Profile of Giorgio Parisi

Physicist Giorgio Parisi has noproblem stretching the conven-tions of mathematics if it helpshim solve a difficult physics

problem. Once, while struggling to de-velop a mathematical model for a com-plicated system, he found that turningbasic concepts inside-out was the onlyway to crack the puzzle. Parisi explainsthat it was as if he needed to figureout the number of distinct ways thata handful objects could be placed ina row. The mathematics would notbudge, until he decided to introducethe idea of a ‘‘half-object.’’ ‘‘Now, ahalf-object is something that does notmake sense,’’ Parisi admits. ‘‘Whenphysicists use mathematics, they use itin a looser way.’’

Parisi’s idea of the half-object broughthim acclaim in the field of disorderedsystems, and nearly a quarter of a cen-tury later, mathematicians agreed thathis innovation was correct. Parisi’saccomplishments span many fields ofmodern physics, including elementaryparticles, statistical mechanics, mathe-matical physics, and, especially, disor-dered systems. Professor of QuantumTheories at the University of Roma I,‘‘La Sapienza,’’ in Rome, Parisi waselected as a correspondent fellowof the Accademia dei Lincei in 1992, afellow of the French Academy in 1993,and a foreign associate of the NationalAcademy of Sciences in 2003. Hismany honors include the FeltrinelliPrize for physics in 1986, the Boltz-mann Medal in 1992, the Italgas Prizein 1993, the Dirac Medal and Prize in1999, the Enrico Fermi Award in 2002,and the Dannie Heinemann Prize in2005. In his Inaugural Article (1), pub-lished in this issue of PNAS, Parisireviews recent advances in the studyof spin glasses and structural fragileglasses and discusses problems remain-ing in the field.

Bringing High Energy to PhysicsIf Parisi had followed in his family’s foot-steps, he might be working with steel andconcrete instead of string theory andquarks. His father and grandfather wereboth construction workers, and the youngParisi was encouraged to become an engi-neer. Instead, Parisi was drawn to thecomplicated abstractions he read in booksof popular science and mathematics. ‘‘Ifelt I wanted to do something scientificbecause it was challenging,’’ he recalls.

Parisi was torn between majoring inphysics and mathematics at the Univer-sity of Roma, ‘‘La Sapienza.’’ At thetime, he could see that the field of

physics had made remarkable progressover the first half of the 20th century,he says, but mathematics’ analogousaccomplishments were more mysteri-ous. Parisi decided to study physics.As soon as he started classes, he knewhe wanted to do research in physics.The highest degree offered in Italy atthe time was equivalent to a singleyear of doctoral studies, and Parisitook full advantage of the year afterreceiving his bachelor’s degree at theUniversity of Roma. He worked withNicola Cabibbo, a high-energy physicistwho was ‘‘by far the most brilliant the-oretician in Rome at that time,’’ Parisisays.

Cabibbo’s and Parisi’s research in-volved high-energy particle physics,widely considered ‘‘the most challeng-ing and most important thing’’ to studyat the time, Parisi says. He graduatedfrom the University of Roma in 1970and went immediately to work at theLaboratori Nazionali di Frascati, a lab-oratory with a particle accelerator nearRome. He worked there for 10 yearsbefore returning as a full professor oftheoretical physics at the University ofRoma II, ‘‘Tor Vergata,’’ in Rome. In1992, he returned to the University ofRoma I, ‘‘La Sapienza,’’ as a professorof quantum theories, where he remainstoday.

Frustration in Spin GlassesParisi’s arguably best work involves aspecial type of magnetic alloy calledspin glass, a field he stumbled upon in

December 1978. At the Frascati labo-ratory, he studied an ‘‘exotic problem’’that cropped up in his work in high-dimensional gauge theory, a field ofstudy that describes how certain physi-cal theories share special mathematicalproperties. For this particular problem,Parisi wanted to use the replica tech-nique, a mathematical tool that re-searchers can sometimes use to reducethe complexity of physical system mod-els. But Parisi learned that the replicatechnique gave inconsistent resultswhen applied to spin glass systems (2).‘‘I decided before I use the techniquefor myself, I would like to understandwhy it did not work for spin glasses. Istarted to study, and the first inquiryconvinced me that there was somethingwrong,’’ he says.

Spin glasses are particularly interest-ing to theoreticians because they are‘‘magnetically frustrated’’ and repre-sent the ultimate disordered system.Spin glasses consist of a crystalline ma-terial (such as copper) into which asmall number of magnetic atoms (suchas manganese) have been placed atrandom. Magnetic atoms always have a‘‘spin,’’ or orientation. However, unlikea pure ferromagnet, which has orderlymagnetic atoms, the spins in a spinglass are frozen in random directions.

This is a Profile of a recently elected member of the NationalAcademy of Sciences to accompany the member’s InauguralArticle on page 7948.

© 2006 by The National Academy of Sciences of the USA

Giorgio Parisi

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Because these materials representsuch complex systems, researchers inthe 1970s struggled to develop a simplemathematical approximation for spinglasses in the way that they had devel-oped mean field approximations forferromagnets. When David Sherringtonand Scott Kirkpatrick attempted to ap-ply mean field theory developed bySamuel Edwards and Philip Andersonto infinite range spin glasses, they real-ized that it yielded inconsistent solu-tions showing negative entropy atextremely low temperatures (2, 3).‘‘I found the problem very, very inter-esting,’’ Parisi says. He began to studyspin glasses ‘‘in a concentrated way,’’he says, hoping to resolve the crisissurrounding the models.

Breaking the Replica SymmetryThe mathematical details behind spinglasses are admittedly ‘‘very bizarre,’’says Peter Young, professor of physicsat the University of California, SantaCruz. Applying the mean field theory tospin glasses first requires creating ‘‘repli-cas’’ of the system and then breakingthese replicas into groups in a way thattakes advantage of some mathematicalsymmetry among them. The originalapplications of the mean field theory tospin glasses ‘‘broke the symmetry’’ in asimple way that yielded a single-orderparameter, but this method led to theinconsistent entropy results.

A novel way of breaking the symme-try thus was needed. A complicationwas that the method required the num-ber of replicas be forced to approachzero. ‘‘It’s a strange mathematicaldevice. Very improper mathematics,really,’’ says Young. As a result, thesymmetry could potentially be brokenin an infinite number of ways, and itwas not clear how to proceed.

Parisi then entered the spin glassfray with an idea that was ‘‘a mark ofgenius,’’ Young says (4). ‘‘He dividedthe replicas into groups, and then hedivided these groups into subgroups,and those subgroups into smaller sub-groups, and so on. And the net resultis that you get not one but an infinitenumber of order parameters,’’ Youngexplains. This infinite subgroupingsolved the problem of breaking thesymmetry so that the number of repli-cas tended to zero, and it turned outthat a mathematical function charac-terized the infinite number of orderparameters. ‘‘At the end you get some-thing that is mathematically respect-able. It’s just amazing that it works,’’says Young.

Parisi also showed that the entropyof the system seems to go to zero atzero temperature and so was more

consistent than were previous results.Further work revealed that the func-tion characterizing the infinite numberof order parameters was related to theprobability distribution of the system’sorder parameter of the system as itf luctuates through its complicated en-ergy landscape (5). Most researchersnow believe that the Parisi solutionis in fact an exact solution and notmerely an approximation. Parisi’s cita-tion for the award of the Boltzmann

Medal in 1992 stated that his work‘‘forms one of the most importantbreakthroughs in the history of dis-ordered systems.’’

Disordered systems crop up in morethan just magnetic alloys, and the les-sons learned from spin glasses haveextended to other fields. In particular,using replicas and breaking the symme-try among them has been useful in thecomputer science field of combinatori-cal optimization, in which researchersneed to maximize or minimize a func-tion of many variables subject to con-straints (6). Many of the papers writtenon spin glasses have been compiledinto the book Spin Glass Theory andBeyond, which Parisi coauthored (7).

In his PNAS Inaugural Article (1),Parisi reviews recent theoretical re-sults for spin glasses and for the actualglasses known as structural fragileglasses. He discusses unsolved prob-lems for which he sees a great needfor new research, including gainingbetter quantitative predictions and amore precise comparison betweentheory and experiments. Parisi also re-views other applications, including neu-ral networks and constraint satisfactionproblems in computer science.

Physics for the BirdsAlthough noteworthy, Parisi’s work onspin glasses is only one part of his re-search palette. ‘‘I have a tendency towork on different subjects at the sametime, because in order to get an idea,it takes time. You have to digest theconcepts,’’ he says. He has made ad-vances in a number of fields, includingelementary particles, statistical me-chanics, string theory, biophysics, andcomputer design (both software andhardware).

Parisi’s diverse research contribu-tions include the study of scaling viola-tions in deep inelastic processes [theAltarelli–Parisi equations (8)], a simpleexplanation for quark confinementbased on the superconductor’s f luxconfinement model (9), the introduc-tion of multifractals in turbulence andin strange attractors (10), the study ofidiotypic network theory for antibodiesin theoretical immunology (11), andthe Array Processor Expansible (APE)project (12).

Graphical elaboration of three subsequent photographs of starling flocks.

‘‘When physicistsuse mathematics,they use it in alooser way.’’

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More recently, Parisi has studied highlycomplex, disordered systems that evennonscientists can appreciate: whirlingflocks of birds. ‘‘Starlings are very inter-esting because they make very fast move-ments in the air. They move very straight,very fast, and this is done by thousandsand thousands of them,’’ Parisi says. ‘‘Oneof the problems is how do they communi-cate in order to have this collective move-ment done together?’’

Parisi and 20 colleagues spent thepast winter studying starlings in action.Parisi estimates having taken approxi-mately 100,000 photographs of flocks inthe air. Now the group is writing com-puter programs to create a 3D recon-struction of the flocks and hopes tohave results soon. Starling flocks pro-vide a convenient, measurable exampleof a complex system. ‘‘They may seemvery far from spin glasses, but there is

something in common,’’ Parisi says.‘‘What they share, and what is very in-teresting, is how complex behaviorsarise. This is a theme recurrent in phys-ics and biology, and most of the re-search that I have done is to get at thisthing: how complex collective behaviormay arise from elements that each havea simple behavior.’’

Regina Nuzzo, Science Writer

1. Parisi, G. (2006) Proc. Natl. Acad. Sci. USA 103,7948–7955.

2. Sherrington, D. & Kirkpatrick, S. (1975) Phys. Rev.Lett. 35, 1792–1796.

3. Edwards, S. F. & Anderson, P. W. (1975) J. Phys.F Met. Phys. 5, 965–974.

4. Parisi, G. (1980) J. Phys. A Math. Gen. 13, 1101–1112.

5. Mezard, M., Parisi, G. & Virasoro, M. A. (1986)Europhys. Lett. 1, 77–82.

6. Mezard, M. & Parisi, G. (1985) J. Physique Lett. 46,L771–L778.

7. Mezard, M., Parisi, G. & Virasoro, M. (1987) SpinGlass Theory and Beyond: World Scientific LectureNotes in Physics, Vol. 9 (World Scientific, Singapore).

8. Altarelli, G. & Parisi, G. (1976) Nucl. Phys. B 126,298–318.

9. Parisi, G. (1975) Phys. Rev. D 11, 970–971.10. Frisch, U. & Parisi, G. (1976) in Turbulence and

Predictability of Geophysical Fluid Dynamics and

Climate Dynamics, Resoconti della Scuola Inter-nazionale di Fisica Enrico Fermi, Corso LXXX-VIII, Varenna, ed. Ghil, M. (North–Holland, NewYork).

11. Parisi, G. (1990) Proc. Natl. Acad. Sci. USA 87,429–433.

12. Parisi, G., Rapuano, F. & Remiddi, E. (1987) inLattice Gauge Theory Using Parallel Processors,eds. Li, X., Qiu, Z. & Ren, H.-C. (Gordon &Breach, London).

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