Equilibrium magnetization of a spin glass : is mean-field theory valid ?

2014 Measurement of field cooled magnetization of a AgMn 10.6 % alloy as a function of temperature between 4.2 K and 50 K and magnetic field from 26 to 9 000 gauss is performed in order to link the low field and high field regimes of a spin glass. This can possibly help to define a phase diagram in the (H, T) plane as proposed by De Almeida and Thouless and Parisi and Toulouse. A closer investigation of the vicinity of Tg, above Tg, allows to compare the magnetization curves with the specific prediction of the mean-field model of Sherrington and Kirkpatrick. A qualitative good agreement is found. J. Physique LETTRES 43 (1982) L-45 L-53 15 JANVIER 1982, Classification Physics Abstracts 75 . 30H 75 . 30K Introduction. In this letter [1] we present some systematic measurements of the equilibrium magnetization of a spin glass alloy above and below the spin glass temperature Tg as a function of temperature and magnetic field The first purpose of these experiments stems from the need of a detailed survey, for a given system, of the « low field » behaviour of the magnetization together with the « high field » one : we define these two regions below. Indeed previous work on spin glass alloys has very often been confined in one of these two regimes : for instance the work of Nagata et al. [2] or Guy [3] is certainly a low field study; on the contrary the work of Smit et al. [4] offers a very high field determination of M. A notable exception is the early work of Hirschkoff et al. [5] continued by Thomson and Thompson [6]. However these authors work within a concentration and temperature range two orders of magnitude smaller, and the necessary connection between their results and those aforementioned rely on the validity of scaling laws for the spin glass parameters [7]. Finally the work of Tholence and Toumier [8] spans a quite large area of field and temperature but is not systematic enough. The theoretical motivation of our work is based on the possible existence of a phase diagram Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:0198200430204500 L-46 JOURNAL DE PHYSIQUE LETTRES for the spin glass phase in the (H, T) plane as originally proposed by De Almeida and Thouless [9]. Specifically, at any temperature T below Tg, there exists a magnetic field 7~(T) above which the system can be described by the mean-field equations of Sherrington and Kirkpatrick [10], whereas below that field a specific hypothesis concerning the structure of the spin glass phase must be done. This line of thinking is entirely contained in the work of Parisi and Toulouse [11] ] and rests on the existence of a well defined line 77e(T) ~ the (H, T) plane separating the spin glass phase from the paramagnetic phase. To what extent can this be made apparent in a real spin glass alloy ? The second purpose of our work is to look very carefully at the vicinity of Tg, above T g, in order to be able to compare the measured magnetization with the mean-field prediction from the Sherrington-Kirkpatrick model [10] as analysed by Suzuki [12]. This author has shown that the phase transition at Tg which manifests itself in small finite field by a divergence of the parameter q(H, T) implies also a divergence of the non linear magnetization i.e. the term beyond the (non divergent) susceptibility. By measuring the magnetization above Tg one avoids experimentally the problem of the occurrence of irreversible effects and one also avoids theoretically the question linked to the symmetry breaking of the order parameter as it is well established [13] that above Tg a single value of q(H, T) is obtained. It is thus very tempting to test the meanfield theory for spin glasses in this region. Curiously enough very little work has been devoted to this specific issue in the past : only the recent data of Miyako et al. [14], Berton et al. [15], Barbara et al. [16] are pertinent although a lot of older data could also be used in the same spirit [17]. The final aim of this type of measurement would be to establish experimentally the existence of an equation of state for the spin glass transition as originally proposed by Suzuki [ 18]. 1. Equilibrium magnetization results. In order to compare the magnetization with some specific predictions [11, 12, 19] we want to determine what is the equilibrium magnetization of our spin glass below Tg. From lack of a better criterion we take as a definition of equilibrium the time independent magnetization i.e. the field cooled magnetization below Tg. By doing so we do not prove it to be at equilibrium in the thermodynamic sense : this question is precisely being raised by the calorimetric high precision work of Fogle et al. at Berkeley [20]. However we do not know of any other means to obtain an other time independent magnetization state. We note that the results of Knitter and Kouvel [21] who addressed exactly this question by a different technique are not inconsistent with our definition : their result is similar to the extrapolation to infinite time of a time varying magnetization. The fact that the field cooled state is time independent and history independent has been established by many people : Tholence and Toumier [8], Guy [3], Nagata [2] and more recently Malozemoff et al. [22] over a longer period of time. Our measurements were done on a single crystal of AgMn 10.6 % previously used for the X-ray determination of the short-range order [23]. It was obtained after quenching in water from below the melting temperature and after spark cutting a cylinder 8 mm in height and 5 mm in diameter. The homogeneity of the Mn concentration was checked by an X-ray microprobe analysis carried out on a separate piece. The magnetization was obtained with a vibrating sample magnetometer with an accuracy better than 1 % and a sensitivity of about 10-4 emu. It was equipped with a He gas flow variable temperature cryostat The He gas flow temperature was stabilized to better than 0.1 K with reference to a platinum thermometer attached to the cryostat tail near the sample position. The sample temperature was continuously recorded with another platinum thermometer located a few millimeters above the sample position on the sample holder. In this way very reproducible results could be obtained although no absolute calibration of our temperature was carried out The main source of systematic error came from small contributions to either M or H which tend to blow up the ratio M/77 as 77 -~ 0. The magnetic field of our magnet was determined by a fit to a Curie law at high temperature. L-47 EQUILIBRIUM MAGNETIZATION OF A SPIN GLASS Figure 1 displays the ensemble of our results for our fixed field, variable temperature measurements. In order to exhibit the Curie-Weiss behaviour of the high temperature part (T > Fg) we have plotted the inverse of the apparent susceptibility, /7/M, as a function of T, showing a Curie-Weiss temperature of + 13 + 0.5 K for this alloy. The field dependence of this apparent susceptibility is remarkable : indeed a very flat ( 1 %) « plateau » appears below Tg for fields less than 1000 gauss and a small « cusp » like minimum (5 % deep at most) is present right near Tg for the lowest field used (26 gauss). At higher fields the transition between the CurieWeiss regime and the spin glass plateau is progressive and a smooth rounding is obtained. None of these magnetization data are time dependent provided that they are obtained with field cooling through Tg. Once the lowest temperature was obtained (4.2 K) the magnetization upon subsequent warming was observed to be identical to that measured during cooling (provided that the field was kept fixed at all times). This fact further stresses the equilibrium nature of the magnetization. These observations allow one to link the low field results obtained [2] to the « high » field ones, especially those of Hirschkoff, Symko and Wheatley [5]. A qualitative fit of these results can be obtained for the region T > T g with the prediction of Fig. 1. Inverse of the apparent susceptibility HIM for AgMn 10.6 % as a function of magnetic field as indicated on each set of point symbols (in gauss). These data points have been obtained at fixed field and variable temperature ie. by field cooling through Tg. As explained in the text these represent an equilibrium state of the spin glass phase. The insert shows an attempt to determine a phase diagram in the (H, T) plane. We define (arbitrarily) the boundary of the spin glass phase by the point of the M(T ) curve departing by 3 % from the low temperature value (arrows). L-48 JOURNAL DE PHYSIQUE LETTRES a simple Sherrington-Kirkpatrick set of equations including external field [26]. Such a qualitative fit is presented in figure 2 but no attempt is made to make this into a quantitative fit In particular we have left the parameter Jo = 0 whereas the experimental data clearly show a Curie-Weiss behaviour. The salient features from figure 2 are the extreme field sensitivity to the departure from the Curie law near Tg and the location of the De Almeida-Thouless instability point which occurs very near the maximum of magnetization of the S. K. solution as noted by Vannimenus, Parisi and Toulouse [19] (which corresponds to a minimum of the quantity HIM). If we further Fig. 2. Calculated [26] Sherrington-Kirkpatrick magnetization for fixed field and variable temperature expressed in units of Tg. The curves represent eqs. (1) and (2) until the De Almeida-Thouless point (open circle) below this point the magnetization is represented as independent of T according to the Parisi-Toulouse hypothesis [11]. A better qualitative fit can be obtained with figure 1 by introducing the S. K. parameter Jo = 0.33 to account for the finite Curie-Weiss temperature observed assume, according to the Parisi-Toulouse hypothesis, that the magnetization is temperature independent below Hc(T) then it becomes a quite difficult challenge to pin-point experimentally the De Almeida-Thouless line because the point where the magnetization plateau ends is experimentally ill defined As a working criterion we have taken those points where the magnetization departs by 3 % from the low temperature plateau [24] : this is indicated by an arrow on figure 1 and enables one to draw a set of points in the (H, T) diagram (insert of figure 1) with admittedly increasing error bars along the T axis as H becomes larger. From this data no serious test of the De Almeida-Thouless relation is possible. We note however that the scale of fields is certainly wrong as the data presented for 9 kgauss mimic approximately the mean field result for H = I in reduced units ie. 99B H = kB Tg for a spin of 1/2. In our case Tg = 37.5 K so this would correspond to a field of the order of 250 kgauss. At this stage we can only list the factors we should take into account to reduce this seemingly large discrepancy : i) the fact that Mn has a spin S ~ 2 [1]; ii) the slope of the Curie law indicates that the effective moment corresponds to L-49 EQUILIBRIUM MAGNETIZATION OF A SPIN GLASS 3 Mn atoms because of the presence of short-range order as revealed by X-ray diffuse scattering [23]; iii) the finite Curie-Weiss temperature : we feel that these factors taken together will go in the right direction but will not be sufficient to account for all the discrepancy. 2. Non linear magnetization near Tg. The Sherrington-Kirkpatrick equations [10] in presence of a field H and for Jo = 0 are : with J = kTg and Mo the saturation magnetization. A straightforward expansion of these equations for small values of the arguments above Tg gives :