REFERENCE FRAME

SPIN GLASS VII: SPIN GLASSAS PARADIGMPhilip W. Anderson

In my last column on spin glasses(September 1989, page 9) I tried toshow you that the exact solutions of aparticular spin glass problem, byGiorgio Parisi and Gerard Toulouse,gave us great insight into the theoryof complex optimization problems, aswell as an algorithm for solving someof them. One such problem, whichhas been exhaustively studied bymethods of spin glass theory, is thegraph partition problem. This is thequestion of how to divide an arbitrarygraph into two pieces, cutting thefewest possible bonds. My studentYao-Tian Fu (now at WashingtonUniversity, St. Louis) initiated thestudy of the graph partition problemby replica theory. This classic prob-lem of complexity theory was difficultto solve for sparse graphs by thosemethods, but another of my students,Wuwell Liao, seems to have done it.

Even more interesting than theseapplications to complexity theory isthe way apparently unrelated areasof science have been stimulated intoparallel growth by the spin glasswork. John Hopfield (Caltech), whowas instrumental in bringing me toPrinceton in 1975, became interestedin models for neural networks andbrain function about 1979-80. It wasnatural for him to realize that com-plex, interconnected systems of sim-ple units could have the "ruggedlandscape," multistable properties ofspin glasses. Using, very ingeniously,an ad hoc and apparently unrealisticassumption of symmetric couplingbetween neurons, John got the follow-ing results:> For a given set of coupling synap-ses (interactions) JtJ between neurons(spins) i and j , the conventionalMcCulloch-Pitts model of neuronalinteractions maps onto a "greedy"algorithm for finding the local groundstate of a corresponding spin glass.("Greedy" is the computer scientists'

Philip Anderson is a condensed mattertheorist whose work has also hadimpact on field theory, astrophysics,computer science and biology. He isJoseph Henry Professor of Physics atPrinceton University.

self-evident jargon term for jumpingdirectly to the lowest local energy foreach spin.)> Modifying the couplings, or choos-ing the Jtj, in such a way as to form theso-called Hebb synapses makes theneural network into a model for"content addressable" memory: amemory like our own, which canreproduce full detail from fragmen-tary information. A system of Nneurons connected by N(N— l)/2symmetric synapses can rememberabout N/6 N-bit messages in the formof locally stable "spin" (that is, neuronfiring rate) configurations. (Johnmade this conjecture about the capac-ity of a neural network on the basis ofsimulations, but it later turned out toagree with exact analytic theory.)Thus, in exchange for a capacityreduction of a factor of 3 relative to theinformation-theoretic maximum, onegets the content-addressable feature.> Finally, several other brain func-tions, such as pattern recognition,could be modeled with the spin glasstype of neural network.

Many of you may be aware of thegigantic growth of neural networkscience in recent years. In 1979,however, when I tried to whip upinterest in John's ideas among com-puter scientists at Bell Labs, therewas little response; and he, AlanGelperin and John Connor got nearlyequally short shrift among neurosci-entists. Nowadays the neuroscien-tists and computer scientists like topoint to prior claims for each compo-nent of John's achievements. I canhardly believe, however, that suchfurther developments in neural net-works as the revival of the perceptronwould have occurred except as aresponse to John's beautiful demon-strations that, after all, one suchsystem—the Hopfield neural net-work—does work and has a rigorous,mathematically respectable basis. Inparticular, John's work has generateda very healthy trend toward rigorousmathematical demonstrations of lim-its on capacity, accuracy and so on inneural networks and perceptron-likemodels, using the statistical mechan-ics methods provided by Toulouse,Haim Somopolinsky, Miguel Vira-

soro, John Hertz, Richard Palmer(who was John's associate at Prince-ton in 1975-78) and many others. Itturns out that statistical mechanicscan be applied to realistic, asymmet-ric networks as well, and that there isno real difference between the capa-bilities of symmetric and asymmetricnetworks.

I promised I would close my serieson spin glasses with this column, so itmust be descriptive, not detailed. ButI must also mention how the "ruggedlandscape" of spin glasses relates totheories of biological evolution. In1981 I visited John Hopfield at Cal-tech and helped with the course on"physics of information" that he,Richard Feynman and Carver Meadwere giving. Stimulated by John'swork on neural nets, I came back toPrinceton with the realization that Icould put my own rugged-landscapeideas into a theory of prebiotic evolu-tion that Daniel Stein (now at theUniversity of Arizona) and I werealready working on. The genome ofan organism can be thought of as a setof Ising spins—two for each base in theDNA because there are four types ofbases and the Ising spin has twopossible values. The fitness, or repro-ductive capacity, of the genome can bemodeled by a frustrated, quenchedrandom function of this list of spins,and the simplest random functionthat satisfies the requisite plus-minussymmetry is a spin glass Hamiltonian

•function. (The plus-minus symmetryis imposed by the complementarity ofbase pairing.) With a senior thesisstudent, Dan Rokhsar, Stein and Imade a simple model of primitiveevolutionary processes using this idea.

Related ideas, but without the sta-tistical mechanics insights, had al-ready occurred to Gerard Weisbuch atthe Ecole Normale in Paris and to StuKauffman at the University of Penn-sylvania. Nonetheless the extra un-derstanding those insights providehas encouraged us, and especially Stu,to go on and attack all kinds ofevolutionary—and other—problemswith spin-glass-like random, rugged-landscape models. This approach hasbecome an important part of theprogram at the Santa Fe Institute, of

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REFERENCE FRAMEwhich Stu and I are members. Unfor-tunately I cannot discuss here themany other ramifications of this wayof thinking. From my point of view,its attractiveness lies in that it allowsus to explain simultaneously the con-tradictory aspects of variety and sta-bility of certain special forms andpatterns. Life exhibits these contra-dictory aspects: Among the countlesspossible mutations, many lead to sta-ble species. In the language of spinglass theory, there are many "basinsof attraction," but each is stable (or atleast metastable). It is also clear, asGerard Weisbuch first pointed out,that evolution in such a landscapewill exhibit "punctuated equilibri-um," or sudden changes from onedeep maximum of fitness to another, afeature that has been emphasizedrecently as characteristic of much ofevolution.

I realize that I never returned to"real" spin glasses, even though itwas studies of the low-temperatureproperties of those dilute magneticalloys that led to the theoretical ideasI have been discussing. There is areason: In spite of much beautifulexperimental, computational andtheoretical work, a complete and con-sistent understanding of those mate-rials is not yet at hand. HeleneBouchiat and Pierre Monod, amongothers such as Laurent Levy, havebeautifully demonstrated that realspin glasses have divergent nonlinearmagnetic susceptibility at TSG, verify-ing that there is a real phase transi-tion, albeit one without a visiblespecific heat singularity. Spectacularsimulations carried out on specialpurpose machines by Peter Young(University of California, Santa Cruz)and Andrew Ogielski (AT&T BellLabs) have also verified the existenceof a phase transition in three dimen-sions. Daniel Fisher (Princeton) andDavid Huse (Bell Labs), among others,speculate, however, that real spinglasses are really not ultrametric orreplica-symmetry breaking. The the-ory is still under development. Someof it was explored in the December1988 PHYSICS TODAY special issue ondisordered solids.

For further information on randomlandscapes and evolution, Stu Kauff-man's forthcoming book Origins ofOrder: Self-Organization and Selec-tion in Evolution (Oxford U. P., NewYork) is perhaps the best source.John Hopfield has written (with Da-vid W. Tank) a Scientific Americanarticle on his neural network ideas(December 1987, page 104). An arti-cle on neural networks by HaimSompolinsky appeared in the Decem-ber 1988 PHYSICS TODAY (page 70). •

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