On a classical spin glass model

Z. Phys. B - CondensedMatter 50, 311-336 (1983) Condensed Zeitschrift for Physik B Matter 9 Springer-Verlag 1983 On a Classical Spin Glass Model J.L. van Hemmen, A.C.D. van Enter, and J. Canisius Sonderforschungsbereich 123, Universitit Heidelberg, Federal Republic of Germany Received December 23, 1982 A simple, exactly soluble, model of a spin-glass with weakly correlated disorder is presented. It includes both randomness and frustration, but its solution can be obtained without replicas. As the temperature T is lowered, the spin-glass phase is reached via an equilibrium phase transition at T - - T I. The spin-glass magnetization exhibits a distinct S-shape character, which is indicative of a field-induced transition to a state of higher magnetization above a certain threshold field. For suitable probability distributions of the exchange interactions. (a) A mixed phase is found where spin-glass and ferromagnetism coexist. (b) The zero-field susceptibility has a flat plateau for 0_<T_< T~ and a Curie-Weiss behaviour for T > TI. (c) At low temperatures the magnetic specific heat is linearly dependent on the temperature. The physical origin of the dependence upon the probability distributions is explained, and a careful analysis of the ground state structure is given. I. Introduction A spin glass is a disordered, magnetic system with a well-defined freezing temperature Ts such that for T < TI the magnetic moments are frozen in random orientations without a conventional long range order. Archetypical spin glasses are AuFe and CuMn, where Fe and Mn are the magnetic impurities (the spins) randomly distributed in a non-magnetic, metallic host [1]. There is some, at least experimental, consensus [2-4] that one has an equilibrium phase transition at T = Ty. In this paper we aim at providing a soluble model [5] to describe the spin-glass transition and the spin-glass phase, with particular emphasis on metallic systems. We will first review some characteristic features of archetypical spin glasses, emphasizing their response to an external field h. (a) T>> TI. The susceptibility )~ has a typical CurieWeiss behaviour, i.e., )~=C/(T--O) with 0 > 0 (or 0<0). The earliest experiments, which date back to the mid-fifties [6], mainly consider CuMn in a temperature range between room temperature and TI, and give 0 < 0 < TI. A positive Curie-Weiss temperature 0 would indicate that a net ferromagnetic exchange is present in C guMn. These early results on C__u_uMnhave been confirmed recently by Morgownik and Mydosh [7] through a series of detailed measurements, which also showed a linear dependence of 0 upon concentration (c between 1 and 6 at. % Mn). (b) T ~ TI. Experimental evidence for the spin-glass phase was first provided by a sharp cusp [1] in the zero-field (initial) susceptibility Zo. If the spins were frozen for T < T I , it should be hard to magnetize them; hence the cusp. Subsequent experiments [8] pointed out the possibility that the sharp cusp is not an equilibrium phenomenon and one should have a flat plateau for 0_<T~< Tz. For a slightly different view-point, see Mulder et al. [91. The cusp in X0, as a point of non-analyticity, is not in doubt however. So we are left with the question whether we have an equilibrium phase transition at T = T p If so, we would have a new thermodynamic phase at T < T f , the spin glass, and not a nonequilibrium phenomenon. At the moment there is increasing experimental evidence [2-4, 45] that we do have a 312 conventional phase transition. Consequently we will assume that using the formalism of equilibrium statistical mechanics is legitimate. (c) T< TI. In the absence of an external field (h = 0) there seems to be no discernible long-range spatial ordering of the spins, at least not in the pure spin glass phase. The time-independent magnetization re(h) exhibits, as a function of the external field h, a distinct S-shape character [10, 11] which begins at TI and becomes more pronounced for further decreasing T. That is, re(h) is convex if O<_h<_ht and concave on the right of ht, which acts as a point of inflection. This behaviour is indicative [10] of a fieldinduced transition to a state of higher magnetization above a certain threshold field, ht, whose value increases as the temperature is lowered. In addition, once the external field has been larger than ht, the high-magnetization state does not return isothermally to the initial low magnetization state upon subsequent removal of the field [-10] - at least not within the time available. It is to be noted that hysteresis phenomena have been observed quite frequently in spin glasses below T~. Within the context of equilibrium statistical mechanics we will interpret hysteresis as the occurrence of a metastable state. (d) Whatever the temperature, thermodynamic quantities like TI itself, the magnetization re(h), the susceptibility X, and the magnetic part of the specific heat are reproducible, i.e., two alloys whose microscopic structures differ but whose macroscopic constitution and preparation are identical (same concentration) give the same experimental outcomes, So one can simply take a specific sample and need not average over an ensemble of them. These findings have long represented a challenge to the theory, but to meet this challenge we have to pay careful attention to the interactions between the magnetic moments and isolate the characteristics we believe to be essential in a typical spin glass like A__~uFe. First, the concentration of the magnetic moments (Fe) in the metallic host (A____~u)is rather low, so the distance between them is fairly large, and their locations are only weakly correlated. Second, the spins interact mainly indirectly via the RKKY-interaction [12] which is to be noted (i) for its tong range and (ii) for the strongly oscillating sign as a function of the distance. Finally there is a direct, ferromagnetic coupling [-2]. Since the anisotropy considered by Walstedt and Walker E13] is also ferromagnetic, it may be included here as well. One may assume the spins are on a regular lattice and take the interaction as random [141. The simplest assumption, a _+J or Gaussian nearest-neighbour interaction without external field, is unlikely to produce a phase transition at positive temperatures J.L. van Hemmenet al.: Classical Spin Glass Model [15]. There is, however, yet another reason why it is advantageous to consider long range interactions: a finite range interaction does not allow metastable states [16]. A comment might be appropriate. We are working within the context of equilibrium statistical mechanics. An equilibrium state is specified through the Gibbs prescription: the canonical ensemble. Taking the thermodynamic limit we may imagine an infinite system (so as to get a phase transition) and say it is in thermal equilibrium if each finite subsystem is in thermal equilibrium with its surroundings. Let us take the conventional free energy with a minus sign. Then the true equilibrium state maximizes the free energy. A metastable state is such that each of its finite subsystems is in thermal equilibrium with its surroundings, but it does not maximize the free energy. The assertion is that metastable states do not exist in finite-range models. Moreover, a metastable state as a smooth continuation, say in the temperature, of an equilibrium phase corresponding to different thermodynamical conditions cannot exist either. Only long-range interactions allow metastable (equilibrium) states and hysteresis. This being so, why not take an infinite-range interaction? Though intuitively reasonable, the infiniterange Gaussian interaction proposed by Sherrington and Kirkpatrick [,17] does not yet allow a simple analytical treatment - in spite of considerable efforts [18] which, however, did show a spin glass transition and pointed out the possibility of a "mixed" phase. We, therefore, propose a new Ansatz, which is consistent with the randomly oscillating, long-range, RKKY-interaction and incorporates a ferromagnetic anisotropy. The model itself [5] is introduced in Sect. II and the nature of the disorder is analyzed in some detail. Solving the model means finding a closed expression for the free energy per spin. This is done in Sect. III by using a new technique which has been developed by Donsker and Varadhan [34]. We illustrate the method with a simple discrete probability distribution in Sect. IV, postponing further discussion of the general case to Sect, VI, and evaluate the phase diagram, which includes a mixed phase and already predicts the S-shape of the spin-glass magnetization. Continuous probability distributions give rise to new phenomena: in the Gaussian case, there is no mixed phase and the model predicts a flat plateau for the initial susceptibility )(o if 0 < T_< To. We extensively comment on these phenomena in Sects. VI-X. With the temperature T as a parameter one may consider a phase transition as a bifurcation of states which maximize the free energy (once again notice our convention with a minus sign). It turns out that J.L. van Hemmen et al.: Classical Spin Glass Model 313 we can use the ground states (T=0) to label the various branches which extend to positive temperatures and represent unstable, metastable, and stable equilibrium states. We, therefore, start our bifurcation analysis by studying the ground states in Sect.VII. The ground states rank among the extremal points of a certain convex set in the plane: the extremal set. Here the novel effects of the randomness become particularly apparent. Section VIII is devoted to the bifurcation analysis proper. In Sect. IX we deal with magnetization, linear and nonlinear susceptibility, and specific heat. Some comments are to be found at the end of this paper. II. The Model Ferromagnetic bonds favour a parallel alignment of the spins whereas antiferromagnetic bonds favour an antiparallel alignment. The competition of ferromagnetic and antiferromagnetic bonds is inherent to a spin glass, and a realistic model should incorporate this frustration [19, 20], which is so closely related to the RKKY-interaction. Furthermore, we should include a ferromagnetic anisotropy which breaks the spin-glass order. Let us, therefore, start with the Hamiltonian HN - Jo ~ S(i)S(j)- ~ 4jS(i)S(j)-h ~ S(i). (2.1) N (i,j) (i,j) Its height is determined by the mean free path of the conduction electrons which mediate the RKKY-interaction. Let Prob {A} denote the probability that the event A occurs. If we take ~ and t/ as _+1 with equal probability, we get a probability distribution of the coupling-constants which already has the required form, Pr~ = 88 Pr~ = 89 (2.3) Prob {J/j > 0} _1_ --4' i.e., the central peak dominates. For more complicated distributions of { and ~/, in particular the Gaussian case, one may take advantage of [22]. We now turn to the collective behaviour of the Jij s. Suppose for the moment that { and i/ are Gaussian. If the N-l{irly, l<i,j<N, were also independent Gaussians, we would find FN(~)=lexP{N~ChJ~itlJ}> =exp{-2~ ~2 } . (2.4) The angular brackets denote an average over the randomness. Because of (2.2), e =(~u) must be a real symmetric matrix. The characteristic function Fu determines the stochastic behaviour completely [23], so it suffices to compute FN as N--* oo. Instead of (2.4) an approximate relation holds. To see this let i This describes N Ising spins (S(i)= + l ) interacting with each other in pairs (i,j), and with an external magnetic field h. A direct ferromagnetic coupling has been incorporated via J0 [17]. The 4j's contain the randomness, A 4~ = J { ~ , t/j + ~j,h} which is the right side of (2.4). The key to understanding (2.6) is that for jq=k we have to take the weighted inner product (2.5) of two different vectors which are randomly chosen from an N-dimensional space with N ~ oo. For a large class of ='s the Ajk , j :t=k, then approach zero, and (2.6) holds. Thus the Ju's are nearly independent (weakly correlated). On the other hand, using the same approximation as before we would get in the case ~=r/i [24] (2.2) where the {~'s and t/fs are independent, identically distributed random variables with an even distribution around zero and a finite variance, say one. We first check the stochastic behaviour of the Ju's and show in Sect.V that about half of the spins belong to a fully frustrated configuration. The stochastic behaviour of the J~fs has two aspects. First, we have to consider a single 4j and check that its distribution has the required form. Second, we have to study the correlations between many, different Ju's and show that they are weak (cf. Sect. I). The form of the distribution of the RKKY-coupling constants has been indicated by Binder and Schr6der [21]. It is indeed even, and highly peaked at zero because of the long range of the potential. 1S1 Lcq, J k = N ~ N i=l and a = } O[ik (2.5) (Ajk). Then FN is given by FN(r0 = det (]1 + A)- ,/2 ~ exp { - 89Wr A}, FN(~) ~exp { - N ~. cqj} (2.6) (2.7) and we would have neither frustration nor independence. Turning to the other extreme we take ~ and i/as + 1 with equal probability and compute FN(e) once again, 314 J.L. van Hemmen et al.: Classical Spin Glass Model FN(~)=2-N ~ cqjrlij . 1~ COS (2.8) {r/j= _+ 1} i = 1 The vast majority of the q-configurations having approximately as many positive as negative t/'s, the weighted sums N - I ~ ~ijtlj should be small ( N ~ oo), J - fif (fi) = lim 1 In Tr exp { - fiHN}, so that we are left with 1 2 2 (2.9) exp{- 89 jk {r/j= _+ i} (3.2) N -, oo N FN(~)~2-Nr/j_~_+_,exp{-- 89 (N~C~zJrlJ) } =2-N and {~i, t/j} is afixed random configuration. One has to keep in mind that [26]: "No real atom is an average atom, nor is an experiment ever done on an ensemble of samples." We will show that the free energy per spin f(fi), which is defined by As before we keep only the diagonal terms in the exponent as N ~ oo and obtain FN(~) ~ exp { -- 89 (2.10) But what is the gist of what we have done in deriving (2.6), (2.9) and (2.10)? Whatever the probability distribution, by chopping the cumulant expansion [25] of the characteristic function for both the ~and the tt-distribution after the second-order term, invoking the smallness of the various averages, we will be left with the very same exp{- 89 as in (2.6) and (2.10). Apparently the approximation (2.10) has some universality. Accordingly some properties of our spin glass model will not depend on the details of the probability distribution of ~ and t/. For instance, the transition paramagnet-spin glass is only determined by their second moment, whatever their distribution (i.e., flcJ~ 2> 1 where fi~ is the inverse critical temperature; see in particular Sect.VIII). Higher order correlations do contribute however, and determine finer details of the phase diagram like the occurrence of a stable mixed phase. In a similar vein one shows [cf. Eq. (2.7)] that the random variables [24] N - 1~i ~j, 1 __<i, j <_N, are not appropriate as weakly correlated coupling-constants in a frustrated spin glass - in contrast to our choice (2.2). : exists with probability one and does not depend on the specific sample of ~'s and r/'s we have taken. Moreover we will derive a closed expression for the right side of (3.2). In so doing we first consider the physics, then the mathematics of our method of solution, and finish this section with a simple example. Let us rewrite Hu in the form - - f i l l N -~ N[ 89 om2 + KqlNq2u + HmN] = NQ(m) (3.3) with flJo = K o , fiJ=K, and fih=H. Instead of the N spins we would like to use the three-vector m =(m N, qlN, q2u) as a new set of variables in doing the trace in (3.2). But to do so we need something like a "Jacobian". More precisely, fixing m=(mu, qlN, qZU), which still depends on all the spins {S(1), ..., S(N)}-S, we fix the energy E=NQ(m). For the time being we write E =E(S), showing its ultimate dependence upon S. Now exp { - fiE(S)} is the quantity we have to sum, so fixing E we need the degeneracy of the level E. This degeneracy is determined by the entropy [27] SN(E) through the expression exp{SN(E)}. As the entropy is extensive we may write SN(E)=Ns where s is the mean entropy. Suppose we could write s=s(m). Then we expect that, as N - , o% we could transform the trace, which is a sum over the 2N spin configurations, into a threedimensional integral over m = (m~N,q 1N, q 2 N ) , Trexp{-fiHN} ~ ~ dmexpN{Q(m)+s(m)}, (3.4) N3 so that by using the Laplace method [28] we would end up with - flf(fl) = max {Q(m) + s(m)}. (3.5) Ill IIL Calculation of the Free Energy In calculating the free energy we have to take the thermodynamic limit N--,oo. Only then the model becomes exactly soluble. It is a mean-field model where, in the limit N ~ o o , the "means" (or order parameters) are given by 1 N m~ = V • S(i), *~ i = 1 1 1 N qlN = ~ ~ ~i S(0, i=1 N q2s = ~ j~=ltljS(j), (3.1) Equation (3.5) seems just a rewriting of the wellknown thermodynamic relation f = u - Ts. Indeed it is. But we still have to show that the limit (3.2) makes sense and gives a non-random answer with probability one, and we still have to prove the existence of an entropy function s(m) which does the job (3.4). Finally, and most importantly, we still have to calculate s(m) explicitly. To do all this we will appeal to the theory of large deviations. What is a "large deviation"? Suppose the simplest possible case: we have a sequence of independent, J.L. van Hemmenet al. : Classical Spin Glass Model 315 identically distributed stochastic variables 0.(1), 0-(2), 0.(3),..., and consider their arithmetical means. As N--~ o% N SN-~ 1 /=~10.(i)__~(0.) (3.6) with probability one [29]. The event { ] S N - @ ) [ > e } is called a large deviation since it becomes more and more improbable as N - - , ~ ( e > 0 and fixed). There is a classical result, obtained by Cram6r in the late thirties and extended by Chernoff [30], which may be used to calculate the probability of a large deviation. We introduce two functions, c (t) = In (exp (t 0.)) (3.7) and c*(m) = sup {rot- c(t)). (3.8) --(X)<t < +~ -~HN= 89 ~ ~ S(i) . [. c(t) is convex and so is its Legendre transform c*(m) [31]. We may also assume ( a ) = 0 The function by subtracting the mean from 0.. Take e > 0 and let Prob{A} denote the probability of the event A. Then we have, according to Cram6r-Chernoff, lira 1 In Prob {SN_>_e} = - c*(e) (3.9) where c*(m)>O, with equality if and only if m = 0 (under some mild restrictions). For the events {SN<e } with e < 0 an analogous formula holds. The limit N ~ ov is essential. One may say that the probability of a large deviation goes to zero exponentially fast. But more can be said. Let IE{X} denote the mathematical expectation of the observable X. For the moment it is just (X}, but the point is that later on we will use IE{. } for the (configurational) average with respect to the spin variables and ( - } for the random variables and ~/. Note IE(~I)= 1; so IE{. } is normalized. Let Q be a smooth function. We would like to evaluate, say, IE{expNQ(SN) } for large N. Now (3.9) suggests that, as N---, o% Prob{m<=SN<m+dm}~exp{-Nc*(m)}dm (3.10) so that +oo IE{expNQ(SN)}~ ~ dmexpN{Q(m)-c*(m)}, (3.11) -oo and thus lim 1 has been developed by Varadhan [32] in a slightly more general context. Before proceeding we will apply this result to Ising spin systems and introduce some useful notation. The Ising spins S(i) may be considered as independent, identically distributed stochastic variables which are ___1 with equal probability. Given N we divide the trace by 2 N so as to get a normalized trace: IEs{X}=2-NTr{X}; it is the mathematical expectation with respect to the probability distribution of the spins. We will use this convention throughout what follows. It is completely harmless except for the fact that one has to add In 2 to get the physical entropy. As a simple, even trivial, application of the above formalism we now evaluate the free energy of the Curie-Weiss model. Its Hamiltonian is given by lniE{expNQ(SN)}=max{Q(m)_c,(m)}. (3.12) Let us write S N for the expression between the curly brackets. For the present purposes we may identify S(i) and 0.(i); cf. Eq. (3.6). Then the partition function may be rewritten IEs{expNQ(SN) } with Q(m) =l(/~Jo)m2. The function c(t) is readily obtained, c(t)=ln[cosh(t)], and so is its Legendre transform c*(m), which we denote for future use by g(m): #(m)= 89 +m)ln(l +m)+(1-m)ln(1-m)] (3.14) if Iml < 1, and #(m)= + oe elsewhere. The model is solved by invoking (3.12). It has a phase transition at /~Jo = 1. There are of course much cheaper ways to solve it [33]. We now turn to more difficult problems which require more powerful methods. In a series of remarkable papers Donsker and Varadhan [34] tackled the problem of developing an infinite-dimensional Laplace method and finding the corresponding entropy functional for stochastic processes of increasing complexity - their levels I, II, and III. What we need here is their level I, and we will present it on the basis of an analogy with the Cram6r-Chernoff theory. We return to (3.1), and define WN=(NmN, NqlN, Nq2N). The three-vector Wu contains the Ising spins as stochastic variables, while the ~i's and t/fs have been chosen randomly according to their distribution and will remain fixed throughout what follows. Specifically, 0.(0 is taken to be S(i), ~iS(i), and tliS(i) respectively. As before we define a c-function, c(t) = lim 1 In IEs{exp(t-W,)}, N~oo Equations (3.10)-(3.12) represent the main idea behind an infinite-dimensional Laplace method which (3.13) i=l (3.15) X where t = ( t l , ta,t3) is also a three-vector. For Ising spins we directly obtain 316 J.L. van Hemmenet al.: ClassicalSpin Glass Model c(t) = (ln[cosh {tl + t z ~ + t s ~ } ] ) , (3.16) with probability one [29]; we have to average over one ~ and one ~1. The function c(t) is convex and so is its Legendre transform c*(m) [31]. Taking advantage of (3.3) we then find, as N ~ o% - fif(fl) = max {Q(m) - c*(m)}. (3.17) m The maximum in (3.17) is realized for a certain m =(m, qDq2), does not depend on the specific configuration of ~'s and t/j's we started with, and is such that the negative of c*(m) is the system's mean entropy s(m). The relation (3.17) is basic to all that follows. It will be analyzed further in Sect.VI. Though the generalization of (3.15) and (3.16) to the case of n-vector models is immediate, we will not pursue this idea here. The present model offers a real "mean-field" description of a spin glass - in contrast to SK [17]. The point is that SK also requires a theory of fluctuations instead of means like those in (3.1), as is brought out by SK's scaling by N -I/2 instead of N-1. We remind the reader of the simplest example where fluctuations play a nontrivial role, the central limit theorem. One may apply the central limit theorem only if one averages [35]. The large number of solutions of the TAP-equations [36], for instance, indicates that a probability-one-behaviour might be illusory if T < T~, so that here too one must average outside the logarithm to get a unique answer. If so, this would contradict the experimental fact that thermodynamical data are reproducible (Sect. I) [2]. On the other hand the quantity c(t) does not depend on the randomness any more and is given by the expression (3.16) for almost every sample, i.e., the samples where the fimit (3.15) does not exist or (3.16) does not hold have probability zero and may be discarded (they do not occur). The final part of this section may be skipped if the reader desires so, though reading it will be helpful in understanding Sect. IV. Some years ago Luttinger proposed an unfrustrated spin glass model [24], where he took ~ = r h and ~ = _+1 with equal probability while Q(m) was a certain quadratic form in ms and qlN, say Q(m)= 89 Qm; q2u did not play any role yet. The corresponding c-function is readily obtained by putting t 3 = 0 in (3.16), c(t) = 89 1 +t2) ] + l n [cosh(t I - t2)]} , (3.18) Defining new independent variables x~ and x 2 through x l = t l + t 2 and x 2 = t l - t 2 , we may write x = M t for a real symmetric 2 x 2 matrix M such that M -1 = 89 We then replace the supremum over t in (3.19) by a supremum over xl and x2, so that the problem decouples, c*(m)=sup 89 ( ~=~ln[cosh(x~)]},l that is, c*(m) = 89 {C(al)+ g(a2)}, c*(m) = sup { m . t - c(t)). (3.19) (3.21) where use has been made of (3.14), with a l = (ml+m2) and a 2 = ( m l - m 2 ) as Luttinger's a~'s. Their natural domain is ]a~]<l, i=1,2. Luttinger's Eqs. (16)-(18) are easily recovered once the maximization problem (3.17) has been rewritten in terms of the ai's , - flf(fl) = max a. ( 89189 lail < 1 g(ai) 9 (3.22) i Note that the Legendre transform (3.19) has become trivial by introducing new independent variables. This little trick will be used again in the next section. IV. A Simple Discrete Probability Distribution Let us take ~ and ~ as _+1 with equal probability. This case will be assumed for now, postponing further discussion of the continuous case until Sect.VI. The random variables ~ and q have mean zero, variance one, and give rise to a sensible discrete probability distribution for the coupling constants J~.j (cf. Sect. II). To apply the large deviations result (3.17) we need the Legendre transform c*(m). We, therefore, use a little trick. We introduce a fourth order parameter, 1 N q3N = ~ ~=~~ tl~S(i), (4.1) whose physical significance will soon become apparent (Sect.V). The quadratic form Q, which represents the energy, may be considered as a function of q3N also: Q simply does not depend on q3n. The new order parameter only serves to calculate efficiently c*(m), the second constituent of (3.17). Defining the four-vector WN=(NmN, Nq~N, Nq2s, Nq3N), we first have to evaluate c(t) = lim t In IEs {exp(t. WN)} and we have to calculate its Legendre transform (3.20) N~oc (4.2) l'q where t=(tl, t2,t3, t4) is also a four-vector. With probability one we find J.L. van Hemmen et al.: Classical Spin Glass Model 317 c(t) = ( l n [cosh(t~ + t 2 ~ q- t 3 r] q- t4. ~ r/)]). (4.3) where The Legendre transform of c(t) is given by [cf. (3.16)] Mq)M=K c*(m) = sup 88 {4m. t - l n [cosh(t 1 + t 2 + t 3 + t4.)] t \ct-2 a-2 ~+2 c~ a+2 ~-2 c~ 9 (4.10) ~+2/ - In [cosh(t 1 + t 2 - t3 - t4.)] - I n [cosh(t 1 - t 2 + t 3 - t4) ] - In [cosh(t I - t 2 - t 3 + t4.)]}. (4.4) With new independent variables x l , . . . , x 4 which are defined through x 1 = ( t 1 + t 2 + t 3 + t 4 ) , x2=(t 1 + t z - t 3 -t4.), etc., we m a y write x = M t for a real s y m m e t r i c 4 • 4 matrix M. Explicitly, M= 1 1 -1 1 -1 1 1 -1 (4.5) ' -1 The matrix M is such that M -1--- 88 We now replace the s u p r e m u m over t in (4.4) by a s u p r e m u m over Xl,..., x4., so that the p r o b l e m decouples, I c*(m)=suP 88 (Mm).xxde,4 { ~ ln[cosh(x/)] i= 1 } , (4.6) is fixed and K varies with the temperature. Stepping back for a first overview we note that the a-representation has several advantages. (a) T h e a's are independent. W e only have to require [ai[<=1. (b) T h r o u g h a = M m the four a's are closely related to the four physical order p a r a m e t e r s m, ql, q2, and q3, while conversely m = 8 8 (c) They enable a straightforward analysis of the free energy functional (4.9). The points which maximize the free energy functional are a m o n g those a's where the partial derivatives of first order vanish. Using the identity 89 in [(1 + a)/(1 -a)]=tanh-l(a), and subtracting the equations for a 2 and a3, t a n h - 1(a2) = 88 tanh-l(a3)= 88 1 + (c~- 2)a 2 + (e + 2)a 3 + ca4} + H, q-(~ + 2 ) a 2 + ( ~ - 2 ) a 3 + ~a J + H , that is, (4.11) 4- c*(m) = 882 g(ai), i=1 (4.7) we arrive at an interesting relation, t a n h - 1(a2) - tanh l(a3) = - K(a 2 - a3). (4.12) where use has been m a d e of (3.14) and a=Mm. (4.7a) Since c * ( m ) = + o o if [ a i [ > l for a certain % the natural d o m a i n of the ai's is the hypercube {la~[ < 1, 1 < i < 4}. Jensen's inequality implies o~(a) > 0; equality holds if and only if a = 0. W e assigned the negative of c*(m) to be the entropy. The conventional e n t r o p y is always nonnegative, and so is the e n t r o p y derived from g(a) by adding in 2. One simply has ln2-g(a)>0 if [ a [ < l . Except for a trivial constant (ln 2) there is no h a r m in using a normalized trace. It is to be anticipated from later sections that the conventions f l J = K , /3h=H, and o~=Jo/J are convenient. T h e n fiJ = e K . In addition, we m a y assume J > 0 ; otherwise put ~i--+-~i- T h e H a m i l t o n i a n m a y be written -fill N= 89 Qm+Hmu, 1 a - / ~ f ( f l ) = m a x xI ~ a1 . (~M~)M) l + H(a 1 + a 2 + a 3 + 7-14.} a4.) --i~C(ai) K al = t a n h ~ { ( ~ + 2 ) a 1 +2C~az+(Ct-2)a4+4h }, K a 2 = tanh ~- {c~(aa + 2a 2 + a4) + 4h}, (4.13) a4. = tanh K {(~ - 2) al +2~az+(~+2)a4.+4h } . (4.8) Q being symmetric, and the free energy is given by [all<=1 By a s s u m p t i o n K > 0 . If K = 0 , then a z = a 3. So let K > 0 . If a z > a 3 ( a z < a 3 ) , the right side of (4.12) is negative (positive) and the left side is positive (negative), which is a contradiction. H e n c e a2=a 3. The relation a 2 = a 3 implies and is implied by ql =qz=-q: there is no breaking of the s y m m e t r y between ql u n d qzSince a 2 equals a3, there remain three equations which determine the stationary points of the free energy functional. (4.9) T h e units have been chosen in such a way that J = 1. These nonlinear equations m a y be considered as fixed point equations because a solution a has to be such that a = g ' ( K ; a ) where g' is the right-hand side of (4.13); K is a p a r a m e t e r which is at our disposal. W e now discuss the various phases which occur as solutions of the fixed point equations. 318 J.L. van Hemmen et al.: Classical Spin Glass Model o.o -05 ~ ' " ' " 0.0 0.5 two values whose squares equal one. The ensuing analysis is presented in the next section. Here we summarize the results obtained so far. The phases of the system may be plotted in a phase diagram, like Fig. 1, whose horizontal axis is the eaxis (o:=Jo/J) and whose vertical axis is the K -1axis ( K - l = T/J). If the temperature is high enough, the system is in the paramagnetic phase. Lowering the temperature we may expect to enter the spin glass phase if we cross the line K - t = l [cf. (4.15)], or the ferromagnetic phase if we cross the line K = a [cf. (4.14)]. If 0<c~<1, we first reach the horizontal line K - z = 1 ; see Fig. 1. A mixed phase, if any, is not to be expected before we enter the intersection of the domains K - z < 1 and K - 1 < c~. Note, however, that we still have to decide which of the phases (SG, F, or II) maximizes the free energy. FEIRRO Jo 1.0 "~J Fig. 1. Phase diagram for ~ and i/= _+1 with equal probability. II is the mixed phase. Note that (1, 1) is a triple point [41]. There is no external field. The critical line SG-II and its continuation, the broken line, represent the curve (8.23) where the spin-glass fixed point bifurcates. The clot-dashed line indicates a (secondary) bifurcation of the ferromagnetic fixed point; see Sect. VIII V. The Mixed Phase In a ferromagnet we expect m ~ 0 , but q t = q 2 = 0 . The relations q z = q a = 0 imply and are implied by a~ = a 4. Plugging a 1 = a 4 into (4.13) gives aa=a2, so that we infer that the ferromagnetic phase (F) is characterized by a 1 =a2=a3=a4--~m and m = tanh K { e m +h}, (4.14) where m is the magnetization. If h--0, which we will assume from now on, a nonzero m occurs only if c~K> 1. In the pure spin-glass phase we expect m = 0 but q~ = qa = q 4=0. The magnetization m vanishing, at+2a2+a4=O, hence a 2 = 0 via (4.13), and thus ax = - a 4. Conversely, a 2 = a~ = 0 and a~ = - a 4 imply m--q~ = 0, so that the spin glass phase (SG) is characterized either by re=q3 = 0, or through a2=a3=0, a1 -a4=b , and b = tanh {K b}. (4.15) Moreover, q= 89 A nonzero q occurs only if K>I. If h = 0 and K is small enough, i.e., the temperature is high enough, a = 0 is the only solution to (4.13). Hence m = 0 , q = 0 , and we are in the paramagnetic phase (P). But what about a mixed phase? If there were a mixed phase (II), both m and q would have to be nonzero. To see how this might occur, without solving (4.13), we exploit the fact that ~ and t/ assume only Suppose we have N spins ( N ~ ) and take, as before, ~ and ~ as _+ 1 with equal probability. The N lattice points can be divided into two disjoint subsets according to the sign of ~ . We call the points with ~ f ~ = + 1 blue and the remaining ones, where ~ g ~ h = - l , red. Since { ~ r / j + ~ f i i } = ~ / 2 { l + ~ r h ~ f i i } = 0 whenever i and j have a different colour, the r a n d o m interaction only connects points of the same colour. The ferromagnetic interaction is "colourblind". Which one wins depends on c~=Jo/J and K- 1= T/J. The blue and red picture offers us a direct interpretation of the order parameter q3. To see this, let B denote the set of blue points, mN(B) its magnetization, and R the set of red points with mN(R) as its magnetization. Then 1 N 1 I~S. ' So q3 may be interpreted i.e. q3N= 89 as half the difference between the magnetization on blue and the magnetization on red. The total magnetization being m~= 89 it is easy to calculate the sublattice magnetizations on blue and red as N--* oo once q3 and m are known (Sect. IV). If the ferromagnetic interaction is absent (a=0), the system breaks up into two decoupled subsystems, blue and red, which both contain about 89 points, and the Hamiltonian may be written _flHSN=K [ (N~B~iS(i)) ( ~ r l j S J.L. van H e m m e n et al.: Classical Spin Glass Model 319 The Mattis [37] transformation S ( i ) ~ S ( i ) (the model is classical) transforms blue into ferromagnetic and red into antiferromagnetic; explicitly, 1 2 2 1.0 ' ~ ~ I I , nn /" 0.5 At sufficiently low temperatures ( K - l < 1) blue will thus [-38] be ordered, whereas red remains disordered (uncorrelated) down to T = 0 because of the large amount of frustration - all the simple closed loops (i.e., all triangles) are frustrated. We now undo the Mattis transformation, and note in passing that we have also shown that about half of the original lattice is fully frustrated as a Mattis transformation is' frustration preserving. Blue has spin-glass order with order parameter qlN, whereas red remains uncorrelated. To see why qlN is the right order parameter we return to (5.3). As yet there is no coupling between blue and red (Jo =0). The blue points are ferromagnetic, so their order parameter is qB=N -~ ~,S(i); the sum is over blue only and a factor 2 has already been dropped. Through the inverse Mattis transformation qn is transformed into q'B=N-1 ~iS(i). The red points remaining uncorrelated, they do not contribute to N - ~ ~iS(i) and thus q~ equals qzNAn extension of the above arguments deserves mention: whatever the (even) probability distribution of the ~'s and the r/'s, about half of the original lattice is fully frustrated. For the proof we simply replace ~i and r/j by sign (~i) and sign (r/j), and note that Prob {sign(~) = + 1} = Prob{sign(~i)= - 1} = 89 with a similar expression for r/j. As the exceptional case P r o b { ~ i = 0 } =t=0 requires only a trivial modification, it is discarded. We now return to the main object of this section, showing the possibility of a mixed phase. If 0 < e ~ 1 and K - ~ < 1 the random interaction dominates and, as before, we find blue in a spin-glass phase. Whatever its strength, the random interaction can never order red. Nevertheless ferromagnetic order will appear on red provided fiJo=Ko>2, because in this way red gains extra energy via the J0term [39], i.e., we obtain a mixed phase (II). The temperature being quite low, the entropy does not yet play a significant role. At fiJo ~2, which is near the line K-~-- 89 red has a ferromagnetic phase transition and becomes uncorrelated above this line. We enter the pure SG-phase, which will persist up to K -a =1. On the a-axis of Fig. 1 we find a II-F transition at a =2/3. Here T = 0 , so an energy argument suffices: with S(i)=~ i on blue and S(i)= + 1 on red the ground state energy is - ~ J o - 8 8 whereas it is - 89 when all the points are ferromagnetic. Equating the I /J r ,/ it/Ill i 0.0 -- I o5 i h ~.0 Fig. 2. The spin-glass magnetization m as a function of the external field h, with e = 0 . 2 and T / J = 0 . 6 where the units are such that J = l . The r a n d o m variables ~ and ~ are +_1 with equal probability. At h t there is a field-induced transition to a state of higher magnetization. The magnetization is convex on the left of h t and concave on the right; i.e., h t acts as a point of inflection. The broken line indicates the metastable continuation of the high-magnetization state; cf. Fig. 2 of Ref. 10 energies we obtain e = 2/3. If e > 1, the Jo-term ought to dominate, whatever the temperature. This is indeed the case. If e <0, there is no ferromagnetic feedback to favour a ferromagnetic order on red. In fact, e < 0 favours a spin glass order on blue and, hence, this is all we can get for K - 1 < 1 1-54]. A mixed phase cannot exist. At T = 0 the physical entropy is positive because of the disorder on red. Analytically one has to solve the fixed point equation (4.13) and choose the solution that maximizes (4.9). Since the phase transition at fiJo~2 is second order, we anticipate that the mixed phase a p p e a r s through a secondary bifurcation, as a twig on the main SG-branch (see Sect. VIII). It is acceptable if e < 2/3. Furthermore, the analytic characterization o f the spin-glass phase which was found in Sect. IV, Eq. (4.15), is also easily understood: r e = q 3 = 0 implies and is implied by m(B)=m(R)=O; cf. Eq. (5.1). In the pure SG-phase red is uncorrelated, so the sublattice magnetizations re(B) and m(R) are bound to vanish. As seen in Fig. 2, the spin glass magnetization m(h) exhibits a distinct S-shape character. It starts linear in h, turns convex, and becomes concave after hi; so ht acts as a point of inflection. The system has a fieldinduced transition to a state of higher magnetization above a certain threshold field, ht, whose value increases as the temperature is lowered. We can continue the high-magnetization state below ht as a metastable state, whose magnetization (dashed line) resembles quite well the experimental hysteresis loop 320 J.L. van H e m m e n et al.: Classical Spin Glass Model 0.50 1.o ht ~.=0.2 0.25 m SG o.o 05 0.5 T 1.o Fig. 3. The threshold field ht of Fig. 2 as a function of the temperature T. Note that ht vanishes at T = T s and that its value increases as the temperature is lowered. The arrow indicates the transition SG-II at T/J =0.1. The picture is symmetric w.r.t, the Taxis and represents a part of the phase diagram in the h - T plane; the corresponding phases, SG and P, have been indicated 0.0 O.5 1.0 T 1.5 0.5 1.0 T 1.5 1.0 m 0.5 o.5 Amt 0.0 ,L 0.5 T 1.o 0.0 Fig.4. Am, the j u m p of the magnetization at h , as a function of T. The threshold field h, is the one of Figs. 2 and 3. At low temperatures Amt~0.5. This is readily understood if one realizes that the external field has to flip the "blue" spins which are frozen in the spin-glass state (Sect. V) Fig. 5a and h. The magnetization m as a function of T for fixed values of the external field: h = 0 , 0.05, 0.1, and 0.2. The r a n d o m variables ~ and rl are _+1 with equal probability. At a fixed T the magnetization increases as a function of h. Due to the "red" spins (Sect. V) rn=0.5 at low temperatures, a c~=Jo/J=0.2, b ~ = 0 . 4 (curve B in Fig. 2 of Ref. [10]). If on the other hand h starts at zero and assumes only moderate values, the magnetization curves resemble very narrow lancets [403 . This also is consistent with the present model, as we remain in the domain where re(h) is still linear in h. In Fig. 3 ht has been plotted as a function of the temperature. The decrease of ht to zero at Tf constitutes a connection between the isothermal field-induced transition at h t and the zero-field transition associated with the Zo-CUSp at Ts. In our opinion Knitter and Kouvel [10] correctly inferred that the high-magnetization state below T• is an e x t e n s i o n of the state at and above TI. A further indication is provided by our numerical calculations, which show that if the magnetization jumps from convex to concave at ht it goes from a state with both rn and q nonzero to a state where only m is nonzero. Finally, the explanation in terms of blue and red is immediate. The blue points leave the spin-glass state, where q ~ 0, and become paramagnetic (q--0). If the temperature is lower, it is harder to flip them. Hence the increase of h~. Let A m t denote the j u m p of the magnetization at ht. We have plotted A m t as a function of T in Fig. 4; A m t - - , O as T-~ TI. The magnetization itself has also been evaluated for various temperatures and fixed h. The ensuing functional dependence upon T is shown in Fig. 5. The initial susceptibility Zo (see Fig. 6) has a cusp at the P - S G boundary and a divergence at the SG-II boundary; both lines are critical. The specific heat has its main singularity (a jump) at the P - S G boundary if 0 < ~ < 1. Due to the ferromagnetic feedback the model has an essentially unique ground state with zero entropy. The last three observations easily follow from the blue and red picture. Evaluating this section: A simple model has been studied that reproduces several spin-glass features quite well (the cusp in Zo, J,L. van H e m m e n et al.: Classical Spin Glass Model 321 VI. General Probability Distributions i a An old heuristic device for understanding when a phase transition will take place is to consider the balance of energy versus entropy [42]: a phase transition occurs when the randomizing thermal effects, controlled by temperature and estimated by entropy, are overcome for sufficiently low temperature by the energy gain. A neat illustration of this principle is provided by (3.17). 0~=0.2 4 X o 3 - flf(fi) = max {Q(m) - c* (In)} (6.1) nl where Q measures the energy and c*(m) the entropy; c*(m) is the Legendre transform of the c-function, I 0.0 015 i.O i t T 1.5 viz. c(t) = (ln[cosh{q + t 2 ~ + t 3 ~}]). In general, c*(m)>0 with equality if and only if m =0. The negative of c*(m) would realize a maximum at m = 0 . On the other hand, Q(m) can assume large positive values for large ]m[, = . 4 Xo 3 Q(m) = 89 o m 2 + K q l q2 +Hm. J 0.0 (6.2) b J, . 0.5 1.0 T 1.5 Fig.6a and b. The zero-field susceptibility Zo as a function of the temperature T. As in Fig. 5, the arrow indicates the transition SGII, where Zo diverges. We have a Curie-Weiss behaviour for T>Tf=J. a e=0.2, b c~=0.4 S-shape of the magnetization, hysteresis). But one might wonder whether at low temperatures and in a metallic spin glass such a simple discrete probability distribution of the Ju still remains a good approximation. In fact, we expect it is not. Decreasing the temperature means increasing the mean free path of the conduction electrons, which mediate the RKKYinteraction, hence the effective range of the interaction itself [46]. A long range implies that we sample many Ju-values in the neighbourhood of the origin, so that a continuous probability distribution should yield a much better description. We, therefore, turn to an analysis which is valid for this type of distribution also. (6.3) As before, K0=flJ0, K = f i J , and H=flh. Q becomes more important as the temperature is lower. Though conceptually quite illuminating, as it stands (6.1) is not very useful. The main formulae to be derived in this section are (6.15) and (6.19). Unless stated otherwise, we assume Jo and J to be positive. Both c and Q are symmetric in their second and third argument, and so is c*(m), c*(m) = sup {m. t - c(t)}. (6.4) t One, therefore, may ask whether the symmetry between ql and q2 will be broken. It will not, as we now show. More precisely, if m=(m, ql,q2 ) maximizes the free energy functional (6.1), then ql =q2- For the time being we denote this m by (r~,c~1,~2). It is to be shown that c~1=g/2. Suppose c~1#c~2. By assumption max {Q(m)- c*(m)} = 89 K 0/77/2~ - H ~ / ~ - K c~1 q2 m + ( - c* (r~, c~1, q2)). (6.5) The function - c * is concave and symmetric in ql and q2, so that - c* (rh, q- l , q-2 ) _1 - g { - c * ( N , ql,~2)}+ 89 =<--Cg (/~/'~1 -}~22 ' ql -}) 2q2 " (6.6) 322 J.L. van Hemmenet al.: Classical Spin Glass Model Moreover, l, (6 i and equality holds if and only if 01 =02- Combining (6.5)-(6.7) we find for the right-hand side of (6.5) 89 --C* q2 -- c*(m, 01,02) (m,01+02~ , ql ~-t- 02 ). (6.8) The inequality is strict if 01 =#q2' So we have a contradiction, and thus 0 1 = 0 2 = q . This confirms the special result obtained via (4.12). Since m and q are the only remaining order parameters, (6.1) may be rewritten - flf(fi) = max { 89 K 0 m 2 + K q2 + / . / m - c* (m, q, q)}. "'q (6.9) Let I (m, q) = c* (m, q, q). Since c* is convex, so is I. The first order partial derivatives of the free energy functional (6.9) should vanish at a point (m, q) where the maximum is realized 81 Kom+H=~m, 81 2Kq=~q, (6.10) (6.11) We now exploit the convexity once again [43J, V, I = (81)(p) = {(8I*)-a } (#) (6.15 a) q = ( t a n h { K o m + H + K q ( ~ + t l ) } ( ~ + r l ) / 2 ). (6.15b) Putting /-/=0 we quickly recognize three phases as special solutions of (6.15). The trivial solution m = q --0 represents a paramagnet. If q = 0 and m=~0, we have a ferromagnet and when m = 0 and q =~0, a spin glass phase appears. But what about a mixed phase? If there were a mixed phase, both m and q would have to be nonzero. As seen in the previous section, for a suitable probability distribution a stable mixed phase may indeed occur. Specifically, we had m=q ---2 • at T = 0 (without external field). Equation(6.15) is to be compared with (4.13)-(4.15). However, as will soon become apparent, the existence of a mixed phase is not always guaranteed for a continuous probability distribution. If H = 0 , there is no harm in assuming m > 0 and q > 0 . With (re, q) as a solution of (6.15) the evenness of the distribution of ~ and t/ implies that (-re, q), ( m , - q ) , and thus ( - m , - q ) also are solutions of the fixed point equations (6.15). The model has a symmetry group consisting of four elements. Among the solutions of (6.15) we choose the one that maximizes the free energy functional, +Hm+Kq2-c*(m,q,q). (6.16) Note that Q(p) is a convex function, whereas Q(m) = 89 is not. I(p) being convex too, we now may apply the analogue of Fenchel's duality theorem [44] so as to find (see the appendix) (6.12) where 8 denotes the subdifferential and all convex functions in sight are assumed to be smooth and closed. For a differentiable convex function the subdifferential is nothing but the gradient [-47]. We have already met a relation like (6.12) in (4.11). There we had c(t)=ln[,cosh(0], (Sc)(O=c'(t)=tanh(t) and (8c*)(m) = {(8c)- 1} (m) = t a n h - l(m), since c** = c. Equations (6.11) and (6.12) imply a fixed point equation, p = (8I*)(K o m + H, 2Kq). m = (tanh {K 0 m + H + K q (3 + r/)}), Q(p)-I(p)= 89 or succinctly, with p = (m, q), (Kom+ H, 2 K q ) = V,I. where the second equality follows from (6.6). Combining (6.2), (6.13), and (6.14) we find that the order parameters m and q satisfy the equations (6.13) max {Q(p) - I(p)} = max {I* (t) - Q* (t)}, p t (6.17) so that, with (6.14), -fif(fl)=max{lln[cosh{tl+t2(~)}]) - 8 9K o (t1-1t) 2 - 88 K- I ) . (6.18) One easily verifies that the last equation is equivalent to We still have to compute the Legendre transform I*, - flf(fl) = max {(In [-cosh {K o m + H + K q (3 + t/)}]) m,q I*(tz, t2) = sup {mt 1 + q t 2 - c* (m, q, q)} 89 o m 2 - K q2}, m,q (6.19) = sup {mr l+ql( 89189 Fgt, q l , q 2 ----c**(t 1, ~1 t2, ~1 t2) = c(tl, ~1 t2, ~1 t2) , (6.14) where m and q are the usual order parameters, which satisfy the fixed point equations (6.15). It is J.L. van Hemmen et al.: Classical Spin Glass Mode1 323 1.0 PARA , I I ' / 1.0 I m i a 0.5 SPINGLASS / ' ~ /'" 0.5 RRO / , / , / / , ~ ~ t ,, h t , , J 0.0 I - 0.5 1.o 05 0.0 Fig.7. Phase diagram for ~ and t/ Gaussian. Here too (1,1) is a triple point [41]. There is no external field 0.0 Fig. 9a. The O.5 h 1.0 magnetization m as a function of the spin-glass external field h. The random variables ~ and I/ are Gaussian, c~ =0.2 and T/J=0.6. As for the rest the caption of Fig. 2 applies i 0.O9 m I I I m I 0.13 1.5 O.O5 X 1.0 0.10 0.5 0.0 ~ i 0.5 I ~.0 I 15 3.20 . I T 2.0 0.07 0.0 0.15 2.5 o.0 I 0.5 I 1.o T 1.5 Fig.8. The zero-field susceptibility Z0 as a function of the temperature T. The random variables ~ and ~/ are Gaussian, and ~ = 0.2. We have a Curie-Weiss behaviour for T > T/=.I Fig. 9b. The spin-glass magnetization rn as a function of the temperature T in the Gaussian case with c~= 0.2 and for fixed values of the external field: h=0.05, 0.10, and 0.20. Note the difference in scale: on the right h=0.05, on the left h=0.1, and in the lower left-hand corner h=0.20. The field dependence of the "knee" temperature is in fair agreement with Ref. 4 h a n d y to choose the units in such a way that J = 1. The Eqs. (6.15) and (6.19) are quite suitable for numerical calculations since they avoid the nasty evaluation of c* (m). As an illustration of both the universal aspects which are c o m m o n to all probability distributions and the new p h e n o m e n a which occur if continuous distributions are allowed, we consider the Gaussian Since T = 0 , an energy a r g u m e n t suffices. By symmetry we m a y assume q > 0 . W i t h H = m = 0 (6.15) case: 3, t/, and (~+r/)/1/2 are Gaussians with m e a n zero and variance one. F o r the m o m e n t we do not include an external field. Once m and q are k n o w n as solutions of the fixed point equations (6.15), the numerical evaluation of the Gaussian integral (6.19) is easily done. As shown in Fig. 7, the phase diagram has the same P - S G and T - F phase boundaries as the diagram in Fig. 1; both lines are critical. Apparently these features are universal. However, in the Gaussian case there is no mixed phase. On the eaxis of Fig. 7 we find a S G - F transition at c~= 2/7c. then gives q ( 0 ) = 8 9 as the value of q in the spin-glass phase at T = 0. Equating the g r o u n d state energies - J / n and - J o / 2 we find e=2/7c, as announced. N o t e that the S G - F transition is on the left of c~=l. This is also generally true (Sect. VIII). The initial susceptibility Zo, which has been plotted in Fig. 8, exhibits a flat plateau between T = 0 and Ty and a Curie-Weiss behaviour for T > T I. Both features are in agreement with experiment [-4, 7, 8] as is the spin-glass magnetization re(h), which displays a distinct S-shape character [-10]; cf. Fig. 9a. Finally the zero-temperature entropy of the spin-glass phase vanishes. In the remaining sections we will study the mechanisms which underlie the similarities and differences between models with discrete and continuous probability distributions. 324 J.L. van Hemmen et al.: Classical Spin Glass Model VII. Ground States q The basic idea of this section is that the various bifurcating solutions (re, q) of the fixed point equations (6.15), which move away from (0,0) as the temperature decreases, can be labeled by means of the solutions which are found as K ~ oo ( K = f l J ) . The ground states minimize the energy and are stable, but we will also find metastable and unstable solutions. Furthermore we will present a geometrical criterion for determining the (meta)stability at zero temperature. Specifically, we show that the closure of the set C = {(m, q)l(m, q) solution to (6.15), - oo <c~< + oo,0<K_< ~ } (7.1) is convex and that the ground state energy u o may be obtained from (J = 1) u0=- max { 89 (7.2) (m,q)eC If so, the solution of (6.15) corresponding to K = o o is to be found on ~C, the boundary of the convex set C, which we call the extremal set. Suppose c~>0. Then { 89 2} is a convex function of m and q, and the maximum in (7.2) is realized for one of the extremal points of C, a subset of ~?C. The geometry of the extremal set, which can be obtained without solving (6.15), determines which phase is stable at zero temperature. Moreover it will turn out that the stability or metastability of the stationary points of the convex optimization problem (7.2) gives us a considerable amount of information about the stability of the branches which emanate as the temperature is raised. In this section we take h = 0 and 0 < ~ < 1 , unless stated otherwise. If a point in C gives rise to a maximum, we mean the maximum in (7.2); the ground state energy itself is then a minimum. VII.A. A Simple Probability Distribution We start with ~ and r/as _+1 with equal probability. Then we have, according to (6.15), m = 88 [tanh {K(c~m + 2 q)} + 2 tanh {c~K m} + tanh {K(e m - 2 q)}], (7.3 a) q = 88 [tanh {K(e m + 2 q)} - tanh {K(e m - 2 q)}]. (7.3 b) By symmetry (Sect.VI) we may always assume m > 0 and q__>0. To determine C we have to vary m and q over all allowed values ( - o o < c ~ < + o o , 0 _ < K _ < o o ) . However, m and q are dependent. Suppose namely #//// {sOG/2) "Fr (112,112) co,ol/ / / / / / /~\(~ m ,o Fig. 10. The convex set C (hatched area) and the extremal set OC (the boundary) for the case where the random variables ~ and t/ are 4-1 with equal probability; see Sect.VII.A. The points (1,0), 1 (>1 3), and (0, 89 are the ferromagnetic, phase II, and spin@ass fixed points respectively at T = 0 they were independent. Then C would be determined by - l _ < m _ < l and - 8 9 1 8 9 As seen in Fig. 10, this is wrong. The dependence is most easily brought out by noting that in terms of the a-variables which we introduced in Sect. IV m= 88 + 2 a 2 + a 4 ] , q = 88 1 - a 4 ] , (7.4) where la~l<l and the a~ may be chosen independently. The representation (7.4) immediately gives the whole allowed domain of (m, q), which is shown in Fig. 10. Plainly C is convex. In fact, it is polyhedral. A convex set is called polyhedral if it is the intersection of a finite number of closed halfspaces. We now probe C by means of the ellipses m + (q/1) 2 = r 2. (7.5) That is, we vary r and choose a point (re, q) in C such that r is maximal. The maximum is realized for an extremal point of the polyhedral set C [48]. Two extremal points are relevant to us: (1,0) and (7,17), 1 the ground state of the ferromagnetic phase (F) and the mixed phase (II) respectively. Whether F or II 0 1 which is dominates depends on c~. The point (,3), the ground state of the spin-glass phase (SG), never gives a maximum if c~>0 - as we already know from Fig. 1. These results are easy to understand. If 0 < c ~ 1, the ellipses (7.5) are long horizontal ovals whereas for large c~ they turn vertical. Accordingly II wins if c~ is small enough whereas later on (e >2/3) F takes over and maximizes r. Plainly (0,-12) never gives rise to a maximum since it is in the middle of a horizontal line segment, which has zero curvature. For e < 0 , J.L. van Hemmen et al.: Classical Spin Glass Model 325 however, we have hyperbolas instead of ellipses to probe C, so that (0, 89 and (0,- 89 remain the only points in C which produce the maximal r. Hence the spin-glass phase gives the only ground state in this a-domain, as is also confirmed by Fig. 1. We now check the results for c~> 0 analytically. Let us consider several cases separately, taking m > 0 and q > 0 . (a) c~m-2q>O. Suppose we have found a solution (re, q) of (7.3) with e m - 2 q > 0 and K ~ o o . We have to determine (re,q) self-consistently, so we take K ~ o o in (7.3) with e m - 2 q > 0 . We then find m = l and q--0, while consistency requires e > 0 . Here we used the condition e m-2q>O once again. We end up with the ferromagnetic phase. (b) e m - 2 q < 0. Proceeding as before we now find re=q= 89 i.e. the mixed phase. Consistency requires e < 2. For e > 2 there is no mixed phase. (c) It is tempting to put e m - 2 q = 0 , but we have to be more careful and specify the way in which e m - 2 q approaches zero. We, therefore, make the An- Tr Fig. 11. See also Fig. 10. We have indicated the routes which the various branches related to the fixed point equations (6.15) follow through C as the temperature is lowered. Note that both the spin-glass and the ferromagnetic fixed point have a (secondary) bifurcation. III is the unstable saddle point H o w to interpret the new solution III? Consider the function { 89 2} on the line m + q = l and look for an extremum. We find a minimum of 8 9 ( l - m ) 2 at m = 2 / ( e + 2 ) , i.e., at 2 satz re=e+2' K ( a m - 2 q) = c, q c~+2' (7.10) (7.6) for an arbitrary real c which is fixed in the limit K ~ oo. Then we obtain, as K ~ o% m= 88 +tanh(c)], q= 88 - tanh(c)], (7.7) so that m + q = l , whereas (7.6) implies e m - 2 q = 0 . We have found a third solution (Ill) to (7.3) which exists at low temperatures and is given by the intersection of the two lines m + q = 1 and q = (e/2) m provided 0 < e < 2. If c = + ~ , we find (a) and with c = - o o we obtain (b). In general, 89 and 0<q__< 89 (d) Finally we make the Ansatz e K m = c, which is the intersection of the lines re+q=1 and q/m=e/2. The point III is a "saddle" point in the convex domain C: on the line m+q=l it gives a minimum EII and F give rise to (local) maxima in C] whereas it gives a m a x i m u m if we move toward it along a line perpendicular to re+q= 1 but in C. By continuity, a solution of (7.3) has to start at the point III as fi comes down from + oo. The present argument also explains why this solution is unstable at finite ft. It will merge into the ferromagnetic fixed point F at a certain/31, and below fil F itself is unstable (subcritical bifurcation); see Fig. 11. Full details are given in Sect.VIII.B. (7.8) VII.B. Discrete Probability Distributions and find, as K ~ 0% m = 89 tanh (c), SO q 1- ~ , (7.9) so that -7__<rn<~ 1 1 and q= 89 whereas (7.8) implies m =0. The point (0,3) is the only solution of (7.3) on this line segment. It is also unstable with respect to the "'dynamics" induced by the right-hand side of (7.3): if one takes (m, g )1 on the line segment, with m > 0, and let K ~ oo in the right-hand side of (7.3), one always moves in the direction of II, away from 1 In fact, this should not be a surprise since for (0 , ~). 0 < e < 1 the stable II-phase branches away from the spin-glass phase before we reach T = 0 . In Fig. 11 we have indicated the routes which the various branches follow through C as the temperature is lowered. For a general discrete probability distribution of and t/we may assume that (~ + r/) assumes the values X_n<X_n+ 1~ ... <Xo=O< ... < X n _ 1 < X n with probabilities pj=p_j, O<j<__n. For the sake of definiteness we take n finite. Equation (7.3) is now given by +n m= ~ p~tanh{K(em+qx~)}+Potanh{~Km}, J2*o" (7.11 a) +m q= 89 ~ pjxjtanh{g(~m+qxj)}. j= m (7.11b) 326 J.L. v a n H e m m e n et al.: Classical Spin G l a s s M o d e l Putting m = 0 we have O < q = 8 9 as K ~ c o . If and so on, until finally J we start with this q-value but take m slightly positive, we immediately move to m = p o as K--r co. Specifically, with m__>0, q > 0 , and c~K m = c we obtain m=potanh(c), q= 89 ~ pjlxj[, (7.12) IJl ~ 1 where - p o < m < p o is the allowed domain of m. Equation(7.12), which may be compared with (7.9), gives us the first line segment of the extremal set of C. We will determine the convex polyhedral set C by constructing all the line segments of its extremal set. Putting c = + co we find the first extremal point, which we call P1. A line segment is a convex set and a convex function attains its maxima at the boundary. Hence the spin-glass ground state is a local minimum; in C it is a saddle point and, hence, unstable. Moving m slightly further away from zero we make the Ansatz K(c~m-qlx ll)=c, (7.13) and find, as K ~ co, m =P0 + P l [tanh(c)+ 1], q= 89189 ~ pjlxjl, = - (qk -- qk + 1)(qk + qk + 1), + 89 ~ (7.15) K(~z m - - q lx_k + l [)=c (7.20) (qk+%+l)/2 [ qk--qk+l ] C~--(ink +mk + 1)/2 [ i m ~1+1~mm~/2_[" (7.21) Using (7.17) with c = + c o one easily verifies that (qk--qk+l)=PkXk and (mk+t--mk)=2Pk SO that the fraction between the brackets in (7.21) equals x k. Equation (7.21) then asserts that q=(C~/Xk) m bisects the line segment PkPk+l. The sequence of arguments may be reversed. The above observation has two simple corollaries. First, the angle Ok between the line segment PkPk+l and the positive m-axis is given by (7.16) tancPk_ qk + l --qk _ and, as K --, oe, l Xk ' (7.22) mk+ 1 --m k pj+pk_l[tanh(c)+l], (7.17a) while the length of the line segment PkPk+l is given by (7.17b) 1 2 Ik=2pk(l +~xk) ljl >=k- 2 q= 89 ~ pjlx~l, [j[>=k with m-r- l)(mk + mk+ ~) so that By varying c in (7.14) we obtain the second line segment of the extremal set. If c = - co, we start at /]1 and for c = + co we find the second extremal point P2. For (re, q) to be a saddle point (7.13) requires q = ~ re~x l, i.e., this line should intersect the line (7.15). Whether this really happens depends on ~. If we continue this way, moving further outwards with m, we find for the kth line segment the Ansatz ~. which goes through (1,0), as it should. Equation(7.16) implies q = ~ m / X k _ 1. It again depends on whether this line and (7.18) intersect. The kth extremal point is given by (7.17) with c = + co. To obtain the convex domain C we can restrict ourselves to the polyhedral set C1, in the first quadrant, which contains (0,0) and is bounded by the lines (7.18). Then C is obtained by symmetry. Equation(7.19) shows that for ~ large enough [cf. (7.5)] the ferromagnetic phase dominates at low temperatures, whereas (7.12) implies that the spin-glass ground state is always unstable. For c~<0 it is just the other way around. The spin-glass ground state is a saddle point if ~ >0. Let us suppose, somewhat more generally, that we have another saddle point between Pk=(mk, qk) and Pk+l=(mk+~,qk+l) and let us denote by u0(Pk) the energy evaluated at Pk. We assert that u0(Pk) =u0(Pk+l) if and only if the line q=(ct/Xk)m bisects the line segment PkPk+l . For the proof we note that u0(Pk)=u0(Pk+ 1) may be rewritten (7.14b) ]jl_> m= (7.19) IJl __<n 89 so that =P~ = Y" p j = l , (7.14a) IJl > 2 m + ~-i m+ i/2 . (7.23) Second, Pk = (ink, qk) gives rise to at least a local maximum if and only if q= 2 [j[ ~<k-- 1 Pj+ 89 E Pj--, [jl>=k Xk (7.18) 1 Xk - 1 < O~mk/qk < X k. (7.24) 327 J.L. van Hemmenet al. : Classical Spin Glass Model certain convex C as n ~ o0. C is not polyhedral. We will derive a parametric representation for 0C. To this end we return to (7.17) and let n--.oo, taking advantage of (7.26) with A j ~ 0 and the fact that p is even. We then find, with x > 0 as a parameter, x Fig. 12. Pk gives rise to a local maximum as indicated in (7.25). The broken line is a part of the ellipse through Pk re(x) = 2 ~ dy p (y), (7.27 a) 0 co q(x) = ~ dy p(y) y. The direction of the line tangent to the ellipse 89 c~rn2 + q 2 = r 2 at the point Pk=(mk,qk) is given by - ~ ink~2 qk" A local maximum means that, if we decrease r, the ellipse intersects both Pk-1 Pk and PkPk+I. As seen in Fig. 12, we obtain a local maximum at Pk if tan Ok < -- C~mk/2 qk < tan (Pk- 1- (7.25) Combining (7.25) and (7.22) we find (7.24). An application of this formula is provided by the previous subsection where x 1 = 2 , and m i = q , = l / 2 . According to (7.24) the extremal point/]1 =(1, 89 which represents the ground state of the mixed phase (see Fig. 11), gives rise to at least a local maximum if and only if 0<c~<2. This is in agreement with a previous observation that in this case the mixed phase does not exist anymore if a > 2. Summarizing this subsection: In the case of a discrete probability distribution the set C, as defined by (7.1). is a convex polygon. The extremal set is the boundary of this polygon. Its geometry, which can be obtained without solving the fixed point equations (6.15), already determines which phase is stable at T = 0 and which phases are metastable or unstable. The ground state of the spin glass phase is never stable if c~> 0. VII.C. Continuous Probability Distributions Using the results of subsection B we can also treat the case that { and t/ and, hence, { + ~/ have an absolutely continuous probability distribution. More precisely, we assume that (3 + r/) has a well-behaved probability density p(x); the function p is even. An expectation value with respect to p may then be obtained as a limit of expectation values with respect to a sequence of discretized probability distributions, +11 p(xj)Aia(x-xi) p,(x)= ~ j = (7.27b) x Eliminating x from (7.27) we obtain q as a function of m. Equation(7.27) represents a curve in the first quadrant; it may be extended to the whole plane by symmetry. Thus we have found 0C and, hence, C. As before, C and 0C only depend on the probability distribution of ~ and t/. As a limit of convex domains C is convex itself. However, one can also prove this directly by showing that the curve (7.27), with rn and q positive, is con- d2q cave, i.e. ~m2<0. Let us denote the differentiation with respect to the parameter x by a dot. We have rh=2p, rh=2/5, 4=-xp(x), 0"=-(p+xtS), dq and thus, since ~ = 4/ph, d 2 q fft O"- q rii dm 2 rn 3 4p(x) <0. 1 - - (7.28) As an extra check on may compare (7.22) with d q = - x/2. dm Since the curvature of 0C at Po, the point where OC intersects the positive q-axis, is nonzero the spin glass ground state will always dominate if c~ is small enough (c~>0). As ~ increases the ferromagnetic phase will take over at a certain ~o- One might wonder, however, when the spin-glass ground state (P0) becomes unstable. This usually happens at some c q > % . We want to determine ~ . The argument is simple. P0 is represented by (7.27) with x =0. It becomes unstable if the negative of the curvature of the ellipse (7.2) at Po, d 2 q _ 0~ ~mSm2Po 2q(0)' (7.29) (7.26) n where Aj is the length of the jth interval, which goes to zero as n-+ oo. For each p, we have a polyhedral set C, with an extremal set ~C., and ~ C , ~ O C for a exceeds 1/4p(0); cf. (7.28). At the borderline we then have oo cq = S dy p (y) y/2 p (0). 0 (7.30) 328 J.L. van Hemmenet al.: ClassicalSpin Glass Model 2 For example, in the Gaussian case we find eo = - < 1 =cq. In view of (7.30) and the strict loss of stability of the spin-glass ground state for c~>e~ the following question is natural: is there a new phase (II) branching away from the spin-glass phase? To answer this question the extremal set will be of great help (Sect. VIII.C). Apparently the geometry of the extremal set already contains much information about the occurrence of bifurcations and the stability of bifurcating solutions. The most surprising new feature of continuous distributions is that they always have a stable spin-glass ground state if ~ is small enough but still positive. We think that this dependence upon the nature of the distribution is quite natural and reflects the physics of the problem. For instance, in metallic spin glasses at low temperatures the long range of R K K Y interaction samples many small Jijvalues [21] so that in the neighbourhood of J = 0 their distribution is well-approximated by a strongly peaked continuous curve. Analogous considerations hold for other long-range (anisotropy or dipole-dipole) interactions. On the other hand, a short-range interaction usually samples only a few J~j-values so that in this case a discrete probability distribution might be more appropriate. Finally we note that the replacement of a discrete probability distribution with many dense values by a continuous distribution is a widely used device in probability theory. The oldest and best known example is the de MoivreLaplace theorem [49]. VIII. Bifurcation Mean field models, like the present one or that of Sherrington and Kirkpatrick [17], have the great advantage that one can study both stable and metastable states simultaneously and in such a manner that they are analytically tractable. Therefore, hysteresis also becomes amenable to an exact analysis. This section is devoted to a careful exposition of the underlying mechanism: bifurcation. In our mean field theory we have to maximize the expression between the curly brackets in (6.19), which we call q0(/~) with /~=(m,q). We have to look for the stationary points of (p. A necessary and sufficient condition for a point /~=(m,q) to be stationary is that it obeys the fixed point equations (6.15), viz. m=(tanh{Kom+ H + K q(~ +~l)}) , q=(tanh{Kom+H+Kq(~+tl)}(~+tl)/2 ) . (8.1a) (8.1 b) Then /~ is said to be stable if (p has a global maximum at /~. It is metastable if/~ gives rise to a local maximum, and unstable if ~o has a (local) minimum or saddle point at /~. The stationary points of the free energy functional are associated with different phases of the model, which may become stable, metastable, or unstable as the temperature, or another parameter, varies. Below we assume K0>0. Let us rewrite the fixed point equations (8.1) in vector form, =g'(~), (8.2) where g' maps IR2 into itself. We note that, by construction, g' is the derivative of a function g. The precise form of g will be specified later. There is no hope for solving (8.2) analytically, so one must have recourse to numerical iteration: starting with a wellchosen/% one generates a sequence t,1 =~'(~,o), ~,2 =g'(~l), ~,3=g'(~2), . . . . The question is whether /~, converges to a fixed point/~o and whether/l ~ gives rise to a (local) maximum of (?. We define/~ to be a stable fixed point if there exists a neigbbourhood U ~ of/~0 such that any sequence which starts at /~0 in U ~ converges to /~o. Otherwise/~0 is called unstable. A criterion for the (in)stability of a solution to (8.2) is readily obtained. We linearize g' about/~o, g'(/l) =/l ~ + g,,(/~0)(/1_/~o) + .... (8.3) If all the eigenvalues of the symmetric matrix g,,(/~0) are in absolute value smaller than one, then g' restricted to a neighbourhood of/~o is a contraction mapping and t ~ is a stable fixed point. If, however, one of the eigenvalues exceeds one in absolute value, then g' is expanding in a certain direction and/~o is unstable. We may phrase the stability criterion succinctly by - 1 < g,,(~o) < 1. (8.4) We want to prove that (8.4) implies and is implied by/~o being a (local) maximum of (p. To this end we return to (6.19) and perform a scale transformation, ]/~oorn~m, 1/~q~q, (8.5) which preserves convexity. Then (6.19) may be rewritten -/3f(/3) = max {g (m, q) - 89 m 2 - 89 q 2}. (8.6) m,q We have a (local) maximum at/~o= (m, q) if/~o satisfies the fixed point equation #o =g,(po), (8.7a) J.L. van Hemmenet al.: Classical Spin Glass Model 329 and the Hessian is negative-definite, g,,(p0)_ 11<0. (8.7b) Now (8.2) and (8.4) imply and are implied by (8.7) since g is convex and smooth so that [50] g"(p)>0. This finishes the proof. Throughout what follows we will use the original order parameters m and q as they appear in (6.15) and (8.1). Numerical iteration of (8.1) gives all the (local) maxima. In the three subsections below we study the various phases which appear through bifurcations of the fixed point equations (8.1). They look simple but have a rather intricate bifurcation pattern. It might be that there exist even more solutions to (8.1) which cannot be found by means of a bifurcation analysis, but as yet we have not localized them numerically. As a general reference for this section we recommend Chap.V of Iooss and Joseph [51]. In this paper we will pay special attention to the domain 0 < = J o / J < l ; a complete analysis will be given elsewhere [52]. For the sake of convenience we do not include a magnetic field (H = 0) and write K 0 = c~K. VILLA. From Paramagnet to Spin-Glass and Ferromagnet We start by rewriting the fixed point equations (8.1) in the form F(K, p) = g'(,,) -~, = 0. (8.8) K is a parameter proportional to the inverse temperature fl, which is still at our disposal. Since H = 0 , we may assume m > 0 and q>0. The advantage of the form (8.8) is that the requirement (8.7b) agrees with the usual stability criterion [51] that all eigenvalues of the derivative F'(K,p ~ are negative. From now on we will use this criterion for discussing stability. We have the following picture in mind. If one of the simple eigenvalues of the matrix F'(K,p~ becomes positive at a certain Ko, then we get a bifurcation into the direction of the corresponding eigenvector of F'(Ko, po). In the high-temperature regime K is small and (0,0) is the only solution to (8.1). To see this we observe that tanh(x)=< x for positive x and thus O<-m<~Km+Kq([~+tl[)<=c~Km+Kq, O<=q<~Km(]~+tl[)+Kq<c~Km+Kq, (8.9a) (8.9b) with 7<1 if K is small enough. Thus we are left with r e = q = 0 , which corresponds to the paramagnetic phase (P). We may expect a bifurcation from (0, 0) only if the derivative F'(K, (0, 0)) is not invertible, i.e., if one of its eigenvalues vanishes. For general p the derivative is given by F'(K, p) /c~K (cosh 2 { . . . } ) - 1 = ~0~g (cosh-2 {...} \ K <c~ 2 {"'} ({ + r/l))) K(cosh-2{...} . (~ + ~)/2> 9( 3 + ~ ) ~ / 2 ) (8.11) where {...} stands for {~Km+Kq(~+tl)}. The distribution of ~ and t/ is assumed to be even. In the case of p = 0 we get F'(K'(0'0))=( ~ K - 1 K-10 ), O<m+q<2K(am+q)<7(m+q) (8.10) (8.12) which is not invertible if either K = 1 or ctK = 1. We have strict loss of stability [51]. Since the eigenvectors of F'(K(O, 0)) are trivial, the directions of bifurcation are easily checked. The q-direction gives rise to a spin-glass (SG) phase and the m-direction corresponds to ferromagnetic behaviour (F). These are the main branches which bifurcate from (0, 0). In phase diagrams (Figs. 1 and 7) we find the paramagnetic phase in the domain above the lines K - I = I and K - l = e . As soon as we cross one of these lines, the paramagnetic phase loses its stability and either the spin-glass or the ferromagnetic phase appears through a second order phase transition. (A bifurcation is continuous.) This behaviour does not depend on the probability distribution of ~ and t/. VIII.B. Bifurcation fi'om the Ferromagnetic Fixed Point A ferromagnetic fixed point exists below the line e K = 1 and is given by q = 0 and m = tanh {c~Km}. (8.13) Simply substitute q = 0 and /-/=0 in (8.1). Here we take 0 < c~< 1. The matrix F'(K, (m, 0)) is diagonal, so its two eigenvalues are readily obtained 21(K ) = c~K cosh- 2(~K m) -- 1, 22(K) = K cosh- 2(c~K m) - 1, since ( [ ~ + t / ] ) < ( ( ~ + t / ) 2 ) l / 2 = l by the CauchySchwarz inequality. Adding (8.9 a) and (8.9b) we obtain - (8.14) whatever the probability distribution of ~ and r/. Below K -1 =c~ the first eigenvalue is always negative, as is most easily seen geometrically by noting that 330 J.L. van Hemmenet al.: Classical Spin Glass Model ~Kcosh-2(c~Km) is the derivative of tanh(eKm) at the point m satisfying (8.13) for e K > 1. We now rewrite the second eigenvalue, using (8.13), x2(K) =K(1 -m (8.15) ~) - 1. When the ferromagnetic fixed point branches away from (0,0), and 0<c~< 1, we cross the line K -~ =c~ with K > I and m=0. Hence 2 1 < 0 and 2 2 > 0 just below K - l = cr so that the ferromagnetic fixed point is unstable into the spin-glass direction. As K -z decreases further, m increases and ( 1 - m 2) approaches zero exponentially fast. Accordingly 22(K ) decreases and becomes negative as soon as we are below the curve whose parametric representation is given by m = tanh(~ K m), K(1 - - m 2 ) ---- 1, (8.16) with 0_<m_<l as a parameter. Below this curve, which is the dot-dashed line in Fig. 1, the ferromagnetic fixed point is stable. How should we interpret the above result? Because of its universality we can take ~ and ~/ as _+1 with equal probability. Let us start at zero temperature and lower K (or /3) from + oo. We get a subcritical bifurcation where the stable ferromagnetic fixed point F "swallows" two unstable branches and becomes unstable itself. See Fig. 11 and note that the unstable branches originate from the saddle points III on the boundary. Finally we give a more convenient representation of the curve (8.16). Let K -~ = y and ~=x. Then y=(l_m2), x = 2 _ ~ l n [1 + m ] . \1 - m ! )LI(K)=c~K (cosh-2 {K q(~ + rl)}) -1, )'2 (K) K (cosh- 2 {K q'(~ q- t])}(~ q- t/)2/2) -- 1. (8.19) = If K -a passes through one and 0 < a < 1 , we have a supercritical bifurcation with strict loss of stability of (0, 0) and, hence, a stable spin-glass fixed point: both 2~ and 22 start negative. It is to be expected that 22(K)<0 for all K > I . The question is what will happen to 21(K ) as we lower the temperature. Specifically, if 21(K ) becomes positive, then (0,q) loses its stability and we get a secondary bifurcation into the direction of the corresponding eigenvector, i.e., the ferromagnet: the mixed phase appears as a twig on the main SG-branch. To study this type of bifurcation we consider two typical cases. Suppose ~ and ~7 are _+1 with equal probability. Then 2q = tanh {K(2q)} ~ p = tanh {Kp} (8.20) with p = 2 q, and 21(K)= 89 + 1] - 1, 22 (K) = K cosh- 2 {K p} - 1. (8.21) As in the previous subsection one showes that ,~2(K)<0 for all K > I . In fact, the function cosh-2{Kp} decreases to zero exponentially fast as K increases. We, therefore, expect (8.17) 21(K)=0 One first chooses y with 0__<y < 1, then determines m, and ends up with x. Summarizing this subsection: Let 0<c~<l. Whatever the probability distribution of ~ and r/, the ferromagnetic fixed point F has a secondary bifurcation as the temperature is lowered and K - l = T/J reaches the curve whose parametric representation is given by (8.16) or (8.17). Below this line F is stable. The twigs which bifurcate from the main F-branch into the SG-direction are unstable and terminate at saddle points of type III, which have been described in Sect.VII.A (Fig. 11). See also Sect. VII.C (Fig. 13). VIII.C. From Spin-Glass to Mixed Phase In the pure spin-glass phase m = 0 and q satisfies q = (tanh {q K(~ + r/)}(4 + I/)/2). For low temperatures q reduces to q(0)=f 89 as given by (7.27b) with x = 0 . By (8.11) the derivative F'(K,(O,q)) is a diagonal matrix with eigenvalues (8.18) if 89 1, (8.22) and a stable mixed phase (II) branching off since 2a(K)>0 below the line K - l = l c ~ (strict loss of stability). As Fig. 1 shows, this is indeed the case. The critical line SG-II and the dashed line together constitute the curve 21(K)=0 whose parametric representation may be given by y=x[-1 - gp 1 2], p=tanh{y-a p}, (8.23) with x = ~ , y = K -i, and 0__<p__<1 as the parameter. In our second example the random variables ~ and t/ have a well-behaved continuous probability distribution. Let p(x) denote the probability density of (~+r/). In Sect.VII.C we have seen that, for small positive ~, the spin-glass phase is stable at zero temperature - in contrast to our previous example. The advantage of T--0 is that q(0) is known explicitly. We, therefore, look for a condition on ~=Jo/J which implies that the spin-glass ground state becomes unstable and a new fixed point (phase) branches off. J.L. van H e m m e n et al.: Classical Spin Glass Model As K ~ 0% the case of )t2(K ) 331 is easy to handle, SG +oo ,~2(K)+ l= 89 S dxp(x) x~ c~ x} -oo +oo = 89 ~ dyp(y/Kq)y2cosh 2(y) - - CX3 ~2 ~ p ( O ) / K 2 q(O)3 ~0. (8.24) Hence 2 2 ( o o ) = - 1 , whatever c~. We now turn to 21(K), the eigenvalue associated with a bifurcation into the ferromagnetic direction. Using the same arguments as in (8.24) we obtain, as K ~ o% 21(K) + 1 ~ 2 ctp (O)/q(0), (8.25) and we conclude that the spin-glass fixed point becomes unstable at c~1 = q (0)/2 p (0), (8.26) whereas for ~<cq it is stable. Fixing the temperature T in a neighbourhood of zero and increasing the parameter c~ we expect to find a supercritical bifurcation SG-II at c~= ~1, where el agrees with (7.30). The type of bifurcation needs some checking however. In general )~I(K) cannot be evaluated analytically. There is, however, an important exception: the Gaussian case. As will be shown in Sect. IX, Eq. (9.21), 21(K)=c~-1 , K>I. (8.27) Fixing an arbitrary K > 1 and increasing c~ we find a subcritical bifurcation at c~= 1, where SG looses its stability. To understand this behaviour we turn to Fig. 13 and look for the stable and unstable ground states, i.e., look for stationary points of the energy 2 on the extremal set ~C; cf. function 89 Eq.(7.2). Using the parameter representation (7.27) for 0C we easily find an equation for the parameters x which make the energy stationary on 3C, p(x) [~ i dy p(y)-x ! dy p(y) y] =O. (8.28) ~C/ Fig. 13. The convex set C and its extremal set ~?C (the boundary) for the Gaussian case; see Sect.VIII.C. At F = ( 1 , 0 ) the slope of ~C is continuous and the curvature infinite. We have indicated the routes which the various branches related to the fixed point equations (6.15) follow through C as the temperature is lowered and 0 < c ~ < l . Note that III, which is unstable, approaches SG = (0, 1/1~-) as c~~ 1. At c~= 1 SG looses its stability For 0 < c ~ < l this equation gives rise to a third solution III with 0 < x < ~ , whereas no such solution exists for c~> 1. The solution III is an unstable saddle point of the type we have found in Sect.VII. With 0 < a < 1 the points SG=(0, 1/1~-) and F = ( 1 , 0 ) correspond to (relative) maxima and III is a minimum on ~C. At c~=l the solution III reaches SG, which now turns into a minimum, and F is left as the only maximum. By symmetry another solution III' approaches SG from the left. Both III and III' are unstable and, when they merge at ~=1, SG is left unstable. That is, we have gotten a subcritical bifurcation. Phase transitions associated with a bifurcation are expected to be second order since the new order parameters branch off continuously from the old ones. But in the present model also first order transitions may occur when the free energies or, at zero temperature, the energies of two different phases become equal. For example, the transitions SG-F and II-F are first order. Here we consider the border between spin glass and ferromagnet at T - - 0 and without external field, where we have uo(SG) = - J q(0) 2, uo(F) = - 8 9 (8.30) The ground state energies u o are equal at e = ~o, Equation (8.28) always has two solutions, x = 0 and x = 0% which correspond to the spin-glass and the ferromagnetic ground state respectively. [Note that k +oc dyp(y)y2< oo]. In the Gaussian case the second _1 -oo integral of (8.28), with p(y)=l/l~exp(be done easily and we are left with 88 can x e o =xe . (8.29) c%=2q(0)2 =3(I 1 ~ +~1) 2 <1 =~{(~ + q)2) = 1. (8.31) However, ~o should occur on the right of ~ = 1 [53]. Though our model reproduces many spin glass properties quite well, the ferromagnetic phase dominates too much at low temperatures. Sometimes the spin glass still exists as a metastable state, for example in the Gaussian case for c~o < c~< 1, but from a thermodynamic point of view this is not completely satisfying. 332 J.L. van Hemrnenet al.: Classical Spin Glass Model Experimentally the picture looks slightly different. Take, for instance, the Gaussian case (Fig. 7). The spin-glass phase, which reaches down to T--0, is unstable for ~ > 1 but gives rise to at least a local maximum for the free energy functional (6.19) and, hence, a local minimum for the free energy itself provided c~< 1. The line which goes downward from (1, 1) to (2/t/z-,0) separates the spin-glass phase from the ferromagnet and represents a first order phase transition. The only way to reach a point below this line experimentally is by lowering the temperature at a fixed ~. When we cross the line, no bifurcation is involved. It, therefore, is not to be expected that the system will jump spontaneously to the ferromagnetic phase. It simply remains in the spin-glass phase down to T=0. IX. Specific Heat and Susceptibility At low temperatures the experimental magnetic specific heat c(fi) is linearly dependent on the temperature T [55, 56]. Discrete probability distributions give an exponential decrease to zero, which is not in accordance with experiment, whereas continuous probability distributions do reproduce the linear dependence upon T. Here we see very clearly the effect of the long range of the R K K Y interaction, which samples many small values of the coupling constants so as to make a continuous distribution appropriate. Experimental data also indicate that the linear susceptibility and in particular the initial susceptibility Z0 have a Curie-Weiss behaviour z = C / ( T - O ) for T > TI [-6, 7]. However, if 0 < T < Ts, the initial susceptibility has a flat plateau [-8]. In the Gaussian case )~o can be evaluated explicitly and it will be shown that Zo does have the required behaviour. We will consider the magnetic specific heat first and then turn to the linear and nonlinear susceptibility. The following definition will be to advantage: dn(fi) = (cosh-2 {K q (3 + t/)}(4 + tl)"), (9.1) where n is a nonnegative integer. IX.A. The Magnetic Specific Heat Whatever the inverse temperature fi, the energy per spin is given by u(fi) = - 89 rn2 - J q2 _ h m (9.2) where m and q satisfy (8.1), or (6.15). We take h=0. In the spin-glass phase, which exists for fi>fic with tic J = l, the order parameter m vanishes, and we have 1 0 c(fi)- J •q T2 ~ u ( f i ) = ~ 2 q ~ f i . (9.3) Differentiating (8.1b) with respect to fl we find ~-fi= 89d2(fi) J q [ 1 - 89 fi J d2(fi)] -1, (9.4) where use has been made of (9.1), and thus c (fl) = kB(fiJ q) 2 d2 (fl) [ 1 - 89 fi J d2 (fl)] -1 (9.5) with kB Boltzmann's constant. For a discrete probability distribution d2(fl) goes to zero exponentially fast as fi--, oo. Suppose now that ~ and t/have a well-behaved continuous probability distribution, and denote by p(x) the probability density of (3 + t/). Then we obtain, by a calculation analogous to (8.24), +~ d2(fi)=_~ dxp(x)x2c~ n2 p(O) 6 [,fiJq(O)] 3 (9.6) as K = f l J ~ oo. Hence the specific heat at low temperatures is linearly dependent on the temperature, Tg2 c(fi) ~ ~ k2s Tp (O)/J q (0). (9.7) Apart from the factor Jq(O), (9.7) is just Eq.(1) of Anderson et al. [-55]. Note that in this argument the continuity of p(x) at x = 0 is essential. Calorimetric experiments have been done mainly on C___uuMn [-56]. Increasing the Mn-concentration c means increasing the ferromagnetic interaction [-7], i.e., Jo. This interpretation is confirmed by the linear dependence of the Curie-Weiss 0 upon the concentration c, as will be shown in the next subsection. Since, by normalization, p (0) and q (0) = 89 (13 + r/I) do not depend on the concentration, the linear temperature specific-heat coefficient is essentially concentration independent [-56]. This independence has also been predicted by Anderson et al. [55], who used a scaling argument [-57]. The high-temperature behaviour of c(p), as deduced from (9.5), is characterized by a discontinuity determined by c(fi~-)=0 and c(fl~+)=3kJ((~+tl)4). In our opinion this is not a serious drawback since the inclusion of a short-range interaction presumably gives the high-temperature behaviour required by experiment [56]; see e.g. Refs. [58] and [59] for numerical simulations. However, a short-range interaction is not expected to contribute significantly J.L. van Hemmen et al.: Classical Spin Glass Model 333 to (9.7) since only finitely many spatial configurations are relevant and, hence, a discrete probability distribution of the coupling constants seems appropriate. IX.B. Linear Susceptibility To calculate the zero-field susceptibility in the paramagnetic (P) and spin-glass (SG) phase we return to (8.1), with Ko=fiJo, K=gJ, H=fih and h4=0, and differentiate both equations with respect to h, ~~m h - = ~/ c o s h - 2 {...}fi~[do ~0m - + l + d ( 4 + t / ) ~ ] )c?q ,(9.8a) 0q / [ ~m ~q] 0 ~ = cosh 2{...}fi Jo0h_+l+d(4+t/)~_mm and (h ~ 0) ;g0(T) = I T - J 0 ] - 1. (9.14) We obtain a pure Curie-Weiss behaviour with 0 =J0. Increasing, for instance, in C_~uMn the Mnconcentration c we also increase the ferromagnetic (short-range) interaction between the spins and, therefore, Jo. In fact, it is natural to assume Jo proportional to c. Then also 0 is proportional to c in agreement with experiment [-7]. See also [54] 9 (b) T~TI: "The Cusp". At T I = J we have a secondorder phase transition, so we expect a discontinuity in the derivative of z0(T) with respect to T. The right derivative is easy, Z• (Ty+) = - l - J - Jo] -2 (9.15) The left derivative equals (~. 9 + r/)/2), (9.8b) where {...}={Kom+H+Kq(~+~) }. If we let h go to zero and take advantage of the fact that d~(fi)=0 since m = 0, we find z;(Ty-) = - [ - J - J 0 ] -2 [1 +fi 0d~ Off r = r ; ] (9.16) so that the jump of Z~ at TI is given by AZ,o(Tf)=)~,o(T/)_Z,o(T]-)= _Z,o(T/)fi ~ (9.17) T= T~" Zo = fl (cosh - 2 {K q (4 + r/)}) 9[ 1 - f l J o ( c o s h - 2 { K q ( ~ + r l ) } ) ] -~ (9.9) A rather lengthy calculation shows 12 Az;(T~)- ((~ +,7~,4,2 z ; ( r / ) < o . Using (9.1) we may rewrite this succinctly (9.10) Zo = fl do (fi) [' 1 - fi do do (fl)] - 1, or, equivalently, (9.11) Zo = Zo (Jo = 0) [, 1 -- Jo )(o (Jo = 0)] - 1, since q satisfies (8.1b) with r e = h = 0 . Before proceeding we note that, in the spin-glass phase, the denominator of (9.9) is precisely the negative of the eigenvalue 2,(K) which we introduced in (8 9 As soon as 21(K ) becomes positive, we obtain a bifurcation SG ~ II. Plainly we do need this bifurcation since otherwise Zo would become negative. Let us now discuss the behaviour of Z0 for the three temperature domains T > TI, T ~ TI, and 0 < T < TI separately. (9.18) (c) O<T<Tr: The Plateau. In general Zo(T) as given by (9.9) cannot be calculated explicitly since we do not know do(fi). We now show that Zo(T) is constant for 0 < T < T j . if 4 and t/ have a Gaussian distribution 9 Then (~+r/)/]/2 is also Gaussian, with mean zero and variance one. According to (9.9), J o z o ( r ) = _ l + [ l _ ~ K (cosh-2{Kq(~ +~l)})] 1 (9.19) with e =Jo/J and q = (tanh {K q (4 + t/)} (4 + ~/)/2). (9.20) In the spin-glass phase q 4=0. We wish to prove K @osh-2 {K q(~ + r/)}) = 1, (9.21) (a) T>Ti: The Curie-Weiss Regime. In the paramagnetic region q = 0 and we have, as one easily verifies, for 0 < T < T I , i.e., K > I . To this end we turn to (9.20) and perform a partial integration with respect to x, z(h, T) = [Tcosh 2 {fi(do m + h)} - a o ] +~ dx q= Soo~ e 1 (9.12) with m = tanh {fi(Jo m + h)}, _X2/2 X ]~tanhtKq]f2x} 2 (9.13) +oo dx _ 2,2 =Kq S e x/ cosh-2{Kq]/2x} (9.22) 334 J.L. van Hemmen et al.: Classical Spin Glass Model which we rewrite q [1 - K (cosh - z {K q(~ + t/)})] = 0. (9.23) Since by assumption 0 < T < T/, the order parameter q is nonzero, the assertion (9.21) follows, and hence Zo(T)=[J-Jo] -1, O<_T<_rf. (9.24) As an extra check one may verify that (9.18) already implies )(;(Tf-)=0 (see also Fig. 8). We think it is satisfying that the rather subtle interplay of equilibrium formalism and randomness is responsible for this remarkable result. IX.C. Nonlinear Susceptibility The nonlinear susceptibility in the neighbourhood of TI is an interesting quantity which is, however, hard to measure [-4]. It is defined as follows. Fix 0 < e < 1 and pick a T near Ts. The phase diagram in the h - T plane looks like Fig. 3 and we may develop the magnetization re(h) into a power series in h around h=0, 2 h3 h . (0)+~(m . . . . m(h)=m(O)+hm ,(0)+2~.Tm (0)+ .... (9.25) where the radius of convergence depends on T. Since m(0)=0 in the paramagnetic and the spin-glass phase, and m"(0)=0 because re(h) is odd, we are left with (X0 =m'(0)) m(h)-hZo ~h2 m,,,(0) + ... (9.26) h which is the nonlinear susceptibility. We find that m'"(0) assumes finite but different values as we approach Ty from above and below. More specifically, lira m'"(0)= r ~r, 2J (J -- Y0)4' lim m'"(0)= [ - 2 + T~s 72 (9.27) ] J (9.28) <(~7-~)') ( J - yo) 4" Note the fourth power in the denominator. X. Comments We have obtained an exactly soluble spin-glass model with frustration. The long range of the R K K Y interaction has been incorporated via the infinite range of the J~j-interaction in the model Hamiltonian (2.1). Statistically the coupling constants J~j are only weakly correlated, and about half of the spins belong to a fully frustrated configuration. In the context of equilibrium statistical mechanics solving the model means calculating the free energy per spin f(fl, h) as a function of the inverse temperature fl and the external field h. Taking advantage of a large deviations result we obtained an explicit expression for f(fl, h), which does not depend on the specific random configuration of the Jij. Two aspects of this result deserve mention. First, we need not resort to the replica method. Second, the probability-one result is important because it implies the reproducibility of thermodynamic observables, i.e., two alloys whose microscopic structures differ but whose macroscopic constitution and preparation are identical (same concentration) give the same experimental outcomes. As yet it is not clear whether the Sherrington-Kirkpatrick model [17] satisfies this criterion. The phases of the model are determined by means of two order parameters, m and q, which are solutions of the fixed point equations (6.15); m is the magnetization and q is the spin-glass order parameter. When both m and q are nonzero, we have a mixed phase. A stable phase (re, q) corresponds to a global maximum of the free energy functional (6.19), whereas a metastable phase gives rise to a local maximum. If one lowers the temperature, new phases occur through bifurcations. The model's bifurcation pattern, though quite intricate, is often symmetric with respect to the behaviour of the two main branches, spin glass and ferromagnet, which split off from the paramagnetic fixed point (0, 0). If the temperature is low enough and the parameter c~=Jo/J is large enough, a (meta)stable mixed phase appears, through a secondary bifurcation into the ferromagnetic direction, as a twig on the main spin-glass branch. On the other hand, two unstable twigs bifurcate from the main ferromagnetic branch into the spin-glass direction; cf. Fig. 11. After the secondary bifurcation the ferromagnetic branch is stable, whereas the spin-glass branch is unstable. Whatever the probability distribution of the coupling constants J~j, we have a paramagnet-spin glass transition at flJ= 1, provided c~< 1, and below this critical line we find the spin-glass phase (Figs 1 and 7). Studying the spin-glass magnetization as a function of the external field h we obtain a distinct Sshape character which is indicative of a field-induced transition to a state of higher magnetization at a certain threshold field ht which acts as a point of inflection of the magnetization curve. Figures 2 and 9a also explain the hysteresis which shows up experimentally as the field is lowered again and the system remains in the high-magnetization state. We interprete hysteresis as the occurrence of a metastable state. Since mean-field models, like the present one J.L. van Hemmen et al.: Classical Spin Glass Model 335 and the Sherrington-Kirkpatrick model, allow both stable and metastable states at the same time, they are quite suitable for studying and simulating hysteresis. The equilibrium susceptibility exhibits a Curie-Weiss behaviour for T > T s. For a Gaussian probability distribution the initial susceptibility )~0 has a flat plateau below Ts (Fig. 8). In the case of a continuous probability distribution the low-temperature magnetic specific heat is predicted to be linearly dependent on T. In short, the model reproduces many spin-glass properties quite well. It is to be noted, however, that a rather subtle interplay of equilibrium formalism and randomness is responsible for this performance. Nevertheless the replica formalism could be dispensed with entirely. The extremal set, a novel and purely geometrical notion, enables one to study the ground states and their stability without solving the fixed point equations (6.15). A parametric representation of the extremal set is available; cf. Eq.(7.27). Analyzing the stability of the ferromagnetic phase we found that it is slightly too dominant at low temperatures. So here some work remains to be done, in spite of the model's satisfying overall performance. We gratefully acknowledge stimulating discussions with Th. Eisele, I. Morgenstern, and J.A. Mydosh. This work has been supported by the Deutsche Forschungsgemeinschaft (SFB 123). Appendix Leaving aside some technicalities we prove sup {/(x) - g(x)} = sup {g* (y) - f * (y)} x (A. 1) y for f and g convex. We have sup{g*(y)-f*(y)} Y = sup {sup [y- x - g ( x ) ] - sup [y. x - f(x)]} y X X < sup {[-y. x - g(x)] - [y. x - f(x)] } y,X = sup { f ( x ) - g(x)} = sup { f ( x ) - g(x)}. y,x (1.2) x Analogously, using (A.2) and f---f** readily obtain [31], we sup { f ( x ) - g(x)} = sup {f**(x**)- g**(x**)} x x** < sup {g*(Y)- f * (Y)}. Y (A.3) Combining (A.2) and (A.3) we find (A.1) and, thus Eq.(6.17). In fact, the relation also holds for a normed linear space X instead of ~". Then y ranges through the dual space X* and (A.3) may be rewritten sup { f ( x ) - g(x)} = sup {f**(x**)-g**(x**)} xEX x**~X < sup {f**(x**)-g**(x**)} x**~X _-<sup {g* (x*) -f*(x*)}. x*~X* (A.4) In obtaining the second inequality we have used (A.1); the first inequality follows from X c _ X * * . We thank Prof. E.M. 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Vol. I, 3rd Edn., Sect.VII.3. New York: Wiley 1970 50. Ref. 31, Theorem42.F 51. Iooss, G., Joseph, D.D.: Elementary stability and bifurcation theory. Berlin, Heidelberg, New York: Springer-Verlag 1980 52. Eisele, Th., Enter, A.C.D. van, Hemmen, J.L. van: (Manuscript in preparation) 53. See, for instance, Verbeek, B.H., Nieuwenhuys, G.J., Stocker, H., Mydosh, J.A.: Phys. Rev. Lett. 40, 586 (1978) 54. Shell, J., Cowen, J.A., Foiles, C i . : Phys. Rev. B 25, 6015 (1982) 55. Anderson, P.W., Halperin, B.I., Varma, C.M.: Philos. Mag. 25, 1 (1972) 56. Wenger, L.E., Keesom, P.H.: Phys. Rev. B 13, 4053 (1976) 57. See also Joffrin, J.: In: Ill-condensed matter, Les Houches 1978. Balian, R., Maynard, R., Toulouse, G. (eds.), Sects. 1.1 and 1.2. Amsterdam: North-Holland 1979 58. Binder, K.: In: Fundamental problems in statistical mechanics. Cohen, E.G.D. (ed.), Vol.V, pp. 21-51. Amsterdam: North-Holland 1980 59. Soukoulis, C.M., Levin, K., Grest, G.S.: Phys. Rev. Lett. 48, 1756 (1982) J.L. van Hemmen A.C.D. van Enter J. Canisius Sonderforschungsbereich 123 Universit~it Heidelberg Im Neuenheimer Feld 294 D-6900 Heidelberg 1 Federal Republic of Germany