H}^{\text{eff}}$ from equation (16) is delicate because the C_peak(t_w) are implicit functions of the rescaled time t_new. In order to solve equation (16), we calculate the ${t}_{H}^{\text{eff}}$ values through a quadratic interpolation. First, we calculate the original discrete time t_n such that:

$$\begin{equation}C\left(t+{t}_{n+1},{t}_{\text{w}};H\right){< }{C}_{\text{peak}}{\leqslant}C\left(t+{t}_{n},{t}_{\text{w}};H\right).\end{equation} \tag{ A.11 }$$

Then, we solve the three-equation system:

$$\begin{equation}C\left(t+{t}_{n-1}\right)={\alpha }_{0}+{\alpha }_{1}{x}_{n-1}+{\alpha }_{2}{x}_{n-1}^{2},\end{equation} \tag{ A.12 }$$

$$\begin{equation}C\left(t+{t}_{n}\right)={\alpha }_{0}+{\alpha }_{1}{x}_{n}+{\alpha }_{2}{x}_{n}^{2},\end{equation} \tag{ A.13 }$$

$$\begin{equation}C\left(t+{t}_{n+1}\right)={\alpha }_{0}+{\alpha }_{1}{x}_{n+1}+{\alpha }_{2}{x}_{n+1}^{2},\end{equation} \tag{ A.14 }$$

where x_n = ln t_n and, for brevity's sake, we omit the arguments t_w and H. The solution generates the α_i coefficients:

$$\begin{align}\hfill {\alpha }_{2}& =\frac{C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n-1}\right)-C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n}\right)}{{x}_{n-1}-{x}_{n}}\hfill \\ \hfill & \quad -\frac{C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n+1}\right)-C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n}\right)}{\left({x}_{n-1}-{x}_{n+1}\right)\left({x}_{n+1}-{x}_{n}\right)},\hfill \end{align} \tag{ A.15 }$$

$$\begin{equation}{\alpha }_{1}=\frac{C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n+1}\right)-C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n}\right)}{{x}_{n+1}-{x}_{n}}-{\alpha }_{2}\left({x}_{n}+{x}_{n+1}\right),\end{equation} \tag{ A.16 }$$

$$\begin{equation}{\alpha }_{0}=C\left({t}_{\text{w}},{t}_{\text{w}}+{t}_{n}\right)-{\alpha }_{1}{x}_{n}-{\alpha }_{2}{x}_{n}^{2}.\end{equation} \tag{ A.17 }$$

We can then calculate the ${t}_{H}^{\text{eff}}$ solving the equation:

$$\begin{equation}{C}_{\text{peak}}={\alpha }_{0}+{\alpha }_{1}\enspace \mathrm{ln}\left({t}_{H}^{\text{eff}}\right)+{\alpha }_{2}{\left[\mathrm{ln}\left({t}_{H}^{\text{eff}}\right)\right]}^{2},\end{equation} \tag{ A.18 }$$

where only the solution verifying ${t}_{n}{\leqslant}{t}_{H}^{\text{eff}}{< }{t}_{n+1}$ is physical.

We shall explain here our computation of the spin-glass correlation length in the presence of a magnetic field. For the simpler case of H = 0, we refer the reader to [2, 65].

The most informative connected correlator we can construct with four replicas is the replicon propagator [60, 66]. Extending the replicon propagator to the off-equilibrium regime we have:

$$\begin{equation}{G}_{\text{R}}\left(\boldsymbol{r},{t}^{\prime }\right)=\frac{1}{V}\sum\limits _{\boldsymbol{x}}\bar{{\left(\langle {s}_{\boldsymbol{x},{t}^{\prime }}\enspace {s}_{\boldsymbol{x}+\boldsymbol{r},{t}^{\prime }}\enspace \rangle -\langle {s}_{\boldsymbol{x},{t}^{\prime }}\rangle \langle {s}_{\boldsymbol{x}+\boldsymbol{r},{t}^{\prime }}\rangle \right)}^{2}},\end{equation} \tag{ A.19 }$$

where t' = t_w + t. To compute G_R( r , t'), we calculate the four-replica field

$$\begin{equation}{{\Phi}}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(a,b;c,d\right)}=\frac{1}{2}\left({s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(a\right)}-{s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(b\right)}\right)\left({s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(c\right)}-{s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(d\right)}\right),\end{equation} \tag{ A.20 }$$

where indices a, b, c, d indicate strictly different replica. Notice that

$$\begin{equation}\langle {{\Phi}}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(a,b;c,d\right)}{{\Phi}}_{\boldsymbol{y},{\mathrm{t}}^{\prime }}^{\left(a,b;c,d\right)}\rangle ={\left(\langle {s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}{s}_{\boldsymbol{y},{\mathrm{t}}^{\prime }}\rangle -\langle {s}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}\rangle \langle {s}_{\boldsymbol{y},{\mathrm{t}}^{\prime }}\rangle \right)}^{2}.\end{equation} \tag{ A.21 }$$

Therefore, we obtain G_R( r , t') by taking the average over the samples

$$\begin{equation}\mathrm{E}\left({{\Phi}}_{\boldsymbol{x},{\mathrm{t}}^{\prime }}^{\left(a,b;cd\right)}{{\Phi}}_{\boldsymbol{y},{\mathrm{t}}^{\prime }}^{\left(a,b;c,d\right)}\right)={G}_{\text{R}}\left(\boldsymbol{x}-\boldsymbol{y},{t}^{\prime }\right).\end{equation} \tag{ A.22 }$$

With 512 replicas at our disposal, there are 3 × 512!/(508! × 4!) ways of choosing the replica indices a, b, c, and d. We have found an efficient way for averaging over (roughly) one third of this astronomic number of possibilities [67]. We define the correlation function, C( r , t') , as the replicon propagator, G_R( x − y , t'), where we consider the difference x − y as the lattice displacement r = (r, 0, 0). Of course, we can align the lattice displacement vector r along any of three coordinate axes, so we average over these three choices. The replicon correlator G_R decays to zero in the large-distance limit. We, therefore, computed the correlation length ξ(t, t_w; H) exploiting the integral estimators [7, 8]:

$$\begin{equation}{I}_{k}\left({t}^{\prime };T\right)={\int }_{0}^{\infty }\mathrm{d}r\enspace {r}^{k}C\left(r,{t}^{\prime };T\right),\qquad {\xi }_{k,k+1}\left({t}^{\prime };T\right)=\frac{{I}_{k+1}\left({t}^{\prime };T\right)}{{I}_{k}\left({t}^{\prime };T\right)}.\end{equation} \tag{ A.23 }$$

In the main text, we always refer to the ξ₁₂(t'; T) correlation length except for section 5.6.2, where we evaluate ξ₂₃(t'; T).