Spin-glass dynamics in the presence of a magnetic field: exploration of microscopic properties
I. Paga, Q. Zhai, M. Baity-Jesi, E. Calore, A. Cruz, L.A. Fernandez, J. M. Gil-Narvion, I. Gonzalez-Adalid Pemartin, A Gordillo-Guerrero, D. Iñiguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Muñoz-Sudupe, D. Navarro, R. L. Orbach, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S.F. Schifano, D. L. Schlagel, B. Seoane, A. Tarancon, R. Tripiccione, D. Yllanes
SSpin-glass dynamics in the presence of a magnetic field:exploration of microscopic properties
I. Paga , , ∗ , Q. Zhai , ∗ , M. Baity-Jesi , E. Calore , A. Cruz , ,L. A. Fernandez , , J. M. Gil-Narvion , I. Gonzalez-Adalid Pemartin ,A Gordillo-Guerrero , , , D. Iñiguez , , , A. Maiorano , , ,E. Marinari , , V. Martin-Mayor , , J. Moreno-Gordo , ,A. Muñoz-Sudupe , , D. Navarro , R. L. Orbach , G. Parisi , ,S. Perez-Gaviro , , , F. Ricci-Tersenghi , , J. J. Ruiz-Lorenzo , , ,S. F. Schifano , D. L. Schlagel , D. Seoane , , A. Tarancon , ,R. Tripiccione , D. Yllanes , Dipartimento di Fisica, Sapienza Università di Roma, INFN, Sezione di Roma 1, I-00185Rome, Italy Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA Eawag, Überlandstrasse 133, CH-8600 Dübendorf, Switzerland Dipartimento di Fisica e Scienze della Terra, Università di Ferrara e INFN, Sezione diFerrara, I-44122 Ferrara, Italy Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50018 Zaragoza, Spain Departamento de Ingeniería Eléctrica, Electrónica y Automática, U. de Extremadura,10003, Cáceres, Spain Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura,06006 Badajoz, Spain Fundación ARAID, Diputación General de Aragón, 50018 Zaragoza, Spain Dipartimento di Biotecnologie, Chimica e Farmacia, Università degli studi di Siena, 53100,Siena, Italy INFN, Sezione di Roma 1, I-00185 Rome, Italy Dipartimento di Fisica, Sapienza Università di Roma, and CNR-Nanotec, I-00185 Rome,Italy Departamento de Ingeniería, Electrónica y Comunicaciones and I3A, U. de Zaragoza, 50018Zaragoza, Spain Centro Universitario de la Defensa, 50090 Zaragoza, Spain Departamento de Física, Universidad de Extremadura, 06006 Badajoz, Spain Dipartimento di Scienze Chimiche e Farmaceutiche, Università di Ferrara e INFN Sezionedi Ferrara, I-44122 Ferrara, Italy Division of Materials Science and Engineering, Ames Laboratory, Ames, Iowa 50011, USA Chan Zuckerberg Biohub, San Francisco, CA 94158, USA ∗ These authors contributed equally to this work. a r X i v : . [ c ond - m a t . d i s - nn ] J a n pin-glass dynamics in the presence of a magnetic field Abstract.
The synergy between experiment, theory, and simulations enables a microscopicanalysis of spin-glass dynamics in a magnetic field in the vicinity of and below the spin-glass transition temperature T g . The spin-glass correlation length, ξ ( t, t w ; T ) , is analysedboth in experiments and in simulations in terms of the waiting time t w after the spin glasshas been cooled down to a stabilised measuring temperature T < T g and of the time t after the magnetic field is changed. This correlation length is extracted experimentally fora CuMn 6 at. % single crystal, as well as for simulations on the Janus II special-purposesupercomputer, the latter with time and length scales comparable to experiment. The non-linear magnetic susceptibility is reported from experiment and simulations, using ξ ( t, t w ; T ) asthe scaling variable. Previous experiments are reanalysed, and disagreements about the natureof the Zeeman energy are resolved. The growth of the spin-glass magnetisation in zero-fieldmagnetisation experiments, M ZFC ( t, t w ; T ) , is measured from simulations, verifying the scalingrelationships in the dynamical or non-equilibrium regime. Our preliminary search for the deAlmeida-Thouless line in D = 3 is discussed.PACS numbers: Submitted to:
J. Stat. Mech.: Theor. and Exp.
ONTENTS Contents1 Introduction 42 Experimental details 73 Some details of the simulations 84 Measurements and computations of the relaxation rate 9 S ( t, t w ; H ) . . . . . . . . . . . . . . . . . . . 94.2 A different approach for the computation of t eff H in simulations . . . . . . . . . . 10 H and H = 0 + . . . . . . . 245.5 Non-linear scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6 Overshooting phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.6.1 Dynamical scaling close to T g . . . . . . . . . . . . . . . . . . . . . . . . 275.6.2 Overshooting in a ferromagnetic system . . . . . . . . . . . . . . . . . . . 29 D = 3
317 Conclusions 35Acknowledgements 36A Technical details about our simulations 37
A.1 Smoothing and interpolating the data . . . . . . . . . . . . . . . . . . . . . . . . 37A.2 Over-fitting problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.3 Time discretisation and the calculation of the relaxation function S ( t, t w ; H ) . . 41A.4 The t eff H calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.5 The construction of ξ ( t, t w ; H ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 B The Josephson length 43C Sample dependence of the non-linear scaling results 43References 44
ONTENTS
1. Introduction
This paper examines in detail the dynamics of spin glasses in the vicinity of and below theircondensation temperature T g in the presence of a magnetic field, an exploration relevant to manyother condensed-matter glassy systems: fragile molecular glasses, polymers, colloids, super-cooled liquids, and now even social science through “Resource dynamics on species packingin diverse ecosystems” [1], to name a few. The advent of realistic time and length scales onthe Janus II dedicated supercomputer generates a synergy between theory, experiment, andsimulations that encompasses a thorough examination of spin-glass dynamics reaching fromthe low-temperature regime to the vicinity of T g [2–4].The special nature of this approach is the use of the spin-glass correlation length, ξ ( t, t w ; T ) as the primary factor in the analysis [4], where t is the time after a magnetic field change,when measurements of the magnetisation take place; t w is the waiting time after the systemis quenched from above T g to temperatures within the condensation regime, and before thechange in magnetic field; and T is the temperature. This correlation length can be extractedfrom experiment and simulations, under dynamical or non-equilibrium configurations [5–8].As such, it will be used for a detailed study of a new powerful scaling law for the non-linearmagnetisation near to and below T g . We shall show that this law not only accounts for theexperiments presented in this paper, but also clears up some historical controversies about thenature of the Zeeman, or magnetic-field, energy in the spin-glass state [5, 9].To be specific about the temperature and magnetic field protocol, we present an analysis ofthe zero-field-cooled magnetisation, M ZFC ( t, t w ; H ) , as a function of t, t w and magnetic field H at prescribed temperatures T ≤ T g . The protocol is one where the “sample” is quenched froma temperature T > T g to a measuring temperature T ≤ T g in zero magnetic field. The word quench means different things experimentally and in simulations. In the former, there is a finite cooling rate as the system is brought from above T g to the measurement temperature T . It istypically of the order of one to tens of seconds per degree of cooling. In the case of simulations,it is instantaneous. Though on the surface this would seem a difficult issue, in fact temperaturechaos [10, 11] (which we now know to be present in non-equilibrium dynamics as well [12])makes the two approaches similar if not identical. Experimentally, though the cooling rate isfinite, lowering the temperature sufficiently ( δT larger than milli-Kelvins) creates new spin-glassstates without knowledge of previous history (a process termed rejuvenation [13]). This is thereason that the magnetic susceptibility is reproducible from one experiment to another, withoutrecourse to the cooling rate. Thus, the final state reached upon an experimental temperaturequench is as fresh as the state arrived at in simulations upon instantaneous quenching.After the measurement temperature T is reached, the system is held for a time t w , the waiting time , after which a magnetic field H is turned on. The resulting magnetisation, M ZFC ( t, t w ; H ) , is then measured over a time interval t . The response consists of two terms: aninstantaneous increase in magnetisation (the so-called reversible magnetisation ), and a slowlyincreasing part termed the irreversible magnetisation . The latter is found to depend upon allof the factors t, t w , H . The rise of the irreversible term is typically very slow, taking literallytimes of the order of the age of the universe to reach equilibrium. For this reason, a spin glass,once perturbed from a quasi-equilibrium state, never reaches equilibrium, and an experimentis always in a dynamical or non-equilibrium regime. ONTENTS field-cooled magnetisation, M FC , for which the measuring protocol is theopposite of the zero-field magnetisation. Namely, at T > T g , a magnetic field H is turned on,and then the temperature is quenched to T ≤ T g . Typically, M FC is relatively constant, butnot without its own dynamics. If the magnetic field is suddenly removed, the magnetisationimmediately decays by its reversible part (the same as in the zero-field case), followed by aslow decay termed the irreversible part or M TRM ( t, t w ; H ) , the thermo-remanent magnetisation dependent upon the waiting time t w . In general, it is found that M FC = M ZFC ( t, t w ; H ) + M TRM ( t, t w ; H ) . (1)This is known as the extended principle of superposition [14]. There is an immense literaturecovering both M ZFC ( t, t w ; H ) and M TRM ( t, t w ; H ) measurements, and the physical insightsgained from them [14–16].Our approach is rather different, in that we choose to represent the dynamics in terms of thespin-glass correlation length ξ ( t, t w ; H ) . This quantity was first extracted from experiment inthe work of Joh et al. [5], who developed a protocol based on the relaxation function S ( t, t w ; H ) defined by S ( t, t w ; H ) = d (cid:104) − M TRM ( t, t w ; H ) H (cid:105)(cid:46) d ln t . (2)It was known that S ( t, t w ; H ) peaked at what is termed an effective waiting time, t eff H , whichis usually of the order of t w [17]. This time is characteristic of the decay of M TRM ( t, t w ; H ) ,or, through Eq. (1), of the increase of M ZFC ( t, t w ; H ) with time t . As noted by Hammann etal. [18, 19], for states distributed according to ultrametric symmetry, the dynamics is controlledby a largest free-energy barrier height, ∆ max , associated with the state that has the smallestoverlap with the initial state, q min . Thus, t eff H can be associated with ∆ max through the usualArrhenius law: ∆ max = k B T ( ln t eff H − ln τ ) , (3)where τ is a characteristic exchange time, τ ∼ (cid:126) /k B T g .In order to extract a correlation length, Joh et al. [5] used the notion that the free-energybarrier heights were reduced in the presence of a magnetic field by the Zeeman energy, E Z ,[5, 20–23] and that, for small magnetic fields H , E Z = ( V corr /a ) χ FC H , (4)where χ FC is the field-cooled magnetic susceptibility per spin, V corr is the correlated volume,and a the average spatial separation of the magnetic ions, so that the number of correlatedspins is N corr = V corr /a . Joh et al. took N corr = V corr /a ≈ π ξ , (5)where ξ is in units of a . We now know that a more appropriate relationship would be N corr = V corr /a = 4 π ξ − θ (˜ x ) / ≡ b ξ − θ (˜ x ) / , (6)where b is a geometrical factor, and θ (˜ x ) is the replicon exponent [23]. Using Eqs. (2) through(4), the experiments in [5] produced the plot in Fig. 1. The data in the limit of small magneticfields H were clearly linear in H , allowing Eqs. (4) through (6) to set a value for ξ . ONTENTS . . . . .
00 20 40 60 80 100 ( × ) l n (cid:18) t e ff H / s (cid:19) H (Oe ) Figure 1: A plot of ln t eff H , extracted from the Zeeman energy E Z as in Eq. (4), vs H for Cu:Mn6 at. % ( T /T g = 0 . , t w = 480 s) at fixed t w and T . Data taken from Ref. [5].The deviation from linearity in H was puzzling, leading to the authors’ stating: “We donot have a satisfactory explanation for this change in slope. A different description predicts alinear dependence of E Z upon H , which can be made to fit the data (. . . ) but with a significantdeviation at small field changes”. It is the purpose of this paper to analyse the entirety of thedata considering non-linear terms in the spin-glass magnetisation according to a new scalinglaw. In addition to analysing magnetisation data from new experiments, we shall also showthat the data of Fig. 1, and subsequent experiments of Bert et al. [9] on the Ising spin glassFe . Mn . TiO , fit the new scaling law well, obviating the need to question Eq. (4), and puttingto rest the controversy over the nature of the Zeeman energy in spin glasses.This paper brings together a complete set of magnetisation measurements of a single crystalof the prototypical spin glass, CuMn 6 at. %. The beauty of the experimental results is thatthe correlated volume is most certainly spherical (as opposed to thin films where the correlatedvolume is of pancake geometry [6]), and unlimited by finite-size crystallites separated by grainboundaries [24, 25]. Accompanying these measurements are the remarkable simulations of theJanus II dedicated supercomputer, which, for the first time, yield spin-glass correlations thatapproach experimental time and length scales. Indeed, the range of correlation lengths thatwe are able to achieve both experimentally and numerically is itself a breakthrough. Thecurrent manuscript reports results up to ξ ≈ . a ( a is the typical Mn-Mn distance), whichrepresents a step forward by a factor of three from previous work [23]. On the experimentalside, we reach a correlation length four times larger than in Ref. [5].The synergy between these two approaches, combined with theory, opens up a new vistafor spin-glass dynamics. A direct outgrowth of this collaboration is the introduction of the newmagnetisation scaling law, which encompasses the full range of magnetic fields for temperaturesin the vicinity of the condensation temperature T g [4]. This scaling law successfully describesboth experimental and simulation results and, as noted above, will resolve a nearly three-decade-old controversy concerning the nature of the magnetic state.Although our simulations were not designed to that end, we take the occasion as well to ONTENTS T m and waiting times t w for the fourexperimental regimes, the respective correlation lengths at times t w (in units of the averageMn-Mn spacing a ), and the effective replicon exponent θ (˜ x ) obtained from Eqs. (41) and (42)below (see also Appendix B). T m (K) t w (s) ξ ( t w ) /a θ (˜ x ) Exp. 1 28.50 10 000 320.36 0.337Exp. 2 28.75 10 000 341.76 0.344Exp. 3 28.75 20 000 359.18 0.342Exp. 4 29.00 10 000 391.27 0.349 attempt a preliminary search for the de Almeida-Thouless (dAT) line in the phase diagram ofthe 3D spin glass.The paper is organised as follows. Section 2 details the experimental measurements ofthe non-linear magnetisation in the CuMn spin glass. Section 3 describes the nature of ournumerical simulations. Section 4 introduces the response function, Eq. (2), and its extractionfrom experiment and simulations. Section 5 develops the new scaling law, and applies it to bothexperimental and simulation results. In addition, Section 5.6 shows the nature of the growthof the numerical correlation length ξ in the presence of a magnetic field at temperatures closeto the critical temperature T g . We observed interesting overshoot phenomena that we proveto be general, as they are observed even in ferromagnetic systems. Section 6 investigates thedAT phase boundary in D = 3 . Important technical details are provided in the appendices.Finally, Section 7 summarises our results, and points to future opportunities stemming fromthe synergy expressed in this paper between theory, experiment, and simulations.
2. Experimental details
The experimental measurements were made with a CuMn ∼ T g = 31 . K, was determined from the temperature at which M ZFC ( T ) first began to departfrom M FC ( T ) .The magnetisation measurements were made using a commercial DC SQUID. The samplewas quenched from 40 K at 10 K/min to the measuring temperature T m in zero magnetic field.After stabilisation of the temperature, the system was aged for a waiting time t w before amagnetic field was applied, and the magnetisation M ZFC ( t, t w ; T m ) recorded as a function oftime t . The temperatures T m were chosen as 28.5 K, 28.75 K, and 29 K, so that T m ≥ . T g .The magnetic fields ranged from 16 Oe to 59 Oe. Table 1 displays the relevant experimentalparameters, including the effective replicon exponent θ (˜ x ) . ONTENTS T , t w , ξ ( t w ) , the longestsimulation time t max , the replicon exponent θ (˜ x ) (see Appendix B) and the value of C peak ( t w ) employed in Eq. (16). Here and in the rest of the paper, error bars are one standard deviation. T t w ξ ( t w ; H = 0) t max θ (˜ x ) C peak Run 1 0.9 . . . . . . . . . . . .
3. Some details of the simulations
We carried out massive simulations on the Janus II supercomputer [26] to study the Ising-Edwards-Anderson (IEA) model in a cubic lattice with periodic boundary conditions and size L = 160 a , where a is the average distance between magnetic moments, see Table 2 for thesimulation details. The N = L D Ising spins, s x = ± , interact with their lattice nearestneighbours through the Hamiltonian: H = − (cid:88) (cid:104) x , y (cid:105) J xy s x s x − H (cid:88) x s x , (7)where the quenched disordered couplings are J xy = ± with probability. We name aparticular choice of the couplings a sample . In the absence of an external magnetic field, H = 0 ,this model undergoes a spin-glass transition at the critical temperature T g = 1 . [27].As we explained in the Introduction, in order to simulate the experimental zero-field-cooling (ZFC) protocol the following procedure was performed: We place the initial randomspin configuration instantaneously at the working temperature T and leave to relax for a time t w at H = 0 . At time t w , we turn on the external magnetic field and we start recording themagnetic density, M ZFC ( t, t w ; H ) = 1160 (cid:88) x s x ( t + t w ; H ) , (8)as well as the correlation function, C ( t, t w ; H ) = 1160 (cid:88) x s x ( t w ; 0) s x ( t + t w ; H ) . (9)The non-equilibrium dynamics was simulated with a Metropolis algorithm; the numerical timeunit being the lattice sweep, roughly corresponding to one picosecond of physical time [28].For each temperature and waiting time, see Table 2, several magnetic fields were simulated.For computational reasons, one single independent sample was simulated for each case. Wechecked, however, the robustness and the sample independence of our results in a single case,studied in detail in Appendix C.According to Ref. [23], the value of the dimensionless magnetic field H used in the numericalsimulation can be matched to the physical field. This relation was estimated from experimental Fe . Mn . TiO data [29]. We found that H = 1 in the IEA model corresponded to ≈ × Oe ONTENTS . (cid:46) H (cid:46) . in the IEA model. However, the signal-to-noise ratio, which scaleslinearly in H for small fields [30], limited our simulation to H ≥ . , equivalent to a physical H = 250 G.In order to match the experimental and numerical scales, we exploited dimensionalanalysis [31] to relate H and the reduced temperature ˆ t = ( T g − T ) /T g through the scaling ˆ t num ≈ ˆ t exp (cid:18) H num H exp (cid:19) ν (5 − η ) , (10)where ν = 2 . and η = − . are H = 0 critical exponents [27], while subscripts exp andnum stand for experiment and simulation, respectively. According to Eq. (10), and mindingsignal-to-noise limitations, we can match the experimental and numerical scales by increasing ˆ t num , resulting in . (cid:46) T num (cid:46) . . Given our pre-existing database of long simulations at H = 0 [2], it has been convenient to work at temperatures T num = 0 . and T num = 1 . . Table 2displays the relevant numerical parameters, including the effective replicon exponent θ (˜ x ) andthe C peak ( t w ) values that will be introduced and explained in Section 4.2.Let us finally remark that Section 5.6 uses a different set of simulations from the rest ofthe paper.
4. Measurements and computations of the relaxation rate
In this section we describe the relaxation function S ( t, t w ; H ) (Section 4.1) and explain how t eff H is extracted from simulations (Section 4.2). S ( t, t w ; H ) The main quantity used in the experiments of [5] is the relaxation function S ( t, t w ; H ) : S ( t, t w ; H ) = d M ZFC ( t, t w ; H )d ln t , (11)which exhibits a local maximum at time t eff H . 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 ofdata for T m = 28 . K and t w = 10 s is displayed in Fig. 2. Numerically, the calculation of S ( t, t w ; H ) is sensitive to the relative errors of the magnetisation density, which increase as δM ZFC ( t, t w ; H ) M ZFC ∝ H . (12)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 (FDR) [32–36] TH M
ZFC ( t, t w ; H ) = F ( C ; H ) , (13)where F ( C ; H ) behaves at large C ( t, t w ; H ) as F ( C ; H ) = 1 − C ( t, t w ; H ) . We report the detailsin Appendix A.1. On the other hand, we define S ( t, t w ; H ) as a finite-time difference S ( t, t w , t (cid:48) ; H ) = M ZFC ( t (cid:48) , t w ; H ) − M ZFC ( t, t w ; H )ln ( t (cid:48) /t ) . (14) ONTENTS . . . . . ( × − ) d M Z F C ( t ) / d l n t × / H t w H = 24 . H = 32 Oe H = 40 . H = 47 Oe H = 56 . Figure 2: Example of S ( t, t w ; H ) measurements for different magnetic fields. The sample is asingle crystal CuMn 6 at. %, and the measurements were taken at a waiting time of t w = 10 000 sand at T = 28 . K. The time at which S ( t, t w ; H ) peaks defines t eff H , the effective responsetime. The shift to shorter times as H increases is the measure of the reduction of ∆ max withincreasing Zeeman interaction, and is used to extract the linear and non-linear terms in themagnetic susceptibility.In simulations, time is discrete and we store configurations at times t n = integer-part-of n/ ,with n an integer. Let us write explicitly the integer dependence of times t and t (cid:48) as: t ≡ t n , t (cid:48) ≡ t n + k , (15)where k is an integer time parameter. The reader will note that there is a trade-off in the choiceof k . On the one hand, the smaller k is the better the finite difference in Eq. (14) represents thederivative. On the other hand, when k grows the statistical error in the evaluation of Eq. (14)decreases significantly. In this section we report only the case for k = 8 (more details abouttime discretisation are provided in Appendix A.3). The numerical S ( t, t w , t (cid:48) ; H ) are exhibitedin Fig. 3, where a local maximum in the long-time region can be seen. t eff H in simulations As explained in the Introduction, we are interested in the evaluation of the time when therelaxation function S ( t, t w , t (cid:48) ; H ) peaks, namely t eff H . Two problems arise:(i) The reader will note two separate peaks in the S ( t, t w , t (cid:48) ; H ) curves of Fig. 3: namely thepeak at microscopic times t ∼ , 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 ≈ ,(non-linear susceptibilities grow very fast with ξ , see next section).(ii) We are most interested in the limit H → , which is extremely noisy, as we have explainedabove. ONTENTS . . . Run 1: T = 0 . , t w = 2 . . . Run 2: T = 0 . , t w = 2 . . . .
03 2 Run 3: T = 0 . , t w = 2 . Run 4: T = 1 . , t w = 2 . Run 5: T = 1 . , t w = 2 . Run 6: T = 1 . , t w = 2 . S ( t , t w , t ; H ) H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . t Figure 3: Time evolution of the relaxation rate S ( t, t w , t (cid:48) ; H ) of Eq. (14) for the six runs ofTable 2. All plots have the time parameter k = 8 in Eq. (15).An interesting possibility emerges when plotting the relaxation function S ( t, t w , t (cid:48) ; H ) interms of the correlation function C ( t, t w ; H ) , rather than as a function of time (see Fig. 4). C ( t, t w ; H ) is a decreasing function of time, the long-time region corresponding to small ONTENTS . . . Run 1: T = 0 . , t w = 2 . . . Run 2: T = 0 . , t w = 2 . . . .
03 0 . . . . Run 3: T = 0 . , t w = 2 . Run 4: T = 1 . , t w = 2 . Run 5: T = 1 . , t w = 2 . . . . . Run 6: T = 1 . , t w = 2 . S ( t , t w , t ; H ) H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . C ( t new , t w ; H ) Figure 4: S ( t, t w , t (cid:48) ; H ) as a function of C ( t, t w ; H ) . The peak region is enlarged in Fig. 5. Thephysically relevant peak is the one for small C , corresponding to long times. We consider thereparametrised t new with k = 8 in Eq. (17). C ( t, t w ; H ) , and vice versa. Hence, the physical peak in which we are interested is the onethat appears at small C ( t, t w ; H ) (see Fig. 4). Analogously to Fig. 3, we report only the case ONTENTS . . . . . Run 1: T = 0 . , t w = 2 . . . Run 2: T = 0 . , t w = 2 . . . . Run 2: T = 0 . , t w = 2 . . . . . .
024 0 .
48 0 .
50 0 .
52 0 .
54 0 . Run 3: T = 0 . , t w = 2 . Run 4: T = 1 . , t w = 2 . Run 4: T = 1 . , t w = 2 . Run 5: T = 1 . , t w = 2 . .
36 0 .
40 0 .
44 0 . Run 6: T = 1 . t w = 2 . H = 0 . , k = 4 H = 0 . , k = 8 H = 0 . , k = 2 H = 0 . , k = 8 H = 0 . , k = 4 H = 0 . , k = 2 S ( t , t w , t ; H ) S ( t , t w , t ; H ) H = 0 . , k = 8 H = 0 . , k = 4 H = 0 . , k = 2 C ( t new , t w ; H ) Figure 5: Enlargement of the peak region of S ( t, t w , t (cid:48) ; H ) as a function of C ( t, t w ; H ) for severalvalues of time parameter k in Eq. (15). The dashed black lines indicate the C peak ( t w ) positions.for k = 8 in Fig. 4.The simulation data strongly suggest that, when H → , the correlation function C ( t, t w ; H ) approaches a constant value C peak ( t w ) at the maximum of the relaxation function. ONTENTS t eff H in simulations as the time when C ( t, t w ; H ) reaches thevalue C peak ( t w ) : C ( t eff H , t w ; H ) = C peak ( t w ) . (16)As the reader can see, Eq. (16) is applicable also at H = 0 , solving the problem of the vanishingmagnetisation in this limit. The crucial point for our new t eff H definition, see Eq. (16), is, hence,the computation of C peak ( t w ) . Two problems arise:(i) The constant-value C peak ( t w ) is well defined only for small magnetic field H .(ii) The relaxation function as a function of the correlation, S ( C ; H ) , is an implicit functionof a reparametrised time, t new = 12 ln (cid:18) t n + k t n (cid:19) , (17)(see Appendix A.3 for details).Our strategy has been to study, for each run, the behaviour of S ( C ; H ) for the two smallestmagnetic fields H and for three different time parameters k . We report in Fig. 5 a closeup ofthe peak region for S ( C ; H ) , used for the evaluation of C peak ( t w ) . We report our estimates for C peak ( t w ) in Table 2.The relaxation function S ( t new , t w ; H ) depends on the correlation length ξ ( t w ) , and on theapplied magnetic field H , Eq. (2). We observe, however, that S ( t new , t w ; H ) has a temperaturedependence, which we extract by comparing Runs 4 and 2 in Fig. 5. These two cases arecharacterised by(i) A similar starting correlation length ξ ( t w ; H = 0) ≈ . , (see Table 2),(ii) The same set of applied magnetic fields, namely H = 0 . and H = 0 . .Yet, there appear to be two different scenarios in the data plotted in Fig. 5. In Run 2, thepeak of S ( C ; H ) is almost the same for all the rescaled time curves. In Run 4, however, thepeaks separate for different k . As will be explained in Section 5.5, this difference in behaviouris caused by increasing non-linear effects in the magnetisation, M ZFC ( t, t w ; H ) .In conclusion, Eq. (16) solves our two problems at once. We no longer need to resolvethe short-time and the long-time peaks in Fig. 3 and it bypasses the problem of the vanishingmagnetisation as H goes to zero.
5. Scaling law
We address here three different aspects of the scaling law. The assumptions that led us toour scaling law are given in Section 5.1. Next, in Section 5.2, we use the scaling law in theanalysis of our experimental data (previous data are also reanalysed in Section 5.3), withthe corresponding analysis for our simulations given in Section 5.4. Section 5.5 shows ourexperimental and numerical results together, according to the new scaling law. In Section 5.6we address the nature of the growth of the numerical correlation length ξ ( t w ) in the presenceof a magnetic field at temperatures close to the condensation temperature T g . ONTENTS Scaling laws for the spin-glass susceptibility in the vicinity of the condensation temperaturehave been proposed and analysed for decades. We first recall an important early approach, andthen 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 firstintroduced by Malozemoff, Barbara, and Imry [37–39], who introduced the relation for thesingular part of the magnetic susceptibility, χ s = H /δ f ( t r /H /φ ) , (18)where f ( x ) is a constant for x → ; f ( x ) = x − γ for x → ∞ ; φ = γδ/ ( δ − ≡ βδ ; and t r is thereduced 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 Eq. (18)were evaluated as β = 0 . , γ = 2 . , δ = 3 . , ν = 1 . , and z = 5 . . They differ substantiallyfrom Monte Carlo simulations for short-range Ising systems: β = 0 . , γ = 6 . , ν = 2 . from [27]. The discrepancy in the value of γ is very large, and most probablyarises from the lack of an exact value for T g in the experiments. This illustrates the value ofand need for a different approach for scaling the non-linear magnetisation of spin glasses in thevicinity of T g .Our approach is to express the non-linear components of the magnetic susceptibility interms of ξ ( t, t w ) , the spin-glass correlation length in a magnetic field H . ‡ This approach givesthe non-linear magnetisation a direct connection to a measurable quantity and obviates theneed for an accurate value of T g .The argument goes as follows. Let M ( t, t w ; H ) be the magnetisation per spin, whereexplicit attention is paid to the waiting (aging) time t w in the preparation of the spin-glassstate. The generalised susceptibilities χ , χ , χ , . . . are defined through the Taylor expansion M ( H ) = χ H + χ H + χ H + O ( H ) . (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 ascaling theory for the magnetic response to an external field H [42, 43]. Our main hypothesisin this work is that this scaling theory holds not only at equilibrium, but even in the non-equilibrium regime for a spin glass close to T g and in the presence of a small external magneticfield H : M ( t, t w ; H ) = [ ξ ( t + t w )] y H − D F (cid:18) H [ ξ ( t + t w )] y H , ξ ( t + t w ) ξ ( t w ) (cid:19) . (20)According to full-aging spin-glass dynamics (see, e.g. , [44]), Eq. (20) tells us that ξ ( t + t w ) /ξ ( t w ) will be approximately constant close to the maximum of the relaxation rate (see Fig. 2), so weshall omit this dependence. Hence, combining Eq. (19) and (20), one can express the generalisedsusceptibilities χ , χ , χ , . . . in terms of the spin-glass correlation length ξ ( t, t w ; H ) : χ n − ∝ | ξ ( t w ) | n y H − D , (21) ‡ The correlation length ξ ( t, t w ) is of course also a function of the temperature T , but here we are only interestedin the non-linearity of the magnetisation. ONTENTS t, H for convenience, and y H = D − θ (˜ x )2 , (22)with θ (˜ x ) the replicon exponent [23].The first term of M ( H ) in Eq. (19) is χ , which contains the linear term as well as thefirst non-linear scaling term [4], so we write χ = ˆ ST + a ( T ) ξ θ (˜ x ) / . (23)where ˆ S is the function appearing in the fluctuation-dissipation relations [45] and a ( T ) is someunknown constant (hopefully smoothly varying with temperature).The free-energy variation per spin in presence of a magnetic field can be obtained byintegrating the magnetic density, Eq. (19), with respect to the magnetic field: ∆ F = − (cid:104) χ H + χ H + χ H + O ( H ) (cid:105) . (24)Substituting the scaling behaviour from Eq. (21) and Eq. (23), the free energy ∆ F can bewritten as (we drop the ˜ x dependence of θ for brevity) ∆ F = − (cid:104) ˆ S T H + a ( T ) ξ θ/ H + a ( T ) ξ D − θ H + a ( T ) ξ D − (3 θ/ H + O ( H ) (cid:105) , (25)where again the a n ( T ) are unknowns and (again, hopefully) smoothly varying functions oftemperature. We use the effective response time, t eff H , to reflect the total free-energy change atmagnetic field H and H = 0 + : ln (cid:104) t eff H t eff H → + (cid:105) = N corr ∆ F , (26)where N corr is the number of correlated spins, N corr = V corr /a , with V corr the correlated spinsvolume and a the lattice constant or average distance between magnetic moments. CombiningEq. (26) with Eqs. (6) and (24) leads to ln (cid:104) t eff H t eff H → + (cid:105) = − b (cid:104)(cid:16) ˆ S T + a ( T ) ξ θ/ (cid:17) ξ D − ( θ/ H + a ( T ) ξ D − (3 θ/ H + a ( T ) ξ D − θ H + O ( H ) (cid:105) , (27)where coefficient b is a geometrical factor, see Eq. (6), and we have absorbed the k B T term inthe a n ( T ) coefficients. The correction term a ( T ) /ξ θ (˜ x ) / is small compared to ˆ S/T , so it willbe dropped in subsequent expressions. Eq. (27) shows that the higher-order terms have thefunctional form χ n − H n (2 n )! = a n − ( T ) ξ − θ (˜ x ) / [ ξ y H H ] n , (28)where y H = D − θ (˜ x )2 . (29)This leads to the new scaling relation, ln (cid:104) t eff H t eff H → + (cid:105) = ˆ S T ξ D − θ (˜ x ) / H + ξ − θ (˜ x ) / G ( T, ξ D − θ (˜ x ) / H ) , (30)where the geometrical factor b has been absorbed into the scaling function G . Comparisonwith the previous, more classical, relation, Eq. (18), evinces the simplicity and power of ourapproach to scale the non-linear magnetisation in the vicinity of T g . ONTENTS l n (cid:18) t e ff H / s (cid:19) H (Oe ) Exp . . . . Figure 6: A plot of the peak times t eff H for the single-crystal CuMn 6 at. % vs H for the fourvalues of T m and t w listed in Table 1. The slope for small H is used to extract ξ ( t w ) in Table 1,and the lines come from the fits to the scaling law introduced in Section 5.1. We extract the effective waiting time t eff H in Eq. (3) from the time at which S ( t, t w ; H ) is amaximum, as before. Our results for all four conditions in Table 1 are exhibited as a functionof H in Fig. 6. The slope of the data in Fig. 6 at small values of the magnetic field H generatesthe spin-glass correlation length ξ ( t w ; T m ) from Eqs. (4) and (5), see Table 1, which also liststhe employed values for the replicon exponent θ (˜ x ) . These results will allow us to express thenon-linear susceptibility in terms of ξ ( t w ) .An example of the measured relaxation function S ( t, t w ; H ) is plotted for T m = 28 . Kand t w = 10 000 s in Fig. 2 for five different magnetic fields, while the effective response times, ln t eff H , are plotted in Fig. 6 for all four experiments listed in Table 1. Note the remarkablesimilarity in shape of the original experimental results for ln t eff H in Fig. 1 with our results inFig. 6. Also, note the fits of all four of our results for ln t eff H in Fig. 6 to the scaling relationshipfor the non-linear magnetisation, Eq. (27), which will be described in more detail below.Because the scaling relationship, Eq. (30), depends upon the magnitude of the waitingtime in ξ ( t, t w ; T m ) , two different values of t w were used at the same intermediate temperature T m = 28 . K, among the three temperatures (28.5 K, 28.75 K, and 29.0 K) listed in Section 2and in Table 1, to test Eq. (30) at a given temperature. This allows us to discriminate betweenthe influence of temperature and of waiting time on ξ ( t, t w ; T m ) . In this way, we are able todemonstrate explicitly that ξ ( t, t w ; T m ) is the control parameter.It is useful to display t eff H against H individually for each of the four values of T m and t w .They are exhibited below in Fig. 7. The data for ln t eff H is fitted to the function f ( x ) = c + c x + c x + c x + O ( x ) , (31)where x ≡ H and the c n coefficients, according to Eq. (27), correspond to: c = ln (cid:0) t eff H → + (cid:1) , (32) ONTENTS Exp. 1 Exp. 2 Exp. 3
Exp. 4 T = 28 . , t w = 10 000 s H term contribution t e ff w ( s ) T = 28 .
75 K , t w = 20 000 s H term contribution T = 28 .
75 K , t w = 10 000 s H term contribution H (Oe ) T = 29 K , t w = 10 000 s H term contribution Figure 7: Plots of the peak time t eff H for the single-crystal CuMn 6 at. % vs H for the fourexperimental regimes of Table 1. The straight lines are extrapolations of the linear term in themagnetisation, and the dashed lines are fits to Eq. (27). c = (cid:34) ˆ S T m (cid:35) ξ D − θ (˜ x ) / , (33) c = a ( T m ) ξ D − θ (˜ x ) / , (34) c = a ( T m ) ξ D − θ (˜ x ) . (35)Notice that we have absorbed the geometrical prefactor b of Eq. (27) in the non-linear coefficients a n ( T m ) and in the linear coefficient ˆ S , and we neglect the sub-leading coefficient a ( T m ) /ξ θ (˜ x ) / .The effect of increasing temperature with waiting time held constant can be seen in thedifference between the measured t eff H and the extrapolated value of the linear magnetisationterm (quadratic in H ) for the largest magnetic field ( H = 59 Oe) in Exps. 1, 2 and 4 in Fig. 7.Non-linear effects grow for larger t w , hence larger ξ ( t, t w ; T m ) at the same temperature, whichcan be seen by comparing Exps. 2 and 3. The linear and non-linear coefficients of Eq. (27) canbe extracted from fits of the data in Fig. 7 (dashed lines), whose resulting coefficients are listedin Table 3.To test the scaling relationship of Eq. (30) we first consider the fits of the data at T m = 28 . K for the two waiting times, t w = 2 × s and t w = 10 s. The linear term ONTENTS ln t eff H , as a functionof T m and t w (data from Fig. 7). The uninteresting fit parameter c is not included in the table. T m (K) t w (s) Coefficient Numerical value28.5 10 000 c − . × − c . × − c − . × − c − . × − c . × − c − . × − c − . × − c . × − c − . × −
29 10 000 c − . × − c . × − c − . × − is proportional to ξ D − θ (˜ x ) / . The ratio of the two correlation lengths from Table 3 is, hence, ξ ( t w = 20 000 s) ξ ( t w = 10 000 s) = (cid:104) c ( t w = 20 000 s) c ( t w = 10 000 s) (cid:105) / [ D − θ (˜ x ) / ≈ . . (36)Should scaling hold according to Eq. (30), then consistency requires that the ratios of thecorrelation lengths from the non-linear terms be the same as that for the linear term. Theyare: ξ ( t w = 20 000 s) ξ ( t w = 10 000 s) = (cid:104) c ( t w = 20 000 s) c ( t w = 10 000 s) (cid:105) / [2 D − θ (˜ x ) / ≈ . , (37)and ξ ( t w = 20 000 s) ξ ( t w = 10 000 s) = (cid:104) c ( t w = 20 000 s) c ( t w = 10 000 s) (cid:105) / [3 D − θ (˜ x )] ≈ . . (38)The equality (within experimental error) of Eqs. (36)–(38) is an impressive experimentalverification of scaling relationship (30). Another check is the growth of the correlation lengthitself. At temperature T m = 28 . K and for the two waiting times, it is possible to calculatethe ratio of the two values of the correlation length directly, using the expression for power-lawgrowth [5, 46], ξ ( t w ; T m ) = a ˆ C (cid:16) t w τ (cid:17) ˆ C ( T m /T g ) ≡ a ˆ C (cid:16) t w τ (cid:17) T m / ( z c T g ) , (39)where ˆ C and ˆ C are constants, by definition ˆ C ≡ /z c , and τ is a characteristic exchangetime , here taken as (cid:126) /k B T g .Using the growth-rate parameter z c = 12 . [2, 3] one finds ξ ( t w = 20 000 s) ξ ( t w = 10 000 s) ≈ (cid:16) × (cid:17) T m / (12 . T g ) = 2 . / (12 . × . ≈ . . (40)Comparing the ratio of ξ ( t w ; T m ) for the two different waiting times, Eq. (40), from thegrowth law, Eq. (39), with the ratios from fitting to the scaling relationship, Eqs. (36)–(38), ONTENTS T g .The lingering issue from Eqs. (6) and (30), according to Zhai et al. [3] “is that the repliconexponent [ θ (˜ x ) ] (. . . ) depends upon both the temperature and ξ through the crossover variable[ ˜ x ]”, with ˜ x = (cid:96) J ξ ( t w ; T m ) . (41)From the notation of Table 3 and Eq. (6) the number of correlated spins is N corr = k B T m c χ FC = ξ D − θ (˜ x ) / = ξ D − [ θ ( (cid:96) J /ξ ) / . (42)The left-hand side is a number, the right-hand side is an implicit function of ξ and θ . Usingthe definition of the Josephson length (cid:96) J and of the replicon θ (˜ x ) , § 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 inTable 1.The aging rate z c varies as a function of ξ . Using the data at T m = 28 . K, the approximateaging-rate factor is z c = 12 . at ξ ∼ lattice spacings a [3]. Although the correlationlength extracted at 28.75 K is larger than that at 28.5 K, the higher temperature sets thecrossover variable x = (cid:96) J ( T m ) /ξ = 0 . for t w = 20 000 s and x = 0 . for t w = 10 000 s,in the range of the crossover variable obtained by us previously at T m = 28 . K [47]. Bothmeasurements have exhibited slowing down of the spin-glass correlation growth rate near thecritical temperature at large correlation lengths.Using the average value of θ from Table 1, θ = 0 . , and setting z c = 12 . , the valuesexhibited in Eqs. (36)–(40) are altered to 1.053, 1.048, 1.052, and 1.051, respectively. Usingthe values from Table 1, the temperature-dependent coefficients a ( T m ) and a ( T m ) of Eq. (27)can be calculated for each of the four values of T m and t w . They are displayed in Fig. 8.From Fig. 8, one sees that the “hope” expressed after Eq. (25), i.e. , that the temperaturedependence of the coefficients a n ( T m ) appearing in Eq. (25) be weak, is realised in this set ofexperiments. For a ( T m ) , within the experimental error bars, there is little or no change withtemperature. The situation for a ( T m ) is not as nice, but there appears to be little change withtemperature at the two highest temperatures.It is interesting to test the scaling relationship [4] χ n − ( t w ; T m ) ∝ a n − [ ξ ( t w ; T m )] ( n − D − nθ (˜ x ) / . (43)Thus, χ ∝ ξ D − θ (˜ x ) a , χ ∝ ξ D − θ (˜ x )2 a . (44)The measured non-linear susceptibilities are exhibited below for the three temperatures . K, . K and . K.One can test the scaling relationships (43) and (44) by using the measured values for thespin-glass correlation length ξ , the replicon exponent θ (˜ x ) from Table 1, and the values of c and c from temperatures . K and K and t w = 10 000 s. For T m = 28 . K, we have ξ = 320 . a , θ (˜ x ) = 0 . (from Table 1), and c = 4 . × − (from Table 3, note that we have § The reader will find all the necessary details in Appendix B.
ONTENTS . . . . . a ( T ) × t w = 10 000 s t w = 20 000 s a ( T ) × T (K) t w = 10 000 s t w = 20 000 s Figure 8: Non-linear coefficients a and a , as defined in Eq. (27), calculated using the extractedvalues of ξ and θ for the different measuring temperatures T m and waiting times t w in ourexperiments. . . . . . . . . . . . . . . . . . χ ( T ) × ( a . u . ) t w = 10 000 s t w = 20 000 s χ ( T ) × ( a . u . ) T (K) t w = 10 000 s t w = 20 000 s Figure 9: Non-linear susceptibilities χ ( t w ; T m ) and χ ( t w ; T m ) from Eq. (42), plotted as afunction of temperature for the four experimental regimes of Table 1.ignored the error bars). Similarly, for T m = 29 . K and t w = 10 000 s we have ξ = 391 . a , θ (˜ x ) = 0 . , and c = 10 . × − . Using Eqs. (31) and (34) and the just quoted values of c ( t w ; T m ) one finds χ ( t w = 10 000 s; T m = 28 . χ ( t w = 10 000 s; T m = 29 . ≈ . . (45)This ratio is well within the error bars of the measured non-linear susceptibilities in Fig. 9. Asimilar result is also found for χ .With these scaling observations in hand, it is interesting to wonder about using them to ONTENTS T g . In principle, determination of T g would require aninfinite 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 onlaboratory time scales could establish a lower bound for T g .The non-linear susceptibility χ diverges as χ ( t w → ∞ ; T m ) = χ T g ( t w → ∞ ) | T g ( t w → ∞ ) − T m | γ , (46)where χ is a constant independent of temperature, and γ = 6 . [27]. For finite t w , χ ( t w ; T m ) only has a maximum as a function of temperature. A way of arriving at thismaximum would be to fit the data to the function χ ( t w ; T m ) = χ T g ( t w ) | T g ( t w ) − T m | γ , (47)and then use the data points from just two or three temperatures to extract T g ( t w ) . Forlarger and larger t w , one could in principle extrapolate to the true T g . We emphasise that,though Eq. (47) suggests χ ( t w ; T m ) diverges at T m = T g ( t w ) , it does not, arriving only at amaximum value for finite t w . Nevertheless, Eq. (47) is a way of estimating T g ( t w ) for use in anextrapolation procedure.To test whether this trick has any validity, consider the data exhibited in Fig. 9. Here, t w = 10 000 s and, taking χ ( t w ; T m ) at the centre of the error bars for the two temperatures . K and K, one finds T g ( t w = 10 000 s) = 32 K. This value is too high, as magnetisationmeasurements suggest T g ( t w → ∞ ) = 31 . K. More accurate determination of the parametersin Table 3 would diminish the error in T g ( t w ) , but it does suggest a feasible process for takinglaboratory 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 H or to H (alternatively, to the number of correlated spins or to the square root of the total number ofspins, respectively). We have already alluded to the results of these works as displaying theeffect of magnetisation non-linearity. We now explore this assertion in detail using the analysisof Subsection 5.2.Fig. 1 of Joh et al. and Fig. 3 of Bert et al. are reproduced in Fig. 1 and Fig. 10 in thispaper. Both exhibit significant deviations from an H dependence of the ln t eff H with increasingvalues of H . Bert et al. [9] go on to assert a linear dependence, as exhibited in their Fig. 3,reproduced here in Fig. 10. The magnetic fields in [9] are quite large, and the scale of their plotdoes not cover the dependence on H for small H . Nevertheless, they claim their data fits alinear dependence of ln t eff H on H . A glance at the left panel of Fig. 10 suggests how they couldrationalise their conclusion.Yet, as noted by the authors of the experiments in Fig. 1 [5], a linear dependence on H isa poor fit to the data at small H . Further, the argument for the magnetisation’s growth with ONTENTS Fe . Mn . TiO . T g . . t e ff w ( s ) H (Oe) t w = 1 000 s t w = 3 000 s t w = 10 000 s t w = 30 000 s H (kOe) cubic polynomials Figure 10:
Left:
Effective waiting times (in log scale) derived from field-change experiments onan Ising sample (Fe . Mn . TiO ) as a function of the magnetic field H . The plot reproducesFig. 3 of Bert et al. [9] (solid lines are linear interpolations to data with same t w ). Right:
Samedata plotted against H . The dashed lines are fits to Eq. (31), with fit parameters listed inTable 4. √ N corr is supposedly valid at small H [48], while the dependence on the number of correlatedspins is argued to be proportional to √ N corr , rather than linear in N corr , as from Eq. (4). Onthe other hand, the data exhibited in Fig. 10 uses magnetic fields that are substantially largerthan those considered in our experiments.We assert that the departure from linearity in H as H increases observed in [9] is simplythe effect of non-linearity. To prove this, we apply the scaling relation, Eq. (30), to their data,doing our best to extract their measured values from their figure. Our fit to Eq. (31) is shownin the right panel of Fig. 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 ofreproducibility. The fitting quality for t w = 10 000 s and t w = 30 000 s is better than for theother two waiting times. Note that the coefficients c n listed in Table 4 are considerably smallerthan in our Table 3 for our current experiments on a CuMn 6 at. % single crystal. We believethis is because our measurements are for T m ≈ . T g whereas Bert et al. [9] worked at . T g ,where non-linear terms are expected to be much smaller.Using the fitting coefficients from Table 4 and θ (˜ x ) = 0 . , we obtain ξ ( t w = 30 000 s) ξ ( t w = 10 000 s) = (cid:20) c (30 000 s) c (10 000 s) (cid:21) / [ D − θ (˜ x ) / ≈ . , (48) ξ ( t w = 30 000 s) ξ ( t w = 10 000 s) = (cid:20) c (30 000 s) c (10 000 s) (cid:21) / [2 D − θ (˜ x ) / ≈ . , (49) ξ ( t w = 30 000 s) ξ ( t w = 10 000 s) = (cid:20) c (30 000 s) c (10 000 s) (cid:21) / [3 D − θ (˜ x )] ≈ . . (50)The three ratios, Eq. (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 ONTENTS t w , the parameters from fits to Eq. (31) of the data obtainedby Bert et al. [9] for ln t effH . Their data correspond to Fe . Mn . TiO at T m = 0 . T g (seeFig. 3 of Ref. [9]). The fits are shown in our Fig. 10. The uninteresting fit parameter, c , is notincluded in the table. t w (s) coefficient value1 000 c − . × − c . × − c − . × − c − . × − c . × − c − . × −
10 000 c − . × − c . × − c − . × −
30 000 c − . × − c . × − c − . × − that the assessment of Ref. [9] that their data is evidence for E Z ∝ H is in error. Rather,we believe the departure they observe from linearity in H arises from non-linear terms in themagnetisation as a result of the large magnetic fields utilised in their study. H and H = 0 + We have exploited our proposed relation (16) to extract effective times t eff H , as explained inAppendix A.3. Our results are displayed in Fig. 11. In the subsequent analysis in Section 5.5,we shall need the derivative of ln (cid:0) t eff H /t eff H → (cid:1) with respect to H , evaluated numerically at H = 0 . Our main scope here will be evaluating this derivative.An obvious strategy would be to fit the numerical data for ln (cid:0) t eff H /t eff H → (cid:1) as we did for theexperimental data in Eq. (31). Note that our sought derivative at H = 0 is just the c coefficientin the fit. A welcome simplification in the analysis of the numerical data is that we can explicitlyput c = 0 in the fit to Eq. (31) [indeed, we are able to carry out the fit for ln (cid:0) t eff H /t eff H → (cid:1) thanksto Eq. (16)]. Our fitting parameters are reported in Table 5. Unfortunately, as the reader willnote from Fig. 11, these fits predict unphysically wild oscillations that are not observed in thenumerical data. A plausible explanation for these oscillations relies on the very large magneticfields used (recall that H = 1 for the IEA model roughly corresponds to × Oe in physicalunits). These huge magnetic fields probably exceed the radius of convergence of the Taylorexpansion of Eq. (30). At any rate, the oscillations cast some doubts on the determination ofthe derivative at H = 0 . This is why we have turned to a different strategy in order to validateour computation. Our starting point, recall [2] and Eq. (27), is the expected scaling behaviourfor the coefficient c ( t w ; T ) listed in Table 5. The non-linear coefficient c ( t w ; T ) , reported in ONTENTS − − −
20 0 .
002 0 .
004 0 .
006 0 .
008 0 .
01 0 .
012 0 . − − −
20 0 .
002 0 .
004 0 .
006 0 .
008 0 .
01 0 .
012 0 . l n (cid:18) t e ff H / t e ff H → + (cid:19) H Run6Run5Run4 l n (cid:18) t e ff H / t e ff H → + (cid:19) H Run3Run2Run1
Figure 11: The numerical time ratio ln( t eff H /t eff H → + ) for the six runs of Table 2. The data werefitted as a polynomial of H as reported in Table 5. Continuous lines are fits for data at T = 1 . ;dashed lines correspond to the data at T = 0 . .Table 5: Results of the fits to Eq. (31) of the numerical data for the time ratio ln( t eff H /t eff H → + ) .Note that, in order to stabilise the fits, we needed to include an extra terms in Eq. (31) for twocases. In the table, ξ stands for ξ ( t = 0 , t w ; H = 0) and the fitting range is ≤ H ≤ H . T t w ξ c ( t w ; T ) c ( t w ; T ) c ( t w ; T ) c ( t w ; T ) H − . × . × − . × . × . . − . × . × − . × . × . . − . × . × − . × . . − . × . × − . × . . − . × . × − . × . . − . × . × − . × . Table 5, behaves as [4] c ( t w ; T ) = ξ D − θ (˜ x ) / (cid:32) ˆ S T + a ( T ) ξ θ (˜ x ) / (cid:33) , (51)using the scaling of the susceptibility χ from Eq. (23). Here, ˆ S is the function appearing inthe fluctuation-dissipation relation [45] and a ( T ) is a smooth function of temperature, andwe have absorbed the geometrical prefactor b of Eq. (27) in a ( T ) and ˆ S ( T ) . Notice that the a ( T ) ξ − θ (˜ x ) / term is sub-leading compared to ˆ S/ (2 T ) and it was neglected in the previousanalysis.We rewrite Eq. (51) as: c ( t w ; T ) ξ D − θ (˜ x ) / = ˆ S T a ( T ) ξ − θ (˜ x ) / . (52)and we study this quantity as a function of [ ξ ( t w )] − θ (˜ x ) / in Fig. 12. Note that in the above ONTENTS − . − . − . − . . .
55 0 . − −
100 0 . . . . c ( t w ; T ) T [ ξ ( t w ) ] D − θ ( x ) / ξ ( t w ) − θ ( x ) / T = 0 . T = 1 . l n (cid:18) t e ff H / t e ff H → + (cid:19) H T = 1 . , t w = 2 . Figure 12:
Left:
Behaviour of the rescaled quantity c ( t w ; T ) T / [ ξ ( t w )] D − θ (˜ x ) / as a function of ξ ( t w ) − θ (˜ x ) / , see Eq. (52). Right:
An enlargement of Fig. 11 in the small-field regime, for thecase t w = 2 . at T = 1 . (Run 6 of Table 2) and its fit reported in Table 5.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 Ref. [7]). The data exhibit aconstant value, except for the point correspondent to t w = 2 . at T = 1 . (Run 6). Therefore,we shall accept the numerical estimation of the derivative at H = 0 through the coefficient c for all cases but for our Run 6. In order to clarify what is going on with Run 6, we report inthe right panel of Fig. 12 an enlargement of Fig. 11 in the small-magnetic-field regime for thiscase. As it could be guessed from the left panel, the fitting procedure clearly underestimatesthe slope of the curve at H = 0 . Therefore, in order to estimate the derivative for Run 6, wehave instead relied on Eq. (52) by averaging the constant value found in Fig. 12 over all otherruns and by multiplying this averaged constant value by [ ξ ( t w )] D − θ (˜ x ) / [see Eq. (52)]. In order to test the scaling form, Eq. (30), the data for all of the non-linear contributions tothe magnetisation, experimental and numerical, are plotted according to the functional form ξ − θ (˜ x )2 G ( T, ξ D − θ (˜ x )2 H ) (53)in Fig. 13-14. The fit to scaling relationship (30) is remarkable and testimony to the agreementfor both the experimental and numerical data. We first address the dynamical scaling law for a system in presence of a magnetic field attemperatures close to T g in Section 5.6.1. Then, in Section 5.6.2, we analyse the dynamicalscaling for ferromagnetic systems, either Ising or Heisenberg, in the presence of an externalmagnetic field. ONTENTS .
001 0 .
01 0 . l n t e ff H t e ff H → + − c ( t w ; T ) H ξ θ ( ˜ x ) / (cid:20) ξ D − θ (˜ x ) / H (cid:21) Exp . . . . Figure 13: The non-linear parts from the experimental response time data, [ln t eff H − c ( t w ; T m ) H ] ξ θ (˜ x ) / , plotted against ( ξ D − θ (˜ x ) / H ) . The deviations of the data at T m = 29 Kmay be caused by a shift in T g as the temperature begins to approach T g ( H ) . The small- x range is enlarged in the inset . .
001 0 . l n t e ff H t e ff H → + − c ( t w ; T ) H ξ θ ( ˜ x ) / (cid:20) ξ D − θ (˜ x ) / H (cid:21) Run 3Run 2Run 1 Run 6Run 5Run 4
Figure 14: The non-linear parts from the numerical response time data, [ln( t eff H /t eff H → + ) − c ( t w ; T m ) H ] ξ θ (˜ x/ , plotted against ( ξ D − θ (˜ x ) / H ) . The abscissa of the main panel is in linearscale and shows a closeup for small values of ( ξ D − θ (˜ x ) / H ) . The abscissa of the insert is inlog scale in order to report all our numerical data. 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 thetemperature 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 ONTENTS z ( T ) used in Fig. 15. T z ( T ) T = 1 . T = 1 . T = 1 . T = 1 . ξ ( t , t w ) H / y H tH z ( T ) /y H H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . ξ ( t , t w ) H / y H tH z ( T ) /y H Figure 15: Critical dynamical scaling according to Eqs. (54) and (57). We show data for T = 1 . ≈ T g [27] and for T = 1 . . T g = 1 . [27] (in IEA units) and at T = 1 . for several external magnetic fields and samples. A protocol equivalent to FC, but convenient for simulations, is to place a randomspin configuration instantaneously at the working temperature T , and turning on the externalmagnetic field at the same instant, so that t w = 0 .According to Eq. (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 : [ ξ ( t, t w ; H ) H /y H ] ∝ const . (54)The correlation length ξ ( t, t w ; H ) grows as ξ ( t w ) ∝ t /z ( T )w , (55)with an exponent that, in a first approximation, is expected to behave near the criticaltemperature as [2]: z ( T ) (cid:39) z c T g T , where z c = z ( T g ) = 6 . . (56)Hence, using Eq. (55) and Eq. (56) in the scaling argument of Eq. (54), we have equivalently, [ t × H z ( T ) /y H ] ∝ const . (57) ONTENTS
System
L H
Number of samplesIsing 256 0 22000.001 2 6000.003 1 7000.005 1 200Heisenberg 200 0 20000.003 1 0000.004 1 6000.005 2 400
In Table 6 we list the aging-rate factors z ( T ) used in our analysis. We plot our rescaled datain Fig. 15.The agreement with the scaling prediction, evinced by the data collapse, is striking. Ourplots also exhibit overshooting, as evidence for the paramagnetic phase when the magnetic fieldis turned on.The reader could wonder why we have used Eq. (54) for the scaling analysis at thetemperature T = 1 . < T g , and whether this implies evidence of the absence of the deAlmeda-Thouless (dAT) line in finite dimension. We shall address these questions in Section 6. By studying two ordered systems we can showthat the overshooting phenomenon is, in fact, general. To demonstrate generality, we havesimulated the three-dimensional Ising and Heisenberg models in a cubic lattice with periodicboundary conditions and size L at the critical point T c . The N = L D ( D = 3 ) Heisenberg spinsinteract with their lattice nearest neighbours through the Hamiltonian H = − (cid:88) (cid:104) r , r (cid:48) (cid:105) S r · S r (cid:48) + H · (cid:88) r S r . (58)where S r are unit vector spins and H is an external magnetic field. The connected correlationfunction is C ( r, t ) = 1 L (cid:88) x S x ( t ) · S r + x ( t ) − [ m ( t )] , (59)with m ( t ) = 1 L (cid:88) x S x ( t ) . (60)We only write equations for the Heisenberg model, Eqs. (58)–(60), but the Ising analogues canbe obtained trivially by just dropping the vector symbol in the spins. We report the simulationdetails in Table 7.The Ising and Heisenberg model have different symmetry properties, so they belong totwo distinct universal classes. In other words, each model has a distinct value for the criticaltemperature and exponents. The Ising model has η = 0 .
036 297 8(20) [49], z = 2 . [50]and β c = 0 .
221 654 626(5) [51]. Instead, for the Heisenberg ferromagnet η = 0 . [52, 53], z = 2 . [54] and β c = 0 . [55] ( β c ≡ /T c ). ONTENTS Ising
100 1000 10000
Heisenberg ξ ( t ) H = 0 H = 0 . H = 0 . H = 0 . t H = 0 H = 0 . H = 0 . H = 0 . Figure 16: The log-log plots show the behaviour of ξ versus time for the Ising ( left ) andHeisenberg ( right ) models for three magnetic fields. All simulations were performed at thecritical temperature T c appropriate to each model. The saturation at long times exhibited bythe Ising model at H = 0 is a finite-size effect. . . .
01 0 . Ising . Heisenberg ξ ( t ) H / y H H = 0 . H = 0 . H = 0 . tH z c /y H H = 0 . H = 0 . H = 0 . Figure 17: Critical dynamical scaling for ferromagnetic models. The data from Fig. 16 forthe Ising ( left ) and Heisenberg ( right ) models for the three non-vanishing magnetic fields arerescaled following the predictions of the Renormalisation Group [see Eqs. (62)–(63)]. In thiscase the relevant variables are ξH /y H and tH z c /y H , with y H = ( D + 2 − η ) / with D = 3 .As explained in Appendix A.5, the correlation length ξ ( t, t w ; H ) can be calculated withintegral estimators [7, 8], I k ( T, t w ) = (cid:90) ∞ d r r k C ( r, t w ; T ) , ξ k,k +1 ( t w ; T ) = I k +1 ( t w ; T ) I k ( t w ; T ) . (61)In this Section, we evaluate the correlation length ξ ( t, t w ; H ) . As the reader can notice, thegrowth of ξ ( t ) overshoots before reaching equilibrium for any external magnetic field for bothferromagnetic models, see Fig. 16.According to Eq. (20), at the critical temperature T c , and for small external magnetic fields ONTENTS H , there exists a scaling law that connects ξ ( t, t w ; H ) with the external magnetic field H inferromagnetic system: [ ξ ( t, t w ; H ) H /y H ] ∝ const . (62)As the reader can notice, Eq. (62) differs from Eq. (54) in the power of the magnetic field. Inthe ferromagnetic system, the relevant external variable is H and not H as it would be forspin glasses [42, 43]. Analogously to Eq. (57), we can rescale time as [ t × H z ( T ) /y H ] ∝ const . (63)We plot our rescaled data in Fig. 17. The agreement with the scaling prediction, both for theHeisenberg and for the Ising model, is remarkable.In conclusion, the overshooting phenomenon is general and we have observed it both inferromagnetic systems, Figs. 16–17, and in disordered ones, Fig. 15.
6. Investigation of the dAT line in D = 3 The existence (or not) of the spin-glass condensation in the presence of a magnetic field remainsthe subject of some controversy (see, e.g. , [56–59]). In a mean-field treatment, de Almeida andThouless [60] showed that, for the Sherrington-Kirkpatrick infinite-range mean-field model [61],there would be a phase transition according to the following relationship for Ising spin glasses, (cid:16) − T g ( H ) T g (0) (cid:17) = 34 h , (64)with h = µHk B T g (0) , (65)where µ is the spin magnetic moment. Conversely, the droplet model [62, 63] would predict nophase transition except exactly at H = 0 . This dispute was addressed by Lefloch et al. [64].Their final conclusion bears repetition: “Thus, even if the spin glass does not exist in a magneticfield, at least it looks the same as in zero field , as far as we experimentalists can see”.In finite dimension and for T very close to the critical temperature T g ( H = 0) , the deAlmeida-Thouless (dAT) line, provided it exists, should be governed by the Fisher-Sompolinsky[31] relation: (cid:18) − T g ( H ) T g (0) (cid:19) ∝ H /ν (5 − η ) , (66)where we have specialised to D = 3 . (cid:107) Rather than through T g ( H ) , we are interested indescribing the dAT line geometrically by the inverse function of T g ( H ) , namely H c ( T ) . Hence,we rewrite Eqs. (64) and (66) asMean-Field: H c ( T ) ∝ (cid:18) − TT g (cid:19) a MF , a MF = 1 . , (67)
3D : H c ( T ) ∝ (cid:18) − TT g (cid:19) a , a = ν (5 − η )4 → a = 3 . , (68)where we have taken the 3D critical exponents ν and η from Ref. [27]. The followingconsiderations, based on Eqs. (67)–(68), will be useful: (cid:107) Notice this is the same relation used for matching the numerical and experimental scales in Section 3
ONTENTS T = 1 . t w = 2 . t w = 2 . t w = 2 . T = 0 . t w = 2 . t w = 2 . t w = 2 ξ ( t , t w ; H ) H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . t H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . Figure 18: Growth of ξ ( t, t w (cid:54) = 0; H ) in simulations that mimic the experimental zero-field-cooling protocol. Plots are in log-log scale. • H c ( T ) is a decreasing function of T (remember T ≤ T g ) and H c ( T g ) = 0 . This means that,upon approaching T g from below, one eventually crosses the dAT line for any H > , nomatter how small H is. • When
H > H c ( T ) the system is above the dAT line, in its paramagnetic phase: thecorrelation length, ξ ( t, t w ; H ) , reaches asymptotically its equilibrium value ξ eq ( H ) for verylong time t . • When
H < H c ( T ) we are in the spin-glass phase and one expects to observe a power-lawgrowth of the correlation length, see Eq. (55). • The a exponent is much larger than the mean-field (MF) one, a ≈ . × a MF . Thisimplies that, in D = 3 , the dAT line is very flat when one approaches the criticaltemperature T (cid:119) T g .In particular, our last item above suggests an interpretation of the somewhat surprisingresults in Fig. 15, where data for T = 1 . were successfully scaled with the scaling lawappropriate for T g [recall that . < T g = 1 . and that at T = T g we are in theparamagnetic phase for any H > ]. Assuming that the proportionality coefficient is of orderunity, let us estimate the critical magnetic field at T = 1 . exploiting Eq. (68): H c ( T = 1 . ∼ × − . (69)Considering, now, that the smallest magnetic field in Fig. 15, namely H = 0 . , is larger than H c ( T = 1 . by a factor of 1000 or so, there is little surprise in that a scaling law assuming H c ( T = 1 .
05) = 0 works with our data.Our focus in this section will be an exploration of the growth of the spin-glass correlationlength, ξ ( t, t w ; H ) , under conditions that mimic the experimental protocol for measurement ofthe zero-field-cooled magnetisation, M ZFC ( t, t w ; H ) for t w (cid:54) = 0 , recall Section 3.In Fig. 18 we plot ξ ( t, t w (cid:54) = 0; H ) as a function of time for different magnetic fields H . Wecompute ξ from the microscopic correlation function C ( r ) (see Appendix A.5), which requiresthat we compute error bars from the sample-to-sample fluctuations. We have simulated different ONTENTS H and t w because of the enormous computational effort involved.We show error bars in Fig. 18 only in those cases where they can be computed.The time evolution of the spin-glass correlation length ξ ( t, t w ; H ) depends markedly on theinterplay between the waiting time t w and the value of the magnetic field H , see Fig. 18. Thesystem needs several time steps before responding to the switching on of the magnetic field.Different scenarios appear.On the one hand, for the largest magnetic fields, namely H > . both at T = 0 . and T = 1 . , the correlation length displays a non-monotonic time behaviour, just as we found inSection 5.6.1 for the dynamics in the paramagnetic phase (recall that t w = 0 in Section 5.6.1).In particular, for those cases when the starting correlation length, ξ ( t = 0 , t w ; H ) , is larger than the equilibrium value ξ eq ( H ) , the correlation length decays. Otherwise, we observe anovershooting phenomenon reminiscent of our findings in Section 5.6.1, see Fig. 18.On the other hand, for H < . , we observe that the correlation length ξ ( t, t w ; H ) appearsto follow the same power-law growth for all the different waiting times. Here we must distinguishbetween the mean-field H MFc ( T ) and the Fisher-Sompolinsky scaling H ( T ) , i.e. , betweenEqs. (67) and (68). Using Eq. (64) for the former, one finds H MFc ( T = 0 . ≈ . H MFc ( T = 1 . ≈ . . (70)Interestingly, the scaling result, Eq. (68), yields H ( T = 0 . ≈ .
003 and H ( T = 1 . ≈ . . (71)For the magnetic fields used in our simulations, therefore, one is presumably in the condensedstate for . ≤ H < . from the perspective of the mean-field solution of the Sherrington-Kirkpatrick model [61], while from the perspective of the Fisher-Sompolinsky scaling [31] one isalways in the paramagnetic state since H > H ( T = 0 . , . . Though this latter region is notaccessible experimentally through magnetic measurements, one can argue that the simulationresults should be symmetric around T c ( H ) . This is the basis for the comparison betweenexperiment and simulations contained in Section 5 of this paper.Let us now attempt a scaling analysis similar to the one in Section 5.6.1 for those magneticfield values for which the asymptotic ξ eq ( H ) can be at least guessed from Fig. 18. We start bymodifying scaling relationship (54) to ξ ( t, t w ; H ) | H − H ( T ) | /y H ∝ const . (72)Next, using Eq. (55) in the scaling argument of Eq. (72), we have, equivalently, (cid:0) t × | H − H ( T ) | z ( T ) /y H (cid:1) ∝ const . (73)We replot our rescaled data in Fig. 19 for the mean-field values of H MFc ( T ) , see Eq. (70).As seen in the panel for T = 0 . , there is nearly perfect scaling for H ≥ . but not for H = 0 . , though the curves do seem to coalesce for the three different waiting times.It is tempting to suggest that, for this value of magnetic field, one is in the condensedphase. Glancing at Fig. 18, however, the growth of ξ ( t, t w ; H ) for H = 0 . breaks away fromthe curves for the larger magnetic fields, so that it is very possible that it would join theequilibrium curves ( i.e. , the paramagnetic regime) at times longer than those accessible in oursimulations. This ambiguity softens the interpretation that we have broached the dAT line inour simulations, as would be predicted from a mean-field approach. ONTENTS . .
01 1 100 10000 T = 1 . . .
01 1 100 10000 T = 0 . ξ ( t , t w ; H ) | H − H c ( T ) M F | / y H t | H − H ( T ) MF | z ( T ) /y H H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . Figure 19: Search for the dAT line in D = 3 using mean-field scaling. Plots are in log-log scaleand show the behaviour of the rescaled quantities defined in Eqs. (72)–(73) for the mean-fieldestimators H c ( T ) MF , see Eq. (70). The aging rates z ( T ) used in this scaling are listed in Table 6,to be found in Section 5.6. . .
01 1 100 10000 T = 1 . . .
01 1 100 10000 T = 0 . ξ ( t , t w ; H ) | H − H c ( T ) D | / y H t | H − H ( T ) | z ( T ) /y H H = 0 . H = 0 . H = 0 . H = 0 . H = 0 . Figure 20: Search for the dAT line in D = 3 with Fisher-Sompolinsky scaling. Plots are inlog-log scale and show the behaviour of the rescaled quantities defined in Eqs. (72)–(73) for theFisher-Sompolinsky estimators H c ( T ) , see Eq. (71). We report the aging rates z ( T ) used inthis scaling in Table 6 to be found in Section 5.6.However, if we replot our data using the scaling result of Eq. (71), as exhibited in Fig. 20,for the values of H ( T ) for T = 0 . , the data appear to collapse for all of the magneticfields, including H = 0 . [ H ( T = 0 . (cid:28) . ]. The Fisher-Sompolinsky scaling would,therefore, support the conjecture that at T = 0 . , our simulation results for H = 0 . are inthe paramagnetic regime.As to the rescaled T = 1 . data in Figs. 19 and 20, they are of low quality, limiting themagnetic fields to relatively large values. The three values ( H = 0 . , . , . ) for whichit is feasible to rescale are all above the H MF , ( T = 1 . values given by Eqs. (70) and (71). ONTENTS T = 0 . , ξ ( t, t w ; T ) may be growing as a power law, and thus be in the condensed phase, our limitedtime scale for the simulations is unable to conclude that we have, in fact, straddled the dATline. If one assumes Fisher-Sompolinsky scaling, Eq. (71), all of our simulation results would bein the paramagnetic region. Until much longer times scales become reachable (either at lowertemperatures, or smaller magnetic field), even our powerful Janus II simulations are unable toarrive at a definitive conclusion regarding the existence, or non-existence, of the dAT line for D = 3 Ising spin glasses.
7. Conclusions
This paper demonstrates the unique and powerful combination of experiment, theory, andsimulations addressing complex dynamics. The use of single crystals enables experiments toexhibit the consequences of very large spin-glass correlation lengths. The power of a special-purpose computer, Janus II, in combination with theory, is sufficient to extend simulationtime and length scales to values explored experimentally. Together, these approaches unite todevelop new and important insights into spin-glass dynamics.Previous work [5, 23] explored the reduction of the free-energy barrier heights responsiblefor aging in spin glasses by the Zeeman (magnetic field H ) energy. Observations for smallmagnetic fields, proportional to H , were used to extract a quantitative value for the spin-glass correlation length and its growth rate with time. As the magnetic field was increased,however, departures from proportionality to H were observed. This paper presents detailedexperimental observations of this behaviour and, together with theory [4], is able to demonstratethe applicability of a new non-linear scaling law for the magnetisation in the vicinity of thespin-glass condensation temperature T g . Remarkably, Janus II simulations were able to generatecomparable values for the magnetisation dynamics, with the added value of direct measurementof the characteristic response time.The combination of these two approaches has put to rest a decades-old controversyconcerning the nature of the Zeeman energy. We have shown that the departures fromproportionality to H are caused by non-linear terms in the magnetisation, and not byfluctuations of the magnetisation that lead to a Zeeman energy proportional to H . Further,the departure from an H behaviour that was used to justify the proportionality to H is shownto be a consequence of non-linear behaviour of the magnetisation in H , and fully accounted forusing the new scaling law. This is an important finding, because otherwise the extraction ofthe spin-glass correlation length from the Zeeman-energy reduction in the barrier height wouldhave been in error.One of the most interesting findings in the paper is the extraction of the characteristicresponse time for spin glasses, t eff H , from simulations. It has been made possible by noting thatthe spin-glass correlation function reaches a peak at the response time . That is, C ( t eff H , t w ; H ) = C peak ( t w ) . (74)Thus, by extracting C peak ( t w ) one can determine the characteristic response time t eff H . It is this ONTENTS T g , the Fisher-Sompolinsky scaling relation holds under out-of-equilibrium conditions (see Section 5.5.) This will enable us in future simulations to comparethe magnitude and growth of the spin-glass correlation length under two experimental protocols:the dynamics of zero-field-cooled and thermo-remanent magnetisations. The important pointhere is that this paper shows that our analysis will be valid under these non-equilibriumconditions.We have discovered an overshooting phenomenon that is shown to be general for bothordered and disordered magnetic systems. And finally, we have explored the nature of thespin-glass condensation at T g as a function of the external magnetic field, the so-called deAlmeida-Thouless line. We have presented preliminary evidence for its existence as a truecondensation transition, but this conclusion should be regarded as provisional.In conclusion, this paper has explored the nature of the spin-glass state in the vicinityof its condensation temperature T g . We displayed the power of combining insights fromboth experiment and simulations, coupled together by theory. We look forward to continuedinvestigation of spin-glass dynamics using this relationship as we examine the microscopicnature of such phenomena as rejuvenation and memory. Finally, because spin-glass dynamicshas applications in many diverse fields (ecology, biology, optimisation, and even social science),our work demonstrates that modelling complex systems is feasible in finite dimensions. Acknowledgements
We are grateful for helpful discussions with S. Swinnea about sample characterisation. Thiswork was partially supported by the U.S. Department of Energy, Office of Basic Energy Sciences,Division of Materials Science and Engineering, under Award No. DE-SC0013599, and ContractNo. DE-AC02- 07CH11358. We were partly funded as well by Ministerio de Economía,Industria y Competitividad (MINECO, Spain), Agencia Estatal de Investigación (AEI, Spain),and Fondo Europeo de Desarrollo Regional (FEDER, EU) through Grants No. FIS2016-76359-P, No. PID2019-103939RB-I00, No. PGC2018-094684-B-C21 and PGC2018-094684-B-C22, bythe Junta de Extremadura (Spain) and Fondo Europeo de Desarrollo Regional (FEDER, EU)through Grant No. GRU18079 and IB15013 and by the DGA-FSE (Diputación General deAragón – Fondo Social Europeo). This project has also received funding from the EuropeanResearch Council (ERC) under the European Union’s Horizon 2020 research and innovationprogram (Grant No. 694925-LotglasSy). DY was supported by the Chan Zuckerberg Biohuband IGAP was supported by the Ministerio de Ciencia, Innovación y Universidades (MCIU,Spain) through FPU grant No. FPU18/02665. BS was supported by the Comunidad de Madridand the Complutense University of Madrid (Spain) through the Atracción de Talento program(Ref. 2019- T1/TIC-12776). The simulations of Section 5.6.2 were carried out at the ICCAExsupercomputer center in Badajoz. We thank the staff at ICCAEx for their assistance.
ONTENTS A. Technical details about our simulations
A.1. Smoothing and interpolating the data
Our numerical data for the magnetisation at small magnetic fields are rather noisy, whichcomplicates the process of taking its derivative with respect to ln t . This derivative is theresponse function S ( t, t w ; H ) [recall Eq. (11)]. This is why, before differentiating, we havefollowed a de-noising method first proposed in Ref. [45]. Because we work at larger correlationlengths (and closer to T g ) than in that work, however, we have found it preferable to changesome technical details. We explain below the precise de-noising method that we have followedin this work.Our starting observation is that the derivative of both M ZFC ( t, t w ; H ) and T M
ZFC ( t, t w ; H ) /H peak at exactly the same time t eff H . However, T M
ZFC ( t, t w ; H ) /H enjoysthe advantage of being a very smooth function of the correlation C ( t, t w ; H ) . This smoothfunction is named the fluctuation-dissipation relation [32–36]. The key point is that, at vari-ance with the magnetisation, C ( t, t w ; H ) can be computed with high accuracy for any value ofthe field H , including H = 0 . Thus, we follow a simple two-step de-noising algorithm:(i) We fit our data for T M
ZFC ( t, t w ; H ) /H as a function of C ( t, t w ; H ) , see Eq. (A.1).(ii) We replace our data for T M
ZFC ( t, t w ; H ) /H by the just-mentioned fitted function evaluatedat C ( t, t w ; H ) .Our chosen functional form is as follows. Let the quantity T M
ZFC ( t, t w ; H ) /H beapproximated by f (ˆ x ) , where for notational simplicity we do not write the explicit dependenceon t, t w and H , f (ˆ x ) = f L (ˆ x ) 1 + tanh[ Q (ˆ x )]2 + f s (ˆ x ) 1 − tanh[ Q (ˆ x )]2 (A.1)with Q (ˆ x ) = (ˆ x − x ∗ ) /w . The function f (ˆ x ) has distinct behaviour for large and small ˆ x . Thecrossover between the two functional forms is smoothed by the tanh[ Q (ˆ x )] functional term,where x ∗ is the crossover point and w is the crossover rate. The functional form for small ˆ x is f s (ˆ x ) = a + N (cid:88) k =1 a k (ˆ x − ˆ x min ) k k ! . (A.2)For the large- ˆ x region, we choose a polynomial expansion in terms of (1 − ˆ x ) : f L (ˆ x ) = (1 − ˆ x ) + N (cid:48) +1 (cid:88) k =2 b k (1 − ˆ x ) k k ! . (A.3)The polynomial expansion in − ˆ x is quite natural in the large- ˆ x region [45], as a deviationfrom the fluctuation-dissipation theorem. This theorem, which holds only under equilibriumconditions, predicts N (cid:48) = 0 for Eq. (A.3) and x ∗ = w = 0 for Eq. (A.1) [so that, in equilibrium,one would have f (ˆ x ) = (1 − ˆ x ) in Eq. (A.1)]. In the small- ˆ x region, there is not a strongjustification (other than convenience) for our choice of f s (ˆ x ) . In fact, the choice of Ref. [45]for f s (ˆ x ) was a Padé approximant. The quantity T M
ZFC ( t, t w ; H ) /H turns out to be affectedby strong non-linear effects that increase with increasing external magnetic field H and uponapproaching the glass temperature T g (see Fig. A1). ONTENTS N , the number of fitted parameters in Eq. (A.3) N (cid:48) , [ N (cid:48) = 0 means f L (ˆ x ) = (1 − ˆ x ) ], thecrossover parameters x ∗ and w , and the fit’s figure of merit χ / d . o . f . (d.o.f. stands for degreesof freedom). Note that we can only compute the so-called diagonal χ , which takes into accountonly the diagonal elements of the covariance matrix. Because of this limitation, we find a valueof χ significantly smaller than the number of degrees of freedom for many of our fits. t w H N N (cid:48) x ∗ w χ / d . o . f . . / . / . / . / . / . / . / . / . / . / . . / T = 0 . . / . / . / . / . / . / . / . . / . / . / . / . / . / . / . / . . / . / . / . / . / . / T = 1 . . . / . / . / . / . / . / . . / . / . / ONTENTS . . . . . T = 1 . , t w = 2 . . . . . . . . . . . . T = 0 . w = 2 . M Z F C ( t , t w ; H ) T / H H = 0 . H = 0 . H = 0 . C ( t, t w ; H ) H = 0 . H = 0 . − x Figure A1: The behaviour of
T M
ZFC ( t, t w ; H ) /H is exhibited as a function of C ( t, t w ; H ) . The top plot is for T = 1 . , t w = 2 . . The bottom plot is for T = 0 . , t w = 2 . . We do notreport all the magnetic values for simplicity. . . . . . . .
35 0 . .
45 0 . . T = 1 . , t w = 2 . M Z F C ( t , t w ; H ) T / H C ( t, t w ; H ) H = 0 . H = 0 . H = 0 . H = 0 . smoothedFDT Figure A2: Comparison between the original and smoothed data at T = 1 . and waiting time t w = 2 . . One can clearly see the advantage of the de-noising method for the lowest magneticfield.As we discuss in Appendix A.2, it is necessary to select the appropriate order for thepolynomials in Eqs. (A.2) and (A.3). Our preferred choices are given in Table A1.Errors are computed using a jackknife procedure. We perform an independent fit foreach jackknife block, and compute errors from the jackknife fluctuations of the fitted f (ˆ x ) . InFig. A2 we show a comparison between the original and smoothed data for the case T = 1 . and ONTENTS . . . . T δ M Z F C ( t , t w , H ) / H t original dataover-fittedpreferred choice Figure A3: Comparison between the errors for the original and the de-noised data for
T M
ZFC ( t, t w ; H ) /H (with T = 0 . , t w = 2 . and H = 0 . ), for the two different fittingfunctions f (ˆ x ) reported in Table A2. t w = 2 . . As expected, the de-noising technique is most important for the smallest magneticfields. A.2. Over-fitting problem
A difficulty in our fits to Eq. (A.1) is that we can use only the diagonal part of the covariancematrix in the computation of the goodness-of-fit indicator χ . This is the reason underlying thevery small values for χ that we show in Table A1. As a consequence, we cannot trust the χ test for selecting the appropriate order for the polynomial expansions in Eqs. (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 T M
ZFC ( t, t w ; H ) /H with different choices of the polynomial expansion (remember that theseerrors are not computed from χ , but from the jackknife fluctuations). As an example, considerthe case at T = 0 . for a waiting time t w = 2 . and H = 0 . , which is exhibited in Fig. A3.The figure compares the statistical errors of the original, non-de-noised data with the errorsfound with two possible choices for the polynomial fits in Eqs. (A.2) and (A.3). Although bothfits are indistinguishable from the point of view of the χ test, see Table A2, the resulting errorsare very different. In one case, we find statistical errors that evolve rather smoothly with t . Forthe 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 wesaid above, our final choices are reported in Table A1. ONTENTS
Case
N N (cid:48) x ∗ w χ / d . o . f . Over-fitted 1 1 0.547(6) 0.132(1) . / Our choice 2 0 0.550(7) 0.0335(3) . / A.3. Time discretisation and the calculation of the relaxation function S ( t, t w ; H ) As explained in the main text, the quantity used in experiment [5] to extract t eff ( H ) is therelaxation function S ( t, t w ; H ) of Eq. (11). We calculate S ( t, t w ; H ) as a finite-time difference: S ( t, t w , t (cid:48) ; H ) = M ZFC ( t (cid:48) , t w ; H ) − M ZFC ( t, t w ; H )ln (cid:0) t (cid:48) t (cid:1) . (A.4)In simulations, time is discrete and we have stored configurations at t n = integer-part-of n/ ,with n an integer. Let us write the integer dependence of times t and t (cid:48) explicitly as: t ≡ t n , t (cid:48) ≡ t n + k , (A.5)where k is an integer time parameter. Hence, time is rescaled as t new = 12 ln (cid:18) t n + k t n (cid:19) . (A.6)We expressed our observables as functions of t new : S ( t, t w , t (cid:48) ; H ) → S ( t new , t w ; H ) , (A.7) C ( t, t w ; H ) → C ( t new , t w ; H ) . (A.8)The relaxation function S ( t new , t w ; H ) is trivial to construct, see Eq. (A.4). However, thecorrelation function C ( t new , t w ; H ) needs to be calculated using a linear interpolation. For anygiven value of t new , we looked for our original discrete time t n such that ln( t n ) < ln( t new ) ≤ ln( t n +1 ) . (A.9)Using a linear interpolation, we obtain C ( t new ) = ln( t new ) − ln( t n +1 )ln( t n ) − ln( t n +1 ) C ( t n ) − ln( t new ) − ln( t n )ln( t n ) − ln( t n +1 ) C ( t n +1 ) . (A.10)Finally, one can express the relaxation function, S ( t new , t w ; H ) , as a function of thecorrelation function, S ( C ; H ) , in much the same manner as Eq. (A.10). A.4. The t eff H calculation As explained in the main text, the extraction of t eff H from Eq. (16) is delicate because the C peak ( t w ) are implicit functions of the rescaled time t new . In order to solve Eq. (16), we calculatethe t eff H values through a quadratic interpolation. First, we calculate the original discrete time t n such that: C ( t + t n +1 , t w ; H ) < C peak ≤ C ( t + t n , t w ; H ) . (A.11) ONTENTS C ( t + t n − ) = α + α x n − + α x n − , (A.12) C ( t + t n ) = α + α x n + α x n , (A.13) C ( t + t n +1 ) = α + α x n +1 + α x n +1 , (A.14)where x n = ln t n and, for brevity’s sake, we omit the arguments t w and H . The solutiongenerates the α i coefficients: α = C ( t w , t w + t n − ) − C ( t w , t w + t n ) x n − − x n − C ( t w , t w + t n +1 ) − C ( t w , t w + t n )( x n − − x n +1 )( x n +1 − x n ) , (A.15) α = C ( t w , t w + t n +1 ) − C ( t w , t w + t n ) x n +1 − x n − α ( x n + x n +1 ) , (A.16) α = C ( t w , t w + t n ) − α x n − α x n . (A.17)We can then calculate the t eff H solving the equation: C peak = α + α ln (cid:0) t eff H (cid:1) + α (cid:2) ln (cid:0) t eff H (cid:1)(cid:3) , (A.18)where only the solution verifying t n ≤ t eff H < t n +1 is physical. A.5. The construction of ξ ( t, t w ; H ) We shall explain here our computation of the spin-glass correlation length in the presence of amagnetic field. For the simpler case of H = 0 , we refer the reader to Refs. [2, 65].The most informative connected correlator we can construct with replicas is the repliconpropagator [60, 66]. Extending the replicon propagator to the off-equilibrium regime we have: G R ( r , t (cid:48) ) = 1 V (cid:88) x ( (cid:104) s x ,t (cid:48) s x + r ,t (cid:48) (cid:105) − (cid:104) s x ,t (cid:48) (cid:105)(cid:104) s x + r ,t (cid:48) (cid:105) ) , (A.19)where t (cid:48) = t w + t . To compute G R ( r , t (cid:48) ) , we calculate the -replica field Φ ( a,b ; c,d ) x , t (cid:48) = 12 (cid:16) s ( a ) x , t (cid:48) − s ( b ) x , t (cid:48) (cid:17) (cid:16) s ( c ) x , t (cid:48) − s ( d ) x , t (cid:48) (cid:17) , (A.20)where indices a, b, c, d indicate strictly different replica. Notice that (cid:104) Φ ( a,b ; c,d ) x , t (cid:48) Φ ( a,b ; c,d ) y , t (cid:48) (cid:105) = ( (cid:104) s x , t (cid:48) s y , t (cid:48) (cid:105) − (cid:104) s x , t (cid:48) (cid:105)(cid:104) s y , t (cid:48) (cid:105) ) . (A.21)Therefore, we obtain G R ( r , t (cid:48) ) by taking the average over the samples E (cid:16) Φ ( a,b ; cd ) x , t (cid:48) Φ ( a,b ; c,d ) y , t (cid:48) (cid:17) = G R ( x − y , t (cid:48) ) . (A.22)With replicas at our disposal, there are × / (508! × ways of choosing the replicaindices a, b, c , and d . We have found an efficient way for averaging over (roughly) one third ofthis astronomic number of possibilities [67]. We define the correlation function, C ( r , t (cid:48) ) , asthe replicon propagator, G R ( x − y , t (cid:48) ) , where we consider the difference x − y as the latticedisplacement r = ( r, , . Of course, we can align the lattice displacement vector r alongany of three coordinate axes, so we average over these three choices. The replicon correlator ONTENTS 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]: I k ( t (cid:48) ; T ) = (cid:90) ∞ d r r k C ( r, t (cid:48) ; T ) , ξ k,k +1 ( t (cid:48) ; T ) = I k +1 ( t (cid:48) ; T ) I k ( t (cid:48) ; T ) . (A.23)In the main text, we always refer to the ξ ( t (cid:48) ; T ) correlation length except for Section 5.6.2 ,where we evaluate ξ ( t (cid:48) ; T ) . B. The Josephson length
For the reader’s convenience, we reproduce here the interpolation proposed in Ref. [3] of thedata obtained in Ref. [2] for the replicon exponent as a function of the Josephson length andthe correlation length.The Josephson length, (cid:96) J ( T ( J ) ) , scales as (cid:96) J ( T ( J ) ) = b + b ( T ( J )c − T ( J ) ) ν + b ( T ( J )c − T ( J ) ) ων ( T ( J )c − T ( J ) ) ν , (B.1)where T ( J ) is the temperature in IEA units, T ( J ) = TT c T ( J )c T ( J )c = 1 . , (B.2)and we have included analytic ( b ) and confluent ( b ) scaling corrections with ω = 1 . and ν = 2 . [27]. The numerical coefficients are: ω = 1 . , ν = 2 . , b ≈ . , b ≈ . , b ≈ . . (B.3)The replicon exponent θ ( ξ ; T ) depends on both the temperature and the correlation length ξ through the crossover variable ˜ x = (cid:96) J ( T ) ξ ( t w ; T ) . (B.4)In fact, θ (˜ x ( ξ ; T )) can be well interpolated as, θ ( x ) = θ + d (cid:16) x e x (cid:17) − θ + d (cid:16) x e x (cid:17) − θ , (B.5)where θ ≈ . , e ≈ . , d ≈ . , (B.6) e ≈ . , d ≈ − . . (B.7)(B.8) C. Sample dependence of the non-linear scaling results
We demonstrate that the non-linear scaling results are sample independent for the case H = 0 . , t w = 2 . , at T = 0 . . We simulate four independent samples and, for eachone, we build the relaxation function S ( C ; H ) , see Section 4.2 and Appendix A.3. We exhibitthem in Fig. C1 where we report only the case for k = 8 . To compare peak regions of therelaxation function, S ( C, H ) , we shift the lowest curves, namely S and S , vertically. Anamplification of the peak region is shown in the inset of Fig. C1. As one can observe, there EFERENCES . . . .
024 0 . . . . . . . .
023 0 .
48 0 . . S ( t n e w , t w ; H ) C ( t new , t w ; H ) S S S S Figure C1: Plots show the behaviour of S ( C ; H ) for four independent samples for H = 0 . , t w = 2 . at T = 0 . . The peak area is enlarged in the inset after a vertical shift.Table C1: Values of t eff H and C peak ( t w ) for four independent samples in the case H = 0 . , t w = 2 . and T = 0 . . Sample log (cid:0) t eff H (cid:1) C peak ( t w ) S S S S is a sample dependence of the peak position. We report the estimates of C peak ( t w ) for eachsample in Table C1. Note that the sample S is the one analysed in the main text. We extractthe effective time t eff H for each sample, according to Eq.(16). They are listed in Table C1. Thesample dependence found for the C S i peak ( t w ) values is seen in the t eff H values too. Accordingly,we repeat the analysis of Section 5.4 using, as input parameter for extracting t eff H , the C S peak ( t w ) value shown in Table C1. We analyse the effective time ratio ln t eff H t eff H → according to Eq. (30).We then compare the scaling behaviour for the two values of C S i peak ( t w ) . The two sets of dataare statistically compatible, see Fig. C2. This implies that the physical scenario is not affectedby the small uncertainty in the determination of C peak ( t w ) . We therefore assert that the scalingresults are sample independent. References [1] Cui W, Marsland R and Mehta P 2020
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