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 , 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 Group and Critical Phenomena. I. 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 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 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. 1. Potential V(x) with minima at x& and x& and second-order equation one should write down (this a maximum at pc. 3176 KENNE TH G. %II SON lead to scaling laws. $(Kz„hz, ) is the correlation length of the block The idea that blocks of spins act as a unit near Hamiltonian, in units of the block spacing. For the the critical temperature does not stand up to close two to agree, one must have examination; in fact only very near zero tempera- = ture (K»Kc) is it true. ' The reason for discussing $ (K, Iz) L )(Kz„ lzz, ). (5) the Kadanoff hypothesis is that it leads to a very The Kadanoff picture is, in summary, that there simple set of dnferential equations. By studying exists effective coupling parameters KI, and h~ such these differential equations, one builds an under- that Eqs. (4) and (5) hold, for any L. Kadanoff al- standing of how critical singularities can emerge so requires that correlation functions for large from a set of equations of motion. In Paper II we distances be calculable through the block Hamil- will discuss a generalization of the differential tonian, but this will not be assumed here. equations which can be more realistic than the Kadanoff's picture requires that L be an integer, Kadanoff block hypothesis. The intuitive picture but we shall assume that L can be a continuous of how the critical singularities arise from the variable, in order to be able to write differential differential equations will not be changed by the equations in L. Kadanoff restricts L to be much generalization, although a wider range of critical less than g(K, Iz); we shall allow L to be arbitrary. singularities is possible with the generalized The differential equations of the renormalization equations. group will be, in the Kadanoff picture, equations In short the Kadanoff block picture, although for EI, and h~. So far nothing has been said about absurd, will be the basis for generalizations which how to compute KI, and h~. Kadanoff proposed are not absurd, and it is helpful to understand the definite forms for the dependence of KI. and h~ on Kadanoff picture in differential form before study- L, namely, ing these generalizations. Imagine an infinite cubic lattice divided into cubic Kz =Kc &L blocks L lattice sites on a side. Each block con- h, =aL', tains L lattice sites. The total spin in a block is the sum of the L spins in the block. According to where the Kadanoff block picture, all the spins in the &=K, —K block are aligned together, so the total spin has (6) only two values, +L or -L3. The blocks of spins and Eqs. (6) and (V) are valid only for L «$( ,K)I.z themselves define a lattice of spins, but the lattice Here we shall first derive differential equations for spacing of the blocks is L times the spacing of in- KI, and h~ and show later that the solution of the dividual spins. Introduce a spin variable s'- differential equations has Kadanoff 's form. associated with block m. Normalize s'» so s@=+1, To obtain the general form of the differential s„'- i. e. , is L x(total spin in blockm). The equations for KI, and h&, we note the following. The interactions between blocks involve only nearest- constants K» and h» are functions of K~ and AI. neighbor blocks, and the magnetic field couples to but not of L separately. The change from KI. and each block separately. This suggests that the in- hr, to K» and h» is equivalent to making new teraction energy of the blocks can be expressed in blocks of size 2L out of old blocks of size L. Each Ising form [Eq. {1)]except that one must substitute new block is a cube containing eight old blocks. new constants K~ and h~ for the original parameters But in writing an effective Hamiltonian with con- Ka d I. stants K~ and 0&, one has substituted a lattice for Kadanoff proposes in particular that the total free the old blocks; having made this substitution the energy of the original Ising model is the same as Hamiltonian does not know what the size L of the the free energy of the blocks calculated using the old blocks was. Regardless of the value of L, the block parameters. In practice this equivalence is change to 2L is simply a matter of combining eight expressed in terms of the free-energy density lattice sites to make the new block, so K» and rather than the total free energy. Let F(K, h) be h» must be the same function of K& and h& for any L. the free energy per lattice site of the original Ising This continues to be true if one goes from L to Hamiltonian of Eq. {1). The free energy per block SL, L to 4L, etc. Generalizing to the continuous of the block Hamiltonian is simply F (Kz, , hz, ). If case, we assume this is true also for going from the total free energy is the same for both, then L to (1+5)L, for small 5. This means 6LdK~/dL can depend on K~ and h~ but not L separately: F(K& lz) = L F(Kz& hz) . (4) In the Kadanoff picture one can also compute the correlation length using the block Hamiltonian. Let $(K, fz) be the correlation length for the original Vfe expect u to depend only on hr, owing to the sym- Hamiltonian in units of the lattice spacing. Then metry of the Ising Hamiltonian for hr, --h~. The CRITICAL PHENOME NA. I. . .

analogous equation for h& is back to the analogy of a ball on the hill, the func- tion V(x) can be analytic 111 x and still have x{~,xo) -~ = I.-'a, ') be discontinuous ln Analogously we wouM like dJ v(K„ I, . (10) xp u(K, h ) and v(K, JP) to be analytic at the critical Equations (9) and (10) are the renormalization- point. The trouble with Eq. (13) is that it express- group equations suggested by the Kadanoff block es u(K, h2) in terms of F(K, h) and $ (K, h), both of picture. Because of the questionable validity of which are singular at the critical point, implying this picture one would expect the differential equa- that u is singular also. Most previous formula- tions to be equally questionable. Actually this is tions of the renormalization group used equations not so; there is another way to derive the differen- such as Eq. (13) to define the functions u and v tial equations which involves only minimal assump- which appear in the renormalization-group equa- tions, such that the differential equations become tions. This has been the cause of much confusion essentially a, tautology. Namely, let us define K~ about the purpose of the renormalization group. and h~ to be the solutions of Eqs. (4) and (5). That The Kadanoff block hypothesis suggests that is, we assume that one has found the exact solution u(K, h ) and v(K, h ) will indeed be analytic at the of the Ising model, as a function of K and h, and critical point. In the block picture one should be regard Eqs. (4) and (5) as giving implicit definitions able to construct E~ and hr just by adding up inter- of EI, and hr, for any I. Assume that these equa- actions of individual spins within a block or across tions have a unique solution for any I. The dif- the boundary between two blocks; it is hard to see ferential equations are now trivial to obtain: Dif- how this simple addition over a finite region can ferentiate both Eqs. (4) and (5) with respect to L, . lead to singular expressions for K~ and hL, as a One gets function of K and h, if I. is fixed. More generally this process should give K„I.andh„r. as analytic func- 3 dKI, 8F(KI„ki) ( ) tions of Kz, and hr for fixed n; specializing to n= 1 L. " ''dz, eZ, + 0, the functions u and v should be analytic. How- ever, in the spirit of the Kadanoff approach one dhr 8F(K~, h~) (11) does not try to get specific forms for u{K,h') and dI ~h v(K, h') because this would require that one take literally the idea that all spins within a block act 0=L Kr„hr der, 8 EJ. h~ dh~ 8 KI, h~ , as a unit. An explicit realization of the renorma- lization- group equations will be presented in Paper (12) II. ' These equations can be solved for dK~/dI. and Another feature of the renormalization-group de/dI, to give Eqs. (9) and (10), with equations suggested by the Kadanoff block picture is the following: In classical physics it is usually easier to write down the equation of motion, such ) ) '„' », ) .(»., a, =(s»(»„a,y Lehh (»,Lj a,E as Eq. (3), than to write down the solution of the equation. Typically the potential V(x) is a simple a, '„' (»„a, function, or easily approximated by a simple func- .((»., ), )) tion. The solution of the equation ean be much more complicated, if one has ajar especially coupled (Kz„hr„) „(K~,h~) differential equations to solve. The Kadanoff Mock hypothesis suggests that the renormalization- — &$ BF„(»„a,))' group differential equations will also be simpler „(»„a.), than their solution. This is because the smaller I. is, the fewer the interactions that have to be 9 and a similar formula for v(K~, g~~). Note that summed to give I|".I. and hL, . It is not immediately u(K~, h~) does not depend on I, (except through K~ obvious that I.= 1+5 is easier to compute than and &~), a,s expected. This result means that one I.= 2, but this is not the point. The point is that has an explicit formula for the function u(K, /P) in I.= 2, 3, or 4 is much easier to compute than terms of F(K, h) and ((K, h) and their derivatives. 1.=10000; one expects I =1+0 also to be easier The differential equations thus obtained may not than I = 10000. be very interesting. The reason is this. The idea If it is true that the renormalization-group equa- of converting the Ising problem into a group was tloQ ls easier to write down than its solution then that the singularities at the critical point should it is natural to try to solve the Ising model by first result from solving the group equations. This obtaining the renormalization-group equations and makes sense only if the differential equations are then trying. to integrate them. 8peclallzlng to the themselves free of singularities at K,. Referring problem of critical behavior, what one thinks of do- 31V8 KENNETH G. WIL SON ing is integrating the equations until L is of order predicted from the Kadanoff block picture. Using When L is of order $, $(KI,, hz„) is of order 1 the linearized equations of the renormalization group [by Eq. (5)], which means K~ and h~ must be well is analogous to replacing V(x) by a quadratic form away from their critical values; then it is easy to when x is near x~. compute $(K~, hz, ) and F(K~, hz ) by other means and The solutions of Eqs. (15) and (16) are the form- reconstruct F(K, li) and $(K, k) from Eqs. (4) and ulas (6) and (V) proposed by Kadanoff. Assume (5). This integration procedure can be carried out that this approximation is valid until K~ = K,/2. for K and h near their critical values, and if N(K, h) Then one can solve for the value of L giving KL and v(K, h) are analytic at the critical point, one = K, /2: obtains immediately the Widom-Kadanoff scaling laws. K, /2=&L' (i9) To make the calculation precise, let us proceed so as follows. Imagine that one is at a temperature L = (K,/2e)'~', (2o) slightly above T,. This means K is slightly smaller than K,. As L increases KL must decrease, so as which is a scaling law for L as a function of c. to go away from K,. This ensures that $(K~, h~) For this value of L, AL is decreases as I. increases, as required by Eq. (5). h, = h(K,/2~)"" . Pick a value for K~ well away from K„say K,/2; let us integrate the renormalization-group equations Hence one can compute F(K, 8) and $(K, h) to be until a value of L is reached for which K~ = K,/2, F(K, a) = (K, /2e) '"F[K,/2, @(K,/2~)""], (22) then stop and compute F(K, 0) and f (K, I) from Eqs. (4) ~d (5). $(K, h) = (K, /2&)' ' &[K,/2, &(K,/2&) ~'l . (23) If K and h have exactly the critical values K, and These formulas are scaling laws; the functions 0, respectively, then one must have KL = K, and F(K, h) and $(K, h) depending on two variables have h~ =0 for all I,. The reason for this is that $(K„O) been reduced to explicit powers of e (i.e., 7 —T,) is infinite; therefore &(K~, h~) must be infinite for multiplying functions depending only on the single all L. For $(K~, h~) to be infinite, K~ and h~ must variable h& " '. For consequences of these laws have the critical values. Hence =-K, and =0 KL hL see Ref. l. must be a solution of the renormalization group To be accurate we should have integrated the equations, which is true only if nonlinear equations once KL —K, became large. u(K„O) = 0 . (i4) What can one say about the solution KL and AL in this case'P One can say the following: Let Comparing this result to the classical analog of the ball on the hill, the point K= K, is analogous K =Q(L, h), (24) to one of the points of equilibrium for the ball =P(L, (25) (x=x„, or xs, or xc). h~ Q) Now let K and h be near the critical values K, be the solution of the exact equations (9) and (10) and 0. For small values of L, namely for L over the range 0& L & ~ satisfying the boundary con- «$(K, h), the effective correlation length $(K~, hl, ) ditions will be large and KL and hL will also be near the critical values. For this range of L one can use Q(l, ho) =K,/2, (26) a linearized form of the renormalization group $(l, h~)=Q, (2V) equations to compute KL and AL. The linearized equations for KL and hL are that is, boundary conditions well away from the critical point. For L «1 this solution should be ' =—(K, -K.)y, (15) near the critical point. Hence for L «1, KL and hL should be solutions of the linearized equations, giving

Lx & (16) p(L, ho) = Q(Q)L'+K, , (26) where x and y are constants: g(L, ho) = p(ho)L" (29)

y= —, (K„o), for I, «1 where P(Q) and $(ho) are constants depend- ing on ho. Now the exact renormalization-group x=v(K,, O) . (is) equations have translational symmetry when written in terms of lnL, just as the classical equation (3) In writing these equations we have assumed that does. Hence the functions u and v are differentiable at the critical point; this is how one uses in practice the arialyticity K =P(aL, h ), RENORMALI ZATION GROUP AND CRITICAL PHENOMENA. I. . . 3179 (31) is analogous to finding the dependence of t»p on xo —x&. Most of the time t»& is spent near the are solutions of the renormalization-group equations top of the hill, and so the dependence of t, &2 on where a is any constant. For al. «1 this solution xo —x~ can be determined to a good approximation reduces to from linearized equations about the unstable equilib- X~ = $(ho)(aL)'+K, , (32) rium point x~. This completes the discussion of the simplest = (33) hI, $(ho)(aL)" . renormalization-group equations following from This solution can be matched to any given initial the Kadanoff block picture. One sees from this condition for K and h; if K and h are near their how the idea of differential equations with analytic critical values then K and h can be matched to the coefficients leads to singularities at the critical asymptotic forms (32) and (33), giving point satisfying the %idom-Kadanoff scaling laws. The critical point corresponds to an unstable equi- ~ = a'Q (II, (s4) ), librium point of the differential equation, and the h= a"P(Q) . (s5) singularity at the critical point is a consequence of the infinite time required to move away from a point This means that of unstable equilibrium. " h~ '=q(h, )/[y(h, )]' ' (35) The problem with the simple renormalization- group equations discussed earlier is that there is which is an implicit equation for depends only ho; ho at present no hope of showing that the functions on the single variable hc " '. Then, from Eq. (34), u(K, h2) and v(K, h ) are analytic at the critical we have point. Without the analyticity, the renormaliza- a = [~/y(a )]'" (3'7) tion-group equations become a tautology, as ex- plained earlier. Instead of trying to prove that But now the value of L for which KI, =K,/2 is L u and v are analytic, one can try to generalize the = a ', from the boundary condition on [Eq. (26)]; Q renormalization-group equations in the hope that and for this value of L, III, = ho (2V)]. Using [Eq. analyticity will be easier to establish for the gen- Eqs. (4) and (5) one again gets scaling laws for F eralization. The generalizations which the author and for example (, has been able to construct are rather complicated, $(K, h) = [&/$(ho)] '~' $(K,/2, ho), (38) involving an infinite number of L-dependent coupling constants. To prepare for these generalizations where IIO on he "~'. This scaling law depends only it is worth discussing the nature of the renormali- is the same as (23) except for a different functional zation-group equations whenthey involve one addi- dependence on the scaling variable hq " '. tional coupling constant q~. The initial variable These results can be related to the classical q will be an "irrelevant" variable in Kadanoff's lan- analogue of the ball on a hill. Consider only the guage. ' It might be the coefficient of a second- of no magnetic field (h= in which =0 case 0) case hI, nearest-neighbor coupling, for example. The vari- for all L and one has left single dependent varia- a able will prove to be irrelevant only in certain con- q ble K~. With h=0, ho is zero also; Q(0) is a respects that will be explained later. stant $0 and the formula for $(K, 0) is Imagine that the renormalization-group equations involve three L-dependent variables K~, ql, , and hl, . %e think of K~ and @ as coefficients of first- and second-nearest-neighbor couplings which are where $0 is $(2K„O) . s-'- —s-' The point K= K, is analogous to the top of the hill even to the exchange while h~ still multi- (x=xc). This is because if K is different from K, plies the spins s-' themselves which are odd to this exchange. Hence one the the then KL, —K, increases as L increases, and this is expects form of re- analogous to the ball rolling away from the top of normalization-group equations to be the hill. In other words the point K=K, is analo- dKI, gous to a point of unstable equilibrium in classical I s qI & ~I ) (40) mechanics. %hat is the analogue of the correla- tion length $(K, 0)? From Eq. (5), $(K, 0) is.pro- dq~ 1 IV(KI, q gI, ~ llI, ) i (41) portional to L where L is chosen to make K~ = K, /2. This is analogous to computing the time t»~ for dhl, the ball to roll half-way down the hill. As the initial kJ V(KJ p qgp kI ) (42) location xo of the ball approaches the top of the hill, this time t»2 increases, becoming infinite as xo The initial values of KI, qI, , and kI, (for L = 1) are —xc. The scaling law for $ is obtained by computing denoted K, q, and h. We assume the same rules the dependence of L on K- K, for K near K,. This for computing the free energy and the correlation KENNETH G. Vf IL 8GN

1 length as before, namely, ~'N y = —(K„q„0), (58) Bq P(E, q, I ) = I.-'P(E„q„a,), ~d t(E, q, a) =r, t (E„q„I,). x =v (K„q„o). We now expect there wiQ be a line of critical There will, in general, be two linearly independent points; for each value of q there should be a criti- solutions of the first two equations behaving as a cal value K, (q) for K. If the initial values of K, power of I, say' q, and h lie on the critical line (K= K,(q), h = 0) E~ -E, =L,', q~ -q, =x„I.' then the solution EI, , qL, ,h& must lie on the critical line for all I., in order that g (E~,q~, hz ) be infinite for aII I.. Hence, if E=K, (q) and A =0 one can z EL K L qL q r I. write K~ =Ko (@)and hl, =0; the equation for qI, becomes One of these solutions must be decreasing as I -~, since if the initial value of E is E, (q) the'n (45) X~ -E, and q~ - q, as L -, and this is not pos- sible if both y and z are positive. I.et z therefore be the negative exponent. The exponent y must be positive so that K~ goes away from K, if EW E, (q). m, (qz, ) =w(E, 0) . (q~), q~, %hat this means is that the point E„q, must be This is a one-dimensional equation of motion, analogous to a saddle point in classical mechanics which is directly analogous to the classical equa- which is stable from one direction but unstable in tion of xnotion for the baQ on a hill discussed eax'- the orthogonal direction. lier. In the limit L - ~ there is the possibility One can set up a classical analog to the coupled that ql, approaches an equilibrium point of the dif- equations fox El, and q&. Consider only the case ferential equation. This is not the only possibility; h& = 0, fox simplicity. A classical analog is the it is alsopossiblethat q~ -~ as L- ~. We shall study pair of equations only the ease that qJ. approaches as equilibrium point, this being the that is easiest to This case study. dt"'=-eK' q' means we wiQ not give a completely general dis- cussion of the solution of the renormalization- —(f) =- (E, q) (58) group equations. In fact a general discussion is dt &q very complicated and becomes hopeless when the equations are genexalized further. This is not as general as Eqs. (40) and (41), since Let the equilibrium point approached by q& be it is not true in general that two functions u and se can be the q„and let g (q, ) be g. If one chooses q =q, and written as gradient of a potential. But we can illustrate the that sad- E=E„ then qJ. -=q, for aQ I,, E-=E, for aQ I. so idea E„q, define a one has an equilibrium point of the full renormal- dle point with this example. Think of Eqs. (5V) and ization-group equations. (58) as describing the motion of a ball on a two- I.et us study the solutions of the renormalization- dimensional terrain, with V(K, q) being the eleva- group equations in linearized form about the equi- tion at the point (E, q). It is convenient to illus- librium point E„q,. The linearized equations are trate V(E, q) by a contour map, as in Pig. 2. Fig- ure 2 shows a depression at the point Pz (the dE~ origin); Ps is the top of a hill and Po a saddle L =su«~-Kc}+via(qi „~ -n}, point. There is a ridge going down the hill to the saddle and a guQy from the saddle to the bottom of &qI. ~ the depression. qo is an ax'bitrary value of q. If d~ =~ai«~ -K.)+~i~(qi-q. ) ~ one starts the baQ at coordinates qo and a small dA, value of E', it wiQ simply xoQ to the bottom of the ~— = I xhL, , depression. If one increases the initial value of E, the ball will still roll to I'&, until one starts at the point P& exactly on the ridge. In this case the eu y =eE(-' " (50) ball rolls to the saddle point I'z and stops. If one starts just short of P&, the ball roQs almost to eu I'~ and then goes down the gully to P&. The baG yi3= —(Ks& qadi o) eq moves very slowly when it is near P& and a large ceo time elapses before it moves an appreciable dis- 8E (K„q„o), tance down the gully. q~ = q, —& r„L' +qxgL', (60)

hl, =hI",

where z and q depend on the initial values K and q: r,(K K,) —(q —q,}

r„(K-K,)-(q-q, )

One is on the critical line if c =h =0 Rnd q 40, for then Kl. K„q& - q, when I - . One is near the critical line away from K, and q, if e «q and k is small. Then for I.- 1, the g term dominates, and since z is negative, KL, and q~ approach K, and q, (i.e., the ball is rolling down the ridge to the saddle point}. For large enough I. the e term doIQlDates RDd Kg RDd qg IQove RwRy fx'oIQ Kc RDd q, . The g term continues to decrease and can be PIG. 2. Potential VN, plotted by contours of con- q) neglected. Since g measures the initial location stant V. The minimum of V is at the origin (P&); the along the critical line, independence of q is maximum is at Pz and Pc, is a saddle point. The ridge line is the trajectory going to Pz,. the gully line is the independence of the initial location on the cx'itical trajectory from Pc to P~. Unmarked lines are contours. line. One can now compute the value of L, for The dashed line marks an arbitrary value (qo) for q. which K~ =K,/2, giving K, /2 = &I' (04) ol The ridge line is analogous to the critical line K= K, (q) and the saddle point P, is analogous to the point K„q,. Note that the initial location can be as before. For this value of L, anywhere along the ridge line, and stiQ the baQ will roll down to the saddle point Pz, or if one starts just short of the ridge line, the path of the and down the line, ball wiQ just miss P~ go guQy The important property of these formulas is that independently of where along the ridge line one q~ is a constant independent of h, e, or q, while started. It is in this sense that q is an irrelevant "~'. hI, again depends only on the ratio h& So To be there is one relevant and variable. fairex, from Egs. (43) and (44) one gets the same form of one irrelevant variable in the pair (K,q) but whether scaling laws for E(K, h, q) and $ (K, h, q) as was ob- the irrelevant variable does not K or q is caQed tained earlier when there was no irrelevant vari- matter. If one looks at the functions Kl, start- ,qr, able q. Since K=K, (q) for e =0, it follows from ing from an initial point close to the critical line, Eq. (62) that & is proportional to K- K,(q), which then L increases approaches the saddle as (K~,q~) is proportional to T —T,(q). Hence no matter what point (K„q,). By L = 5 or 10 one should be close to q is, e is the customary temperature variable = rate of is slow). K„q, (unless z 0 so the approach T -T, apart from a meaningless normalization Then KI, and ql, change very little until I becomes factol o at which of order of the correlation length g (K, q), So far nothing has been said about initial condi- point and move away from the saddle point. KI, ql, tions below T„ i. e. , K&K,(q). This is easily the lax L of the In the classical analog, ge part handled by the same analysis except that when curve is the line, and this is indepen- KJ. ,qJ. guQy Kl. moves away from K, it increases instead of the value of dent of the irrelevant variable. Also decreasing. Hence one cannot compute the value of I at which Kl, =K,/2 should be essentially indepen- I for which Kz, =K,/2. Instead one can find the dent of where along the ridge line one starts; how value of L, for which KL, =2Äsay. In the classical large I. is is determined by how far fxom the ridge anlaog, when K& K, (q) the ball rolls down the line oIle starts oppos1te side of the hlQ fx'oIQ the guIlyy RIll one Returning to the solution of the linearized equa- measures the time required to reach the line tions, the general solution is a linear combination K= 2K, instead of the time to reach K, /2. It must of the three simple power solutions, namely, be noted, however, that for K &K,(q) and h = 0 one K~ =K, —&I."+qI', is on the boundary between the two low-temperature 3182 KENNETH G. %II SON phases (the two phases corresponding to positive described, the other choice being to compute the or negative magnetization). As I. increases K~ partition function directly and extract the critical moves away from the critical value K,(q), but h~ exponents. However there is a difference in the is still zero so one is still on the boundary between two approaches in that the first choice (working the two phases. So for the renormaliztion group from the differential equation) is a well-posed to be useful below T, one must be able to calculate problem, while solving for the partition function is the free energy and correlation length by other not. Solving for a saddle point of the differential methods on the phase boundary well below T, . equation is a well-posed problem in the sense that It was important in the analysis that z be nega- small modifications of the functions u, v, and u tive. If z were positive one would have a point make only small changes in E„q„and x and y; like P~ unstable in all directions instead of a sad- in other words one can get approximate values of dle point. The only way to reach P~ is to sit on it E, and q, by requiring that u and so only vanish ap- to start with which means fixing both E and q. In proximately. In contrast, to get the singular part general a fixed point with both y and z as well as of the free energy requires a calculation to infinite x positive is still a critical point but one which re- precision since the singular part approaches zero quires fixing three thermodynamic quantities in- as K- and one must know how it ' K,(q) approaches stead of two. An exceptional circumstance arises zero; the singular term exists underneath a finite when z = 0; the analysis of this case is complicated nonsingular term which is why infinite precision is but still feasible. The principal results are that the required. critical point may or may not require fixing a third Clearly, if the renormalization-group picture of thermodynamic variable, and there can be logarith- critical behavior is correct, it is important to mic violations of the scaling laws. derive the differential equation of the renormaliza- The above calculations used the liqearized form tion group and try to find the critical saddle point of the renormalization-group equations, but the of the equation. Extrapolating from the analysis of conclusions will not be changed by using the non- this paper, it is clear that the renormalization- linear equations. group equations can involve any number of coupling To summarize the results of this analysis, one constants, not just two or three; one will still get starts with values of E and q near the critical line the Widom-Kadanoff scaling laws provided the K=K, (q). The functions A~ and q~ have an initial solution of the equations approaches a saddle point transient behavior in which El. and q& adjust to the when one is at the critical temperature. If there critical values E, and q, . In other words, while are n coupling constants instead of three, the only the initial Ising Hamiltonian can have an arbitrary change will be that there will be n —2 linearly inde- fraction of second-nearest-neighbor coupling, by the pendent initial transient solutions of the linearized time one has gone to a block Hamiltonian with rea- equations, instead of one, and hence n —2 irrelevant sonably large block size the couplings have become variables. However the equations with n variables essentially fixed at E, and q, . The Hamiltonians are complicated, if n is large, and it is not at all for larger I. are therefore independent of the initial certain that critical behavior would be caused by a fraction of second-nearest-neighbor coupling, and saddle point. The solutions of the renormalization- the critical behavior of the theory is likewise in- group equations might instead approach a limit dependent of the initial fraction. cycle' or go off to infinity or go into irregular In fact the parameters of the critical behavior oscillations (ergodic or turbulent?) as I;~. So such as the exponents x and y are determined by the while generalized renormalization-group equations renormalization-group differential equation, rather are capable of reproducing the Widom-Kadanoff than the initial Hamiltonian. If one knows the form theory, they are also capable of producing more of the functions u, v, and se, one can calculate the challenging types of behavior. exponents x and y by finding the saddle point E„q, I have benefitted from many discussions with of the differential equation and then solving the Professor M. Fisher, Professor B. Widom, Pro- linearized equations about the saddle point. In fessor L. Kadanoff, Professor D. Jasnow, Pro- principle one then has two choices for how to find fessor M. Wortis, and many others in the field the exponents x and y, one choice being as just of critical phenomena.

*Supported by the National Science Foundation. Phys. Acta 26, 499 (1953); ¹ N. Bogoliubov and D. V. ~For reviews of the theory of critical phenomena, see Shirkov, Introduction to the Theo~ of Quantized Eields M. Fisher, Rept. Progr. Phys. 30, 731 (1967); L. P. (Interscience, New York, 1959), Chap. VDI. For a re- Kadanoff et al. , Rev. Mod. Phys. 39, 395 (1967). cent review see K. Wilson, Phys. Rev. D 3 1818 (1971). 2The original references on the renormalization group 3C. DiCastro and G. Jona-Lasinio, Phys. Letters 29A, are M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300 322 (1969). Renormalization-group ideas are also used (1954); E. C. G. Stueckelberg and A. Petermann, Helv. in the program of Migdal and others: See A. A. Migdal, RENORMAI. I ZATION GROU X' AND CRITICAI PHENOMENA. I. . .

Zh. Eksperim. i Teor. Fiz. 59 1015 (1970) fSov. Phys. group first arose, there is a corresponding analyticity 552 (1971)j, and references cited therein. problem; namely, the renormalized amplitudes of quan- K. Wilson, following paper, Phys. Bev. B 4, 3184 tum electrodynamics diverge when the electron mass (1971), referred to as II. goes to zero yet it is assumed that the function (called g L. P. Kadanoff, Physics 2, 263 (1966); and L. P. in Gell-Mann and Low, Bef. 2) appearing in the renor- Kadanoff et al. (Ref. 1). KadanoN assumed that a block malization-group equations has a finite limit for zero of spins of size L would have a magnetization +L with electron mass. The g function was defined by Gell-Mann g unknown. In this paper there is no need to use this pre- and Low in terms of renormalized amplitudes, so the cise form of Kadanoff's hypothesis, so for simplicity g assumption seems questionable. But Gell-Mann and Low will be set equal to zero. The choice g =0 is false, as showed that g does have a zero-mass limit through fourth explained here; but even with g an undetermined param- order in perturbation theory for which g can be calculated eter no one has been able to justify Kadanoff's idea that explicitly. For further discussion see Gell-Mann and a block of spins acts as a unit with only two possible val- Low (Bef. 2) or Wilson (Ref. 2). This analyticity prob- ues for its magnetization. lem is ignored in Bogoliubov and Shirkov (Ref. 2). 6This scaling law can actually be true only for the ' L. P. Kadanoff (private communication). In the an- singular part of E. The reason is that at the critical isotropic Heisenberg model of a ferromagnet an irrele- point one has Kz = K~ and hz = 0 for all L (see below) and vant variable could be the ratio of the o„o nearest- therefore E(K~, 0) =L 3F(K„O),which is true only if E(K„O) neighbor coupling to the o, 0» nearest-neighbor coupling. is zero or ~. In practice the singular part of J' is zero (See, however, Bef. 13.) at the critical point (it is only the derivatives of E which '2In special cases equations like Eqs. (47) and (48) may diverge at the critical point). have solutions behaving as L" and L~ lnL instead of L" I have chosen e to be K~ —K instead of K- K~ so that and L». This cannot occur here (it is shown below that e is positive for temperatures above T~. z &y). The original Ising interaction is invariant to the trans- ~3An example where this is relevant is in the theory of formation K K, g —g, and s~ —sg for all n. This means the Heisenberg ferromagnet, where the fixed point asso- that the free energy J cannot depend on the sign of 5, so ciated with the isotropic ferromagnet is expected to be it could be written as a function of h2. One wouM expect unstable with respect to adding anisotropic interactions this symmetry to be apparent in the effective Ising model; to the Hamiltonian as well as to changing the temperature namely, one would expect that Kz, Kz, , hz, —hz„and or magnetic field. See D. Jasnow and M. Wortis, Phys. s-' ——s~ under this transformation. Hence Kz, should Bev. 176, 739(1968); E. Riedel and F. Wegner, Z. Phy- h2 depend only on but hz should change sign when A, sik 225, 195 (1969), Phys. Rev. Letters 24 730 (1970); changes sign. 24 930(Z) (1970). The function z(Kz, h&) depends only on @& (i.e. , is 4For an introduction to the theory of limit cycles, even in k~) because E(Kz, h J.) and ((Kz, k&) are even see ¹ Minorsky, Nonlinear Oscillations (Van Nostrand, functions of fg&. Princeton, ¹ J., 1962), Chap. 3. See also K. Wilson, In quantum field theory, where the renormalization in Ref. 2.