RE &' IT" KS OF MODERN PHYSIC& VOLUME 39, NUMBER 4 OCTOB F. R & &)ei 7

..—.,istory o): tIze . en' —..sag .V. .oc.e..

STEPHEN G. BRIJSH

DePartment of I'hysi cs and DePartment of History of Science, Harvard University, Cambridge, Massachusetts

Many physico-chemical systems can be represented more or less accurately by a lattice arrangement of molecules withnearest-neighbor interactions. The simplest and most popular version of this theory is the so-called "Ising model, " dis-cussed by Krnst Ising in 1925 but suggested earlier (1920) by Wilhelm Lenz.

Major events in the subsequent history of the Lenz —Ising model are reviewed, including early approximate methodsof solution, Onsager's exact result for the two-dimensional model, the use of the mathematically equivalent "lattice gas"model to study gas —liquid and liquid —solid phase transitions, and recent progress in determining the singularities ofthermodynamic and magnetic properties at the critical point. Not only is there a wide range of possible physical applica-tions of the model, there is also an urgent need for the application of advanced mathematical techniques in order toestablish its exact properties, especially in the neighborhood of phase transitions where approximate methods are un-reliable.

After many years of being scorned or ignored by mostscientists, the so-called "Ising model" has recentlyenjoyed increased popularity and may, if present trendscontinue, take its place as the preferred basic theory ofall cooperative phenomena. Whereas previously itappeared that the greatly over-simplified representationof intermolecular forces on which this model is basedwould make it inapplicable to any real systems, it isnow being claimed that the essential features of co-operative phenomena (especially at the critical point)depend not on the details of intermolecular forces buton the mechanism of propagation of long-range order,and the Ising model is the only one which offers muchhope of an accurate study of this mechanism. Whetheror not it does eventually turn out that gas —liquid criticalphenomena, magnetic Curie points, order —disordertransitions in alloys, and phase separation in liquidmixtures can all be described, to a good first approxima-tion, by the same model, the problem of a generalizeddescription of cooperative phenomena now deservesserious attention. We are just beginning to realizesome of the implications of such a generalized descrip-tion: specialists in the properties of gases and liquidscould not afford to ignore progress being made inresearch on phase transitions in solids, and conversely.While some scientists might not appreciate the burdenimposed by the need for keeping up with the literaturein unfamiliar fields which now suddenly appear to berelated to their own, students should benefit by theprospective unification of different subjects. No longerwould it be necessary to learn a different theory foreach kind of cooperative phenomenon.

The historical development of the "Ising model"also shows the same disregard for traditional boundariesbetween disciplines. Physics, chemistry, metallurgy,and mathematics have all been involved, and some ofthe most recent applications have been in biology.The most striking success in the history of the Isingmodel —the exact solution of the two-dimensionalproblem —involved such difficult mathematics that itstumped all the physicists who attempted it, and was

8

finally accomplished by. . .a chemist. (Just as ironicalis the fact that the supposed inventor of the model,Krnst Ising, gave up research in physics after thinkinghe had proved that his model had no physical useful-

ness, and only discovered twenty years later that he hadbecome famous as a result of work on his model byother scientists. )

THE MODEL

We assume that the physical system can be repre-sented by a regular lattice arrangement of molecules inspace. We are interested in three kinds of physicalsystems: (1) magnets, in which each molecule has a"spin" that can be oriented either up or down relativeto the direction of an externally applied field; (2) mix-tures of two kinds of molecules; (3) mixtures of mole-cules and "holes" (empty spaces) . All three kinds canbe represented abstractly by the same model, if we

simply say that each node of a regular space lattice isassigned a two-valued variable. Depending on whetherthis variable has the value +1 or —1, we say that themolecule at that node (1) has spin up or down, or (2)belongs to one or the other of two species, (3) is presentor absent. Usually the two-valued variable is calledthe spiri o., associated with node i of the lattice.

A coegggraiioe of the lattice is a particular set ofvalues of all the spins; if there are X nodes, there will

be 2~ different configurations. A typical configurationis shown in Fig. 1.

We assume that the molecules exert only short-rangeforces on each other; in particular, we assume that theinteraction energy depends only on the configurationsof neighboring nodes of the lattice. For example, wecould say that the forces are such that when two neigh-boring spins are the same (both +1 or both —1) theenergy is —U, and when two neighboring spins aredifferent (one is +1, the other —1) the energy is +U.In other words, the interaction tends to make neigh-boring spins the same. In the three types of physicalsystems mentioned above, such an interaction couldlead to (1) spontaneous magnetization, with all or

83

884 REvIEws 07 MoDERN PHYsIcs ' OcToBER 196'f

LENZ AND ISING

FIG. 1. A possible configura-tion of a Gnite square lattice.The energy of this conngura-tion is 8= —2U+3IJ,H.

most spins in the same direction even in the absenceof an external field, (2) phase separation, with mole-cules of the same kind clustering together, (3) con-densation of molecules into one region of space, leavingempty space in the rest of the container. Whether theinteraction does in fact lead to such phenomena re-mains to be seen, of course; one must first work out thestatistical mechanical theory of the model.

If we simply reverse the sign of U, so that the energyis positive when two neighboring spins are the same andnegative when they are different, then the interactiontends to produce a regular alternation of up and downspins. Physically, this type of conGguration could cor-respond to (1) antiferromagnetic ordering, (2) super-lattice structure in an alloy, (3) a solid-like arrangementof molecules with repulsive forces.

We have assumed that each nearest-neighbor paircontributes an interaction energy which can be written(as one easily sees) in the form —Uo.,o;, where U iseither positive or negative. In addition, we assume thatthe total energy of a conGguration also includes aterm of the form @II, for each spin. The notation sug-gests that p, is a magnetic moment and H is an externalmagnetic field, and this would indeed be a reasonableinterpretation when we are applying the model tomagnetic systems. In general, pH may be any parameterwhich plays the role of a "chemical potential" indetermining the average number of up and down spins,or average composition of a mixture, or average densityof a molecule —hole system.

The mathematical problem associated with our modelis the following: Find a closed-form analytic expressionfor the statistical mechanical partition function

Q= g exp (—E/frT) (sum over all configurations),conf

where

E= QUo;o;+fr H Qa;. —(sum over all {sum over s)

nearest-neighborpairs)

From the partition function one can then derive allthe thermodynamic functions of the system by theusual procedures of statistical mechanics, and in par-ticular one can Gnd out whether the system undergoesa phase transition.

"In a quantum treatment certain angles n will bedistinguished, among them in any case a=0 and o.=m.If the potential energy 8' has large values in the inter-mediate positions, as one must assume taking accountof the crystal structure, then these positions will bevery seMom occupied, Umklapp processes will thereforeoccur very rarely, and the magnet will Gnd itself almostexclusively in the two distinguished positions, and in-deed on the average will be in each one equally long.In the presence of an external magnetic Geld, whichwe assume to be in the direction of the null position forthe sake of simplicity, this equivalence of the two posi-tions will disappear, and one has, according to theBoltzmann principle, a resulting magnetic moment ofthe bar magnet at temperature T:

&=II(& e ')/(e'+e ') a=fr,H/kT.

For su%.ciently small values of u, this reduces to

p =@'H/k T,

i.e., we obtain the Curie law. . . ."For ferromagnetic bodies, in addition to the tem-

perature dependence of the susceptibility, one has toexplain Grst of all the fact of spontaneous magnetiza-tion, as is observed in magnetite and pyrites. . . .

"If one assumes that in ferromagnetic bodies thepotential energy of an atom (elementary magnet)

~ E. Ising, Z. Physik 31, 253 (j.925).~ W. I enz, Physik. Z. 21, 613 (1920).

Ortwein, Physik. Bla,tter 4, 30 (1948).4 A. Sommerfeld, Z. Naturforsch. 3a, 186 (1948l.I P. Jordan, Physik. Blatter 13, 269 (1957).6 Professor Dr. H. Raether (private communication}.

This model is commonly referred to as the "Isingmodel" although one has only to read Ising's originalpaper' to learn that it was previously proposed byIsing's research director, Wilhelm Lenz, in 1920.' lt israther curious that Lenz's priority has never beenrecognized by later writers. Of three biographical noteson Lenz, ' ' only one4 even mentions this paper, andthen simply summarizes it in one sentence withoutmentioning its connection with the "Ising model. "Lenzhimself apparently never made any attempt later onto claim credit for suggesting the model, and even hiscolleagues at Hamburg University were not aware ofhis contribution. '

Lenz suggested that dipolar atoms in a crystal may befree to rotate around a Gxed rest position in a lattice. Tounderstand the physical basis of his assumptions, itwould be necessary to go into some of the forgottendetails of the old quantum theory, which we will not dohere. Instead, we will simply quote Lenz's own formula-tion (my translation):

STEPHEN G. BRrTsH IIsstory of the Lens I—sing Illodel 885

with respect to its neighbors is different in the nullposition and in the x position, then there arises anatural directedness of the atom corresponding to thecrystal state, and hence a spontaneous magnetization.The magnetic properties of ferromagnetics would thenbe explained in terms of nonmagnetic forces, in agree-ment with the viewpoint of Weiss, who has by calcula-tion and experiment established that the internal 6eld,which he introduced and which generally gives a goodrepresentation of the situation, is of a nonmagneticnature. It is to be hoped that one can succeed in ex-plaining the properties of ferromagnetics in the mannerindicated. "Lenz was at Rostock University in 1920, but the fol-lowing year he was appointed Ordinary Professor atHamburg. One of his first students was Ernst Ising.Since no biographical information has ever been pub-lished about Ising (aside from a brief entry in A meri cartMen of Science), we shall quote here the "Lebenslauf"from his Dissertation (my translation)r:

"I,Ernst Ising, was born on May 10, 1900, the son ofthe merchant Gustav Ising and his wife The1da, atLowe, Koln. Shortly thereafter my parents moved toBochum, %estfall, where I started school in Easter1907. I received my diploma at the Gymnasium there,in 1918.After brief military training I began my studiesof mathematics and physics at Gottingen University inEaster 1919.After an absence of one semester, I con-

Fzo. 3. Ernst Ising. Photograph courtesy of Professor Ising.

tinued my studies in Bonn, where I studied astronomyamong other things. Two semesters later I went toHamburg. There I turned especially to the study oftheoretical physics at the suggestion of Professor Dr.W. Lenz, and at the end of 1922 I began under his

guidance the investigation of ferromagnetism, whichled to the results presented here. "

MIMI I

Fro. 2. Wilhelm Lenz (1888—1957). Photograph courtesy ofProfessor Dr. H. Raether.

~ A copy of the Lebenslauf was kindly sent to me by ProfessorDr. H. Raether; the dissertation itself is hard to obtain outsideof Germany. A copy is now available at the archives of theCenter for History and Philosophy of Physics of the AmericanInstitute of Physics in New York.

In his dissertation, surrimarized in a short paperpublished in 1925,' Ising carried out an exact calcula-tion of the partition function for the model describedabove, for the special case of a one-dimensional lat-tice. His analysis showed that there was no phase tran-sition to a ferromagnetic ordered state at any tem-perature. This result can be understood by a qualitativeargument: suppose one had an ordered state in which,

say, all the spins were up. Then if any random thermalfluctuation should cause spins in the middle of the latticeto Qip to the down position, the ordering would bedestroyed, because there would be nothing to preventall the spins on one side from Gipping simultaneously. .In other words, the ordered state is unstable at anyfinite temperature, because the "communication" be-tween any two parts of the lattice can be broken by asingle defect. However, Ising did not realize that thisargument is valid only in one dimension, and he gavesome approximate calculations purporting to shower thathis model could not exhibit a phase transition in threedimensions either.

According to a letter from Ising to the author:

"At the time I wrote my doctor thesis Stern andGerlach were vrorking in the same institute on theirfamous experiment on space quantization. The ideas

886 REvIEws oF MQDERN PHYsIcs ' OcTQBER 1967

we had at that time were that atoms or molecules ofmagnets had magnetic dipoles and that these di-poles had a limited number of orientations. We as-sumed that the field of these dipoles would die downfast enough so that only interactions should be takeninto account, at least in the first order. . . . I discussedthe result of my paper widely with Professor Lenzand with Dr. Wolfgang Pauli, who at that time wasteaching in Hamburg. There was some disappointmentthat the linear model did not show the expected ferro-magnetic properties. . . . I do not know of any reactionto my paper while I was in Europe, except that Heisen-berg mentioned it in one of his publications. Only afterI had come to this country $U.S.A.$ in 1947 did Ilearn that the idea had been expanded. I have tried toextend my model to more complicated forms, but havenot published anything yet.

"After I got my doctor's degree I worked in the re-search department of the Allgemeine Elektrizitats-Gesellschaft (General Electric Company) in Berlin.I was not satisfied, returned to the University and be-came a teacher. When Hitler came to power in 1933I was dismissed from public schools, and for four yearsI was the head of a private Jewish school near Potsdam.I left Germany in 1939, but was not able to come tothe U.S.A. immediately. We survived the war in a smalltown in Luxembourg, but I was there completely shutoB from scientific and social life. After coming to theU.S.A. I taught for one year at the State TeachersCollege in Minot, N.D., and since 1948 I have beenteaching physics at Bradley University t Peoria,Illinois]. "

Ising remarks that the only contemporary citationof his paper was by Heisenberg. Heisenberg, when heproposed his own theory of ferromagnetism in 1928,'said:

"Ising succeeded in showing that also the assumptionof directed sufficiently great forces between two neigh-boring atoms of a chain is not su%cient to explain ferro-magnetism. "

Thus Heisenberg used the supposed failure of the Lenz-Ising model to explain ferromagnetism as one justifi-cation for developing his own theory based on a morecomplicated interaction between spins. In this way thenatural order of development of theories of ferromagne-tism was inverted; the more sophisticated Heisenberg'Inodel was exploited first, and only later did theo-reticians return to investigate the properties of thesimpler Lenz —Ising model.

W. Heisenberg, Z. Physik 49, 619 (1928). Ising's result isalso cited by L. Nordheim in the article "Quantentheorie desMagnetismus" in Mueller-Pouillet's I.ehrblch der I'kysik (Fred-erick Vieweg und Sohn, Braunschwieg, Germany, 1934), 11th ed. ,Vol. IV, Teil 4, p. 859. (1 am indebted to Professor Ising forthis reference. )

Indeed, as a result of Ising's own rejection of themodel, we might never have heard any more about it,if it had not been for developments in a diferent areaof physics: order —disorder transformations in alloys.Tamman had proposed in 1919 that the atoms in alloys

may be in a definite ordered arrangement, and a num-

ber of workers had developed the idea that order—disorder transformations result from the opposingeffects of temperature and the lower potential energyof order. ' In 1928, Gorsky tried to construct a statisticaltheory on this basis by assuming that the work needed

to move an atom from an "ordered" position to a"disordered" position is proportional to the degree oforder already existing. "Bragg and Williams developedGorsky's theory further in 1934, and the assumption hassubsequently become known as the "Bragg—Williams

approximation. '"' It differs from the Lenz —Ising modelin that the energy of each configuration of an individualatom is assumed to depend only on the average degreeof order throughout the entire system, rather than on

the configurations of neighboring atoms.In 1935, Hans Bethe showed how the Bragg —Williams

theory could be improved by taking account of theshort-range ordering produced by interactions betweenneighboring atoms. " He did this by constructing an

approximate combinatorial factor based on configura-tions of the first shell of lattice sites around a centralone. In the same year, E. A. Guggenheiin. developed atheory of liquid solutions in which nearest-neighborinteractions were taken into account by what is knownas the "quasi-chemical" (QC) method. " In the QCmethod one constructs an approximate combinatorialfactor by counting configurations of neighboring pairsor larger groups of atoms, assuming that these groupscan be treated as independent statistical entities. (Thisis of course not strictly true since each atom belongs toseveral groups, so that the configurations of these

groups cannot really be independent. ) The QC methodwas improved and extended by Rushbrooke" by meansof Bethe's method. Guggenheim" then showed that the

z G. Tammann, Z. Anorg. Chem. 107z 1 (1919):,U. Dehnnger,gontgenforsckzzng zn der Metallkzznde (Julius Springer-Verlag,Berlin, 1930};Z. Physik 74, 267 {1932);79, 550 (1932);83, 832(1933); Z. Physik. Chem. 326, 343 (1934); G. Borelius, Ann.Physik 20, 57 (1934).I W. Gorsky, Z. Physik 50, 64 (1928)."W. L. Bragg and E. J. Williams, Proc. Roy. Soc. (London)A145, 699 (1934); A151, 540 (1935);E. J. Williams, Proc. Roy.Soc. (London) A152, 231 (1935). Bragg and Williams acknowl-edged the priority of Gorsky but claimed that he had made anerror in his formulation; but according to R. H. Fowler and E. A.Guggenheim )Statzstzca/ T/zernzodynarnzcs (Cambridge UniversityPress, Cambrrdge, England, 1939), p. 574j, Gorsky's formulais equivalent to that of Bragg and Williams. (l am indebted toProfessor Guggenheim for pointing this out to me. )

"H. A. Bethe, Proc. Roy. Soc. (London) A150, 552 (1935)."E. A. Guggenheim, Proc. Roy. Soc. (London) A148, 304

(1935).'4 G. S. Rushbrooke, Proc. Roy. Soc. (London) A1M, 296

(1938).'~ E. A. Guggenheim, Proc. Roy. Soc. (London) A169, 134

(1938).

STEpHEN G. BRUsH IIistory of the Lens —Ising Model 887

QC method is really equivalent to the Bethe method,and more convenient to use in many cases such as thoseinvolving complicated lattices. In 1940, Fowler andGuggenheim' published a general formulation of theQC method and applied it to alloys with long-rangeorder. '~

ATTEMPTS TO FIND AN EXACT SOLUTION

The concept of the "Ising model" as a mathematicalobject existing independently of any particular physicalapproximation seems to have been developed by theCambridge group led by R. H. Fowler in the 1930's.Fowler discussed rotations of molecules in solids in apaper published in 1935,' where he stated that theneed for a quantitative theory of such phenomena

".. . was 6rst brought clearly to my notice at a con-ference on the solid state held at Leningrad in 1932.As will appear, however, an essential feature of thetheory is an application of the ideas of order and dis-order in metallic alloys, where the ordered state istypically cooperative, recently put forward by Braggand Williams. As soon as their ideas are incorporatedthe theory 'goes'. "

In 1936, R. Peierls published a paper with the title"On Ising's Model of Ferromagnetism '" in which herecognized the equivalence of the Ising theory offerromagnetism, the Bethe theory of order —disordertransformations in alloys, and the work of Fowler andPeierls" on adsorption isotherms. Peierls gave a simpleargument purporting to show that (contrary to Ising'sstatement) the Lenz —Ising model in two or three di-mensions does exhibit spontaneous magnetization atsufficiently low temperatures. He pointed out that eachpossible configuration of (+) and ( —) spins on thelattice corresponds to a set of "boundaries" betweenregions of (+) spins and regions of ( —) spins. If onecan show that, at sufficiently low temperatures, theaverage (+) area or volume enclosed by boundaries is

only a small fraction of the total area or volume, thenit will follow that the majority of spins must be ( —),which corresponds to a system with net magnetization.Unfortunately Peierls' proof is not rigorous because ofan incorrect step, which was discovered by M. E.Fisher and S. Sherman. " The basic idea is still ofinterest —the analysis of boundaries between magne-tized regions plays an important role in some of themore recent combinatorial methods.

6R. H. Fowler and E. A. Guggenheim, Proc. Roy. Soc. (Lon-don) Ale, 189 (1940).

' See also: E. A. Guggenheim, Mixtures (Clarendon Press,Oxford, England, 1952); J. A. Barker, Proc. Roy. Soc. (London)A216, 45 (1953); S. G. Brush, Trans. Faraday Soc. 54f 1781.(1958). (Extension of QC method to higher approximations. )' R. H. Fowler, Proc. Roy. Soc. (London) A149, 1 (1935).

' R. Peierls, Proc. Cambridge Phil. Soc. 32, 477 (1936).R. Peierls, Proc. Cambridge Phil. Soc. 32, 471 (1936)."S. Sherman (private communication); see R. B. Gri%ths,

Phys. Rev. 136, A437 (1964) .

The next advance was made by J. G. Kirkwood, who

developed in 1938 a systematic method for expandingthe partition function in inverse powers of the tem-perature. "His method was based on the semi-invariantexpansion of T. N. Thiele (1838—1910), used in sta-tistics to characterize a distribution function by itsmoments. "Since only a small number of terms in theexpansion could actually be computed, the result was"essentially equivalent to Bethe's in its degree ofapproximation, but somewhat less unwieldy in form. "In 1939 Bethe and Kirkwood, then both at Cornell,published a joint paper giving a comparison of theirmethods, and including a calculation of the next termin Kirkwood's expansion. '4 Chang (1941) evaluated twomore terms, and Wakefield (1951) determined threeInore."The present status of this expansion, and itsuse in determining the critical point, is reviewed in

survey papers by Bomb and Fisher."The first exact quantitative result for the two-dimen-

sional Ising model was obtained by Kramers andWannier in 1941";they located the transition tempera-ture by using the symmetry of the lattice to relate thehigh- and low-temperature expansion of the partitionfunction. They showed that the partition function canbe written as the largest eigenvalue of a certain matrix;they attribute some of the ideas used in their analysis toMontr oil, who subsequently published a similarmethod. '8 Kramers and Wannier developed a methodthat yields the largest eigenvalue of a sequence offinite matrices and should in principle converge to theexact solution if su%ciently large matrices could beanalyzed. They did not succeed in obtaining an exactsolution in closed form, but they did develop a varia-tional method which is fairly accurate, and which hasbeen used occasionally in later studies. "

Some other exact but incomplete results were ob-tained by other workers at about the same time.Zernike'0 had used the Bethe method to derive a non-linear finite difference equation for the "correlationfunction" (correlation of spin variables at various dis-

s' J. G. Kirkwood, J. Chem. Phys. 6, 70 (1938)."See, for example, M. G. Kendall, Advanced Theory of Sta-

tistics (GrifBn, London, 1947), 3rd ed. , Vol. I, Chap. 3. Kendallprefers to call these quantities "cumulants" but the name "semi-invariant" has survived in statistical mechanics. The history ofsemi-invariants is discussed briefly by H. M. Walker in Studiesin the History of Statistical Method (The Williams and Wilkins Co.,Baltimore, 1929), p. 81.

~4H. A. Bethe and J. G. Kirkwood, J. Chem. Phys. '7, 578(1939)~"T.S. Chang, J. Chem. Phys. 9, 169 (1941);A. J. Wake6eld,Proc. Cambridge Phil. Soc. 4'7, 419, 799 (1951)."C. Domb, Advan. Phys. 9, 150 (1960);M. E. Fisher, J. Math.Phys. 4, 278 (1963): Lectures in Theoretical I'hysics (Universityof Colorado Press, Boulder, Colo. , 1965), Vol. VIIC, p. 1.

"H. A. Kramers and G. H. Wannier, Phys. Rev. 60, 252, 263(1941)."E. Montroll, J. Chem. Phys. 9, 706 (1941);10, 61 (1942) .

2'D. ter Haar and B. Martin, Phys. Rev. '7'7, 721 (1950);B. Martin and D. ter Haar, Physica 18, 569 (1952); T. Oguchi,Phys. Rev. '76, 1001 (1949); J. Phys. Soc. Japan 5, 75 (1950).

30 F. Zernike, Physica 7', 565 (1940).

888 Rzvzzws oz MoDERN' PHYsIcs + OGToBER 1967

tances), Ashkin and Lamb" derived an exact low-tem-perature series for the correlation function of the two-dimensional lattice using the Kramers —Wanniermethod. They compared the results of Zernike's ap-proximation with their own method, and with van derWaerden's exact series expansion" for a three-di-mensional lattice. They also proved that the existenceof long-range order implies degeneracy of the largesteigenvalue of the Kramers —Wannier matrix, as pre-viously remarked by Lassettre and Howe. "Ashkin andTeller'4 located the transition temperature in four-component two-dimensional systems by symmetrymethods.

A contribution by R. Kubo" deserves especial men-tion because it was published only in Japanese and wastherefore unknown to Western physicists. (An Englishtranslation is now available. ) Kubo developed thematrix formulation and showed how a possible phasetransition in the &wo- or three-dimensional systemwould be related to the degeneracy of the largest eigen-value of a matrix, but he did not give detailed calcula-tions except for the one-dimensional case.

ONSAGER'S EXACT SOLUTION

At a meeting of the New York Academy of Science,28 February 1942, Lars Onsager announced his solutionof the two-dimensional Lenz —Ising problem in zeromagnetic field."The details were published two yearslater. "The method is similar to that of Kramers andWannier, and of Montroll, except that Onsager em-

phasized "the abstract. properties of relatively simpleoperators rather than their explicit representation byunwieldy matrices. " We quote Onsager's summary ofhis method:

"The special properties of the operators involvedin this problem allow their expansion as linear com-binations of the generating basis elements of an algebrawhich can be decomposed into direct products of qua-ternion algebras. The representation of the operators inquestion can be reduced accordingly to a sum of directproducts of two-dimensional representations, and theroots of the secular equation for the problem in handare obtained as products of the roots of certain quad-ratic equations. To find all the roots requires completereduction, which is best performed by the explicitconstruction of a transforming matrix, with valuable

3' J. Ashkin and W. E. Lamb, Jr., Phys. Rev. 64, 159 (1943).'2 B.L. van der Waerden, Z. Physik 118, 473 (1941).33E. N. Lassettre and J. P. Howe, J. Chem. Phys. 9, 747

(1941).'4 J. Ashkin and E. Teller, Phys. Rev. 54, 178 (1943); see

also R. S. Potts, Proc. Cambridge Phil. Soc. 48, 106 (1952).sb R. Kubo, Busseiron-kenkyu 1, 1 (1943) /English transl. :

UCRL-Trans. 1030(L), available from Lawrence RadiationLaboratory, Livermore, Californiaj.

36 See T. Shedlovsky and E. Montroll, J. Math. Phys. 4, 145(1963).

~ L. Onsager, Phys. Rev. 05, 117 (1944).

by-products of identities useful for the computation ofaverages pertaining to the crystal. It so happens thatthe representations of maximal dimension, which con-tain the two largest roots, are identified with easefrom simple general properties of the operators andtheir representative matrices. The largest roots whoseeigenvectors satisfy certain special conditions canbe found by a moderate elaboration of the procedure;these results will sufIice for a qualitative investigationof the spectrum. To determine the thermodynamicproperties of the model it su%ces to compute the largestroot of the secular equation as a function of tempera-ture.

"The passage to the limiting case of an infinite baseinvolves merely the substitution of integrals for sums.The integrals are simplified by elliptic substitutions,whereby the symmetrical parameter of Kramers andWannier appears in the modulus of the elliptic func-tions. The modulus equals unity at the 'Curie point'; theconsequent logarithmic infinity of the specific heatconfirms a conjecture made by Kramers and Wannier. "

Onsager's method was subsequently simplified byKaufman" and by Newell and Montroll, ' using ideasfrom the theory of spinors and Lie algebras. A good ex-position of the method is given in the review by Newelland Montrolp'; shorter summaries appear elsewhere. 'The result is that the partition function for a rec-tangular lattice of nrem points can be written in theform

log Qlim = log 2+ (1/2sr') dec dkr

a,marco 0 0

&&log (cosh 2J cosh 21'' —sinh 2X cos cc

—sinh 2E' cos ce),

where E=U/kT, IV= U'/kT (there can be differentinteraction energies U and U' in horizontal and verticaldirections) .

Onsager's formula for the spontaneous magnetizationof the square lattice was

".. . first exposed to the public on 23 August 1948 on ablackboard at Cornell University on the occasion of aconference on phase transitions, . ..To tease a wideraudience, the formula was again exhibited during thediscussion which followed a paper by Rushbrooke at thefirst postwar IUPAP statistical mechanics meeting inFlorence in 1948; it finally appeared in print as a dis-

3' B.Kaufman, Phys. Rev. V6, 1232 {1949).39 G. F. Newell and E. W. Montroll, Rev. Mod. Phys. 25, 353

(1933).4e C. Dornb, Advan. Phys. 9, 13O (1960); D. ter Haar, Ele-

ments of Statsstscet Mechasttcs (Rinehart and Co. , New York,1954); K. Huang, Stctistica/ Mechanics (John Wiley 8z Sons,Inc. , New York, 1963); H. S. Green and C. A. Hurst, Order-Disorder I'henomeea (Interscience Pub/ishers, Inc. , New York,1964).

SrzrHEN G. BRUsa History of the Lens I—sing Mod'el 889

cussion remark. However, Onsager never publishedhis derivation. The puzzle was 6nally solved by C. N.Yang. 4"'

The exact partition functions and other propertiesof several other two-dimensional lattices have beendeduced from the Onsager result by various workers.In a few cases it has been possible to obtain informationabout behavior in a magnetic 6eld for models somewhatsimilar to the Ising model by transforming the variablesin Onsager's partition function. 4' However, althoughOnsager's method has been applied to some closelyrelated problems, no further really new results have beenobtained with it. All the two-dimensional lattices appearto have qualitatively similar properties, at least forthe case of nearest-neighbor ferromagnetic interactions.The problem of calculating the exact partition func-tion in a 6nite magnetic 6eld remains unsolved, thoughsignificant results for the spontaneous magnetizationand susceptibility in a vanishingly small 6eld have beenobtained. And, 25 years after the announcement ofOnsager's original result, no one has yet succeeded insolving the problem in three dimensions. Probably thisis because the abstract algebraic approach of Onsagerand Kaufman involves techniques that are as yet toodificult and unfamiliar to most of the physicists whohave been interested in the Lenz —Ising model; and themathematicians who are competent in this area havenot shown much interest in physical applications.This is regrettable even from the point of view of theprogress of pure mathematics, for research stimulatedby the Lenz —Ising model has revealed a relationshipbetween two separate areas of mathematics whosesignificance is apparently not yet understood by mathe-maticians themselves (see below) .

In an attempt to provide a heuristic explanation ofOnsager's result, Kac and Ward4s constructed a matrixwhose determinant provided a generating function for acertain combinatorial problem related to the Lenz-Ising model partition function. Van der Waerden32 hadshown how the Lenz —Ising problem could be reduced tothe problem of counting the number of lattice graphscontaining closed polygons with various total perime-ters. These polygons may be thought of as boundariesbetween regions of (+) and ( —) spins, as suggested byPeierls, " but van der Waerden and Kac and Wardinvestigated the approach more systematically. Thevalidity of the Kac—Ward method depends on the as-sumption that their determinant is the correct gen-erating function for st(1.), defined as the number ofgraphs of L bonds that can be constructed on the lattice

4~ E. W. Montroll, R. B. Potts, and J. C. Ward, J. Math. Phys.4, 308 (1963); L. Onsager, Nuovo Cimento (Suppl. ) 6, 261(1949); C. N. Yang, Phys. Rev. 85, 809 (1952); C. H. Chang,ibid. 88, 1422 {1952).

n M. E. Fisher, Phys. Rev. 113, 969 (1959); Proc. Roy. Soc.(London) A254, 66 (1960);A256, 502 (1960).'" M. Kac and J. C. Ward, Phys. Rev. 88, 1332 (1952) .

subject to certain conditions. Since "most" graphs (insome asymptotic sense) are indeed counted correctly,and since the determinant does reduce to just the samefunction that occurs in the integrand of the Onsagerpartition function, it was tempting to conclude thatthe method could be made rigorous. It turned out,however, that some graphs are not counted correctly(see the counterexample given by Sherman44) and that asomewhat diferent though closely related method isrequired. The clue to the correct approach was found byFeynman, 4' who conjectured a relation between func-tions of graphs and of closed paths (random walks) on alattice. If one accepts the validity of Feyrunan's rela-tion, he can then make use of methods previouslydeveloped" to count random walks on a lattice, andthereby get to the Onsager partition function by a routesomewhat analogous to that suggested by Kac andWard. '7 The dificult part is to prove the Feynrnanrelation between graphs and paths. This was 6rst doneby Sherman44 and a simpli6ed proof was devised byBurgoyne. 48 The proof uses a result on crossings ofcurves in a plane, originally derived by Whitney. "

What began as an attempt to provide a more com-prehensive derivation of a known result has turned upsome unsuspected insights and conjectures regardingthe borderline of modern algebra and combinatorialanalysis. This is why mathematicians should pay moreattention to the Lenz —Ising model, though it is im-possible to give any more than a brief hint here of someof the technical aspects involved. Here is a morsel forthe experts to chew on: Sherman" remarks that he wasinformed by M. P. Schutzenberger that his (Sherman's)combinatorial theorem, of which Feynman's con-jecture is a special case, involves an identity used toestablish a formula of W. E. Witt" on "the dimensionof the linear space of Lie elements of degree r in a freeLie algebra with k generators over a Geld of charac-teristic zero."Sherman's theorem is a generalization ofWitt's identity "to any planar 1-cycle with sufhcientsmoothness so that winding numbers can be de6ned.It constitutes a relation between the fundamental groupand the 6rst homology group over the integers mod 2of this planar 1-cycle. An analogous relation betweenthe two groups for 1-cycles in 3-space might very wellcrack the long attacked Ising problem.

" Shermanmentions some other mathematical problems whose

44 S. Sherman, J. Math. Phys. 1, 202 {1960).45 Unpublished. An account is given by Harary in the draft of

a chapter of his book on graph theory, dated 1958 but not yetpublished. See also Feynman's lecture notes, Hughes ResearchLaboratory, Malibu, California (1960).

"See, for example, H. N. V. Temperley, Phys. Rev. 103, 1(1956).

4'Harary, Ref. 45; Sherman, Ref. 44; N. Burgoyne, J. Math.Phys. 4, 1230 (1963).~ N. Burgoyne, Ref. 47'.

"H. Whitney, Compos. Math. 4, 276 (1937).~ S. Sherman, Bull. Am. Math. Soc. 68, 225 (1962).+ W. E. Witt, J. Reine Angew. Math. 177, 152 (1937).

890 REvIEws oP MoDERN PHYsIcs ' OcTQBER 1967

solution might be of use in tackling the three-dimen-sional problem.

Domb's address to the Royal Statistical Society ofLondon in 1964 stimulated an interesting discussion onthe relations between professional -statisticians andtheoretical physicists, apropos of "Some statisticalproblems connected with crystal lattices. ""Among otherpertinent remarks was the following by Professor H. E.Daniels: "whereas nowadays nearly every dificultproblem in applied mathematics can be adequatelysolved on a computer, the three-dimensional Isinglattice problem must be one of the rare examples whereultimately only a complete mathematical solution willreally do." But if mathematicians are to participateeffectively in solving problems in theoretical physics,some psychological or at least linguistic barriers mustbe broken down (the varying usages of the term"ergodic" in physics and mathematics is a goodexample from a neighboring 6eld of statistical me-chanics) .

Specialists in quantum 6eld theory should also be in-terested in the Lenz —Ising model, in view of the mathe-Inatical similarities between these two topics." Oneparticular approach, proposed by Hurst and Green, "uses 6eld theory techniques to arrive at expressionssimilar to those that occur in the Kac—Ward method.Each lattice point in a graph in the expansion of thepartition function is associated with a set of noncom-muting operators, chosen in such a way that a product ofcertain combinations of these operators, when ex-panded as a sum over lattice graphs, will be identicallyequal to the partition function. This is sometimes knownas the "Pfaffian" method; it involves triangular arraysof quantities related to antisymmetrical determinants. "Fisher, " Temperley and Fisher, ' and Kasteleyn"applied the PfaKan method to the problem of theconfigurations of dimer molecules on a lattice, andKasteleyn" showed the connection between the dimerproblem and the Ising model. The theory of Toeplitzmatrices turned out to be useful in much of this re-

's C. Domh, J. Roy. Stat. Soc., Ser. B26, 367 (1964) withdiscussion remarks by J. M. Hammersley, H. E. Daniels, D. C.Handscomb, J. F. C. Kingman, D. J. A. Welsh, and E. S. Page.

~ Y. Nambu, Progr. Theoret. Phys. (Kyoto) 5, 1 (1950);R. L. Ingraham, Nuovo Cimento 21, 29 (1961);T. D. Schultz,D. C. Mattis, and E. H. Lieb, Rev. Mod. Phys. 36, 856 (1964);R. Abe, Progr. Theoret. Phys. 33, 600 (1965).~ C. A. Hurst and H. S. Green, J. Chem. Phys. 33, 1059 (1960);A. M. Dykhne and Y. B. Rumer, Vsp. Fiz. Nauk 75, 101 (1961)/English transL: Soviet Phys. Usp. 4, 698 (1962)g; C. A. Hurst,J. Chem. Phys. 38, 2558 (1963);H. S. Green, Z. Physik 171, 129(1962); E. W. Montroll, in A pp/ied Combinatorial 3fathematics,K. F. Beckenbach, Ed. (John Wiley 8z Sons, Inc. , New York,1964); H. S. Green and C. A. Hurst, Order-Disorder Phenomena(Interscience Publishers, Inc. , New York, 1964).

"See, for example, G. Brunel, Mem. Soc. Sci. Bordeaux 5, 165(1895); W. T. Tutte, J. London Math. Soc. 22, 107 (1947)."M. E. Fisher„Phys. Rev. 124, 1664 (1961).

5'H. N. V. Temperley and M. E. Fisher, Phil. Mag. 6, 1061(1961).

ss P. W. Kasteleyn, Physica 27, 1209 (1961).~9 P. W. Kasteleyn, J. Math. Phys. 4, 287 (1963).

search, and has since found applications in other areasof statistical mechanics. "

THE LATTICE GAS

Though it had been recognized earlier" that formulasderived for the Lenz —Ising model could apply equallywell to systems of atoms and holes in a lattice, it wasLee and Yang who 6rst used the term "lattice gas" in apublished paper. " For this reason, and because theydid give a systematic account of the transcription ofvariables and formulas, as well as deriving some newresults, their paper is usually cited as the origin ofwork. on the lattice gas.

Whereas the set of all possible spin values of 2V latticesites corresponds to a canonical ensemble for a magnet,it corresponds to a grand canonical ensemble for alattice gas. The total volume is fixed but the totalnumber of atoms can vary. To calculate the partitionfunction one sums over all possible values of the numberof atoms, from zero up to the total number of latticesites. Thus the magnetization of the magnet (differencebetween number of up and down spins) is directlyrelated to the density of the lattice gas (fraction ofoccupied sites). The role of the external magnetic6eld, which is a controllable parameter for the magneticmodel, is now played by the fugacity for the lattice gas.The case of zero external field —the only one that canbe treated exactly by Onsager's method —now cor-responds to the case in which half of the lattice sitesare filled on the average. Below the transition tem-perature the system may split into two phases ofdifferent densities; these correspond to the two possiblestates of spontaneous magnetization of a magnet inzero field (whether the magnetization is +M or —Mdepends on the direction of the 6eld before it wasturned off).

The lattice gas model with attractive forces betweenneighboring atoms has frequently been studied as apossible model for the liquid —gas transition and thecritical point (see below). However, if the interactionis repulsive, so that con6gurations with alternatingfilled and vacant sites are favored, one has a modelwhich is of interest in connection with the theory ofsolidi6cation. Such a model involving only repulsiveforces was suggested by the results of computer ex-periments on systems of a few hundred hard spheres.These experiments showed that, at least for finitesystems, repulsive forces can produce an ordered phaseat high densities which is reached by a first-order phasetransition from the disordered phase at medium and

6O E. W. Montroll, R. B.Potts, and J. C. Ward, J. Math. Phys.4, 308 (1963);A. Lenard, J. Math. Phys. 5, 930 (1964).

O' Y. Muto, J. Chem. Phys. 16, 519, 524, 1176 (1948); S. Ono,Mem. Fac. Eng. Kyushu Univ. 10, {No. 4), 196 {1947);T.Tanaka, H. Katsumori, and S. Toshima, Progr. Theoret. Phys.(Kyoto) 6, 17 (1951)."T.D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952).

SrEPHEN G. BRUsH Hesiory of the Lewz Isr-'Ng Model 891

low densities. " Temperley and others have thereforepursued the study of antiferromagnetic versions of theLenz —Ising model in the hope of learning more aboutthe liquid —solid transition. '4

represented by those expansions. "His results for thehigh-temperature susceptibility confirmed the estimatesof Domb and Sykes": for temperatures near the criticaltemperature T„the susceptibility behaves as

THE CRITICAL POINT

Until recently, the lack of an exact solution for thethree-dimensional Lenz —Ising model had thwartedattempts to reach definite conclusions about the natureof the phase transition. Many approximate methods hadbeen developed, but comparison of their predictionswith Onsager's exact two-dimensional solution showedthat the approximations generally failed to reproducethe nature of the singularity in the specific heat andother thermodynamic functions at the transition point.It seemed that nothing short of an exact solution couldo6er any convincing evidence about the singularity inthree dimensions.

During the early 1960's, Domb and his colleagues atLondon had been continuing the earlier work" '4" ofcomputing further terms in the series expansions ofthe partition functions and thermodynamic properties.These expansions are either in positive or negativepowers of the temperature, starting from zero or in-finite temperature, respectively. Although the ex-pansions were not expected to give reliable values inthe neighborhood of the transition point, it was thoughtpossible to estimate the location of the critical point byvarious extrapolation methods. "" Then, in 1961,Baker introduced the method of Pade approximants"for extrapolating series expansions and estimating thelocation and nature of singularities of the functions

63 B. J. Alder and T. E. Wainwright, J. Chem. Phys. 2V', 1208(1957); the possibility oi such a transition had previously beensuggested by J. G. Kirkwood and E. Monroe, J. Chem. Phys. 10,394 (1942) and by I. Z. Fisher, Zh. Eksperim. i Teor. Fis. 28,437 (1955) LEnglish transl. : Soviet Phys. —JETP 1, 273 (1955)].For reasons why there should not be a phase transition, see J. E.Mayer, Phys. Today 11, 22 (January 1958).

6'H. N. V. Temperley, Proc. Phys. Soc. London '74, 183, 432,""." (1959);84, 339 (1964); 86, 180 (1965). For earlier work onthe antiferromagnetic model, see J. E. Brooks and C. Domb,A20'7, 343 (1951);Y. Y. Li, Phys. Rev. 84, 721 (1951);J. M.Ziman, Proc. Phys. Soc. (London) 64, 1108 (1951);other recentwork: M. F. Sykes and M. E. Fisher, Phys. Rev. Letters 1, 321(1959);Physica 28, 919 (1962); M. E. Fisher and M. F. Sykes,Physica 28, 939 (1962);D. M. Burley, Proc. Phys. Soc. (London}75, 262 (1960); 7/, 451 (1961); Physica 2'7, 768 (1961); C.Domb, Nuovo Cimento Suppl. 9, 1 (1958);B.Jancovici, Physica31, 1017 (1965); D. S. Gaunt and M. E. Fisher, J. Chem. Phys.43, 2840 (1965); L. K. Runnels, Phys. Rev. Letters 15, 581(1965).

~ C. Domb and M. F. Sykes, J. Math. Phys. 2, 63 (1961);Proc.Roy. Soc. (London) A240, 214 (1957); M. E. Fisher and M. F.Sykes, Physica 28, 939 (1962); M. F. Sykes, J. W. Essam, andD. S. Gaunt, J. Math. Phys. 6, 283 (1965).There has also beensome work on series expansions for models in nonzero magneticfields: C. Domb, Proc. Roy. Soc. (London) A196, 36 (1949);A199, 199 (1949) and later papers.

66H. Pads, Ann. Sci. Ecole Normale Superieure (Paris) 9,Suppl. 1-92 (1892); G. A. Baker, Jr., J. L. Gammel, and J. G.Wills, J. Math. Anal. Appl. 2, 405 (1961); G. A. Baker, Jr., inAdvances in Theoretica/ Physics, K. A. Brueckner, Ed. (AcademicPress Inc. , New York, 1965).

where y=~& in two dimensions, and =~~ in three di-mensions. Baker also showed that the Pade approxirnantmethod could be applied successfully to low-tem-perature series, which previous methods had failed tohandle. For example, Baker's calculations suggestedthat the spontaneous magnetization goes to zero at thetransition temperature in such a way that

Ip( T)~D/1 —( T/To) 1S~

where P =0.30&0.01 for three-dimensional lattices.It was already known4t that P= s in two dimensions.

The Pade approximant method was quickly taken upby the London group and elsewhere; the possibility ofobtaining significant and reliable results for the criticalpoint of three-dimensional systems provided a strongmotivation for grinding out more terms in the seriesexpansions. 's The results have been summarized inpapers by Fisher. "

This recent progress in determining the critical-pointbehavior of the Lenz —Ising model has revealed that themodel may have much greater applicability to realphysical systems than was previously thought. Fisher~'pointed out that according to Van der Waals theory ofthe liquid —gas transition, the difference between liquidand gas phase densities should go to zero as the squareroot of the diGerence between temperature and criticaltemperature, as the critical point is approached frombelow:

pl, —pg~A (T, T) '~' as T~T,—

The compressibility along the critical isochore shouldbehave as

The specific heat at constant volume along the criticalisochore should rise to a maximum and then fall dis-continuously as T increases through T, :

with

The compressibility of the gas and the liquid along thecoexistence curve should also diverge as simple poles.

6 G. A. Baker, Jr., Phys. Rev. 124, 768 (1961).6' J. W. Essam and M. E. Fisher, J. Chem. Phys. 38, 802

(1963); A. V. Ferris-Prabhu, Phys. Letters 15, 127 (1965).69 M. E. Fisher, J. Math. Phys. 4, 278 (1963).re M. E. Fisher, J. Math. Phys. 5, 944 (1964); Lectlres je

Theoretica/ Physics (University of Colorado Press, Boulder,Colo., 1965),Vol. VIIC, p. 1;see also C. N. Yang and C. P. Yang,Phys. Rev. Letters 13, 303 (1964).

892 REvIEws OP MQDERN PHYsIcs OGToBER 1967

These predictions are also made by most other ap-proximate theories, and in fact they are "essentially adirect consequence of the implicit or explicit assumptionthat the free energy and the pressure can be expandedin a Taylor series in density and temperature at thecritical point. "

Experimental results on gas—1'quid systems near thecritical point are definitely in disagreement with thepredictions of the van der Waals theory, however.According to older data, as analyzed by Guggenheim in1945,7' the exponent is about 3 cather than —,'. Morerecent experiments also show that the exponent(usually denoted by P, corresponding to the exponentfor spontaneous magnetization) is between 0.33 and0.36 for most systems. Thus the experimental co-existence curve is much Ratter than the van der Waalscurve. On the other hand, the three-dimensionallattice gas model leads to a value of p between 0.303and 0.318, possibly just T6 =0.3125. While this is some-what outside the range for real systems, it is certainlymuch closer than the van der Waals theory, and createsthe presumption that the lattice gas provides an ex-tremely good first approximation for the behavior offluids at the critical point.

It will be observed, incidentally, that theoreticianshave a strong prejudice towards finding simple fractionsfor these exponents, whatever may be the raw data.This is probably justi6ed by the results of the two-dimensional exact calculations, but as J. F. C. Kingmanhas pointed out, "there must have been a time whenconsiderable plausibility attached to the conjecture

For the singularity in the compressibility (cor-responding to the susceptibility for a magnetic system),the exponent y mentioned above was thought for sometime to have the theoretical value 4. , but expert opinionhas recently tended toward the value ~~."Experimen-tal data, according to Fisher, ~' lead to a value somewhatlarger than 1.1, though there is not sufhcient evidencefor a precise estimate. Furthermore, most of the evi-dence is relevant only below the critical temperature,and. 7 may have a different value (y') above the criticaltemperature. Nevertheless, it seems likely that thelattice —gas value is better than the van der Waals value.

The nature of the specific heat singularity has notbeen settled. Fisher assumes a relation of the form

where A and 8 are constants that may have differentvalues above and below T„there may also be a differentexponent above (n) and below (n'). For the two-dimensional lattice gas, we know that rr=O (repre-senting, in this formula, a logarithmic singularity).

"E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945).» M. E. Fisher (private communication).

For the three-dimensional lattice gas, 0. is positive butless than 0.2; the best guess at present" is o.'=~~.

At this point there is an opportunity for the use ofthermodynamic, as opposed to statistical mechanicalreasoning, in deducing relations among the three ex-ponents. Kssam and Fisher, "on the basis of "heuristicarguments related to the Frenkel-Bijl —Band. pictureof condensation, " conjectured the relation

rr'+2P+v'= 2

This is valid in the two-dimensional Lenz —Ising model,where n =0, P = s, and y =@.Rushbrookers showed thatthe relation is valid as an inequality,

rr'+2P+y'& 2

for ferromagnets, and Fisher appplied the same reason-ing to Quids. GrifIiths74 derived this and several similarinequalities by thermodynamic arguments. Widom,Kadanoff, and others have recently investigated general,equations of state for lattice gases and ferromagnets innonzero magnetic field, in order to discover whatproperties a system must have in order to be con-sistent with information now available about thecritical point behavior. " A new pattern of relationsbetween critical indices now seems to be emerging fromthis research, but it would perhaps be premature toreport here on the tentative conclusions that have beenreached. ~'

For magnets, the Heisenberg model is generally con-sidered to be more accurate than the Lenz —Ising model.Attempts have been made to apply the Pade approxi-mant method to exact series expansions for the Heisen-berg model, but the calculations appear to be morediKcult and the conclusions less certain. It was thoughtfor awhile that the exact value of y for the Heisenbergmodel may be ~, which would agree very well with someof the experimental results though not all of them. 7 Forexample, Kouvel and Fisher analyzed the old data ofWeiss and Forrer (1926) and concluded that y=1.35&0.02 for nickel. 7' But more recent theoreticalwork suggests a value of about 1.42 for y.7'

73 G. S. Rushbrooke, J. Chem. Phys. 39, 842 (1963).r' R. B. Griiiiths, Phys. Rev. Letters 14, 623 (1965);J. Chem.

Phys. 43, 1958 (1965).7' B. Widom, J. Chem. Phys. 43, 3892, 3898 (1965);L. Kada-

noff, Physics 2, 263 (1966);A. Z. Patashinskii and V. L. Pokrov-skii, Zh. Eksperim. i Teor. Fis. 50, 439 (1966) )English transl. :Soviet Phys. —JETP 23, 292 (1966)g."C.Domb and D. L. Hunter, Proc. Phys. Soc. (London) 86,1147 (1965);C. Domb, Ann. Acad. Sci. Fennicae A. VI. 210, 167(1966); L. P. KadanoG et a/. , Rev. Mod. Phys. 39, 395 (1967);M. E. Fisher, J. Appl. Phys. 38, 981 (1967)."C. Domb and M. F. Sykes, Phys. Rev. 128, 168 (1962);J. Gammel, W. Marshall, and L. Morgan, Proc. Roy. Soc. (Lon-don) A275, 257 (1963)."J.S. Kouvel and M. E.Fisher, Phys. Rev. 136, A1626 (1964).

79 G. A. Baker, H. F. Gilbert, J. Eve, and G. S. Rushbrooke,Phys. Letters 20, 146 (1966).

STEPHEN G. BEUsH Btstory of the Lertz Ise-rtg Model 893

In this brief account we have not been able to reviewthe applications of the Lenz —Ising model to liquidmixtures alloys, polymers, and random walk problemsand many other areas of science; we have not evenmentioned all the theoretical work on the model it-self."Research on critical exponents is currently oneof the fastest-moving fields in science, and many ofthe results mentioned here may well be obsolete by thetime this article appears in print. But there does seemto be one general feature of the impact of the Lenz-Ising model on science which is well-enough establishedto be worth noting. " In constructing a theory to in-

terpret a complex physical phenomenon, a scientistfrequently has to choose between two approaches. Onthe one hand, he may want to make his theory con-tain all the possible factors which he knows must in-Quence the eBeets he observes in the laboratory; thetheory must be as "realistic" as possible. But this ap-proach usually leads to formulations that are mathe-matically so complicated that the consequences of thehypotheses cannot be deduced from the theory withoutgross approximations; if the predictions of the theorydisagree with the experimental facts, it is often dificultto know whether to attribute this to defects in theoriginal hypotheses, or to errors incurred by approxi-mations in the calculations. On the other hand, onemay intentionally sacrifice some of the more realistic

0 A bibliography of papers published up to 1964 is given in myreport "History of the Lenz-Ising model, " University of Cali-fornia, Lawrence Radiation Laboratory, Livermore, UCRL-7940(29 June 2964). A supplement covering more recent papers is nowin preparation."'Cf. the opening remarks and views of Frenkel quoted inFisher's Boulder lectures (cited in Ref. 26) .

features of a model in order to obtain a simpler modelthat is exactly, or almost exactly, soluble. Such over-simplided theories are often scorned by empiricallyminded scientists, since they may be completely ir-relevant to the problem of interpreting the subtleeffects that are important in practical problems. It isencouraging, therefore, to those who favor the secondapproach, to have at least one example in which in-sistence on getting an exact solution of a simple modelhas really paid off. (And of course it is also extremelyuseful to be able to test other theoretical methods,such as the currently popular integral equations forpair distribution functions, on a model where the exactsolution is known. sz) The Lens —Ising model providesthis example a many-body problem in which the in-teractions between particles cannot be ignored, oreven treated successfully by perturbation theory;and yet it i s possible, if one works hard and cleverlyenough, to get an exact solution. Here is a modernparadigm for the fruitful application of mathematics tophysical science.

ACKNO WLED GM ENTS

This work was supported by the U.S. Atomic EnergyCommission and the U.S. National Science Foundation.The author has benelted by discussions and corres-pondence with N. Burgoyne, C. Domb, M. Fisher,H. Frisch, E. Guggenheim, E. Ising, S. Katsura, R.Kikuchi, H. Raether, and S. Sherman, and wishes tothank all of them for their valuable contributions tothis article.

"F. H. Stillinger, Jr., and H. L. Frisch, Physics 2/, 752 (2962) .