Problems in Physics with Many Scales of Length

Problems in Physics wi th Many Scales of Length

Physical systems as varied as magnets and fluids are ahke J'n having

fluctuations in structure over a vast range of sizes. A novel method called the renormahzation group has been invented to explain them

by Kenneth G. Wilson

ne of the more conspicuous prop temperature is raised. Near the criti an alloy at the temperature where two O erties of nature is the great di cal point water develops fluctuations in kinds of metal atoms take on an orderly versity of size or length scales in density at all possible scales. The fluctu distribution. Other problems that have a the structure of the world. An ocean, for ations take the form of drops of liquid suitable form include turbulent flow,the example, has currents that persist for thoroughly interspersed with bubbles of onset of superconductivity and of su tho usands of kilometers and has tides gas, and there are bot h drops and bub perfluidity, the conformation of poly of global extent; it also has waves that bles of all sizes from single molecules up mers and the binding together of the ele range in size from less than a centimeter to the volume of the specimen. Precisely mentary particles called quarks. A re to several meters; at much finer resolu at the critical point the scale of the larg markable hyp othesis that seems to be tion seawater must be regarded as an est fl uctuations becomes infinite,but the confirmed by work with the renormali aggregate of molecules whose charac smaller fluctuations are in no way di zation group is that some of these phe teristic scale of length is roughly 10-8 minished. Any theory that describes wa nomena, which superficially seem quite centimeter. From the smallest structure ter near its critical point must take into distinct, are identical at a deeper lev to the largest is a span of some 17 or account the entire spectrum of length el. For example, the critical behavior ders of magnitude. scales. of fluids,ferromag nets, liquid mixtures In general, events distinguished by a Multiple scales of length complicate and alloys can all be described by a sin great disparity in size have little influ many of the outstanding problems in gle theory. ence on one another; they do not com theoretical physics and in certain other municate, and so the phenomena asso fields of study. Exact solutions have he most convenient context in which ciated with each scale can be treated been found for only a few of these prob Tto discuss the operation of the renor independently. The interaction of two lems, and for some others even the best malization group is a ferromagnet, or adjacent water molecules is much the known approximations are unsatisfac permanent magnet. Ferromagnetic ma same whether the molecules are in the tory. In the past decade a new method terials have a critical point called the Pacific Ocean or in a teapot. What is called the renormali zation group has Curie point or the Curie temperature, eq ually important, an ocean wave can been introduced for dealing with prob after Pierre Curie, who studied the ther be described quite acc urately as a dis lems that have multiple scales of length. modynamics of ferromagnets at about turbance of a continuous fluid, ignoring It has by no means made the problems the turn of the century. For iron the entirely the molecular structure of the easy, but some that have resisted all oth Curie temperat ure is \,044 degrees K. liquid. The success of almost all practi er approaches may yield to this one. At higher temperatures iron has no cal theories in physics depends on isolat The renormalization group is not a spontaneous magnetization. As the iron ing some limited range of length scales. descriptive theory of nature but a gener is cooled the magne tization remains If it were necessary in the eq uations of al method for constructing th eories. It zero until the Curie temperature is hydrodynamics to specify the motion of can be applied not only to a fluid at the reached, and th en the material abruptly every water molecule, a theory of ocean critical point but also to a ferromagnetic becomes magnetized. If the temperature waves would be far beyond the means of material at the temperature where spon is reduced further, the strength of the 20th-century science. taneous magnetization first sets in, or to magnetization incr eases smoothly. ' A class of phenomena does exist, a mixture of liquids at the temperature Several properties of ferromagnets however, where events at many scales of where they become fully miscibl e, or to besides the magnetization behave oddly length make contributions of equal im portance. An example is the behavior of

water when it is heated to boiling under MULTIPLE SCALES OF LENGTH characterize the patterns that emerge when a ferromag a pressure of 217 atmospheres. At that netic solid is cooled to the temperature at which it becomes spontaneously magnetized. Each pressure water does not bo il until the square represents the magnetic moment associated with a single atom in the solid, and each temperature reaches 647 degrees Kel moment is assumed to have only two possible orientations, labeled "up" (black sqllares) and vin. This combination of pressure and "down" (opell sqllares). At high temperature (top) the orientation of the magnetic moments is temperature defines the critical point of essentially random, and so there is only short-range order in the pattern. As the temperature is reduced (middle) somewhat larger patches in which most of the magnetic moments are lined water, where the distinction between liq np in the same direction begin to develop. When the temperature reaches a critical value called uid and gas disappears; at higher pres the Curie temperature, or Tc (bottom), these patches expand to infinitesize; significantly, sures there is only a sin gle, undifferenti however, fluctuations at smaller scales persist. As a result all scales of length must be included ated fluid phase, and water cannot be in a th eoretical description of the ferromagnet. This simulation of a ferromagnet was carried made to boil no matter how mu ch the out with the aid of a computer by Stephen Shenker and Jan Toboch.nik of Cornell University.

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© 1979 SCIENTIFIC AMERICAN, INC near the Curie point. Anot her property which defines the direction of the elec other materials is a coupling between of interest is the magnetic susceptibility, tron's magnetic field. nearby spins that makes them tend to or the change in magnetization induced A real ferromag net has a complex line up in the same direction. This ten by a small applied field. Well above the atomic structure, but all the essential dency can be stated more precisely by Curie point the susceptibility is small properties of the system of spins can be pointing out that the total energy of any because the iron cannot retain any mag illustrated by a quite simple model. two adjacent spins is smaller when the netization; well below the Curie temper Indeed, I shall describe a model that spins are parallel than it is when they are ature the susceptibility is small again be includes no atoms or other material par anti parallel. The interaction responsible cause the material is already magnetized ticles but consists only of spin vectors for the coupling of the spins has a short and a weak applied field cannot change arranged in a lattice. For the sake of range, which is reflected in the model by the state of the system very much. At simplicity I shall deal with a two-di specifying that only nearest-neighbor ' temperatures close to 1,044 degrees, mensional lattice: a rectilinear grid of spins are coupled to each other. In the however, the susceptibility rises to a uniformly spaced lines in a plane, with two-dimensional rectilinear lattice each sharp peak, and at the Curie point itself a spin vector at each intersection of the spin is influenced by four nearest neigh the susceptibility becomes infinite. grid lines. Furthermore, I shall assume bors; no other spins have any direct ef The ultimate source of ferromagnet that each spin can point in only two fect on it. ism is the quantum-mechanical spinning possible directions, designated up and of electrons. Because each elec tron ro down. The model lattice is said to be rom the nature of the interaction be tates it has a small magnetic dipole mo magnetized whenever more than half of F tween spins in a ferromagnet one ment; in other words, it acts as a magnet the spins poin t in the same direction. might well predict th at all the spins with one north pole and one south pole. The magnetization can be defined as the would always be parallel and the mate How the spin of the electron gives rise to number of up spins minus the number of rial would always have its maximum the magnetic moment will not concern down spins. magnetization. That is the state of low me here. It is sufficient to note that both Every electron has the same spin est energy, and in the absence of any the spin and the magnetic moment can and the same magnetic dipole moment. perturbing effects it would be the fa be represented by a vector, or arrow, What distinguishes a ferromagnet from vored state. In a real ferromagnet, how ever, there is one perturbation that can not be neglected: the thermal motion of " f\ I f\ f\ the atoms and the electrons. At any tem t perature above absolute zero thermal excitations of the solid randomly flip M = +4 M = +2 M = +2 M =0 ]'IP = .273 P = .037 P = .037 P = .037 some of the spins so that the direction of " " the spin vector is reversed, even when " " reversing the spin puts the magnet in a 1 state of higher energy. Hence it is no surprise that the magnetization decreas es as the temperature increases: that re lation simply reflects increasing thermal disruption. What remains curious is that M =0 the magnetization is not a smooth func P = .005+ tion of the temperature but instead dis appears abruptly at a certain finite tem perature, the Curie point. Competition between the tendency toward a uniform spin orientation and the thermal introduction of disorder can readily be incorporated into a model of a ferromagnet. The strength of the cou M =-2 pling between·adjacent spins is given by P = .037 a number, K. that must be specified in the design of the model. Thermal effects +4-B are included simply by making K in versely proportional to the temperature. With the appropriate units of measure ment the coupling strength can be set equal to the reciprocal of the tempera ture, a relation expressed by the eq ua tion K = 1 IT. What the coupling strength deter mines is the probability that two adja LJ cent spins are parallel. When the tem perature is zero, there are no thermal MODEL OF A FERROMAGNET consists of vectors, or arrows of fixed length, arranged at effects and adjacent spins are certain to the sites of a lattice. Each vector represents the spin angular momentum and the magne tic mo be parallel; the probability is equal to I ment of a single electron, and it can be oriented either up or down. Nearest-neighbor lattice and the coupling strength is infinite. At sites are coupled in such a way that adjacent spin vectors are likelier to be parallel than anti an infinite temperature the coupling parallel. From the strength of the coupling, which declines as the temperature increases, a prob strength falls to zero, so that the spins do ability, P, can be assigned to every possible configuration of the spin vectors. All the configura not interact at all. Hence each spin is tions of a lattice made up of just four sites are shown here. The net magnetization, M, of each free to choose its direction randomly configuration is easily calculated: it is the number of up spins minus the number of down spins. and is independent of its neighbors. The ··The magnetization of the model at any given temperature is found by multiplying the magnetization of. each configuration by the probability of that configuration,then adding up all the re probability that two spins are parallel is sults. The p!'�J1abilities shown were calculated for a coupling strength of .5, which corresponds 1/2, and so is the probability that they to a temperature (in arbitrary units) of 2. The model is called the two-dimensional Ising model. are anti parallel. The region of interest,

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© 1979 SCIENTIFIC AMERICAN, INC I of course, lies between these extremes of I temperature, where the probability of I the adjacent spins' lining up must always I have a val ue between 1!2 and 1. I I uppose there is a large two-dimen I > S sional lattice of spins and that some I 1 :::::; one spin in it is artificially held fixed in l� CD the up orientation. What is the effect I::J� ti:w on the other spins? The effect on the Ia: o spins at the four adjacent lattice sites is l� (fJ easy to imag ine: since they are directly z ::J(fJ o coupled to the fixed spin, they will have I� a greater-than-even probability of point �N I� i= ing up. The extent to which the probabil w IiI z ity is biased depends on the value of K, CJ 1IG which is determined in turn by the tem TEMPERATURE -7 perature. (fJ:Ii O l------"------f------! 0 ::J More distant spins have no direct in o w teraction with the fixed spin, but none Z theless the infl uence of the fixed spin � Z does not end with the immediate neigh o 0.. bors. Because the nearest-neighbor spins (fJ tend to point up more often than down z 3: they create a similar bias in their own o nearest neighbors. In this way the dis a turbance can propagate over a large area of the lattice. The range of influ MAGNETIZATION of a ferromagnet has a sudden onset at the Curie temp erature. Above ence of a single fixed spin can be mea this temperature the average numbers of up spins and down spins are equal and so the magneti zation is zero. At any temperature below the Curie point two states of magnetization are possi sured by observing the orientation of ble, depending on whether the up spins or the down spins are in the majority; in the absence of many spins that are all at the same an external magnetic field the two states are equally likely. The susceptibility of a ferromagnet large distance from the fixed one. If re measures the change in magnetization induced by an arbitrarily small applied magnetic field. versing the orientation of the fixedspin At the Curie point the susceptibility becomes infinite. Near the Curie point a small change in from up to down increases the number either the temperature or tbe external field gives rise to a large change in the magnetization. of down spins in the distant population, then the spins are said to be correlated. The maximum distance over which such cause they have an overall excess of one near the critical point are closely analo a correlation can be detected is called spin direction. Thus an ocean of spins gous to the fluctuations in spin direction the correlation length. Regions separat that are mostly up may have within it an observed in ferromagnets. In fluids, ed by a distance greater than the correla island of sp ins that are mostly down, however, the presence of fluctuations at tion length are essentially independent. which in turn surrounds a lake of up all possible scales of length can be ob In a lattice at very high temperature spins with an islet of down spins. The served directly. When the correlation the correlation length is close to zero. progression continues to the smallest length firstreaches a few thousand ang The distribution of spins is nearly ran possible scale: a single spin. strom units, which is comparable to the dom, and so the average number of up When the temperature is precisely wavelength of light, the fluctuations be and down spins must be equal; in other equal to the Curie temperature, the cor gin to scatter light strongly and the fluid words, the magnetization is zero. As the relation length becomes infinite. Any turns milky, a phenomenon called crit temperature falls (and the coupling two spins are correlated, no matter what ical opalescence. Significantly, when the strength increases) correlations over the distance between them is. Neverthe temperature comes still closer to the larger distances begin to appear. They less, fluctuations persist at all smaller critical point and the maximum scale of take the form of spin fluctuations, or scales of length. The system remains un the fluctuations becomes mu ch larger patches of a few spins each that mostly magnetized, but it is exquisitely sensi (millimeters or centimeters), the critical point in the same di rection. Over any tive to small perturbations. For exam opalescence is not reduced, indicating large area the magnetization is still zero, ple, holding a single spin fixed in the up that the smaller fluctuations persist. The but the struct ure of the lattice is much orientation creates a disturbance that same phenomenon takes place in spin different from what it was near infinite spreads throughout the lattice and gives systems, but because ferromagnetic ma temperature. the entire system a net magnetization. terials are not transparent to light it can As the temperature approaches the Below the Curie temperature the sys not be demonstrated as readily. The crit Curie point the correlation length grows tem becomes magnetized even in the ab ical opalescence of ferromagnets has rapidly. The basic interactions of the sence of an outside perturbation, but been detected, however, in the scattering model have not changed; they still con there is no immediate change in the ap of neutrons from a magnetic material nect only adjacent lattice sites, but long pearance of the lattice. Smaller-scale near the Cu rie temperature. range order has emerged from the short fluctuations persist; they are remnants range forces. What is most significant in of the lakes and islets of opposite spin he model I have been describing is the growth of the cor relation length is direction. Merely by looking at the lat Tnot my own invention. It is a version that as the maxim um size of the spin tice one cannot detect the magnetiza of one introduced in the 1920's by the fluctuations increases, the smaller fluc tion. Only when the system is cooled German physicists Wilhelm Lenz and tuations are not suppressed; they merely further does the bias become obvious, as Ernest Ising, and it is now called the become a finerstr ucture superimposed the increasing coupling strength coerces Ising model. The properties of a syst em on the larger one. The largest fluctua more of the spins into conformity with of Ising spins on a two-dimensional lat tions are not areas of uniform spin align tpe majority. At zero temperature com tice are known in complete detail be ment; they include many smaller fluctu plete uniformity is attained. cause the model was solved exactly in ations and can be distinguished only be- In fluidsthe fluctuations in density 1944 by Lars Onsager of Yale Universi-

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163 © 1979 SCIENTIFIC AMERICAN, INC ty. Since then solutions have al so been determines the macroscopic properties can be evaluated. All that is required is found for several other two-dimension of the system. Instead all possible con to multiply together the separate proba al models (whereas no three-dimension figurations contribute to the observed bilities for every nearest-neighbor pair al model has yet been solved exactly). properties, each in proportion to its prob of spins, in each case taking the value as Nevertheless, the problems of describ ability at a given temperature. p when the spins are parallel and as ing two-dimensional systems are far In principle the macroscopic proper I -p when they are antiparalle l. from trivial. In what follows I shall ties could be calculated di rectly as the apply the methods of the renormaliza sum of all the separate contributions. onsider a spin system that is made up tion group to the two-dimensional Ising First the magnetization would be found C of just four spins arranged at the model as if it were a problem still out for each configuration and then the cor corners of a square. Such a lattice has standing, and Onsager's solution will responding probability. The actual mag four nearest-neighbor couplings, corre serve as a check on the results. netization would be obtained by multi sponding to the four sides of the square. What does it mean to solve or to un plying each of these pairs of numbers Each coupling is considered in turn and derstand a model of a physical system? and adding up all the results. The sus is assigned a probability of either p or In the case of the Ising system the micro ceptibility and the correlation length I -p according to whether the spins are scopic properties are known completely could be found by procedures that are parallel or antiparallel; then the four from the ou tset, since they were speci not much more elaborate. The common separate probabilities are multiplied. In fied in building the model. What is need element in all these calculations is the the configuration with all four spins ori ed is a means of predicting the macro need to determine the probabilities of ented up all four pairs are parallel, and scopic properties of the system from the all possible configurations of the spins. so the relative probability is given by the known microscopic ones. For example, Once the distribution of probabilities is prod uct p X P X P X p. If three spins are a formula giving the spontaneous mag known the macroscopic properties fol up and one is down, the relative proba netization, the su sceptibility and the low directly, bility is p X p X (I -p) X (1 - pl. correlation length of the model as a As I pointed out above, the probabili The calculation must be carried out function of temperature would contrib ty of any two adjacent spins' being par for every configuration of the spins; for ute greatly to understanding. allel is determined solely by the cou a system of four sp ins there are 16 con It is not notably difficult to calculate pling strength K, which I have defined as figurations. A final step is to convert the macroscopic properties of any given the reciprocal of the temperature. If the the relative probabilities into absolute configuration of the spins in an Ising probability of two neighboring spins in ones by adjusting each value so that the model. The magnetization, for example, isolation being parallel is denoted p, total of all 16 values is eq ual to exact can be determined simply by counting then the probability of their being anti ly 1. Since the temperature determines the number of up spins and the number parallel must be 1 - p. From these two the coupling strength and the coupling of down spins and then subtracting. No values alone the relative probability of strength in turn determines the values of one configuration of the spins, however, any specifiedconfig uration of a lattice p and 1 -p, the entire sequence of 16 calculations would also have to be re peated for every temperature of interest. This plan of attack on the Ising model 1 X 10 30 is ambitious but impractical. If the prob ability of every spin configuration could be calculated, the magnetization and the 2 x 10 2• other macroscopic properties could be evaluated for any specified temperature. The problem lies in the number of spin 2 X 10 '. (fl configurations. For a system made up of z 0 11sp ins, each of which can take on two 6 X 10" values, there are 2n possible config ura �a: tions. This exponential function grows � (') rapidly as 11incre ases. As I have men u:: z 7 X 10 '0 tioned, four spins have 24, or 16, config 0 0 urations. A three-by-three block of nine z spins has 512 configurations and a four 0:: 3 X 10' (fl by-four block has 65,536. The practical LL limit of computation is not much larger 0 a: than a six-by-six block of 36 spins, for w 65,536 aJ which there are approximately 7 X 10 to ::;: � con fig urations. z What size lattice would be needed in 512 order to determine the critical proper ties of the two-dimensional Ising model? 16 The array must be at least as large as the largest fluctuations observed at the tem perature of interest. At a temperature 2 reasonably close to the Curie point the I correlation length, in units of the lattice spacing, mi ght be about 100 and the 1 x 1 2 x 2 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 x 8 9 x 9 10 x 10 largest fluctuations would cover about LATTICE SIZE 1002, or 10,000, lattice sites. A block of spins that large has 210,000 possible con NUMBER OF SPIN CONFIGURATIONS rises steeply as the size of a lattice grows. For a figurations, a number that is somewhat system of II spins, each of which has two possible values, the number of configurations is equal greater than 103,00°. The fastest comput to 2n. When the lattice is large, it becomes impractical to calculate the probability of all the con figurations. The limit of practical computation is a lattice somewhat larger than the six-by-six er conceivable could not carry out such array of 36 spins. In order to observe the critical behavior of the system near the Curie temper a calculation. Even if the computer had ature an array of about lOO-by-lOO spins would be needed, which has 210,000 configurations. been working continuously since the

164

© 1979 SCIENTIFIC AMERICAN, INC "big bang" with which the universe be gan, it would not yet have made a signifi � I' cant start on the task. The need to carry out an almost end less enumeration of spin configurations can be circumvented for two special � I' conditions of the lattice. When the tem II perature of the system is zero (so that the coupling strength is infinite), all but I' two of the configurations can be neglect ed. At zero temperature the probability II II that a pair of spins will be antiparallel falls to zero, and therefore so does the probability of any configuration that in cludes even one antiparallel pair. The V only configurations that do not have I at least one anti parallel pair are those I' I' in which all the spins are up or all are V \ down. The lattice is certain to assume one of these configurations, and all oth I er configurations have zero probability. I' At infinite temperature, where the coupling strength is zero, the probability distribution is also much simplified. Ev ery spin is then independent of its neigh f11' bors and its direction at any instant can II be chosen at random. The result is that every configuration of the lattice has I' I' equal probability. Through these two sho rtcuts to the V determination of the probability distri bution it is a trivial exercise to calculate

exactly the properties of the Ising model \ at absolute zero and at infinite tempera ture. Acceptable methods of approxi FORMATION OF BLOCKS mation are also available for any tem I perature low enough to be considered I' I' close to zero or high enough to be con sidered close to infinity. The trouble some region is between these extremes; it corresponds to the region of the crit ical point. Until recently there was no practical and direct method of calculat ing the properties of a system arbitrarily close to the critical point. The renormal ization group provides such a method.

he essence of the renormalization Tgroup method is to break a large problem down into a sequence of small \ er and more manageable stages. Instead

RENORMALIZATION-GROUP approach to a model ferromagnet consists in breaking down an intractable problem with multiple scales of length into a sequence of smaller problems, each of wbicb is confined to a single scale of leng th. One version of the renormali I' zation-group method, calle d the block-spin \ transformation, has three steps. First the lat REPLACEMENT OF INDIVIDUAL SPINS BLOCK SPINS tice is divided into blocks of a few spins each, ey in this case nine. Then each block is replaced by a single spin whose value is the average of I' all the spins in the block; here the average is determined by majority rule. In this way a new lattice is created, with three times the original lattice spacing and one-third the den RESCALING sity of spins. Finally the original scale is re OF LATTICE stored by reducing all dimensions by a factor of 3. The procedure must be carried out for all configurations of a few spins in the origi nal lattice, so that a probability can be found for every configuration of the block spins. II

Lj ------1' L- r-1 P = .3655 P� = .1345 P = .1345� P = .3655

PROBABILITIES OF SIX-SPIN CONFIGURATIONS IN ORIGINAL LATTICE

P = .2943 P = .0147 P = .0020 P = .0007

WA P = .0398 P = .0054 P = .0002 P = .0020 � •1--'\ � � P = .0147 � P = .0020 •� P = .0007 � P = .0054 ¥ :� __.l. VAP = .0054 � P = .0147 P = .0020 � P = .0398

� P = .0054 � P = .0020 � P = .0054 � P = .0020

� P = .0007 � P = .0007 � P = .0007 ViAP = .0007 I P = . 0020 � P = .0020 VA· P = .0020 � P = .0020 P = .0007 � P = .0147 VAP = .0007 � P = .0147 � P = .0147 P = .0007 � P = .0147 I P = .0007 - P = . 0020 � P = .0020 P = .0020 P = .0020 - P = .0007 P = .0007 - P = .0007 � P = .0007 � P = .0020 • P = .0054 � P = .0020 P = .0054 \1&P = .0398 ;:P = .0020 V4P = .0147 � P = .0054 � P = .0054 P = .0007 � P = .0020 P = .0147 � P = .0020 V4 P = .0002 � P = .0054 = P = .0398 V4P = .0007 V4 P = .0020 VAP = .0147 P = .2943 '\\1 -: --;If t t t t '�.4302 .0697 .0697 .4302 = = �= � P � P P � � P = t

PROBABILITIES OF NEAREST-NEIGHBOR BLOCK -SPIN CONFIGURATIONS

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© 1979 SCIENTIFIC AMERICAN, INC of keeping track of all the spins in a re or about 70 billion, possible configura given by the parameter K. the recipro gion the size of the correlation length, tions. After the block-spin transforma cal of the temperature. the long-range properties are deduced tion has been applied the 36 original This guess can easily be checked, be from the behavior of a few quantities spins are replaced by four block spins cause the probability distribution for that incorporate the effects of many with a total of 16 configurations. It is the configurations of at least a small spins. There are several ways to do this. just within the limit of practicality to part of the block-spin system is already I shall describe one, the block-spin tech compute the probability of each of the known; it was computed from the con nique, in which the principles of the configurations of the original 36 spins. figurations of the original lattice in the method are revealed wit h particular From those numbers the probabilities of course of defining the block spins. Sur clarity. It was introduced by Leo P. the 16 block-spin configurations can prisingly, this hypothesis is generally Kadanoff of the University of Chicago readily be determined. The calc ulation wrong: the block spins do not have the and was made a practical tool for cal can be done by sorting all the configu same couplings as the spins in the orig culations by Th. Niemeijer and J. M. J. rations of the original lattice into 16 inal model. Assuming that only adja van Leeuwen of the Delft University of classes according to which configura cent sites interact and that they have a Technology in the Netherlands. tion of the block spins results in each coupling strength equal to K gives the The method has three basic steps, case from applying the principle of ma wrong set of probabilities for the config each of which must be repeated many jority rule. The total probability for any urations of the block spins. times. First the lattice is divided into one configuration of the block spins is blocks of a few spins each; I shall em then found by adding up the probabili f the specifications of the original ploy square blocks with three spins on a ties of all the configurations of the origi I model will not describe the system of side, so that each block in cludes nine nal lattice that fall into that class. block spins, then some new set of cou spins. Next all the spins in the block are It may well seem that nothing is plings must be invented. The guiding averaged in some way and the enti re gained by this procedure. If the com principle in formulating these new inter block is replaced by a single new spin plete probability distribution can be cal actions is to reproduce as accurately as with the value of the average. Here the culated for a system of 36 spins, nothing possible the observed probability distri averaging can be done by a simple pro new is learned by condensing that sys bution. In general the nearest-neighbor cedure: by following the principle of tem into a smaller lattice of four block coupling strength must be changed, that majority rule. If five or more of the orig spins. Near the critical point it is still is, K must take on a new value. What is inal spins are up, the new spin is also up; necessary to consider a much larger lat more, couplings of longer range, which otherwise it is down. tice, with perhaps 10,000 spins instead were excluded by definition from the The result of these two operations is of 36, and the probability di stribution Ising model, must be introduced. For ex to create a new lattice whose fundamen for the block spins generated from this ample, it may be necessary to establish a tal spacing is three times as large as that lattice cannot be calculated because coupling between spins at the opposite of the old lattice. In the third step the there are far too many configurations. corners of a square. There might also be original scale is restored by reducing all As it turns out, however, there is a meth direct interactions among spins taken dimensions by a factor of 3. od for extracting useful information three at a time or four at a time. Cou These three steps define a renormali from a small set of block spins. It is a plings of still longer range are possible. zation-group transformation. Its effect method for observing the behavior of Hence the block spins can be regarded is to eliminate from the system all fluc the system over a large region without as a lattice system, but it is a system tuations in spin direction whose scale is ever dealing explicitly with the configu quite different from the original one. smaller than the block size. In the model rations of all the spins in that region. Notably, because the basic couplings given here any fluctuation of the spins Each block spin represents nine spins have different values, the lattice of block over a range of fewer than three lattice in the original lattice. The complete set spins is at a temperature different from units will be smeared out by the averag of block spins, however, can also be re that of the initial Ising system. ing of the spins in each block. It is as if garded as a spin system in its own right, Once a set of couplings has been one looked at the lattice through an out with properties that can be investigated found that correctly describes the prob of-focus lens, so that the smaller fea by the same methods that are applied to ability distribution for the block spins, a tures are blurred but the larger ones are the original model. It can be assumed lattice of arbitrary size can be construct unaffected. that there are couplings between the ed from them. The new lattice is formed It is not enough to carry out this pro block spins, which depend on the tem the same way the original one was, but cedure for any one configuration of the perature and which determine in turn now the probability for the spin at each original lattice; once again what is the probability of each possible spin site is determined by the newly derived sought is a probability distribution. Sup configuration. An initial guess might be coupling strengths rather than by the pose one considers only a small region that the coupl ings between block spins single coupling of the Ising model. The of the initial lattice, consisting of 36 are the same ones specifiedin the origi renormalization-group calculation now spins that can be arranged in four nal lattice of Ising spins, namely a near proceeds by starting all over again, with blocks. The spins in this region have 236, est-neighbor interaction with a strength the new system of block spins as the starting lattice. Once again blocks of nine spins each are formed, and in some small region, such as an array of 36 spins, the probability of every possible PROBABILITY DISTRIBUTION for a system of block spins is found by adding up the prob abilities for all the configurationsof the original lattice that contribute to each configuration configuration is found. This calculation of the block spins. The calculation is shown for a system of six spins on a triangular lattice. is then employed to define the probabil Two blocks of three spins each are formed from the lattice, and each block is replaced by a sin ity distribution of a second generation gle spin whose orientation is determined by majority rule. The six spins have 64 possible config of block spins, which are once more urations, which are assigned to columns in such a way that all the configurations in each col formed by majority rule. Examination umn give rise to the same block-spin configuration. For example, all the configurations in the of the second-generation block spins column at the far left have at least two spins in each block pointing up, so that they are rep shows that the couplings have again resented by two up block spins. The coupling strength in the original lattice is set equal to .5, changed, so that new values must be which yields the nearest-neighbor probabilities shown at the top of the page. From this set of supplied a second time for each cou numbers a probability is calculated for every configurationof the original lattice; then all the probabilities in each column are added up to give the probability of the corresponding block pling strength. Once the new val ues have spin configuration. The block-spin probabilities are not the same as those specified for the orig been determined another lattice system inal lattice, which implies that the coupling strength is also d�fferent, as is the temperature. (the third generation) can be construct-

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© 1979 SCIENTIFIC AMERICAN, INC ed and the entire procedure can be re the fluctuations at the smallest scale tions up to this size range are averaged peated yet again. have been eliminated, but those slightly out, leaving only those larger than nine The point of this repetitive operation larger, with a scale of roughly three lattice units. The next iteration removes is that it provides information about the times the original lattice spacing, can all fluctuations whose scale is between behavior of distinct but related spin sys be seen more clearly. After the second nine and 27 lattice units, then the fol tems in which the fundamental scale of transformation each block spin repre lowing iteration removes those between length gets larger with each iteration. sents the 81 spins in a nine-by-nine block 27 and 81 units. Eventually fluctuations After the first block-spin transformation of the original lattice, and all flu ctua- at all scales up to the correlation length

T = 1.22Tc ORIGINAL LATTICE

FIRST-STAGE BLOCK SPINS FIRST-STAGE --

SECOND-STAGE BLOCK SPINS SECOND -STAGE BLOCK SPINS

THIRD-STAGE BLOCK SPINS

FOURTH STAGE BLOCK SPINS

l.-<-JII-. �;�.+,-=��

BLOCK-SPIN TRANSFORMATION is applied to a lattice of spins vision of the original lattice into three-by-three blocks. Each block repeatedly, each time elucidating the behavior of the system at a larg is replaced by a single spin whose value is determined by majority er scale. The computer simulation, which was carried out by the au rule; these make up the lattice of first-stage block spins. The proce thor, began with an array of some 236,000 spins; a black square rep dure is then repeated, but with the first-stage block spins serving as resents an up spin and an open square a down spin. The initial tem the starting lattice. The resulting second-stage spins form the initial p�rature was set equal to three values: above the Curie temperature, configuration for the next transformation, and so on. By the time the Te. at T e and just below T e. The transformation begins with the di- third stage is reached the number of spins is small enough for them all

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© 1979 SCIENTIFIC AMERICAN, INC are averaged out. The resulting spin sys Ising spins just below the Curie temper however, the smaller fluctuations disap tem reflects only the long-range proper ature will seldom reveal that the model pear and the long-range magnetization ties of the original Ising system, with all is slightly magnetized. At this tempera becomes obvious. finer-scale fluctuationseliminated. ture there is only a small excess of one Much of the physical meaning of the The value of the block-spin technique spin direction over the other, and the block-spin transformation is to be found can be perceived even through a simple many small-scale fluctuations obscure in the way the couplings between spins visual inspection of the evolving model. the overall bias_ After several applica change. The rules for deriving the new Merely looking at a configuration of tions of the block-spin transformation, couplings from the old ones at each

LATIICE T = .99Tc ORIGINAL LATIICE �1TTTTH u.:..:-:::f::::: U u.::r...... J!I!I �...... __ � �_IIL-..UI•• J.. '------.I..IJ!I�J!I��•• � _ •••• ...... •..IL� _•••• --.--. 1i•• ......

BLOCK SPINS FIRST-STAGE BLOCK SPINS

___ 11.1 •• bM=- _

SECOND-STAGE BLOCK SPINS II II 'ftI.It:I i

THIRD-STAGE THIRD-STAGE 1: BLOCK SPINS BLOCK SPINS

FOURTH FOURTH Ii STAGE STAGE BLOCK BLOCK SPINS SPINS

to be shown, and after the fourth stage there are only 36 spins left, dom in appearance with each iteration and large-scale fluctuations each one representing more than 6,000 sites in the original lattice. In disappear; when the temperature is below Te, the spins become more the first stage any fluctuationswhose scale of length is smaller than nearly uniform and what fluctuationsremain are small in scale. When three lattice units are eliminated by the averaging procedure. The sec the starting temperature is exactly equal to T e. large-scale fluctua ond stage removes the fluctuations between three and nine lattice tions remain at all stages. Because the block-spin transformation units, the third stage those between nine and 27 nnits, and so on. When leaves the large-scale structure of the lattice unchanged at the Curie the initial temperature is above Te, the spins become more nearly ran- temperature, a system at that temperature is said to be at a fixed point.

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© 1979 SCIENTIFIC AMERICAN, INC stage are often complicated, but the ef rapidly approaches zero. With each iter plings between spins. The large-scale or fect of the change can be illustrated by a ation the spin system is converted into a long-range behavior of the new lattice quite simple example. Although the as new system that not only has a thinner is equivalent to the behavior that would sumptions are not realistic, I shall dis lattice but also has weaker coupli ngs be observed in the original lattice at a cuss a model in which no couplings with between the spins. Since K is equal to differenttemperature. a range longer than the original, nearest 1/T, the temperature increases with each There is one initial value of K that neighbor interaction are introd uced. iteration and the lattice approaches the does not diverge either to infinityor to The only change in the coupling is an limit of infinite temperature and ran zero, namely the value K = 1. Since 12 is adjustment in the value of K. which is dom spins. equal to I, K' remains equal to K no equivalent to a shift in the temperature. If the initial coupli ng strength is set matter how many times the transforma Moreover, this adjustment in K will equal to 2 (so that the temperature has tion is repeated. When K is equ al to have a simple form: at each stage in the a value of 1/2), the coupling increases at 1, the system is said to be at a fixed procedure the coupling strength in the each stage in the calculation. After the point, where continued application of new lattice will be set equal to the first block-spin transformation the cou the renormalization-group transforma sq uare of the coupling in the old lattice. pling strength is 4, then 16, then 256; ul tion leaves all essential properties of the If the new coupling is denoted K', it is timately the strength becomes infinite. lattice unchanged. Actually the values given by the equation K' = K2. At the same time, of course, the temper K = 0 and K = infinity also represent Suppose in some initial state K is ature falls and the system approaches fixed points, since zero squared is still equal to 1/2 (which mealls that the tem the state of zero temperature, in which zero and infinity squared is still infinity. perature has been given an initial value all the spins are aligned. Zero and infinity, however, are consid of 2 in the arbitrary units employed It should be emphasized that what is ered trivial fixedpoints, whereas the val here). In the thinned-out lattice formed being observed is not the evolution of ue K = 1 corresponds to the critical as a product of the block-spin transfor any single spin system as the tempera point. mation K will be replaced by K', with a ture changes. Nothing is being heated or In this discussion of the block-spin value of (1/2)2, or 1/4. Repeating the cooled. Instead a new spin system is be technique all the effects of the transfor transformation yields successive values ing created at each stage, a system dis mation have been expressed through a of 1/16, 1/256 and so on, in a series that tinguished by a different set of cou- single parameter: the nearest-neighbor coupling strength K. Actually many oth er parameters are introduced by the K15 = (K'4)' transformation, each one corresponding to a longer-range coupling. All the pos K14 = (K ,,)' sible combinations of these parame

K'3 = (K,,)' ters can be represented geometrically by constructing an imaginary multidimen K" = (K, ,)' sional space in which distance measured along each dimension corresponds to K" = (K lO)' variation in one of the parameters. Ev ery initial state of the spjn system and K,o = (Kg)' every block-spin transformation of it (f) can be represented by a point on a sur z Kg = (K.)' 0 face somewhere in this parameter space. � K. = (K,)' a: w n the geometric description of the re t::: K, = (K.)' normalization-group method the sig z I a: nificance of the fixed points becomes ap (f) K. = (Ks)' � parent. For the two-dimensional Ising u 0 system the surface in parameter space -' Ks = (K4)' CD has the form of a hilly landscape with

K4 = (K3)' two sharp peaks and two deep sinkholes. The ridgeline that connects the peaks K3 = (K,)' and the gully line that connects the sink holes meet in the center at a saddle point K, = (K, )' [see illustration on opposite page]. One sinkhole is the K = 0 fixed point; the = K, (Ko)' other is the K = infinity fixedpoint. The critical fixed point lies at the point Ko . 5 1 of precarious equilibrium in the saddle .

COUPLING STRENGTH (K = 1/ T) The transformation of the system from one state to the next can be repre I I I I I I I I I I sented by the motion of a marble rolling 10 8 7 6 5 4 3 2 1 ,5 ,33 .25 ,2 .17 .13.1 on the surface. One can imagine a time TEMPERATU RE = (T 11K) lapse motion picture that would record CHANGE IN THE COUPLING BETWEEN SPINS is part of the renormalization-group the marble's position at one-second in transformation. The adjustment that must be made to the coupling strength with each itera tervals; then each frame would reveal tion can take many forms, but a simple example is presented here: If the strength of the cou the effect of one iteration of the block pling in the original lattice is given by the number K, then in the new lattice the coupling spin transformation. It is the transfor strength is equal to K2. Any initial value of K greater than 1 must approach infinity when K is mation that allows the marble to move, squared repeatedly; any value less than 1 must approach zero. The special value K = 1 re but the speed and direction of the mar mains unchanged no matter how many times the transformation is repeated. Because the tem ble are determined entirely by the slope perature can be defined (in appropriate units) as the reciprocal of the coupling strength, the of the surface at each point it crosses. renormalization-group transformation can be seen as establishing a correspondence between the original lattice and a new, thinned-out lattice that will generally have a different coupling Suppose the marble is initially placed strength and a different temperature. It is only at the fixed point, which corresponds to the Cu near the top of a hill and just to one side rie temperature, that the coupling and the temperature remain invariant with 'a value of 1. of the ridgeline. At first it moves rapidly,

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© 1979 SCIENTIFIC AMERICAN, INC because the hill is steep near the top, and proceeds in the general direction of the saddle point. As the marble approaches the saddle the slope becomes more grad ual and the marble slows down, but it never comes to a complete stop. More over, because it started to one side of the ridgeline it does not quite reach the sad dle point; instead it is deflected to one side and begins to accelerate again, this time toward a sinkhole. The trajectory of the marble describes the path followed by the point repre senting a system of Ising spins as it is transformed repeatedly by the block spin method. The initial position just off the ridge line corresponds to an initial value of the various coupling parame ters that is equivalent to a temperature just above or just below the critical tem perature. In terms of the simplified ex ample described above, with just one pa rameter, the value of K is either slightly greater than 1 or less than 1. Setting the coupling strength equal to 1 is equi va lent to placing the marble exactly on the ridgeline. It then moves directly to ward the saddle point, or critical fixed point. Again the motion is rapid at first but becomes slower as the saddle is approached. In this case, however, the EVOLUTION OF A SPIN SYSTEM in response to repeated renormalization-group transfor marble remains balanced between the mations can be described as tbe motion of a point on a surface constructed in an imaginary, two descending slopes. Even after a multidimensional space: tbe parameter space. Tbe form of tbe surface is defined by all tbe con large number of iterations it remains at plings between block spins, but only tbe nearest-neigbbor coupling, K, is considered bere. Tbe the fixed point. surface bas two peaks and two sinkboles, wbicb are connected to a saddle point. Tbe trajectory A trajectory on the saddle-shaped sur followed by tbe point tbat represents tbe state of tbe system is determined entirely by tbe slope face in parameter space can be made to of tbe surface. An initial value of K sligbtly greater tban 1 corresponds to an initial position approach the fixed point as closely as sligbtly to one side of tbe ridgeline tbat connects tbe peaks. After several block-spin transfor is wished by setting the initial value of mations tbe point rolls down tbe bill, passes near tbe saddle point and veers off into one of tbe sinkboles, wbere K tends toward infinity. An initial value of K sligbtly less tban 1 leads to a simi K sufficiently close to the critical value. lar trajectory on tbe otber side of tbe ridgeline and terminates in tbe otber sinkbole, wbere K In the example considered here, where approacbes zero. Wben K is equal to exactly 1, tbe point remains permanently on tbe ridgeline, the critical value of K is 1, the initial val approacbing equilibrium at tbe saddle point. Botb of tbe sinkboles are fixed points (since tbe ue of K might be .9999, which can be values of K = 0 and K = infinity do not cbange witb furtber renormalization-group transforma squared several times before it changes tions), but tbey are considered trivial fixed points. Tbe saddle defines tbe critical fixed point. appreciably. As a result the traj ectory comes quite close to the critical fixed point before it veers off toward the high critical points are equivalent. A suitable third exponent, v (the Greek letter nu), temperature sinkhole. quantity is the reduced temperature, t, in a relation of the same form: the length By the examination of many such tra defined as the difference between the ac is proportional to 1/1 tlv. jectories the topography of the surface tual temperature and the critical tem The earliest attempts to for mulate itself can be mapped in the small region perature, divided by the critical temper a mathematical description of critical surrounding the saddle point. The slope ature; thus t is equal to T - Tel Te. On an phenomena were theories of a kind that of the su rface is what determines how ordinary temperature scale such as the are now called mean-field theories. The the system approaches the fixed point Kelvin scale the critical temperatures of first of these was int roduced in 1873 and how it departs from it. Knowing the differentsystems fall at different values, by J. D. van der Waals as an explana slope, then, one can calculate how the but all critical points hav'e the same re tion of phase changes in fluids. A theo properties of the system vary as the ini duced temperature, namely zero. ry of magnetic phase transitions was pro tial coupling and the initial temperature All critical properties are proportion posed in 1907 by Pierre Weiss. In 1937 are changed. That is precisely the infor al to the absolute value of the reduced L. D. Landau of the Academy of Sci mation sought for an understanding of temperature raised to some power. The ences of the U.S.S.R. proposed a more critical phenomena. problem of describing critical phenome general formulation of mean-field theo . na is to determine what that power is, or ry, thereby providing a fr amework in he macroscopic properties of a ther in other words to determine the values which many physical systems could be Tmodynamic system near the critical of the critical exponents. For example, discussed. In all these theories ' the state point are determined by the temper the magnetization, M, of a spin system is of any selected particle is determined by ature. To be more precise, properties given by the proportionality M I till, the average properties of the material as such as the spontaneous magnetization, where {3 (the Greek letter beta) is a crit a whole, properties such as the net mag the susceptibility and the correlation ical exponent and where the vertical netization. In effect all particles in the length are functions of the amount by lines designate the absolute value of t. system contribute equally to the force at which the temperature of the system de The magnetic susceptibility is propor every site, which is equivalent to assum parts from the critical temperature, Te• tional to 1 II flY, where 'Y (the Greek let ing that the forces have infinite range. For this reason it is convenient to define ter gamma) is another exponent. The Mean-field theories are qualitatively the temperature in such a way that all correlation length is associated with a successful. They account for important

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Ko = 1.001 Ko = 1.0001 Ko = 1

SLOPE OF THE PARAMETER SURFACE in tbe vicinity of the tively flat(/e !t ), a trajectory witb an initial value of K sucb as K = 1.01 critical fixed point determines tbe macroscopic properties of the Ising passes close to tbe saddle point. Wben tbe surface is mo re steeply model. If trajectories are plotted for many initial values of K near the curved (right), tbe corre sponding trajectory bends more abruptly critical value (wbicb in tbis case is K = 1) , it is the slope at the saddle toward tbe sinkbole. Because tbe temperature is tbe reciprocal of K point tbat determines bow quickly tbe trajectories veer off toward the tbe slope near tbe fixed point reveals bow tbe properties of tbe sys trivial fixed points at K = 0 and K = infini ty. If tbe surface is compara- tem change as the temperature departs from the critical temperature.

features of the phase diagrams of fluids bility is given by the reciprocal not of 1 spin decimation, instead of assembling and ferromagnets, the most notable of but of 1 raised to the 1.75t h power, blocks of a few spins each, every other these features being the existence of a which makes the divergence near the spin in the lattice is held fixed while a critical point. The quantitative predic critical point steeper and more abrupt. probability distribution is computed for tions, however, are less satisfactory: the The reason for the quantitative failure the remaining spins. These calculations theories give the wrong values for the of mean-field theories is not hard to were much more elaborate than the critical exponents. For {3, the exponent identify. The infinite range assigned to model calculation described here; in my that governs the spontaneous magneti the forces is not even a good approxima own work, for example, 217 couplings zation, mean-field theory implies a val tion to the truth. Not all spins make between spins were included. The crit ue of 112; in other words, the magneti equal contributions; the nearest neigh ical exponents derived from the calcula zation varies as the square root of the bors are more important by far than any tion agree with Onsager's values to with reduced temperature. The exponent as other spins. The same objection can be in about .2 percent. sociated with the susceptibility, 'Y, is as expressed another way: the theories fail signed a value of 1, so that the suscep to take any notice of fluctuations in spin ecause an exact solution is known for tibility is proportional to 1/1 1 1. The ex orientation or in fluid density. B the two-dimensional Ising model, ponent for the correlation length, v, is I.n a renormalization-group calcula the application of the renormalization 112, so that this quantity is proportional tion the critical exponents are deter group to it is something of an academic to 1/ vTtl, mined from the slope of the parameter exercise. For a system of Ising spins in a The exponents calculated from mean surface in the vicinity of the fixed point. three-dimensional lattice, however, no field theory suggest a plausible form for A slope is simply a graphic represen exact solution is known. A method has each of these functions. The magnetiza tation of a rate of change; the slope been devised, by Cyril Domb of Univer tion has two possible values (+ Vi and near the fixed point determines the rate sity College London and many others, -Vi) at all temperatures below the crit at which the properties of the system for finding approximate values of the ical point, and then it vanishes above the change as the temperature (or the cou exponents in the three-dimensional case. critical temperature. Both the suscep pling strength) is varied over some nar First the properties of the system at high tibility and the correlation length ap row range near the critical temperature. temperature are determined with great proach infinity as 1 nears zero from ei Describing the change in the system as a precision, then these properties are ex ther above or below. The actual values function of temperature is also the role trapolated to the critical temperature. of the mean-field exponents, however, of the critical exponents, and so it is rea The best results obtained so far by this are known to be wrong. sonable that there is a connection be method give values fer the exponents of For the two-dimensional Ising model tween the exponents and the slope. {3 = .33, 'Y = 1.25 and v = .63. the critical exponents are known exactly Renormalization-group calculations Although extrapolation from a high from Onsager's solution. The correct for the two-dimensional Ising system temperature solution leads to good ap values are {3 = 1/8, 'Y = 7/4 and v = I, have been carried out by several work proximations for the critical exponents, which differsign ificantly from the pre ers. In 1973 Niemeijer and van Leeuwen it provides little intuitive understand dictions of mean-field theory and imply employed a block-spin method to study ing of how the system behaves near the that the system has rather different be the properties of a system of Ising spins critical point. A renormalization-group havior. For example, the magnetization constructed on a triangular lattice. I calculation gives essentially the same is proportional not to the square root of have applied a somewhat different re values for the exponents, but it also the reduced temperature 1 but to the normalization-group technique, called explains important universal features of eighth root of f. Similarly, the suscepti- spin decimation, to a square lattice. In critical behavior.

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© 1979 SCIENTIFIC AMERICAN, INC Two remarkable facts about the ex zation and the density difference-are the three-dimensional Ising model are ponents in the th ree-dimensional Ising called the order parameters of their re identical in that they have the same sur model should not be overlooked. The spective systems. The susceptibility of face in parameter space. The two sys first is simply that the values are differ the magnet, which is the change in mag tems have different initial positions on ent from those for the two-dimensional netization for a given small change in the surface, but they converge on the model. In mean-field theories the dimen the applied magnetic field, is analogous same saddle point and hence have the sionality of space does not enter the cal to the compressibility of the fluid: the same critical exponents. culations and so the critical exponents change in density that results from a giv have the same values in any space. The en small change in pressure. Like the he similarity observed in the criti second surprise is that the exponents are susceptibility, the compressibility be Tcal behavior of fluids and of ferro not integers or ratios of small integers, comes infinite at the critical point. The magnets is an instance of a more general as they are in mean-field theories. They critical behavior of the fluid and that of hypothesis called critical-point uni ver- may even be irrational numbers. If it is surprising that the spatial di mensionality influences the critical ex PREDICTIONS OF ponents, it is equally remarkable that MEAN-FIELD THEORIES EXACT SOLUTION certain other properties of the model have no effect at all. An example of such an irrelevant parameter is the structure {3 = VB of the lattice. In the two-dimensional Ising model it makes no difference whether the lattice is rectilinear, as in my own work, or triangular, as in the --..... model employed by Niemeijer and van Leeuwen; the critical exponents are the ./ same. By extension, in a real ferromag net the great variety of crystal structures all yield identical critical behavior. There is an intuitive justification for the irrelevance of the lattice structure o + o + and of other microscopic properties. A REDUCED TEMPERATURE (t) REDUCED TEMPERATURE (t) change in the form of the lattice has a large effect on events at the scale of the lattice spacing, but the effect diminishes y = 1 as the scale of interest increases. In a renormalization-group calculation the fluctuations at the scale of the lattice spacing are averaged out after the first few iterations, and so models with many different lattices have the same critical behavior. Through the renormalization group the appearance of the same crit ical exponents in many systems is seen to result from the topography of the sur face in parameter space. Each lattice o + - o + structure corresponds to a different po REDUCED TEMPERATURE (t) REDUCED TEMPERATURE (t) sition in parameter space, but at the crit ical temperature every lattice is rep resented by a point somewhere along v = V2 v = 1 the ridgeline. After repeated renormal ization-group transformations all these systems converge on the same fixed point, namely the saddle point. The idea that certain variables are ir relevant to critical phenomena can be extended to systems other than ferro magnets. A fluid near its critical point, for example, has the same properties as the three-dimensional Ising model of a ferromagnet. In order for this identity to be understood some correspondence o + -0 + REDUCED TEMPERATURE REDUCED TEMPERATURE must be established between the macro (t) (t) scopic properties of the fluid and those CRITICAL EXPONENTS express the dependence of macroscopic properties on the extent of the magnet. The magnetization, to which the temperature of the system departs from the critical temperature. The temperature which is the number of up spins minus is most conveniently given in the form of the reduced temperature, I, defined by the equation the number of down spins, can be identi I = T -Tel Te. All macroscopic properties are then proportional to the absolute value of t raised fied with the density differencein the to some power; the power is the critical exponent for that property. The exponents and power fluid: the density of the liquid phase mi laws in the graphs at the left are those predicted by mean-field theories, which ignore all fluctu nus the density of the vapor phase. Just ations. The exponents in the graphs at the right are derived from an exact solution of the two dimensional Ising model reported in 1944 by Lars Onsager of Yale University. The exponents as the magnetization vanishes at the Cu show how the properties of the system change as the temperature or the coupling strength is rie temperature, so the density differ changed; that is the same information conveyed by the slope of the surface in parameter space ence falls to zero at the critical point of near the critical fixed point. The exponents can be determined from the slope, and calculations the fluid. These quantities-the magneti- by the author and others for the two-dim ensional Ising model give values close to Onsager!s.

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© 1979 SCIENTIFIC AMERICAN, INC sality. According to the hypothesis, only a vector that can point anywhere in that can be expressed as a single num two quantities determine the critical be three-dimensional space has three com ber. Alloys such as brass have a transi havior of most systems: the dimension ponents, so that n eq uals 3. tion between an ordered phase, where ality of space and the dimensionality of For the three-dimensional Ising mod the two metals occupy alternate sites in the order parameter. These quantities el d equals 3 and n equals 1. Ordinary a regular lattice, and a disordered phase, are labeled respectively d and n. All sys fluids belong to the same universality where their dist ribution is less uniform. tems that have the same values of d and class. The space in which the fluid exists The order parameter in this system is n are thought to have the same surface clearly has three dimensions. The order again a concentration difference, so that in parameter space and the same critical parameter-the difference in density be n equals 1. All these systems are expect exponents. They are said to be mem tween the liquid and the vapor phases ed to have the same critical exponents bers of the same universality class. is a quantity that has only a magnitude as the three-dimensional Ising model. The dimensionality of space is seldom and hence only one component; it can So are some real fer romagnets, those difficult to determine, but the dimen be expressed as a single number, just as that are easily magnetized only along a sionality of the order parameter re the value of an Ising spin can be. single axis. The available experimental quires more careful consideration. In Several other physical systems are evidence confirms these predictions. magnetic systems, where the order pa members of this class. A mixture of two The universality hypothesis would be rameter is the magnetization, n is the liquids such as oil and water exhibits trivial if the critical exponents had the number of components needed to define critical behavior near the temperature values of integers or simple fractions the spin vector. The vector of an Ising where the component fluids become such as 11 2. Many physical laws share spin can be oriented only along a single completely miscible in each other, a such exponents, and there is no compel axis, and so it has only one component; temperature called the consolute point. ling reason for postulating a connection for the Ising model n is equal to 1. A At temperatures below the consolute between them. Gravitation and electro spin vector that is allowed to point any point the mixture separates into two magnetism both have an inverse-square where in a plane has two components, phases, and the order parameter is de law (an exponent of -2), but that coin which are customarily drawn along the fined as the concentration difference be cidence does not demonstrate that the two axes that definea plane. Similarly, tween the two phases, another quantity two forces are identical. The correspon dence of exponents does seem remark able, however, when the values are not round numbers but fractions such as .63. UNIVERSALITY CLASS THEORETICAL MODEL PHYSICAL SYSTEM ORDER PARAMETER The convergence of many systems on

Ising model in these values cannot be coincidental. It is d=2 n=1 Adsorbed films Surface density two dimensions evidence that all the details of physical , structure distingu ishing a fluid from a XY model in two Amplitude of n=2 Helium-4 films dimensions superfluid phase magnet are less important than the geo metric plOperties expressed by the val Heisenberg model n=3 Magnetization ues of d and n. in two dimensions The two-dimensional Ising model

d>2 n=oo "Spherical" model None (d = 2, n = 1) typifies a class of systems that are confinedto two-dimensional Conformation of long- Density of d=3 n=O Self·avoiding space. One example is a thin filmof liq random walk chain polymers chain ends uid; another is a gas adsorbed on a solid

Ising model in surface. An ordinary ferromagnet falls n = 1 Uniaxial ferromagnet Magnetization three dimensions into the class with d = 3 and n = 3, that is, the lattice is three-dimensional and Fluid near a critical Density difference each spin has three components, so that point between phases it can point in any direction. When the Mixture of liquids Concentration spins are constrained to lie in a plane, near consolute point difference the class is reduced to d = 3 and n = 2. Alloy near order- Concentration In this same class are the superfluid disorder transition difference transition of liquid helium 4 and the superconducting transitions of various XY model in n = 2 Planar ferromagnet Magnetization metals. three dimensions

Helium 4 near super· Amplitude of ther universality classes have val fluid transition superfluid phase O ues of d and n whose interpretation is somewhat less obvious. The case of Heisenberg model = Isotropic ferromagnet Magnetization � 3 in three dim ensions d = 4 is of interest in the physics of ele mentary particles, where one of the four d,,;;4 n =- 2 None spatial dimensions corresponds to the

Quantum chromo· Quarks bound in axis of time. In a theoretical lattice of n = 2 3 dynamics protons, neutrons, etc. spins callee! the spherical model, where an individual spin can have any magni UNIVERSALITY HYPOTHESIS states that diverse physical systems behave identically near tude and only the total of all the spins is their critical points. In most cases the only factors that determine the critical properties are the constrained, n is effectively infinite. A dimensionality of space, d, and the dimensionality of the order parameter, 11. For magnetic sys self·avoiding random walk through a tems the order parameter is the magnetization, and its dimensionality is the number of compo lattice of points, or in other words a ran nents needed to describe the spin vector. Most systems with the same values of d and 11 are dom walk that never occupies the same members of the same universality class and share the same critical expon ents. For example, lattice site more than once, describes the ferromagnets that resemble the three-dimensional Ising model, fluids, mixtures of liquids and folding up in space of a long-chain poly certain alloys are all members of the class with d = 3 and 11 1;= graphs of their properties near mer; Pierre Gilles de Gennes of the Col a critical point should all have the same form. The interpretation of some values of d and 11 is less obvious, and values such as 11 = -2 can be defined mathematically but correspo nft to no lege de France has shown that this prob known physical system. The XY model and the Heisenberg model are similar to the Ising mod lem belongs to a universality class with n el but describe ferromagnets whose spin vectors have two and three components respectively. equal to zero, In theoretical models n

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© 1979 SCIENTIFIC AMERICAN, INC CONTOURS OF (EXPONENT ASSOCIATED WITH MAGNETIZATION) can even take on the value -2, although {3 "SPHERICAL" MODEL - V2 00 the physical meaning of a negative num : • n = • • ber of vector components is not clear. • • The only values of d and /lthat have • • n=7 a straightforward physical meaning are integer values. This is particularly clear n=6 in the case of d, since a space with a non integer number of di mensions is hard II: even to imagint!. In renormalization W n=5 group calculations, however, d and /I Iii appear in equations where they can be allowed to vary continuously over some � n=4 1£ range. It is even possible to draw a graph II: W in which the values of critical exponents C II: are plotted as continuous functions of d HEISENBERG e n=3 w and /l . The exponents have well-defined MODELS values not only for integer dime nsions j!: u but also for all fractional dimensions be e MODELS tween the integers. Such a graph shows XY ---- n=2 � that the exponents approach the values. �z given by mean-field theory as the num o ISING MODELS -- n = 1 iii ber of spatial dimensions approaches 4. z w When d is equal to exactly 4, and at all � higher values of d, the mean-field values o are exact. This observation has given n=O rise to an important method of perform NONPHYSICAL REGION OF ing renormalization-group calculations. NEGATIVE • The dimensionality of space is ex EXPONENTS n =-1 pressed as being equal to 4 - E, where E (the Greek letter epsilon) is a number that is assumed to be small. The critical 6--'--'--'-...._ ___I __� ---I-..L.__ --- .-.-.... n = -2 2 d=3 exponents can then be calculated as the d=l d = d=4 d>4 DIMENSIONALITY OF SPACE sum of an infinite series of terms includ ing progressively higher powers of E. If E is less than I, a high power of E will have CONTOURS OF y (EXPONENT ASSOCIATED WITH SUSCEPTIBILITY) a small value, and reasonable accuracy - """ERICAL" M OO" n=� can be obtained by neglecting all but the first few terms in the infinite series. This calculation method, which is j: n=7 called the epsilon expansion, was devel oped by Michael E. Fisher of Cornell f fj n=6 University and me. It is a general meth j 'I. od for solving all the problems to which II: mean-field theory can be applied, and it n=5 W I represents the natural successor to Lan W dau's theory. Indced, it supplies answers �II: in the form of corrections to the values n=4 1£ .given by mean-field theory. The block II: W spin method is the more transparent C technique, but the epsilon expansion is HEISENBERG n=3 II: MODELS e the more powerful one. w J: It is not entirely surprising that the I u critical exponents should converge on XY MODELS ---- +._. n = 2 o the mean-field val ues as the number of � spatial dimensions increases. The fun �z damental assumption of mean-field the ISING MODELS --....-:�..... - n = 1 e ories is that the force at each lattice site iii z is influenced by conditions at many oth w � er sites. The number of nearest-neighbor n=O o sites increases along with the number of spatial dimensions. In a one-dimension

al lattice each site has just two nearest n =-1 neighbors, in a two-dimensional lattice four, in a three-dimensional lattice six and in a four-dimensional lattice eight. �----<_---_----.,-...... -...... - ...; n = -2 Hence as the dimensionality increases, d = 1 d=2 d=3 d=4 d>4 the physical situation begins to resemble DIMENSIONALITY OF SPACE more closely the underlying hypothesis of mean-field theory. It remains. a mys V ARIA TlON OF CRITICAL EXPONENTS with the dime nsionality of space (d) and of the order parameter (II) suggests that physical systems in different universality classes should have tery, however, why d = 4 should mark a different critical properties. The exponents can be calculated as continuous functions of d sharp boundary above which the mean and 11,but only systems with an integral number of dimensions are physically possible. In a field exponents are exact. space with four or more dimensions all the critical exponents take on the values predicted by In this article I have discussed main- mean-field theories. The graphs were prepare d by Michael E. Fisher of Cornell University.

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© 1979 SCIENTIFIC AMERICAN, INC Iy the applications of the renormaliza tions between electrically charged par conflict with the measured charge. The tion group to critical phenomena. The ticles and the electromagnetic field. The renormalized the'ory of electrodynam technique is not confined to those prob difficulty encountered in the formula ics does not abolish the infinity; on the lems, however, and indeed it did not be tion of the theory can be understood as contrary, the electron is defined as a gin with them. one of mUltiple scales of length. For point particle whose "bare" charge is in The procedure called renormalization some time it had been apparent that finite. In quantum electrodynamics the was invented in the 1940's as part of the charge of the electron predicted by bare charge has the effect, however, of the development of quantum electrody quantum-mechanical theories was infi inducing a charge of opposite polarity namics, the modern theory of interac- nite, a prediction that was in serious in the surrounding vacuum, which can cels most of the infin'ity, leaving only the small net charge that is observed in ordinary experiments. One can imagine a probe particle that + could measure the electron's charge at UPSPIN arbitrarily close range. At long range I it would findthe familiar finite value, which is the difference between the bare + I I I charge and the induced charge. As DOWN SPIN \ the layers of shielding were penetrated I I the measured charge would increase, and as the range was reduced to zero UNIAXIAL FERROMAGNET the charge would become infinite. The renormalization procedure provides a means for subtracting the infinite shield ing charge from the infinite bare charge so that a finite difference results. In the 1950's it was pointed out by PRESENCE several workers, among them Murray OFAN ATOM + Gell-Mann and Francis E. Low, that the renormalization procedure adopted for + quantum electrodynamics is not unique. ABSENCE They proposed a more general formula OF AN ATOM tion, which is the original version of the renormalization group. In their applica tion of the method to quantum electro FLUID NEAR CRITICAL POINT dynamics a mathematical expression is constructed that gives the magnitude of the charge at some definite distance from the electron. Then the form of the expression is examined as the distance at + which the measurement is made is al ZINC ATOM lowed to approach its limiting value of zero. The arbitrariness of the procedure is in the choice of the initial distance. + Any value can be selected witho ut COPPER ATOM changing the ultimate resul ts, so that there is an infinite set of eq uivalent re normalization procedures. A "group" in mathematics is a set of. BRASS NEAR ORDER-DISORDER TRANSITION transformations that meets a special re quirement: the product of any two trans formations must also be a member of the set. For example, rotations are trans + formations that make up a group, since PRESENCE OF A PARTICLE the product of any two rotations is also a OR AN ANTIPARTICLE rotation. What this means in the case of the renormalization group is that the + procedure can be iterated indefinitely, ABSENCE OF A PARTICLE since applying the procedure twice is OR AN ANTIPARTICLE equivalent to applying the product of two transformations. Actually the re normalization group is properly called a QUANTUM FIELD THEORY semigroup because the inverse of the transformation is not defined. The rea LATTICE SYSTEM can be interpreted as a model not only of a fe;romagnet but also of other son for this can be seen plainly in the physical systems that have fluctuations on many scales. The Ising model describes a nniaxial block-spin technique applied to the two ferromagnet, one with a preferred axis of magnetization. It can also be applied to a fluid near dimensional Ising model. A block of its critical point, where each lattice site either is occupied by an atom or is vacant, so that the nine spins can be condensed into a single fluctuations become variations in density. An alloy such as brass has a similar structure, where average spin, but the original spin con each site is occupied by one kind of metal or the other. In all these systems the fluctuations are figuration cannot be recovered from the thermal; in the quantnm field theories that describe the interactions of eleme ntary particles there are quantum fluctnations of the vacuum, which allow particles and antiparticles to ap average because essential information pear spontaneously. A simple quantnm field theory can be formulated on a lattice by specify has been lost. ing that the particles and antiparticles can be created and annihilated only at the lattice sites. The version of the renormalization

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© 1979 SCIENTIFIC AMERICAN, INC group outlined in this article differs in the coupling is then proportional to the in solid-state physics called the Kondo several respects from the one intro number of lines per unit area that cross effect, after the Japanese physicist Jun duced by Gell-Mann and Low. The ear any surface between the particles. In the Kondo. The effect is observed in non lier version of the techn ique is useful case of electrons when the particles are magnetic metals, such as copper, when only for understanding problems that separated, the lines of force spread out they are contaminated with a small con can be solved by one of the traditional in space, so that there are fewer lines per centration of magnetic atoms. The sim methods of physics: by finding some ap unit area. The density of lines declines plest theories predict that the electrical proximate expression for the behavior as the square of the separation, which resistance of such a metal will fall con of a system and then calculating better yields the familiar inverse-square law tinuously as the temperature is reduced. approximations as a series of pertur for the electromagnetic force. With Actually the resistance reaches a mini bations departing from the original ex quarks, on the other hand, the prevailing mum value at a finite temperature and pression. Moreover, in the original for hypothesis holds that the lines of force then rises again as the temperature is mulation only one quantity is allowed do not spread out in space; they remain reduced further. The anomaly was nev to vary; in the example given above it is confined to a thin tube, or string, that er one of pressing importance because the charge of the electron. As a conse directly links the quarks. As a result the an explanation of it does not illuminate quence the surface in parameter space number of lines per unit area remains more general properties of solids, but it is not a multidimensional landscape but constant no matter what the distance is, tantalized physicists for more than 40 a mere line. The modern version of the and the quarks cannot be separated. Al years, always seeming just beyond the renormalization group, which was in though this account of quark confine reach of the available methods. The root trod uced by me in 1971, gives access to ment has an intuitive appeal, it is a qual of the difficulty is that the cond uction a much broader spectrum of physical itative explanation only. No one has yet electrons in the metal can have any ener problems. What is equally important, it been able to derive the confinement of gy over a range of a few volts, but per gives a physical meaning to the renor quarks from the underlying theory of turbations in that energy are significant malization procedure, which otherwise quantum chromodynamics. down to a level of about 10-4 volt. The seems purely formal. The confinement problem is one with problem was ultimately solved in 1974, many scales of length and energy and when I completed a renormalization n the past few years I have been at hence is a candidate for renormaliza group calculation of the electron ener I tempting to apply the newer version tion-group methods. I have formulat gies at all temperatures down to abso of the renormalization group to a prob ed a version of the problem in which lute zero. lem in the physics of elementary parti the quarks occupy the sites of a lattice A more recent series of renormaliza cles. The problem is how to describe the in four-dimensional space-time and in tion-group calculations is notable in that interactions of quarks, the hypothetical which they are connected by "strings" it makes predictions that have been di elementary particles thought to com that follow the lines connecting sites. rectly confirmed by experiment. The pose protons, neutrons and a multitude The lattice is a strictly artificialstruc calculations concern the lattice-spin of related particles. In one sense the ture with no analogue in real space-time, model in which d equals 2 and IIequ als problem is much like the original re and it must ultimately disappear from 2, or in other words concern a two normalization problem of quantum the theory. That can be accomp lished dimensional lattice of two-component electrodynamics; in another sense it is by allowing the lattice spacing to ap spins. It has been proved that no phase just the opposite. proach zero. with long-range order is possible in this In quantum electrodynamics the As in the study of ferromagnetic sys spin system, but renormalization-group charge of the electron is found to in tems, a renormalization-group transfor studies done by J. M. Kosterlitz of the crease as the electron is approached mation is applied repeatedly to the lat University of Birmingham and David more closely. For interactions of quarks tice of quarks and strings. In this way the J. Thouless of Yale University have the property analogous to electric interaction of the quarks can be exam shown that the behavior of the system charge is called color, and for that rea ined at progressively larger separations. does change abruptly at a critical tem son the theory of quark interactions has The question to be answered is whether perature. These findings have been ap been named quantum chromodynamics. the lines of force remain confined to plied to studies of thin films of super When the color charge of a quark is tubelike bundles or spread out in the lat fluid helium 4, which also fall into the measured at close range, it seems to di tice as the scale of length is increased. universality class of d = 2 and II = 2. In minish as the distance becomes smaller. The calculations are near the limit of particular Kosterlitz and David R. Nel As a result two quarks that are very practicality for the present generation of son of Harvard University have predict close together interact hardly at all: the digital computers. As yet I do not have ed a discontinuous jump in the density coupling between them is weak. On the the answers. of the superfluid fraction of the film. other hand, when the quarks are pulled Such a ju mp has since been ob served apart, the effective color charge in here are many other problems seem experimentally by John D. Reppy of creases and they become tightly bound. Tingly suitable for renormalization Cornell and others and has been found Whereas an electron induces a compen group methods but that have not yet to have the predicted magnitude. sating charge in the surrounding space, a been expressed in such a way that they For all the work that has been invest quark seems to induce a color charge can be solved. The percolation of a fluid ed in the renormalization group it may w.ith the same polarity, which augments through a solid matrix, such as water seem the results obtained so far are rath its own charge at long range. Indeed, it is migrating through the soil or coffee er scanty. It should be kept in mind that a widely accepted hypothesis that the through ground coffee beans, involves the problems to which the method is be effective coupling between quarks in aggregations of fluid on many scales. ing applied are among the hardest prob creases without limit when the distance Turbulence in fluids represents a prob lems known in the physical sciences. If between them exceeds the diameter of a lem of notorious difficulty that has re they were not, they would have been proton, which is about 10 -13 centimeter. sisted more than a century of effort to solved by easier methods long ago. In If that is true, a quark could be torn describe it mathematically. It is charac deed, a substantial number of the un loose from a proton only by expending terized by patterns with many character solved problems in physics trace their an infinite quantity of energy. The istic sizes. In the atmosphere, for exam difficulty to a multiplicity of scales. The quarks would be permanently confined. ple, turbulent flows range in scale from most promising path to their sol ution, One way of visualizing the binding of small "dust devils" to hurricanes. even if it is an ard uous path, is the quarks is to construct imaginary lines One problem that has yielded to the further refinement of renormalization of force between them. The strength of renormalization group is a phenomenon group methods.

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