The Janus Collaboration utilizes a special purpose-built computer [16] to examine the dynamics of the Ising spin glass. They were able to explore the (re-normalized) aging rate [22], z c T , ξ = T T g d l n t w d l n ξ . (9) The re-normalizing factor T/T _g makes z _c (T, ξ) ≈ z _c (ξ) [23].

We can rewrite Eq. 9 as Eq. 10, l n t w = z c T g T l n ξ + const (10) The aging rate, z _c , was found to vary substantially from experiment to experiments, depending upon the temperature and nature of the spin glass sample. For example, for a bulk polycrystalline sample of CuMn (6 at%) [1] found z _c = 5.917 at a reduced temperature of T/T _g = 0.89. For a polycrystalline bulk spinel they found z _c = 7.576 at a reduced temperature of T/T _g = 0.72.

In thin films [24] found for 11.7 at. % that z _c = 9.62 at reduced temperatures of T/T _g = 0.43, 0.59, and 0.78. Working at T/T _g = 0.95 [25], found z _c = 6.80 in bulk polycrystalline CuMn (5 at%).

The Janus collaboration [22] found a hint to reconcile these apparently conflicting values from experiment by computing ξ over a temperature range 0.5 ≤ T/T _g ≤ 1. Figure 2 exhibits their results.

Experimentally, there is a significant confirmation for this variation of z _c . Ref. [22] found that the growth of ξ(t _w , T) slows down as ξ(t _w , T) increases. That is, z _c increases as ξ(t _w , T) increases. In order to achieve large values of ξ(t _w , T) to test this simulation prediction, we were blessed with a single crystal of CuMn, 6 at%, grown by Dr. D.L.Schagel, the description of which is contained in Ref. [26]. By working at 28 K (T _g = 31.5 K) and at long waiting times (up to 10₅ s), the value extracted for ξ(t _w , T) reached 150 nm, the largest coherence length ever reported for a glassy system [26]. The value of ξ(t _w , T) vs. t _w is plotted in Figure 3, from which z _c = 12.37 ± 1.07 can be extracted. This is to be compared with z _c ≈ 9.62 extracted at shorter waiting times for smaller values of ξ(t _w , T).

The coherence length ξ(t _w , T) can be measured precisely for spin glasses both in experiment and through simulations. However, known analysis methods lead to discrepancies either for large external magnetic fields or close to the transition temperature. This problem can be solved through introduction of a scaling law that takes into account both the magnetic field and the time-dependent coherence length. This is especially important because temperatures T ≈ T _g are most relevant for the study of glass formers (ξ is restricted to a very narrow window of variation if one moves away from T _g ).

Historically, non-linear magnetization effects, and their scaling properties in spin glasses, were first introduced by Malozemoff, Barbara, and Imry [27–29] who introduced the relation for the singular part of the magnetic susceptibility, χ s = H 2 / δ f t r / H 2 / ϕ , (11) 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 [30], and Lévy [31] 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. Their critical exponents from Eq. 11 differed substantially from Monte Carlo simulations for short-range Ising systems [32]. The discrepancy in their value of γ was very large, and most probably arose from the lack of an exact value for T _g in their experiments. This illustrates the value of and need for a different approach for scaling the non-linear magnetization of spin glasses in the vicinity of T _g .

The scaling argument goes as follows. Let M(t, t _w ; H) be the magnetization per spin. The generalized susceptibilities χ ₁, χ ₃, χ ₅, … are defined through the Taylor expansion, M H = χ 1 H + χ 3 3 ! H 3 + χ 5 5 ! H 5 + O H 7 . (12) We have omitted t and t _w for brevity. Our hypothesis is that, in the non-equilibrium regime for a spin glass close to T _g in the presence of a small magnetic field, M t , t w ; H = ξ t + t w y H − D × F H ξ t + t w y H , ξ t + t w ξ t w (13) According to full-aging spin-glass dynamics [30] Eq. 13 tells us that ξ(t + t _w )/ξ(t _w ) will be approximately constant close to the maximum of the relaxation rate [i.e., peak of S(t)], so that we omit this dependence. Thus, combining Eqs 12, 13, one can express the generalized susceptibilities χ ₁, χ ₃, χ ₅, … in terms of the spin glass coherence length ξ(t, t _w ; H): χ 2 n − 1 ∝ | ξ t w | 2 n y H − D , (14) where we have omitted the arguments t and H for convenience, and Eq. 15 2 y H = D − θ x ̄ 2 , (15) with θ ( x ̄ ) the replicon exponent [3].

The first term of M(H) in Eq. 12 is χ ₁, which contains the linear term as well as the first non-linear scaling term, so that we write, χ 1 = S ̂ T + a 1 T ξ θ x ̄ / 2 (16) where a ₁(T) is some unknown constant, hopefully varying smoothly with temperature.

The free-energy variation per spin in the presence of a magnetic field can be derived by integrating the magnetic density Eq. 12 with respect to the magnetic field in Eq. 17, Δ F = − χ 1 2 H 2 + χ 3 4 ! H 4 + χ 5 6 ! H 6 + O H 8 . (17) Substituting the scaling from Eqs 14, 16, the free energy ΔF can be written as (we drop the x ̄ dependence of θ for brevity) in Eq. 18, Δ F = − S ̂ 2 T H 2 + a 1 T ξ θ / 2 H 2 + a 3 T ξ D − θ H 4 + a 5 T ξ 2 D − 3 θ / 2 H 6 + O H 8 , (18) where again the a _n (T) are unknowns and hopefully again smoothly varying functions of temperature. Using the effective response time, t H eff , to reflect the total free-energy change at magnetic field H with respect to H → 0⁺, l n t H eff t H → 0 + eff = N s Δ F = − b [ S ̂ 2 T + a 1 T ξ θ / 2 ξ D − θ / 2 H 2 ] + a 3 T ξ 2 D − 3 θ / 2 H 4 + a 5 T ξ 3 D − 2 θ H 6 + O H 8 (19) where the coefficient b is a geometrical factor, and we have absorbed the k _B T term in the a _n (T) coefficients. The correction term a 1 ( T ) / ξ θ ( x ̃ ) / 2 is small compared to S ̂ / T , so it will be neglected in subsequent expressions. Equation 19 shows that the higher order terms have the functional form, in Eq. 20, χ 2 n − 1 H 2 n 2 n ! = a 2 n − 1 T ξ − θ x ̃ / 2 ξ 2 y H H 2 n (20) where, in Eq. 21 2 y H = D − θ x ̃ 2 (21) This leads to the new scaling relation, l n t H eff t H → O + eff = S ̂ T ξ D − θ / 2 H 2 + ξ − θ / 2 G ξ D − θ / 2 H 2 ; T . (22) where the geometrical factor b has been absorbed into the scaling function G ).

Among the many uses of Eq. 22, two can be highlighted. The first is the issue surrounding the magnetization encompassed in ξ _Zeeman (t _w , T). The original introduction proposal, Ref. [1], envisaged the reduction of the barrier heights Δ(t _w , T) by E _Z [Eq. 1] to be caused by the magnetization induced by the magnetic field within the volume subtended by the spin glass coherence length ξ(t _w , T), viz Eq. 1. A subsequent treatment [33] associated E _Z with the magnetization associated with the fluctuations of the entire system of N spins, namely, ∝ N . The magnetic field dependence is very different, the former E _z ∝ H ² while the latter E _Z ∝ H.

A comparison of the two was exhibited in Fig. 10 of Ref. [17], reproduced here as Figure 4. The magnitude of the magnetic fields contained in Figure 4 are quite large. The authors of Ref. [33] state that the proportionality to H fails at low magnetic fields. The reader can judge whether a linear fit to H is obeyed by the left-hand of Figure 4. The right-hand of Figure 4 is the fit to the scaling law, valid over the full range of H, large and small. Again, the reader can judge which fit is preferable.

The second is the value of the condensation temperature, T _g . In principle, determination of T _g would require an infinite t _w because ξ(T) → ∞ when T → 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 an upper bound for T _g .

The non-linear susceptibility χ ₃ diverges as Eq. 23 χ 3 t w → ∞ ; T = χ 0 T g t w → ∞ | T g t w → ∞ − T | γ , (23) where χ ₀ is a constant independent of temperature, and γ = 6.13 (11) from Ref. [32]. For finite t _W , χ ₃ (t _w , T) 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, in Eq. 24 χ 3 t w , T = χ 0 T g t w | T g t w − T | γ , (24) and then use 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 . This is just a suggestion for a feasible process for taking laboratory data for finite t _w and extrapolating to find T _g (t _w → ∞).

Temperature chaos is one of the outstanding mysteries posed by spin glasses. It consists of the complete reorganization of the equilibrium configurations by the slighted change in temperature.” [34] These are the opening lines of a major paper titled “Temperature chaos is a non-local effect” and set the stage for this section. Even the existence of temperature chaos in spin glasses has been questioned [35–38]. Recent experiments [39] and simulations [40] have shed light on its existence and nature, but there are many questions that remain.

From this article’s perspective, the opening gambit was the renormalization group perspective of Bray and Moore [41]. They introduced a length scale associated with temperature chaos that, for all practical purposes, can be simplified to, ℓ c T 1 , T 2 = a 0 T 2 T 1 − T 2 1 / ζ (25) where ζ = d _s − θ, d _s the fractal dimension of the correlated region, and θ is the replicon exponent. The system is in an equilibrium state at a temperature T ₁, after which the temperature is dropped to T ₂. Temperature chaos obtains with a length scale ℓ _c . The reason that ℓ _c is important is that, for a coherence length ξ(t _w , T) not infinite, temperature chaos requires a finite temperature drop. The condiction is, shown in Eq. 26 Temperature chaos : ℓ c T 1 , T 2 ≤ ξ t w , T 1 , Reversible : ℓ c T 1 , T 2 ≥ ξ t w , T 1 , (26) where by “reversible” we mean that the system “remembers where it came from” when the temperature is dropped from T ₁ to T ₂, i.e., no temperature chaos.

Though the relationship Eq. 25 is for a spin glass in equilibrium, realistically, this is never the case. Fortunately, recently an analysis was provided [40] which is titled “Temperature chaos is present in off-equilibrium spin-glass dynamics,” so that we can use the relationship Eq. 25 experimentally. Note that both t _w and ΔT = T ₁ − T ₂ are controllable parameters. Hence, we can probe the onset of temperature chaos by examining spin glass dynamics under difference conditions, and in particular, can control its onset.

Experiments have probed temperature chaos. The first definitive paper [42] defined the length scale for chaos as L _ΔT given by Eq. 27, L Δ T ∼ L 0 | Δ T / J | − 1 / ζ (27) equivalent to our Eq. 25 with L _ΔT ≡ ℓ _c (T ₁, T ₂), and J an exchange energy in units of temperature. They extract and effective chaos length scale, L _eff from the following plot (their Figure 4, our Figure 5): Their plot generates 1/ζ = 2.6 or ζ = 0.38. This value is a factor of nearly 3 below the rather universally accepted value of ζ = 1.1 (see Appendix B in Ref. [39], for a full listing of theoretical values for ζ).

It is difficult to understand why their value for ζ was so far off from what is now regarded as the fairly accepted value near unity. An origin may be lie in their measurement protocol, namely, a zero-field magnetization measurement where the magnetic field is applied after cycling to T ₁. Magnetic field chaos [43, 44] could then be compounded with temperature chaos, and distort the extraction of a value for 1/ζ.

In order to circumvent this possibility, Zhai et al. [39] worked with a protocol where the magnetic field remained constant across temperature cycling. The idea was to use the field-cooled magnetization to investigate temperature chaos in spin glasses. This protocol involved turning on a magnetic field H above the condensation temperature, T _g , keeping it constant throughout the temperature cycling protocol.

The specific steps were as follows. The decay of the field cooled (FC) magnetization, M _FC (t, T ₁, H) is measured at the first temperature stop T ₁ after cooling in the current magnetic field and waiting a time t _w1. The decay curve is denoted as the “reference curve.” Then, temperature cycling is engaged, where one first cools to temperature T ₁, waits a time t _w1, cools to T ₂, waits a time t _w2, then rapidly warms back to T ₁, and measures the decay of the magnetization M _FC (t, T ₁, H).

In order to observe TC, T ₁ was fixed and the temperature T ₂ was gradually lowered in separate experimental runs to T ₂ = T ₁ − ΔT. Following the temperature drop, if, x = ℓ c T 1 , T 2 ξ T 1 , t w 1 ≥ 1 , (28) the coherence length will continue to grow, and one remains in the reversible state. However, in Eq. 29 x = ℓ c T 1 , T 2 ξ T 1 , t w 1 ≤ 1 (29) temperature chaos sets in and one finds a diminished coherence length after heating back to temperature T ₁ (see the discussion below of memory).

As a consequence, under reversible dynamics, Eq. 28, the decay curve of M _FC (t, T ₁, H) after temperature cycling can be superposed on the reference decay curve, allowing for a positive shift in time for M _FC (t, T ₁, H) during the time that T < T ₁. However, after the onset of temperature chaos, The decay curves cannot be superposed for any positive shift of M _FC (t, T ₁, H) in time.

This is seen explicitly in Figure 6 which was reproduced from Figure 1 of Ref [39]. By changing T ₁, t _w1, Eq. 25 can be probed to yield a value for 1/ζ. A value for ζ ≈ 1.1 was extracted in [39], very close to a majority of the theoretical values.

This experimental evidence for the existence of temperature chaos in spin glasses will prove important in our subsequent treatments of rejuvenation and memory. We shall argue that temperature chaos is responsible for the former, and plays an important role in a quantitative treatment of the latter. In any case, the experiments of Ref. [34] have shown that temperature chaos is present in spin glasses, and will be shown to have a profound impact in other dynamical spin glass phenomena.

The singular publication that engendered the attention of both theorists and experimentalists for over three decades was that of Jonason et al. [45] titled “Memory and Chaos Effects in Spin Glasses.” They displayed the remarkable figure (Figure 7 reproduced from Figure 1 in their paper). The system is “aged” at 12 K, becoming “older.” Upon lowering the temperature, it returns to the reference curve, thus becoming “younger”. This is termed “rejuvenation.” It was attributed to temperature chaos, namely, the spin glass “forgot” its previous history of aging when the temperature was lowered beyond the threshold for temperature chaos.

This assignment has yet to be proven unequivocally. A very recent paper [18] by Paga et al. displayed the results of temperature cycling to explore this claim, and indeed to relate rejuvenation to the spin glass coherence length. Figure 8 reproduces their Figure 2. While Figure 8 appears definitive, there needs to be an exploration of t w eff for temperature above and below the temperature for the onset of temperature chaos to arrive at an unequivocal relationship between rejuvenation and temperature chaos. For the moment, Figure 8 seems entirely consistent with that interpretation.

Though rejuvenation in spin glasses is remarkable, the even more remarkable is memory. Once the system has rejuvenated back to the reference curve, upon reheating it traces out the same behavior as it exhibited upon cooling, even with temperature chaos between T ₁ and T ₂. This is explicitly demonstrated in Figure 7. On the surface it seems quite inconsistent. How can the system exhibit memory when it has experienced temperature chaos? There have been a multitude of papers and models that addressed this conundrum. Most involve heuristic models with adjustable parameters that can fit the data represented in Figure 7. A recent treatment [18] gives an interpretation that is free of real space models and the concomitant adjustable parameters, and is based upon the behavior of the spin glass coherence length. The beauty of this formulation is that every term in its interpretation can be tested experimentally, something that previous models lack.

The concept is as follows. Upon cooling the spin glass from above T _g to the first measuring temperature T ₁, the system is aged for a time t _w1. As a consequence, the spin glass coherence length grow from nucleation to ξ(t _w1, T). When the temperature is then lowered to T ₂, the correlations created at T ₁ are essentially frozen. This concept has been introduced by Bouchaud et al. [45]. What is new is that, when the system is aged at T ₂ for a time t _w2, the system has created new coherent regions that have nothing to do with those created at T ₁. Hence, when heating back to T ₂, the two correlated regions interfere, thereby reducing the spin glass coherence length from the native value created at T ₁, t _w1.

This is exhibited explicitly in Figure 8 in the lower part of the figure. The C _n represent three separate temperature and waiting time cycles, and illustrate unequivocally the relationship between memory and competing coherence lengths. Each of the cycles has three steps: 1) the system is “prepared” at T ₁ = 30 K by waiting for the same time t _w1 = 1 h 2) The temperature is then dropped to T ₂ and the system is aged for t _w2. 3) The system is than heated back to T ₁ = 30 K and t w eff measured as rapidly as possible. C ₁ sets T ₂ = 26 K and t _w2 = 1/6 h. C ₂ sets T ₂ = 26 K and t _w2 = 3 h. Finally, C ₃ sets T ₂ = 16 K and t _w2 = 3 h. Memory is quantified by comparing the magnitude of the coherence length measured at step (3) with the native coherence length [the coherence length of the initially prepared state at step (1)]. If the two lengths are the same, memory is perfect. If after step (3), the measured coherence length is smaller than the initially prepared state at step (1), memory is less, a direct result of the interference of the two states. The slopes in Figure 8 are steeper, the larger the coherence length being measured because the volume of the correlated region is larger [Eq. 1].

Consider C ₁. The system has “morphed” from the prepared state into a chaotic regime, but only aged for a short time (1/6 h). The coherence length in the chaotic state has grown during this time, so that its interference with the initially prepared coherence length is significant. Hence, the memory is less, exhibited by the shallower slope as compared to the native slope exhibited in Figure 8. Now, C ₂ increases t _w2 to 3 h, so that the coherence length in the chaotic state can grow beyond it is value in C ₁. This should lead to greater interference, a smaller memory, and a more shallow slope than found for C ₁. This is explicit in Figure 8.

Finally, the C ₃ protocol has the same t _w2 as C ₂, but the temperature drop to T ₂ is much greater, T ₂ = 16 K. At such a low temperature, the growth of the chaotic coherence length is very slow (almost none), so there should be almost no interference, and the memory should be nearly perfect. This again is explicitly exhibited in Figure 8 where the slope of C ₃ is close to the slope for the native slope.

These three temperature cycles, and their properties exhibited in Figure 8, are strong evidence for the interpretation of memory through interfering coherence volumes. This is at odds with Ref. [46] where it is argued that the coherence length returned to the native value upon reheating. By adjusting t _w2 one can change the value of ξ(T ₁, t _w1, T ₂, t _w2) at will, from no loss to complete loss of memory. This have been born out in our Figure 8, and also in independent experiments by Freedberg et al. [47].