31'74 R. A. WE INK R further enhanced, but finite, conserving suscepti- sential before such a calculation can be contem- bility at T=O. plated. Without doing the calculation, all we may do is ACKNOWLEDGMENTS speculate what the results will be. However, our experience with the zero-width calculation indicates We would like to thank Professor M. J. Levine, that the computational problem involved is stupen- Dr. T. V. Ramakrishnan, and Professor H. Suhl dous. Some approximation, such as a I.orentzian for many helpful conversations during the course shape to the low-frequency modes, 4' wiQ be es- of t is research. H. Suhl, Phys. Bev. Letters 19, 442 (1967); 19, 735 generating all diagrams for the vertex Infieldfunction, since (E) (1967). the totally irreducible vertex I must still be obtained 2P. W. Anderson, Phys. Bev. 124, 41 (1961). by a perturbative expansion. 3P. A. Wolff, Phys. Bev. 124, 1030 (1961). A. A, Abrikosov, L. P. Gor'kov, and I.-E. 4M. J. Levine and H. Sihl, Phys. Bev. 171, 567 Dzyaloshinski, in Methods of Quantum Theory in (1968); M. J. Levine, T. V. Bamakrishnan, and R. A. Statistical Physics, edited by B. Silverman (Prentice- Weiner, Phys. Bev. Letters 20, 1370 (1968). Hall, Englewood Cliffs, N. J., 1963). ~D. B. Hamann, Phys. Rev. 186, 549 (1969). '3The Bethe-Salpeter equation for y~~ does not generate 6We shall use the notation of the Anderson model in a narrow low-frequency mode. this paper; the results obtained are identical with those '4The solution to the Suhl vertex equation is roughly based on the Wolff model. proportional to 6~, p at high temperatures (see M. J. M. T. Baal-Monod and D. L. Mills, Phys. Rev. Let- Levine and H. Suhl, Ref. 4). ters 24, 225 (1970). ~This shifts the effective location of the d-state reso- A. A. Abrikosov, Physics 2, 5 (1965); B. Roulet, nance to & =0 (see Bef. 5). J. Gavoret, and P. Nozieres, Phys. Rev. 178, 1072 ~ A similar calculation of the susceptibility in a local- (1969). moment model has been reported by J. A. Appelbaum and 9R. A. Weiner, Phys. Rev. Letters 24, 1071 (1970). D. B. Penn, Phys. Rev. 8 3, 942 (1971). ~06. Baym and L. P. Kadanoff, Phys. Bev. 124, 287 J. B. Schrieffer and P. A. Wolff, Phys. Bev. 149, (1961). 491 (1966). ~~The parquet diagrams are only a formal method of PHYSICAL REVIEW B VOLUME 4, NUMBER 9 1 NOVK MBER 1971 Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture* Kenneth G. 'Wilson Laboratory of Nuclear Studies, Cornell University, Ithaca, Net York 14850 (Received 2 June 1971) The Kadanoff theory of scaling near the critical point fox an Ising ferromagnet is cast in differential form. The resulting differential equations are an example of the differential equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws arise naturally from these differential equations if the coefficients in the equations are ana- lytic at the critical point. A generalization of the. Kadanoff scaling picture involving an "ir- relevant" variable is considered; in this case the scaling laws result from the renormaliza- tion-group equations only if the solution of the equations goes asymptotically to a fixed point. The problem of critical behavior in ferromagnets T is the temperature, and k is Boltzmann's con- (and other systems) has. long been a puzzle. ' Con- stant. The partition function is a sum of exponen- sider the Ising model of a ferromagnet; the parti- tials each of which is analytic in K and k. There- tion function is fore one would expect the partition function itself to be analytic in K and h. In fact, however, the zoic a)=E exp zZEs, s,.;+esp), partition function is singular for K=0, and h=O, (s) where K, is the critical value of K. To be precise, where E= —J/kT, J isa coupling constant, sl is the the singularity occurs only in the infinite-volume limit, in which case one calculates the free-energy spin at lattice site n, P& is a sum over nearest- neighbor sites, and h is a magnetic field variable, density The is restricted to be + means a 1 spin s; 1; 5, ~, & J'(Ã, k) =lim —lnZ(K, k), (2) sum over all possible configurations of the spins. y V RENOBMALI ZATION GROUP AND CRITICAL PHENOMENA. I. 3175 where V is the volume of the system. is true in particular if there are frictional forces Because of the infinite-volume limit there is no which prevent the ball from rolling back and forth formal contradiction in the result that E(K, h) is in the valleys near x„and xp of Fig. 1). Let xc be singular at %=K,. The problem is that the methods the location of the maximum of V(x), i.e. , the top one has for calculating sums such as (1) do not of the hill. If the ball is released at any point easily lead to singular behavior, so it is extremely x & xc (on the left of xc) then the ball rolls down to difficult to get a good understanding of critical the point x& and stops. If it is released to the singularities working directly with the sum of Eq. right of xt.- it rolls to x~ and stops. This means (1). What one would like to do is to transform the the final position of the ball is a discontinuous problem of calculating E(Ã, h) into a form where it function of its initial location. To be precise, let is natural for J' to have singularities at K=K„and the position of the ball at time t be x(t, xp) where where one may hope the nature of the singularity xp is the initial location (at time 0). Then the will be more easily seen than from Eq. (1). function x(~, xp) is a discontinuous function of xp. In this paper it will be suggested that an appro- There is nothing mysterious about this disconti- priate reformulation is in terms of a group, namely, nuity, it is just that a small change in the initial the renormalization group. It has already been condition, from xo slightly less than x~ to xo slight- suggested that the renormalization group is impor- ly greater than x~, can be amplified by the pas- tant for understanding critical phenomena. The sage of time until for very large t the difference in function of this paper is to explain what the re- position is the difference in. xa - x&. With an infi- normalization group is and what the assumptions nite length of time available one can get an infinite are that make it useful. amount of amplification, thus leading to a dis- The renormalization group is a nonlinear trans- continuity in x(~, xp) as a function of xp whereas formation group of the kind that occurs in classical x(t, xp) for finite f is continuous. It is assumed mechanics. The equations of motion of a classical here that the potential V(x) is analytic in x, as in- system with time-independent potentials define dicated by Fig. 1, 'so the discontinuity in x(~, xp) at transformations on phase space which form a xp= xc cannot be blamed on any singularity in V(x) group. The finite transformations of the group are itself. the transformations induced by a finite translation The basic proposal of this paper is that the sin- in time; the infinitesimal transformation is de- gularities at the critical point of a ferromagnet can fined by the equations of motion themselves. It be understood as arising from the t=~ limit of the wi1.1 be shown how a translation group can arise solution of a differential equation. In order to de- in the analysis of critical behavior. This group is velop an understanding of how one relates critical called the renormalization group for historical behavior to a differential equation, we shall set up reasons (the connection with renormalization will Kadanoff's scaling picture in differential form. be explained at the conclusion of paper II4). The Kadanoff 's original hypothesis which led to the infinitesimal transformation of the renormalization Widom-Kadanoff scaling laws was that near the group is analogous to an equation of motion, and critical point one could imagine blocks of spins we shall use the language of differential equations acting as a unit, i.e. , all spins in a block would rather than the language of group theory in the re- be up or down simultaneously. Kadanoff then mainder of this paper. argued that one could treat all spins in a block as a The advantage of a reformulation of Eq. (1) in single effective spin, agd one could write an effec- terms of the differential equations of the renormali- tive Hamiltonian in the Ising form for these effec- zation group is that it allows the singularities of tive spins. He then showed how these assumptions the critical point to occur naturally. Before set- ting up these differential equations we shall show with a simple classical example how singularities can be generated from an equation of motion. Con- sider the equation dx —= ——(x) dx where V(x) is the function shown in Fig. 1. One can think of this equation as describing the motion l t of a ball rolling on a hill with height given by V(x). A xc xB Equation (3) is not strictly speaking the equation of motion for said ball, but qualitatively the solution of this equation is similar to the solution of the FIG.