After olmost a decade of intense research on their unusualphases and even more unusual dynamical behavior, randommagnets have emerged as prototypes for a wide variety ofsystems with frozen-in disorder.
Daniel S. Fisher, Geoffrey M. Grinsrein and Anil Khurana
Much of the enormous increase in our understanding ofcollective phenomena during the past few decades hasarisen from the study of magnetic systems. The formula-tion of concepts such as universality, broken symmetry,and scaling near continuous phase transitions, as well asthe development of the powerful ideas of the renormaliza-tion group, have been strongly influenced by research inmagnetism. This is due in part to the availability of a hostof experimentally accessible magnetic systems, and inpart to the remarkable fact that simple models ofmagnetism capture the essential physics of the phases andordering transitions in more complicated systems.
Until relatively recently, study of the collectivebehavior of magnets and other condensed matter systemshas centered largely on ideal pure materials, such asperfect crystals. Quenched, or frozen-in, structural disor-der is so ubiquitous, however, that it affects the propertiesof virtually all experimental systems to some degree.Experimentalists are often driven to elaborate lengths toreduce the randomness in their samples to negligiblelevels. Only in the past ten years or so has widespread ap-
Daniel S. Fisher was until recently a member of the technicalstaff at AT&T Bell Labs, Murray Hill, New Jersey. He is nowa professor of physics at Princeton University. Geoffrey M.Crinstein is a physicist at IBM Thomas J. Watson ResearchCenter, Yorktown Heights, N. Y. Anil Khurana is anassociate editor of PHYSICS T O D A Y .
56 PHYSICS TODAY DECEMBER 1988
preciation for the fascinating phenomena caused bydisorder itself developed in the condensed matter commu-nity. No longer denigrated as "dirt," "junk" or anunavoidable nuisance, randomness and its consequenceshave become the objects of intense study.
Disordered magnets have emerged as prototypes forcollective phenomena in systems with quenched disorder.Again, this is a result of the simplicity of magnetic modelsand the existence of many convenient experimentalrealizations, some of which have almost ideal, homoge-neous randomness. Ideas from the study of randommagnets have already been applied to structural phasetransitions and charge-density waves in random alloys, tothe melting of intercalates, to dirty superconductors, tofluids and superfluids in porous media and to adsorptionand wetting on disordered surfaces. The renormalizationgroup, conceived by K. G. Wilson to handle problemsinvolving a broad range of length and time scales,1provides, as it does for pure systems, the unifyingframework for the understanding of this marvelouslydiverse class of random systems.
In this article we summarize current understandingof the phases and phase transitions of random magnets,comparing and contrasting them with conventional, puremagnets. We first briefly review the equilibrium phasesand phase transitions of pure magnets, emphasizing therole of symmetries and the spatial dimensionality. Identi-fying three main classes of disordered magnets—randomexchange, random field and spin glass—we then considerthe central issues connected with their equilibriumproperties, such as the existence and nature of theirordered phases. We will see that for spin glasses, a
Droplet of overturned spins (white)in a putative ferromagnetic (up)ground state of the random-fieldIsing magnet. A frozen-in magneticfield acts at each site in the magnet;the field points up on the blue sitesand down on the green sites. Theconfiguration with the overturneddroplet can have lower energy thanthat of the wholly ferromagnetic upconfiguration because the dropletoccurs in a region in which therandom fields point predominantlydown. Y. Imry and S.-k. Maestimated the probability ofoccurrence of such droplets andconcluded that the ferromagneticstate is unstable with respect to theformation of large droplets in fewerthan 2 dimensions even at T = 0.
surprising degree of controversy still surrounds thesequestions. Finally we turn to nonequilibrium effects,which distinguish random magnets strikingly from theirpure counterparts. The existence of a rather sharp"freezing" temperature, below which the system respondsextremely slowly to changes in external conditions, is adramatic signature of quenched randomness on theordering process. Below this temperature, many randomsystems are simply unable to achieve equilibrium in anyreasonable time and so show history-dependent behavior.We will argue that in random magnets, unlike in puresystems, such nonequilibrium manifestations are ofteninextricably linked to equilibrium collective phenomena.This necessitates a refinement of such common notions asequilibrium states and metastability.
Pure magnetsWe focus on the simplest magnetic models, those consist-ing of Ising spins, S,, which can only point "up" (S, = + 1)or "down" (St = — 1). The basic Ising Hamiltonianconsists of exchange interactions JtJ between nearest-neighbor pairs of spins (i,j) on a hypercubic lattice in dspatial dimensions:
H = - (1)
This Hamiltonian exhibits an obvious, global spin-flipsymmetry: The energy is invariant under a simultaneouschange of sign of all the spins.
In a pure Ising ferromagnet all the exchange ini erac-tions have the same value J. Such a system has twophases: At temperatures large compared with J, the
entropy dominates over the energy and the spins fluctuatealmost independently.1 The global spin-flip symmetry ispreserved in this, the paramagnetic phase, each spin beingup on average as often as it is down. Correlations betweentwo spins separated by a distance r decay like e r / f ,where £ is a characteristic correlation length. Conversely,at low temperatures, the energy dominates over theentropy and the spins are almost all aligned, either in theup or down direction. In this, the ferromagnetic phase,then, there are two equivalent equilibrium states—"up"and "down"—related to each other by the spin-flipsymmetry. In choosing one of these states over the otherthe system spontaneously breaks this symmetry, and eachspin develops a nonzero expectation value (S, >, whichindicates that in equilibrium it spends more time pointingup, for example, than down. (Here <. . . > denotes anaverage over very long times.) This "spontaneous magnet-ization density," m = <S, >, vanishes in the symmetric,high-temperature phase and therefore serves as an orderparameter that characterizes the broken-symmetry, orordered, phase. In the ordered phase the correlationbetween two spins approaches the non-zero value mz as thedistance between them goes to infinity. This property iscalled long-range order. The transition between the twophases occurs at a critical temperature T"c of order J, atwhich m vanishes when the system is heated from lowtemperatures. In the pure Ising model of equation 1, £diverges and m vanishes algebraically at Tc, a behaviorcharacteristic of so-called continuous phase transitions.
This simple intuitive picture of the phases of the pureIsing ferromagnet is correct for all dimensions d > 1. Inone dimension, the fact that each spin is connected to the
PHYSICS TODAY DECEMBER 1988 5 7
Frustration in spin glasses arises because loops with an oddnumber of antiferromagnetic bonds do not have a uniquespin configuration of lowest energy. In a simple square withthree antiferromagnetic bonds (green) and a ferromagneticbond (blue), all of equal strength, either of the twoorientations of the spin at the lower right corner gives thesame energy when the orientations of the other three spinsare fixed as shown. This energy is higher than that of thelowest-energy configuration of a square loop with evennumbers of antiferromagnetic and ferromagnetic bonds ofequal strengths.
rest of the lattice by only two bonds means that thermalfluctuations can readily disrupt any ferromagnetic align-ment. Thus, while at zero-temperature the one-dimen-sional Ising model is ordered, at any positive temperaturethe entropy always dominates the energy at long lengthscales and the system is disordered. The principle thatordered states are more easily destroyed by thermalfluctuations as d decreases is very general and gives rise tothe notion of the lower critical dimension, d,, which isdefined as the lowest dimension (not necessarily integer!)above which an ordered phase can exist at nonzerotemperature. For Ising ferromagnets, then, d, = 1.
One can often usefully describe ordered phases interms of their ground states and excitations. In an Isingferromagnet the two obvious ground states have all spinspointing either up or down. (See the box on page 62,however.) The lowest-energy excitations of linear size Lare connected, compact droplets of Ld overturned spins. Itcosts energy, of order J times the surface area ~ Ld 1 , tocreate such a droplet. This simple idea underlies theimportant notion of the "stiffness" of an ordered phase—in many ways its most fundamental characteristic, and aparticularly vital one in our understanding of disorderedsystems. One gets a sense of how basic this notion is by ob-serving that nonphysicists who have never heard ofbroken symmetry have a clear intuitive understandingthat solids are distinguished from liquids by their rigidity,or resistance to distortion. J. M. Kosterlitz and D. J.Thouless have shown that certain ordered phases—forexample, solids in two dimensions—can exist withoutbroken symmetry or long-range order!2 Such phases arecharacterized by their stiffness; for example, a two-dimensional solid is characterized by a nonzero shearmodulus.
The stiffness of a system depends on the length scaleon which it is probed; a rough definition of the stiffness atlength scale L is the cost in free energy of distorting the or-der significantly in a region of volume L'1. In an Ising fer-romagnet, a domain wall separating up and down spins isneeded to distort the order. This costs a free energyproportional to the wall's area L'1 ' , the coefficient ofproportionality being the interfacial tension a. At T= 0,a = J. In general, the free-energy cost of excitations andhence the stiffness of an ordered phase grows with lengthscale L like L", where 0 is the stiffness exponent. If 0 isnegative, large excitations have free energies much lessthan kB Tand so proliferate, destroying the ordered phase.For an ordered phase to exist, therefore, 0, which dependson the dimension d as well as on the system underconsideration, must be non-negative. The marginal case0 = 0 thus corresponds to the lower critical dimension. Inone dimension, for example, ferromagnetic order in pureIsing models is destroyed by droplet fluctuations because
0 = d—l. For solids and superfluids 6 = d — 2, whichimplies that these ordered phases cannot exist in fewerthan two dimensions. The special Kosterlitz-Thoulessphases occur in the marginal case d = 2.
DisorderMost random magnets are substitutionally disorderedmaterials in which several kinds of magnetic or nonmag-netic ions are alloyed together. The qualitative effect is tomake the exchange interactions between pairs of spinsvary randomly from one pair to another, the interactionsfor different pairs being totally uncorrelated in the idealcase. The simplest possibility, called the random-ex-change model, occurs when the couplings are all ferromag-netic (that is, all J^ > 0 in equation 1) but vary in strength.Spins are still fully aligned in the ground states of this sys-tem; for d>d, = 1 a ferromagnetic phase, in which theglobal spin-flip symmetry is spontaneously broken, existsbelow a finite critical temperature. Though early experi-mental data suggested that the sharp transitions in purematerials were somehow broadened, or "smeared," byspatial inhomogeneities, heuristic arguments by A. B.Harris, later confirmed by renormalization group calcula-tions, showed3 that this need not, in fact, be the case, atleast in the absence of macroscopic inhomogeneities, orlong-range correlations in the impurity positions. To seethis, imagine dividing the system into blocks of £d spins,where t, is the correlation length. Spins in different blockscan be thought of as essentially uncorrelated. As S, grows,the statistical variations in the average exchange strength(and hence in the effective Tc) of the various blocksshrinks, thus allowing the system to undergo a perfectlysharp transition at a well-defined Tc. Fabrication of high-quality crystals with almost ideal substitutional disorderhas since allowed experimental confirmation of this idea.4(Indeed, the improvement in fabrication techniques hassparked significant overall progress in the understandingof random systems.) The earlier, "smeared" transitionswere presumably artifacts of macroscopic inhomogene-ities. In fact, the phase transitions in the best randommagnets are virtually as sharp as those in pure magnets(see the figure on the opposite page). Thus the equilibriumbehavior of random-exchange ferromagnets is qualitative-ly the same as that of pure systems.
Far more interesting behavior occurs when thedistribution of the J,, 's includes both positive (ferromagne-tic) and negative (antiferromagnetic) interactions. It isthen possible for the interactions to compete with eachother as shown in the figure above, making it impossible tosatisfy simultaneously all of the exchange interactions ofthe Hamiltonian. This property, called frustration, isresponsible for much of the fascinating behavior of randomsystems. Finding the ground states of a frustrated system
5 8 PHYSICS TODAY DECEMBER 1968
Sharp divergences in the correlation length and thestaggered susceptibility at a critical Tt mark the transition toan antiferromagnetic state in the disordered antiferromagnet
Fe0 5 Zn0 5 F2. The figure shows the inverse of the correlationlength (blue) and of the staggered susceptibility (black).
(Adapted from reference 4.)
is a highly nontrivial optimization problem in which onemust decide which bonds to satisfy.
If the density of negative couplings, and hence thefrustration, is small, only a few spins will be misaligned inthe ground state, and the ordered phase will generally belike that of a normal ferromagnet. When enough of thecouplings are negative to produce strong competition,however, the behavior becomes very subtle, and basicissues such as the existence and nature of an orderedphase become difficult. Such systems are known as spinglasses.5 The system denned by the Hamiltonian ofequation 1 but with the J,/s symmetrically distributedabout zero is the Ising-spin version of the model firstproposed by S. F. Edwards and P. W. Anderson as aprototypical spin glass.6 (Their original model involvedHeisenberg spins—unit-length vectors free to point in anydirection on the unit sphere—which are more realistic.)
Rather than immediately considering spin glasses, wefirst discuss the random-field Ising model, a somewhatsimpler system with a different kind of frustration. Afteryears of sometimes bitter controversy, a rather clearunderstanding of this model has finally emerged. (Spacelimitations prevent us from discussing random-axis mod-els, an interesting class of frustrated systems in whicheach of the Heisenberg spins experiences single-siteanistropy, the orientation of the preferred axes varyingrandomly from site to site. These magnets have featuresin common with both random-field magnets and spinglasses. We refer the interested reader to reference 5 for abrief review and bibliography.)
Random-field Ising modelThe model is defined by the Hamiltonian
(2)
The magnetic fields h, are random variables with no (or atmost, short ranged) correlations between their values atdifferent sites. They are chosen from an even distribution,so the fields point randomly up or down and their averagevalue is zero. Though the random-field model remained atheoretical construct for several years following itsintroduction7 in 1975, it is now understood to describe theessential physics of a strikingly rich class of experimental-ly accessible disordered systems. These include structuralphase transitions in random alloys, commensurate charge-density-wave systems with impurity pinning, binary fluidmixtures in random porous media, and the melting ofintercalates in layered compounds such as TiS2. The mostaccurate data thus far have come from a magneticrealization—disordered Ising antiferromagnets such asFe^Zn, _XF2 in a uniform magnetic field.
The behavior of the random-field model is governed by
0.002.40 41 42 43 44 45 46 47
TEMPERATURE (K)
the competing tendencies of the spins to align ferromagne-tically under the impetus of the exchange Jor to follow thelocal fields h, and so be uncorrelated. This is a competi-tion between two energies—the exchange energy and therandom-field energy—and therefore differs from theenergy-entropy battle that controls the phase transitionsin pure and unfrustrated random systems. Thermalfluctuations play a secondary role in this struggle, since,as we shall see, their disordering effect is considerablyweaker than that of the random fields. One therefore canunderstand most of the equilibrium physics of the model atall temperatures, including the critical properties, bystudying the model at zero temperature.
The random-field model's simplicity relative to spinglasses stems from the fact that its possible phases andorder parameters seem rather straightforward. Eventhough the Hamiltonian of equation 2 has no spin-flipsymmetry, the model is spin-flip symmetric in a statisticalsense: The transformation h, — — h,, S, — — S, showsthat the magnetization m, averaged over the ensemble ofrandom fields, vanishes for any even distribution of theh, 's. Thus </n>av is the natural order parameter for therandom-field model: When the exchange dominates, thesystem orders ferromagnetically, and <m>av acquires anonzero value. When random-field or thermal effectsdominate, the system is a paramagnet, with <m>av = 0, asrequired by the "average" spin-flip symmetry.
This apparent simplicity is deceptive, however. Let usdenote the typical magnitude of the random fields by h.One naively expects that at low temperatures the systemwill be a paramagnet for h/J^l and a ferromagnet forhiJ41- While it is true that for sufficiently large h/Jeachspin will follow its local field, producing paramagnetism atall temperatures and in any dimension, the situation atsmall hIJ is more subtle. This was first appreciated by Y.Imry and S.-k. Ma,7 who studied the stability, with respectto the overturning of a large droplet of spins with linearsize L, of an assumed perfectly aligned ferromagneticground state (say "up") in the presence of random fieldsh4 J. Formation of such a droplet will gain energy fromthe random-field term of the Hamiltonian only if statisti-cal variations have produced a preponderance of down-pointing fields within it (see the figure on page 57). The ex-cess of down fields will typically be of order (Ld )!/2 in d di-mensions, whereupon roughly hL'"2 in random-fieldenergy is gained. The exchange-energy cost of the droplet
PHYSICS TODAY DECEMBER 1988 5 9
0.05;
— 0.01 -
x<
<x
QS<x
0.005 •
0.001 -
42 44TEMPERATURE (K)
46
Hysteretic behavior of a random-field Ising magnet. Thefigure shows the width at half maximum, proportional to theinverse correlation length, of a neutron scattering peak in thedisordered antiferromagnet FeObZnO4F2 placed in a 5.5-Tmagnetic field. When the magnet is cooled in the field, thehalf width (gray) is larger than the experimental resolution(dashed line); but if the field is turned on after the magnet iscooled below Tr = 43 K, then the half width is limited byexperimental resolution (green). A resolution limited halfwidth is evidence for antiferromagnetic ordering. For 7"< 7"cthe half width remains essentially constant at its low-temperature value (blue) when the magnet is heated afterbeing cooled in the field . [Adapted from D. P. Belanger,S. M. Rezende, A. R. King and V. Jaccarino, /. Appl. Phys.57(1)3294(1985).]
is proportional to its surface area, ~JLd '; thus for anyd > 2 it dominates the gain from the random-field term atlarge L, ensuring the stability of the ferromagnetic groundstate. When d <2, however, the random-field energydominates at sufficiently large L even for arbitrarily smallh/J, thereby rendering ferromagnetism unstable withrespect to droplet formation. This domain-wall argumenttherefore predicts that the lower critical dimension is 2 forthe random-field model when h/J is small. For d = 2,where both energies are proportional to L, the exchangeenergy dominates for h/J<il, but a more sophisticatedversion of the argument that treats the meandering (orroughening) of walls in response to local pockets of fieldspointing predominantly up or down (see the figure on page57) shows that the ferromagnetic ground state is actuallyunstable in this marginal case."
Doubt was soon cast upon the conclusion d, = 2 byformal field-theoretic arguments called dimensional re-duction,8 which purported to show that dt = 3. (See thebox on page 67.) Fueling the ensuing controversy wereresults of quantitative experimental studies of random-field magnets made possible by the fabrication of extreme-ly high-quality dilute Ising antiferromagnets such asFe^Zn, _XF2. Application of a uniform magnetic field tosuch systems generates an effective random field thatcouples to the antiferromagnetic order parameter, produc-ing a realization of the random-field model in which thestrength of the effective random field varies with theexternally applied uniform field.9 In principle, high-resolution neutron scattering experiments on these sys-tems ought to have settled the debate about d,, in that thedevelopment of long-range antiferromagnetic order (thatis, a Bragg peak in the static structure factor) at lowtemperatures in three-dimensional samples would con-firm that d, = 2. In practice, this expectation wasconfounded by the occurrence of hysteresis: Samplescooled in even very modest effective random fields showedno Bragg peak and hence no long-range magnetic order,while the long-range antiferromagnetic order developedunder cooling in zero applied (that is, zero random) fieldpersisted under subsequent application of even relativelylarge fields (see the figures above and on the oppositepage).10 The same basic phenomenology was observed intwo- and three-dimensional samples. Difficulties in decid-ing which, if either, of the two modes of measurement wasyielding the true equilibrium behavior extinguished hopesfor a clean experimental resolution of the d, quandary;indeed, these problems intensified the debate.
More recently, rigorous proofs by J. Z. Imbrie and byJ. Bricmont and A. Kupiainen have established thecorrectness of the result d, = 2, and hence of the equilibri-
um phase diagram of the three-dimensional random-fieldmodel shown on the opposite page." Furthermore, anexplicit mechanism for the failure of dimensional reduc-tion, connected with the neglect of statistically rare butphysically significant regions of the sample (see the box onpage 67) has now been identified.12 This still leavesunexplained, however, the apparent inability of theexperimental systems to equilibrate on reasonable timescales. We shall defer treatment of this fundamentalquestion to the section on nonequilibrium phenomena.
Spin glassesTheoretical interest in spin glasses was initiated by theexperimental observation5 l:J of a rather sharp cusp in thetemperature dependence of the low-field, low-frequencysusceptibility in the dilute metallic alloy CuMn (see thefigure on page 63), at roughly 1% concentration ofmagnetic ions (Mn). (See the columns by Anderson in theJanuary, March, June and September issues of PHYSICSTODAY.) Neutron diffraction studies showed that no Braggpeak at any wavenumber arises below the temperature Tfwhere this cusp occurs; thus no long-range ferromagneticor antiferromagnetic order accompanies it. As oneapproaches T( from above, relaxation times becomeextremely long and the system begins to exhibit hystere-sis, which suggests that T( is a kind of "freezing"temperature. The figure on page 63 schematically shows atypical history-dependent effect: The susceptibility ofsamples cooled in a finite field is rather fiat below T{ and isalmost reversible under subsequent heating and cooling.However, application of the same field to a sample cooledin zero field results in a much-reduced susceptibility attemperatures below Tf; this reduced susceptibility in-creases extremely slowly with time for fixed field andtemperature. Magnetic remanence—the persistence of afinite magnetic moment that decays slowly after thesystem is field-cooled below T[ and the field is removed—isalso a standard feature of spin glasses. Although thesephenomena were first observed in dilute metallic samples,the same qualitative features have since been seen inmany very different systems, such as the magneticinsulator Eu^Sr, t S at high concentrations; these char-acteristics are now taken to constitute a loose definition ofspin glass behavior.13
Edwards and Anderson attributed these phenomenato the competition between random ferromagnetic andantiferromagnetic interactions.6 They hypothesized thatthe cusp and apparent freezing at Tf were associated witha true phase transition into a state with broken spin-flipsymmetry. They took the viewpoint that the frustrationinduced by the competing interactions is paramount and
60 PHYSICS TODAY DECEMBER 1986
that the details of how it arises in real systems areperipheral. Thus, even though in dilute metallic alloyslike CuMn competing interactions arise because of theoscillations in sign of the long-range (RKKY) interactionbetween the spins of the randomly positioned magneticions, Edwards and Anderson chose a simple lattice modelwith random, competing short-range interactions—theHeisenberg-spin version of equation 1 with the JtJ'sdistributed symmetrically about zero. They proposed thatthe model orders by spontaneously breaking the spin-flipsymmetry to produce the low-temperature (spin glass)phase. Each spin S, develops a nonzero expectation value,m, =<S,>, in this phase, but the sign and magnitude ofm,'s vary from site to site because of the randomcompeting interactions. The spatial average of the m, 's,that is, the average magnetization density, thus vanishes,as do all other Fourier amplitudes of the m, 's, consistentwith the absence of Bragg peaks in the neutron scatteringmeasurements.
Edwards and Anderson proposed the quantityN
q = (l/N) £ m;i i
where N is the number of spins, as a suitable orderparameter. This "Edwards-Anderson order parameter"vanishes in the disordered phase, where m, = 0 by spin-flip symmetry, and is nonzero in the spin glass phase.Edwards and Anderson went on to construct an approxi-mate mean-field solution for their model, finding that qdid indeed develop a nonzero value at a continuous phasetransition at finite temperature, with an associated cuspin the susceptibility, consistent with the measurements.
This insight left open the crucial question of theexistence of the ordered, spin glass phase in the physicaldimensionalities 2 and 3, that is, of the lower criticaldimension for spin glasses. There are undoubtedly largeclusters of spins in the spin glass ground state that, owingto a delicate balance between ferromagnetic and antiferro-magnetic interactions, can be flipped with a relativelysmall energy cost. This notion that there are manyconfigurations differing substantially from the groundstate in structure but almost identical to it in energysuggests that the spin glass state is particularly fragileand might be readily destroyed by thermal fluctuations.
Unfortunately, computing d, is extremely hard. The spinglass ground state, in which spins point up or down in acomplicated pattern to minimize the exchange energy ofthe random, competing interactions, is hopelessly difficultto compute analytically, and can be computed numericallyonly for very small samples. As in the random-field model,one expects droplet excitations of size L away from theground state to be the chief destabilizing agents of theordered, spin glass state. Unlike in the random-fieldmodel, however, the free-energy cost of such droplets (thatis, the stiffness) cannot readily be determined. In conse-quence, estimates of dt for Ising spin glasses havewandered inconclusively between 2 and 4 for ten years.In the last few years, however, a fairly universal consensusthat 2<d, <3 has emerged.5 Much of this progress hascome from exhaustive Monte Carlo simulations on sam-ples with up to 64x64x64 lattice sites in three dimen-sions. A major advance in the effectiveness of the MonteCarlo method for spin glasses resulted from the decision ofA. P. Young and others to study the behavior of the spinglass susceptibility ^SG in the paramagnetic phase as theputative critical temperature Tt. is approached fromabove. Working in the paramagnetic phase circumventsthe problem of the very long equilibration times encoun-tered below Tc (see the section on nonequilibrium phenom-ena). According to the theory of phase transitions, Ysc, > a n
analogue for spin glasses of the staggered susceptibilitywhose singular behavior characterizes the phase transi-tion in antiferromagnets, should diverge at 7 . if the spinglass transition is continuous. Numerical simulations ofthree-dimensional Ising spin glasses show clear signs ofthis divergence, so one may infer with some confidencethat a continuous phase transition occurs (see the upperfigure on page 64). Similar studies in two dimensions, onthe other hand, indicate that T,. is pushed down to 0.From this we conclude that 2<dt<3. Of course thisinference, based as it is on the analysis of finite systems,will always be somewhat uncertain. But most workers inthe field find the simulations and other numericalevidence5 in favor of this result quite compelling.
Experimenters have used a similar strategy to deter-mine whether real three-dimensional spin glasses undergotrue phase transitions." The nonlinear susceptibility of aspin glass (defined as \ i = diM/dH3\H = n , where Mis the
PARAMAGNET
UJLJ_
oD<
TEMPERATURE
Theoretical phase diagram of three-dimensional random-field magnets. A true equilibrium phase transition occurs atthe black line, whereas the red line marks the onset, oncooling, of irreversible (hysteretic) behavior on macroscopictimescales. A point in the ordered phase may be reachedeither by cooling in a finite field (gray) or by cooling in zerofield and then turning on the field (green). For small randomfields, the phase diagram for one random-field magnet hasbeen mapped out in detail by V. Jaccarino, A. R. King andD. P. Belanger, / Appl. Phys., 57(1) 3291 (1985).
PHYSICS TODAY DECEMBER 1988 6 1
MetastQbility and Equilibrium StatesA system is said to be in a metastable state when itsproperties are virtually independent of time over a widerange of time scales but ultimately change at some evenlonger time. In pure systems such behavior occurs near afirst-order phase transition. If, for example, a smallnegative magnetic field is applied to a pure Ising ferro-magnet in the ordered state with a positive, or "up,"magnetization, a very long time is required to nucleate(critical size) droplets of the "down" state that grow withtime and thus reverse the magnetization. During thistime, the up state is metastable. There is a largeseparation between the time scale for the microscopicrelaxation that occurs immediately after the field isapplied, and the time scale for nucleation of the downstate. (See the figure at right.) Such a wide separation oftime scales is needed for metastability to be a usefulconcept.
Distinguishing between metastable states and equilibri-um states, which are stable for infinitely long times, is asubtle but important problem related to understandingground states of an infinite system. For a finite classicalsystem, the ground state (or states, if there is degeneracy)is simply the configuration of lowest energy. However,for infinite systems, especially in the presence of random-ness, such a statement is not meaningful! A ground stateof an infinite magnet is any configuration whose energycannot be lowered by changing the spin configuration inany finite region. Thus in an Ising ferrmomagnet, theconfiguration with a single planar domain wall separatingup spins from down spins is a truly stable ground state,rather than a metastable one, even though it has a higherenergy locally than the wholly up configuration does.Bona fide equilibrium domain-wall states can also exist atnonzero temperatures. A recent, somewhat surprisingtheoretical prediction is that no equilibrium domain-wall-like states exist in spin glasses.19
10 1CT4 10°TIME (sec)
108
In random Ising magnets domain walls are effectivelyfrozen over long times, local fluctuations being virtuallytime independent. These domain walls thus produce akind of metastability. In any decade, or "epoch," oftime, however, there is always relaxational activity takingplace in some region of a random magnet. Macroscopicproperties of the system thus evolve in every epoch, andthe clean separation of time scales found in pure systemsdoes not occur. The figure shows the time dependenceof a macroscopic variable in a conventional metastablesystem (blue) and that in a random magnet (red). Todistinguish between these two kinds of metastable behav-ior one must probe a system on a wide range of timescales.
The spectrum of time scales in random magnets is sobroad that in any epoch almost all processes in thesystem are either in nearly perfect equilibrium or com-pletely out of equilibrium, with only a few processesactually equilibrating in that epoch. Ideas based on thispartial separation of time scales have led to insights intosome experiments on hysteresis and remanence in spinglasses.19
magnetization produced by an applied magnetic field H)turns out to be closely related to ^SG in the paramagneticphase, and so should diverge at the Tc if the spin glasstransition is continuous. Measurements of this quantityin several spin glass systems, notably CuMn and AgMn,show strong evidence of such a divergence, and hence of aspin glass transition (see the lower figure on page 64).514
One can never in practice see an actual divergence in a fi-nite system, so the case is not airtight, but combined withother evidence from studies of dynamical effects, the datastrongly suggest that at least some real spin glasses doundergo a genuine phase transition in three dimensions.
Equilibrium properties of spin glossesHaving argued that long-range spin glass order occurs inthree dimensions, in both simple models and real systems,we now discuss the equilibrium properties of the spin glassphase itself. Much of the theoretical work on spin glasseshas been based on an infinite-range model introduced byD. Sherrington and S. Kirkpatrick.15 The model is definedby equation 1, except that a random exchange bond J:Jconnects every pair of sites (i,j), not just the nearest-neighbor pairs. Such infinite-range models provide aformulation of mean-field theory, which, at least for puresystems, can be solved exactly (see the box on page 67).Mean-field theory, the first refuge of statistical mechani-cians, has been a source of insight into the ordered phasesand phase transitions of many complex systems.
The SK model is an ordinary paramagnet at hightemperatures. As the critical temperature TQ is ap-proached from above, the spin glass susceptibility ^SGdiverges like (T — Tc)"', signaling a continuous spin glasstransition. The ferromagnetic susceptibility has a cusp atTc and remains constant for all T< Tc, while the Edwards-Anderson order parameter grows linearly from zero for Tslightly less than Tc. All of this is in accord with theoriginal Edwards-Anderson calculations. The spin glassphase, on the other hand, has been argued by G. Parisi andothers516 to have a surprisingly rich structure: For allT< Tc there exist (in a sense that is not yet entirely clear)an infinite number of distinct equilibrium states, orvalleys in the free-energy landscape, into which thesystem may fall when cooled below Tc, much as an Isingferromagnet spontaneously selects one of the two equiva-lent states ("up" or "down") available to it below the Curietemperature. Each such state is separated by infiniteenergy barriers from all the others; having selected one ofthem, the system remains in it forever. The values ofphysical quantities such as the energy and Edwards-Anderson order parameter are independent of the sample(that is, of the particular realization of the JtJ 's) and of thestate. Remarkably, however, some macroscopic quantitiesdepend on the sample even for arbitrarily large systems!5
The spin-glass susceptibility, which diverges as T— Tcfrom above, stays infinite for all temperatures below Tc,indicating that the states are extremely sensitive to an
62 PHYSICS TODAY DECEMBER 1988
Static susceptibility of the spin glass CuMn for 2.02 atomicpercentage of Mn. Below T, ~ 1 5 K, the measured
susceptibility (MIH) is larger when the sample is cooled in asmall magnetic field (gray) than when it is cooled in zero field(green). Arrows indicate the direction of temperature changes
in the course of the two measurements. Such hystereticbehavior is characteristic of spin glasses. [Adapted from S.Nagata, P. H. Keesom and H. R. Harrison, Phys. Rev. B 19,
1633 (1979).]
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applied field. Perhaps most remarkably, the spin glasstransition in the SK model persists even in the presence ofa magnetic field H, with \<SG diverging at a finitetemperature Tc (H) despite the fact that the field explicitlybreaks the spin-flip symmetry of the Hamiltonian.
As yet, the relevance of the SK model to the behaviorof spin glasses with more realistic short-range interactionsremains unclear (see the box on page 67). Building onearlier attempts to calculate the stiffness of short-rangespin glasses,17"4 Fisher and D. A. Huse have proposed avery different picture of the spin glass phase.19 Theyhypothesize that like the ordered phases of simplersystems, the spin glass phase can be characterized by itsstiffness, which they assume grows with length scale L likeL". The argument that ordered phases are stable onlywhen the stiffness exponent is positive requires, given thenumerical results for spin glasses, that 6 > 0 for d = 3 and9 < 0 for d = 2 or 1. The lowest-energy droplet excitationsaway from the ground state, which contain Ld spins,typically have free energies of order FD (L)zz TL", where Yis the (temperature dependent) stiffness modulus of thespin glass. Although such droplets are compact, theirdiameters being typically of order L, their boundaries canbe shown to be fractal,'9 with surface areas proportional toLd", where d — 1 < ds <d. (See the article by Po-zen Wongon page 24.)
Because of the strong frustration, large sections of thewalls separating droplets actually have negative energy(though the total energy must of course be positive). Thissuggests that droplet energies are roughly given by a sumof terms random in both magnitude and sign, producing anexponent 6 much smaller than for ferromagnets, where9 = d— 1. A more careful argument yields the bound0<(d- l)/2; the best numerical estimate1718 gives 6-0.2for d = 3. The frustration also produces droplets ofenergies arbitrarily close to zero; even for large L thedistribution of droplet energies has a nonzero weight downto FD (L) = 0. Clearly only droplets with free energies S Twill be excited at low temperatures; these droplets are the"active" excitations of the system. Although arbitrarilylarge droplets can be thermally active at very lowtemperatures, large active droplets are rare and so tend tobe far apart (see the figure on page 65).
This general picture yields many specific predictionsfor the properties of the spin glass phase. For example, theconnected correlations dr) between a typical pair of spinsseparated by a distance r decay exponentially with r. Inthe rare event that the two spins happen to be part of alarge active droplet, however, their correlations are muchstronger. The mean-square correlation function—anaverage of (C(r)f over all pairs of spins separated bydistance r—reflects the strong correlation of such unusualpairs and decays only algebraically with r.'9 In conse-quence, spin glass and nonlinear susceptibilities areinfinite at all temperatures in the spin glass phase. Thephenomenon of physical quantities (in this case, ^SG, for
example) being controlled by rare, statistically unlikelyregions of the sample is a subtle but important effect thatoccurs frequently in disordered systems.
Another interesting prediction of the droplet pictureof the spin glass phase is the extreme sensitivity of theequilibrium states to changes in temperature. Arbitrarilysmall temperature changes upset the delicate balance thatproduced the equilibrium state at the original tempera-ture, and so result in the reorientation of large regions ofthe system. As a result, the relative orientations, inequilibrium, of spins located sufficiently far apart changerandomly with arbitrarily small changes in temperature!The evolution of the spin glass phase with temperaturemay therefore be thought of as an infinite sequence ofinfinitesimal first-order transitions. As we shall see, thisphenomenon plays an essential role in preventing spinglasses from reaching equilibrium.
Identification of large-scale, low-energy-cost (active)droplets as the dominant excitations of the spin glassphase also gives rise to some rather striking predictionsabout the equilibrium dynamics. For example, the powerspectrum of the equilibrium magnetization noise ispredicted to behave, aside from logarithmic corrections,like \lca—that is, to exhibit equilibrium "\/f" noise at lowfrequencies co. The magnetic susceptibility, a relatedquantity, is argued to vary only logarithmically withfrequency.19 Both predictions are in quantitative agree-ment with recent experiments.20
Although some aspects of this emerging picture of thespin glass phase in systems with short-range interac-tions—for example, the fact that ^SG is infinite—aresimilar to predictions for the SK model, crucial differencesdo exist: First, scaling arguments and the intuitiongleaned from the study of random-field and random-interface models strongly suggest that Ising-spin glassmodels with short-range interactions have only one pair ofground states (and hence only two equilibrium states forT> 0), related to each other by the global spin-flipsymmetry!19 The basic physics underlying this assertionis that a system stiff enough to support an ordered spinglass phase is also stiff enough to resist, in the thermody-namic limit, the overturning of any infinite proper subsetof the spins. Secondly, the Imry-Ma domain-wall argu-ment7 (first used in this context by W. L. McMillan17) canbe used to show that a magnetic field destroys the spinglass phase. The reason for this is that (9<(<i — l)/2, so thefield energy —HLd'2 gained by overturning a dropletalways dominates the exchange cost of ~L° at large L.
These predictions directly contradict two of the mostdramatic features of the spin glass phase suggested by theSK-model results. While there are various possible holesin the arguments leading to these predictions—which arecertainly controversial—it is fair to say that no consistentpicture of finite-range spin glasses incorporating the SK-model result of many equilibrium states has yet beenconstructed.
PHYSICS TODAY DECEMBER 1986 6 3
Nonlinear susceptibilities. Top right: Result of Monte Carlosimulations of three-dimensional Ising spin glasses with 83
(blue), 163 (gray), 323 (green) and 643 (black) spins,performed on a special purpose computer. (Figure adapted
from A. T. Ogielski, Phys. Rev. B 32, 7384 (1988).)Bottom right: Experimental result for the spin glass AgMn
with 10.6 atomic percentage of Mn. Straight lines indicatethe behavior expected if the nonlinear susceptibility divergesat a critical temperature 7"t like (T — Tc) ' with y= 1 or 2.
These data are strong evidence for the existence of a spinglass phase. [Figure adapted from P. Monod and H.
Bouchiat, / Phys. (Paris) Lett. 43, 145 (1982).]
Nonequilibrium phenomenaIn comparing with experiment the results of calculationsfor static, equilibrium properties one implicitly assumesthat the system being investigated achieves equilibriumon time scales short compared with those of the experi-mental probes. This assumption is almost never violatedin typical pure systems, but its conspicuous failure indisordered systems has dramatic consequences for theexperimental phenomenology.
The degree to which static equilibrium theoriesaccurately describe measurements depends on the rate atwhich the system equilibrates following a change in someexternal parameter such as temperature, relative to therate at which that parameter is varied. Since theequilibrium correlation length £ diverges at a continuousphase transition, and since correlations cannot grow toinfinity in a finite time, any large enough system cooledthrough a critical point at a finite rate will drop out ofequilibrium. (The divergence of £ in the vicinity of acritical point is always accompanied by the growth of thecharacteristic relaxation, or equilibration, time, sincemany spins must relax coherently.21 This phenomenon iscalled "critical slowing down.") For typical pure systems,however, the equilibration rate is so rapid that evenmacroscopic systems can equilibrate in reasonable experi-mental times. For example, the characteristic time rrequired for a pure Ising system at its critical point toequilibrate on a length scale L is T~T0(L/O)', where a, atypical lattice spacing, is a few angstroms, r o ~10~ u secand the exponent 2 — 2. Hence in 1 second the system willrelax over distances on the order of ~ 106 A—that is, over amacroscopic length scale. Thus the immense speed ofmicroscopic processes allows pure systems to equilibratequickly even near critical points. For most purposes,therefore, one can ignore nonequilibrium effects.
One can of course force the system out of equilibriumby increasing the rate at which it is perturbed. Let us, forexample, consider quenching a system instantaneouslyfrom a high temperature, where £ is small, to a tempera-ture below Tc and study the "domain growth kinetics"—the rate of growth of the initially small clusters of up anddown spins. In pure Ising-like systems the radius R(t) oftypical domains of correlated spins follows the Lifshitzgrowth law, R(t)~t'/2. This is a consequence of thegrowth's being controlled by surface tension22: The ratedR/dt at which small droplets shrink and large ones growis proportional to the droplets' curvature, 1/R.
In random systems, by contrast, the droplet boundar-ies tend to get pinned by impurities, which retards thegrowth. This is readily seen in the context of the random-exchange model,2' though similar considerations applyequally well to random-field magnets and to spin glasses:
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The domain walls separating up and down spins inrandom-exchange systems minimize their energy cost bypassing between sites connected by weak bonds (see thefigure on page 57), producing- wall meandering, orroughening. The walls must therefore surmount theenergy barriers presented by nearby stronger bonds inorder to move. The possibility of finding weak bonds bymeandering increases with L, and hence so do the energybarriers. One expects the energy barriers opposing theequilibration of random systems of length scale L to scalelike B(L)~LJ' for large L. This result was first derived byJ. Villain and by Grinstein and J. F. Fernandez forrandom-field magnets. The exponent i/> depends on thedimension d and the type of disorder. Since the typicaltime required to surmount a barrier B at temperature Tisproportional to the Arrhenius factor expCB/71), the alge-braic form for B(L) immediately implies that the timerequired for equilibration on length scale L in the orderedphase of a random system is r(L) ~ exp(Z// T). Conversely,the length scale R(t) on which the system can equilibratein time t or, equivalently, the maximum length to whichcorrelations will extend following a quench into theordered phase is
R(t)~(lnt)u*This remarkably slow growth of ordered domains under-lies the hysteresis observed so ubiquitously in randomsystems. For A's of order unity, which is typically the case,domain walls are effectively frozen; many orders of
6 4 PHYSICS TODAY DECEMBER 1968
magnitude of increase in t are required to produce modestchanges in the correlation length and other measurablequantities. Let us now discuss qualitatively some of theimplications of this slow domain growth in each of theclasses of random magnets:
Random-exchange systems ought to exhibit logar-ithmically slow growth of correlations following a rapidquench to a temperature below Tc for both d = 2 andd = 3. But under slow cooling toward Tc the divergence ofcorrelation times r in the critical region is believed to begoverned by conventional critical slowing down withr ~ g r , just as in pure systems. Unless z is anomalouslylarge, such algebraic slowing down allows, as arguedabove, equilibration on macroscopic length scales inordinary experimental times. Hence under normal, gra-dual cooling toward Tc correlation lengths can readilybecome quite large, and remain large for T< Tt., effective-ly masking the domain-wall freezing and concomitantslow growth expected at lower temperatures.
Random fields. Arguments about energy barriers atzero temperature23 suggest that the exponent i/> is unity forboth d = 2 and d = 3. So the slow equilibration followingquenches to low T is qualitatively similar to that ofrandom-exchange magnets. The situation under slowcooling is quite different, however. We remarked earlierthat the fluctuations due to the random fields dominatethose due to thermal effects. Remarkably, this remainstrue even in the critical region, not just at low tempera-tures. Thus the characteristic free-energy scale, andhence the height of typical energy barriers, is set not bythermal energies but by the random fields. Since theequilibrium correlation length |" is the only importantlength in the problem near the critical point, it isreasonable to assume that the barriers due to randomfields scale algebraically with this length: B~^l/'\ where\pc is a critical exponent of order unity. It followsimmediately that the characteristic relaxation time rdiverges exponentially with 4", r—expl^'/T), as oneapproaches Tc from above.2124 This phenomenon, knownas activated dynamic scaling, means that as the tempera-ture is lowered r will increase beyond typical measuringtimes at a temperature somewhat greater than Tc. Atthat point the system will fall rather abruptly out ofequilibrium, and t, will remain essentially fixed at a finitevalue as one cools further. This corresponds closely towhat is observed experimentally in random-field systemswhen a sample is cooled in a magnetic field (see figures onpages 60 and 61). The value at which £ freezes can be quitesmall (~ 100 A, say) even for modest fields.
Thus there is a rather sharply defined "irreversibili-ty" curve in the random-field-temperature plane at whichthe system effectively freezes, even under slow cooling,subsequent growth of correlations in time proceedinglogarithmically slowly. (The position of this freezing linehas now been mapped out rather precisely in experiments;see the figure on page 61.) One simply cannot probe thetrue equilibrium behavior of random-field magnets on anyreasonable experimental time scale; nonequilibrium dy-namics play the dominant role. Note that samples cooledbelow Tc in zero field presumably freeze too: They cannotequilibrate in typical experimental times when the field isapplied. In this case, however, the frozen state does havelong-range order, and so is "closer" to the equilibriumstate of the three-dimensional random-field model than isthe state achieved under field cooling.
Spin glasses. The sharpness of the susceptibilitycusp and the apparent divergence of the nonlinearsusceptibility, found in both numerical simulations andsome experiments, show that spin glass correlationsbecome large as the temperature is lowered to Tc at H = 0.
This is consistent with the relaxation times diverging onlyalgebraically at Tc (as in the random exchange model), abehavior suggested by theoretical arguments. (At present,however, the data are also consistent with activateddynamic scaling, such as occurs in the random-fieldmagnets.) Unlike in either random-exchange or random-field magnets, however, the long correlations achieved inspin glasses during slow cooling to Tc are not retained as Tis lowered below Tc. This is a consequence of thesensitivity of the spin glass state to temperaturechanges.'9 As T is lowered below 2 ., arbitrarily largedroplets of spins must be flipped for the system to be inequilibrium at each new temperature. This processrequires overcoming the barriers, B{L)~L'', and so pro-ceeds logarithmically slowly. Hence the system, which isessentially frozen close to the configuration attained nearTc, drops further and further out of equilibrium as Tdecreases.
While the existence of a true phase transition in spinglasses in a nonzero magnetic field remains controversial,one expects the system effectively to freeze at a ratherwell-defined curve, T = T((H), which passes just above thezero-field critical point in the (T,H) plane. The freezingwill occur whether or not there is an equilibrium orderedphase for nonzero field. If, as in the infinite-range model,a finite-field transition does occur, then freezing will set injust above the thermodynamic phase boundary, as justdiscussed for H=0. If, on the other hand, the dropletscaling picture is correct and there is no transition in afield, T will not diverge for i / # 0 but nonetheless will get
t t t t tt * t t 1
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i i i t * •t I l I ^ t
Excitations in the ordered phase of an Isingspin glass consist of active droplets of spins(blue), which fluctuate nearly independently,so the single-site magnetization (here indicatedby the length of the arrows) is almost thesame for every site in a droplet but is reducedcompared with its value for spins that do notbelong to any active droplet. Active dropletsof all sizes are possible, but small dropletesare more likely than big ones. The shape andlocation of large active droplets changesmarkedly with infinitesimal changes in thetemperature.
PHYSICS TODAY DECEMBER 1988 65
extremely large in the vicinity of the zero-field criticalpoint at H= 0, T= Tc. Thus there will still be a sharply-defined line through this point at which r will exceedtypical experimental times and the system will effectivelyfreeze. Aside from the weak (logarithmic) dependence ofthe position of the freezing line on the time scale of the ex-periment in this latter scenario, the dependence of Tt(H)on H asymptotically close to H = 0 should be independentof the occurrence of a true transition in equilibrium. As inrandom-field magnets, therefore, one finds that thedynamics control the experimentally accessible phenome-na and the thermodynamic behavior plays only a support-ing role.
Summary and outlookWe have seen that strongly competing interactions inrandom magnetic systems produce new equilibrium or-dered phases without counterparts in pure systems.Perhaps more striking is the dominant role played by thedynamics: Even on macroscopic time scales randomsystems below their critical points are often far fromequilibrium; only by understanding their nonequilibriumbehavior can one hope to explain the experimentalobservations.
For random-field magnets, a qualitative understand-ing of the equilibrium phase diagram, critical behaviorand experimental history dependence has been achieved,though quantitative agreement between theory and exper-iment has yet to be attained.
The situation for spin glasses is far less clear. Whileboth model Ising-spin glasses in three dimensions andsome three-dimensional experimental systems apparentlyundergo equilibrium phase transitions, the experimentstypically are performed on systems with Heisenberg spins.The best theoretical estimates for these systems suggestthat di > 3, and hence that a spin glass phase does not existin three dimensions.5 This remains a puzzle. The effect ofthe long-range RKKY interactions present in metallicspin glass systems is likewise poorly understood.5 Indeed,whether all or only some of the experimental spin glasseshave true phase transitions is far from clear. It seemslikely that many of the quintessential spin glass propertiescan occur in systems that do not undergo a thermodynam-ic phase transition.
Even for three-dimensional Ising-spin glass models,crucial equilibrium issues such as the stability of theordered phase with respect to an applied field and thenumber of equilibrium states in zero field remain contro-versial. Given the difficulties of disentangling a truephase transition from a nonequilibrium freezing line andthe absence of a good experimental Ising-spin glass,prospects for settling the former issue experimentally arenot too bright. As for the number of equilibrium states, itis not clear how the existence of many states would bemanifest experimentally. There has nonetheless beenencouraging agreement between theory and experimentfor certain nonequilibrium phenomena (see the box onpage 62). While many of the complicated dynamical andhysteretic phenomena observed in spin glasses over thelast 15 years remain unexplained, progress in the under-standing of important low-energy excitations makes oneoptimistic that theoretical and experimental studies of thespin glass phase will continue to converge.
The principle of universality (see the box on theopposite page) suggests that concepts recently developedfor random magnets should have wide applicability toother quenched random systems. For example, ideasabout random-field magnets have already been used tostudy phase separation of fluids in porous media andwetting on random substrates. The understanding that
certain macroscopic properties are governed by rareregions of a sample has been applied to quantum problemssuch as the onset of superfluidity in dirty superconductorsand superfluids at very low temperature. Spin glassnotions have found their way into neurobiology (see thearticle by Haim Sompolinsky on page 70), molecularbiology, theories of evolution, and a host of optimizationproblems such as designing computer chips, findingoptimum paths for a travelling salesman and partitioninggraphs under specific constraints. Some of these problemshave effective interactions that are long ranged, somethods developed for the Sherrington-Kirkpatrick mod-el are directly applicable. The "simulated annealing"approach to optimization, in which the extremal values ofcomplicated cost functions of many variables are obtainedby what amounts to a series of Monte Carlo simulations atsuccessively lower temperatures, had its genesis in nu-merical attempts to find the ground state of Ising-spinglasses.
There is reason to hope that the mechanisms ofdynamical freezing found in systems with quencheddisorder will prove useful in systems such as real glasses,which apparently generate their own effective random-ness. It is worth mentioning in this connection the recentinterest in systems that exhibit dynamical phase transi-tions without an underlying transition in the equilibriumproperties. The program of analyzing ordered phases interms of stiffness and droplet excitations, the power ofwhich has been so compellingly demonstrated in thecontext of random magnets, turns out to generalize rathernicely to complicated dynamical systems without anunderlying Hamiltonian or energy, such as cellularautomata. This approach is already being fruitfullyemployed in the elucidation of the phases and phasetransitions of these systems.
Theoretical investigations of random magnets beganduring the 1970s with attempts to understand theequilibrium phase transitions and critical phenomena viatechniques, such as formal e expansions, borrowed fromthe study of pure magnets. These tools have provedenormously successful for pure systems, whose orderedphases and dynamics are relatively straightforward, buttheir application to random magnets has produced manyanomalies and misleading results. It has graduallybecome clear that such tools are simply inadequate forrandom systems, whose ordered phases can be intriguinglycomplex and in which subtle interplay between nonequi-librium dynamic phenomena and the underlying equilibri-um behavior is observed.
Though new tools designed specifically for randomsystems are, with a few exceptions, still quite crude, theyhave led to considerable qualitative understanding.Further refinements are certainly needed. These arelikely to require the grand scheme of Wilson's renormal-ization group if they are to capture the many time andlength scales and the statistical variations that contributeto the behavior of disordered systems. The field ofmagnetism has finally reached maturity, however: Thebasic issues and questions are now clear. Research in thisfascinating area is at last beginning to wrest some orderfrom the jaws of disorder.
References1. See, for example, K. G. Wilson, J. Kogut, Phys. Rep. 12, 75
(1974) and references therein.2. J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1983).3. A. B. Harris, J. Phys. C 7, 1671 (1974). For brief reviews, see,
example: G. Grinstein, in Fundamental Problems in Statisti-cal Mechanics VI; Proceedings of the 1984 Trondheim Sum-mer School, E. G. D. Cohen, ed., North-Holland, Amsterdam
66 PHYSICS TODAY DECEMDER 1986
Renormalization Group Techniques in Random SystemsThe revolution of the 1970s in our understanding of thecritical phenomena at continuous phase transitions wasachieved with the concepts and language of the renor-malization group. The essential idea is to focus on thebehavior of a system at successively longer length scales.Universality—the remarkable property that the behaviorof a macroscopic system at large length scales is indepen-dent of microscopic details—is a consequence of theconvergence of the effective Hamiltonians that describethe system at different scales to one of several fixedpoints as the length scale is increased. Progress incalculating critical exponents using these ideas wastriggered by the realization that mean-field theory pro-vides a base about which systematic perturbative renor-malization group expansions could be developed. Forpure systems, mean-field theory is exact in the limit thatthe interactions become infinite ranged, and capturesmuch of the qualitative physics of systems with short-range interactions. It even yields the correct criticalexponents for all dimensions c/above the so-called uppercritical dimension, du, of the system. The renormalizatongroup enabled one to calculate the exponents for d<du
in an expansion in the small parameter e = du — d, withd being treated as a continuous variable. For puremagnets, such as those described by the Ising model, forwhich du is 4, these e-expansion methods give accurateexponent values even for d=3.
It was therefore natural to construct mean-field theo-ries for random systems and then to attempt expansionsabout some appropriate upper critical dimension. Thisprogram worked well for random-exchange magnets,but despite the expenditure of enormous effort, it has sofar failed to provide much insight into the subtleties ofthe random-field and spin glass models with realistic(short-range) interactions. Instead, the approach has
produced some rather grave misapprehensions. Con-sider, for example, the random-field Ising model, forwhich the mean-field theory is rather straightforward.From formal e expansions around the upper criticaldimension, which is 6 for this model, it was concludedthat the critical exponents at the ferromagnetic phasetransition in any dimension d were the same as those forthe pure Ising model in d— 2 dimensions. From thisresult, which is known as "dimensional reduction,"follows the prediction that the lower critical dimensionof the random-field model is 3. This prediction is nowknown to be incorrect. Moreover, the e expansionmisses completely the crucial slow dynamics and thepossibility of history-dependent effects in the model.Both of these failures arise from a difficulty encounteredfrequently when applying formal perturbative methodsto random systems: In these methods the proceduresfor averaging over randomness typically neglect theeffects of statistically unlikely regions of the randomsystem; such regions can make essential contributions tophysical quantities.
For spin glasses, much of the theoretical effort has beenexpended on the infinite-range Sherrington-Kirkpatrickmodel, yielding many interesting results and also insightsinto difficult optimization problems. But so far there is noclear indication that the behavior of this model is relevantto short-range spin glasses. Attempts to expand perturba-tively about it continue to be plagued with severetechnical and conceptual difficulties. Indeed, it is stillunclear that the SK model really represents a high-dimensional limit of short-range spin glasses.
In spite of these difficulties in obtaining quantitativeresults, qualitative renormalization group ideas underliemuch of the progress toward understanding randommagnets that we have sketched in this article.
(1985). A. Aharony in Phase Transitions and Critical Phe-nomena, vol. 6, C. Domb, M. S. Green, eds., Academic Press,New York (1976).
4. R. J. Birgeneau, R. A. Cowley, G. Shirane, H. Yoshizawa, D. P.Belanger, A. R. King, V. Jaccarino, Phys. Rev. B 27, 6747(1983).
5. For a recent review of spin glasses and an extensive bibliogra-phy, see K. Binder, A. P. Young, Rev. Mod. Phys. 58, 801(1986).
6. S. F. Edwards, P. W. Anderson, J. Phys. F 5, 965 (1975). S. F.Edwards, P. W. Anderson, J. Phys. F 6, 1927 (1976).
7. Y. Imry, S.-k. Ma, Phys. Rev. Lett. 35, 1399 (1975).8. For brief reviews of the static equilibrium properties of the
random-field model, the history of their elucidation and ex-tensive bibliographies, see, for example, J. Villain, B. Se-meria, F. Lancon, L. Billard, J. Phys. C 16, 6153 (1983); Y.Imry, J. Stat. Phys. 34, 849 (1984); G Grinstein, J. Appl Phys.55, 2371 (1984).
9. S. Fishman, A. Aharony, J. Phys, C 12, L729 (1979).10. For brief reviews of the experimental phenomenology, see for
example, R. J. Birgeneau, R. A. Cowley, G. Shirane, H. Yo-shizawa, J. Stat. Phys. 34, 817 (1984); P.-z. Wong, J. W. Cable,P. Dimon, J. Appl. Phys. 55 2377 (1984); D. P. Belanger, A. R.King, V. Jaccarino, J. Appl. Phys. 55, 2383 (1984); and refer-ences therein.
11. J. Z. Imbrie, Phys. Rev. Lett. 53, 1747 (1984). J. Bricmont, A.Kupiainen, Phys. Rev. Lett. 59, 1829 (1987).
12. D. S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).13. See, for example, J. A. Mydosh, J. Magn. Magn. Mater. 7, 237
(1978); J. Phys. Soc. Jpn. 52, S-85 (1983).
14. See, for example, P. Monod, H. Bouchiat, J. Phys. (Paris) Lett.43, 145 (1982). B. Barbara, A. P. Malozemoff, Y. Imry, Phys.Rev. Lett. 47, 1852 (1981).
15. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1972(1975).
16. G. Parisi, Phys. Rep. 67, 97 (1980), and references therein.17. See, for example: P. W. Anderson, C. M. Pond, Phys. Rev.
Lett., 40, 903 (1978). J. R. Banavar, M. Cieplak, Phys. Rev.Lett. 48, 832 (1982). W. L. McMillan, Phys. Rev. B30, 476(1984). W. L. McMillan, J. Phys. C 17, 3179 (1984).
18. A. J. Bray, M. A. Moore, in Glassy Dynamics and Optimiz-ation, J. L. van Hemmen, I. Morgenstern, eds., Springer Ver-lag, Berlin (1987).
19. D. S. Fisher, D. A. Huse, Phys. Rev. B. 38, 373 (1988). D. S.Fisher, D. A. Huse, Phys. Rev. B. 38, 386 (1988) and referencestherein. D. S. Fisher, D. A. Huse, J. Phys. A 20, L1005 (1987).
20. M. Ocio, H. Bouchiat, P. Monod, J. Magn. Magn. Mater., 54-57, 11 (1986). S. Geshwind, A. T. Ogielski, G. Devlin, J.Hegarty, P. Bridenbaugh, J. Appl. Phys. 63, 3291 (1988).
21. See, for example, P. C. Hohenberg, B. I. Halperin, Rev. Mod.Phys. 49, 435 (1977), and references therein.
22. See, for example, J. D. Gunton, M. San Miguel, P. S. Sahni, inPhase Transitions and Critical Phenomena, vol. 8, C. Domb,J. L. Lebowitz, eds., Academic Press, London (1983), and refer-ences therein.
23. D. A. Huse, C. Henley, Phys. Rev. Lett. 54, 2708 (1985). J.Villain, Phys. Rev. Lett. 52, 1543 (1984). G. Grinstein, J.Fernandez, Phys. Rev. B 29, 6389 (1984).
24. J. Villain, J. Phys. (Paris) 46, 1843 (1985). D. S. Fisher, Phys.Rev. Lett. 56, 419 (1986). •
