Recent developments in the theory of spin glasses

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Phvsica I!lt~& 1101~t(1'4S2) lt~12 It}14 Norlh. ttolland Publishing ('omp:m\

R E C E N T D E V E L O P M E N ~ I S IN T H E T H E O R Y O F SPIN G L A S S E S

G6rard T O U L O U S E Laboratoire de Physique de l'Ecole Normale Sup&ieure, 24 rue Lhomond. 75_~,~1 Paris ("&le.~ O'T. f"ram'e

A sketch of some recent ideas on spin glasses is presented.

A few months ago, I completed a review article on spin glasses, entitled "Frustrations et desordres: problames nouveaux en mecanique statistique. Histoire des verres de spin" Ill. A more compact survey, entitled "Spin glasses, with special emphasis on frustration erects", ,wets presented in R o m e in May 1981 [2]. The interested reader is advised to consult those texts: this note being only a collection of some basic or recent ideas which in no way do justice to the many accomplishments in the field. There has been a surge of activity on spin glasses during the last five years: experiments, numerical simulation, theory. Noteworthy developments have come from: (i) Extension of the class of "'spin glass" materials, from the archetypal materials (AuFe. CuMn . . . . ) to a much wider category of systems. This has helped a great deal to separate the essential from the contingent, in the interpretation of the characteristic properties of spin glasses. (ii) A new generation of experiments bearing on hysteresis, NMR, ESR, magnetization laws, frequency-dependent susceptibility. Also. the exploration of the transition region between long range order (ferromagnetic or antiferromagnetic) and spin glass order, with the possible existence of mixed phases (coexistence of spin glass and ferromagnetic ordering [3]), and the recent attention brought to magnetocaloric effects (of the adiabatic demagnetization type) 14]. (iii) Predictions of the mean field theory, (137S-4363/82/0000-0000/$02.75

(() 1982 North-Holland

mean field theory being defined as the solution of the infinite ranged Sherrmgton-Kirkpatrick (SK) model. Existence of an upper critical magnetic field for the spin glass phase corresponding to the spontaneous breaking of replica symmetry, the plateau of the magnetization versus temperature in the spin glass phase, the freezing of the transverse spin components for Heisenberg spins in a magnetic field, and the existence of two kinds of mixed phases [3]. These theoretical advances have allowed tile development of a new and fruitful dialogue between theory and experiment during the last year. The reason why mean field theory is so relevant for three-dimensional materials is nol yet satisfactorily understood, however. A long-standing question has concerned the existence or not of a well-defined spin glass transition temperature. Only recently has it been clearly recognized how this question shoukt be answered experimentally, in the paramagnetic phase, the Inagnetization can be developed in odd powers of the magnetic field. When a ferromagnetic transition is approached from high temperatures, the first coefficient (linear susceptibility) diverges. For a spin glass transition, it is the following coefficient (factor of H 3) which diverges. And at the spin glass transition, if there is one, the magnetization cannot be expanded in odd powers of the magnetic field. In practice one should plot M / H versus H. A new generation of careful experiments of this kind is under way [5]. It is remarkable how sharp the transition appears

G. Toulouse / Recent developments in spin glasses

to be in a material like A__gMn. Note that in these measurements made at high temperatures problems arising from time-dependent effects are avoided. The study of the infinite ranged SK model for Ising spin has brought a number of surprises. The replica trick, used as a way to obtain the average of the logarithm of the partition function over disorder, has led to the discovery of a new type of spontaneous symmetry breaking, namely breaking of the symmetry under permutation of the replicas. The most direct physical manifestation of this p h e n o m e n o n is now being understood as a violation (or anomaly) of linear response theory: under the presence of a small external perturbation, the equilibrium state of the system moves far away in phase space, therefore the equilibrium response cannot be evaluated from the fluctuations in the neighbourhood of the initial state. The theory of this broken symmetry phase is not easy: a scheme has been proposed by Parisi [6] and can be solved numerically. Its predictions are compatible with a very simple hypothesis, namely that the entropy S(T,H), in the spin glass phase, depends on only one variable, the temperature [7]. In turn, this hypothesis allows one to derive many predictions for the thermodynamic properties of the spin glass phase. Two kinds of comparison are then in order: (i) Comparison with numerical tests to see whether this hypothesis is verified in the SK model [8]; a new generation of more stringent tests is now under way. (ii) Comparison with experiments on real materials, to see how far mean field theory is relevant for three-dimensional systems. This last comparison invites one to consider Heisenberg spins (and not only Ising spins). In this case a new transition is found in the mean field theory, corresponding to the freezing of the transverse spin components (with respect to an applied field), i.e. a random canting. It is interesting to note that this spin glass transition does not involve replica-symmetry breaking, which

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occurs at lower temperature. Therefore, this transition can be studied via traditional techniques. Some questions concerning the influence of fluctuations, the upper and lower critical dimensions, can presumably be answered more easily for this transition than for the replicasymmetry breaking transition (for which awkward complex critical exponents have been found) (see note added in proof). Experimentally, there have been interesting measurements, concerning the existence of critical magnetic fields, the existence of a magnetization plateau in the spin glass phase, the controversial coexistence of spin glass and ferromagnetic ordering (the coexistence of spin glass order and superconductivity is also topical). The prediction that the thermodynamic entropy is independent of magnetic field in the spin glass phase (in mean field theory) led to the suggestion of performing adiabatic demagnetization measurements. This allows a new determination of the critical magnetic field [4], above which metastability effects become negligible. One important experimental finding concerns the approach into the spin glass phase. When the spin glass phase is entered, by decreasing the temperature under fixed magnetic field (fieldcooling), one finds few metastability effects and the material remains at equilibrium. However, when the temperature (or the entropy) is fixed and the magnetic field is decreased, the system appears to evolve into metastable states, somewhat similar to the original SK solution of the SK mo~.el, which assumed no replica-symmetry breaking. Can we understand these results and develop an interpretation of metastability, in analogy with the Van der Waals theory for fluids? What are the consequences for the observation of the mixed phases? Moreover, for metallic or ordinary glasses, could we find a second physical parameter, besides temperature, which would allow us to reach an "equilibrium" glassy state? Several analogies between spin glasses and ordinary glasses have been noted, e.g. entropy catastrophe

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(3. Toulouse ! Recent det,elopments iH spin ,~lasses

and Fulcher's law. It is tempting to push the analogy further and to bring the advances made in spin glass physics to bear on the case of other amorphous structures. Finally, it should be noted that the problem of spin glasses has led to an impressive refinemenl of numerical techniques: Monte-Carlo. exact enumeration of states for small samples, and optimization processes, hnportant numerical work is now performed to settle some welldefined controversies between different approaches and also to elucidate the connections between mean field theory and the properties of real materials.

Note added in pro~f These last statements must now be taken critically, because Marc Gabay has recenlly found a

replica instability in lhc transverse degrees ot freedom, at this upper transition.

References

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t~roceedings el the I~J~,i / ~ n g i { ' s tic hi Sttcit]t6 Fr~lm,'aisc de P h y s i q u e , i<> be p u b l i s h e d as a stipple'nit'ill i~t< {tit' J t l u l n a [ tic" P h y s i q u e

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Pt~+ccudiilgs ~>i The h l l c r t i a t i < i n t d ('(inlurul+¢c" ,H1 i ) i . ~/rtlcrcd cv~;Ic'lll~, ~li/tI I <~calizatilm. ( ' ( ' a s t u l l a n i . (" l ) i ( ' a M l l ) alld I . P c l i i i . c'ds.. I c c l u i c ~'lRt'~. in P h w i c s 14 lj (~pringor)

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