. Renormalization Group Equations in Critical Dynamics. I

1665 Progress of Theoretical Physics, Vol. 54, No. 6, December 1975

. Renorm alizatio n Group Equatio ns in Critical Dynami cs. I --Isotro pic Heisenber g Ferroma gnet-Kyozi KAWASAKI

Research Instztute for Fundame ntal Physics, Kyoto Universit y, Kyoto (Received June 17, 1975) A renormaliza tion group equation for the inverse propagator is obtained for a stochastic model of isotropic Heisenberg ferromagnet , whose solution at a fixed point yields the dynamical scaling law with the correct dynamic critical exponent. For 6-E dimensions (e>O) we describe an iterative method to obtain an E-expansion for the inverse propagator.

§ I.

Introduc tion

The recent phenomen al successes of the. renormaliz ation group approachv - 4> to static critical phenomen a have stimulated several application s of this approach to dynamic critical phenomen a. 5>-s> In particular, de Dominicis, Brezin and ZinnJustin8> were able to cast the stochastic TDGL models 5> in a field-theor etic language so that the field theoretic renormaliz ation group techniques are directly applicable . However, the stochastic TDGL models are rather special in that important hydrodynamic coupling among modes is· absent, and it does not seem to be possible to extend the method of de Dominicis et al. to more general dynamic critical models. We have recently developed an alternative method of directly constructi ng renor:m,alization group equations (RGE) for general stochastic models of critical dynamics, which will be described in this paper for the case of isotropic Heisenber g ferromagn ets. The many advantage s of the RGE approache s are delineated by de Dominicis et al., 8> which are also shared in our approach. In the next section we describe the model and construct a RGE for the inverse propagato r. The dynamical scaling law with the correct dynamic critical exponent is derived from the RGE for general spatial dimensions . The (sufficient ) condition for the dynamical scaling is that the theory be renormaliz able with the renormaliz ation factors zh Zs and ZL as given by (2. 9) below. In § 3 we study the case .of 6-e dimension s and an iterative method to obtain an e-expansio n of the inverse propagato r is described.

§ 2.

Renorma lization group equation and dynamica l scaling

We consider an isotropic Heisenber g ferromagn et with the Wilson type ( dimensionless) Hamiltoni an H given byv

(2·1)

K. Kawasaki

1666

where Sk a is the Fourier transform of the a-th component of the spin ·density Sa(r):

1-· S k "'=y1;2

Sdr e-lk·r·S"'(r) .

.

and the sum over k is limited by the upper cutoff lkl =A. The dynamical behavior is then described by the Fokker-Planck equation for the probability distrib}ltion function P ( {S}, t) containing the entire set {S} of the variables Sk a, which we write as

~P( {S}, t) = ..i'b ({S} )P( {S}, t),

(2 ·2)

at

The stochastic operator

where the suffix b refers to unrenormalized quantities. is given by

..i'b ( {S})

( 8 + 8H ) 8Sq"' BS':q 8S':q

..[' ({S}.) _"""" 2L0 8 b

-

7' ~ q

"" e _a_8 fJ 8H -~"" !f2 ""--' ""--' afJr O (or tqa arf>':q arf>':q '

CT-

_a_,~.P

aH

afJr U6 where 1J = 0. *> We will see by examining (2·35b) near d=6 that the non-convention al fixed point is seen to be stable-only for d and was explicitly demonstrated in the renormalization group framework by Ma and Mazenko.n

§ 3.

Isotropic Heisenberg ferromagnet in 6-8 dimensions

As we have indicated earlier, the conventional behavior breaks down only for _d The crossover exponent q; in the sense of Ref .. 7) associated with the instability for d ·and ··· represents more complex terms such as vertex correction terms, lll,l 2> which can be expressed again in terms of (C). In this way we obtain a self-consistent equation for q (r;,). A more natural method of dealing with vertex corrections would be to construct renormali.zation group equations for various vertex parts, although we will not discuss this aspec,t here. The normalization condition (2 ·16) now reads

rk

r

r"(O) requiring

=r"

(3·17)

E" (0) = 0. That is,

where we have explicitly indicated the fJ. dependence of Z an4 v. Thus we have a complete self-consistent scheme (3 ·13), (3 ·14) and (3 ·17) or (3 ·18) where k ( r;,) and are simultaneously determined. Since the self-consistent 'scheme obtained here takes the form of a complicated non-linear integral equation, which in fact is basically equivalent to that of the self-consistent mode coupling theory/2> this is not amenable to analytic solution. Thus we here develop an iterative scheme valid near the 6 dimension. We will see that in this case there is a solution of the form

r

z

T (") q "

=

-i" + Z(q) "

Z(fJ.) rq

1-·Q (Z( )") +Z(fJ.) q fJ. " '

(3·19)

where Z(q) is the renormalization factor when the reference wave number is chosen to be q, and the last term is small for small e>O where Qq(O) =0 by (3 ·17). From (2 ·15) we see that (3 ·19) is consistent with the fact that q,b is independent of /}.. Substituting (3 ·19) into our self-consistent set of equations, we find, after expanding in powers of Q, the following:

r

Z(q)-1+_!_Qq(r;,) =Z(fJ.Yv2fJ.a rq

-Z(fJ.)Vf-1."_!_ rc

f !J~,k,q-k . . 1 . Jk rq Z(k)rk+Z(q-k)rq-k+zr::.

s= dr;,' Jkf !J~,k,q-k 1 Qk CC') + .. ·. rq i(r;,' -r;,) +Z(q-k)rq-k [ -ir;,' +Z(k)rkJ -=

(3 ·20) As' the first approximation we ignore Qq (C) and the vertex corrections, which

'K. Kawasaki

1674

will be shown to be of higher order in e.. Setting (=0 and q=p in the resulting equation we have

. Z(p) =1+Z(p)2v2 p 6

.

1 f Uq,k,q-k . . Jk r" Z(k)r~c+Z(q-k)rq'-k

, ((lql) =p). (3-21)

This is nothing but the self-consiste~t mode coupling equation12> for the ratio 'z(p) = L (p) / L (A) where L ( q) is the wave number dependent Onsager kinetic coefficient for the spin diffusion. Hence, up to first order in e we can, noting that Z(p) 2v 2p 6 is in f~ct independent of 11 [see (2·9d)J, convert (3·21) into the following differential equation as in Ref. 13): ' (3·22) where

g2 =v02As/1927f.

(3·23)

On the other hand, the independence of ZZv 2ps on 11 is expressed as

2 dlnZ 11 dp

+ 2 /idlnf + "'" dp

6

=O

(3·24) '

where (3·25) In the following we shall use (3 · 22) and (3 · 25) to yield

f instead of

dlnZ df

~·

Thus we can eliminate 11 from

= -2f/(2f2-e). '

'

(3•26)

The solution Z=Z(f) of this which reduces to unity for f=O is

Z(f)

=

( 1 -2/2)-1/2 -• 6

(3·27)

The functions {3, (which we write as {31 here) and fL are, from (2 · 24b and d), respectively, {31 =f(f'-!e),

(3·28a)

rL= -F.

(3·28b)

The fixed point condition {31 =,0 gives both the conventional (f=O) and non-conventional (f=/=0) fixed' points, and for the latter we have (3·29) and also fL ~TL * = - s/2. The stability of the fixed point is examined from (d/31/ df)* =3(!*) 2 -s/2. In particular, the non-conventional fixed point has (dfii/df)* = e and is stable only below the six .dimension. All these results including (3 · 29)

Renor111;alization Group Equations in Critical Dynamics. I

1675

confirm what are already k:riown for this system.n Although the preceding treatment is based on the self-consistent calculational scheme, the results {3·8) suggest that the first order result in e could have been obtained in a simple power series expansionin f such as Z=l+f2/e if we consistently expand {31 and h in f as in the equilibrium case. 3> The merits of our self-consistent treatment lies in its closer relationship with the mode coupling theory13> as well as in the reduction of number of terms when one attempts higher order calculations in e. Such higher order calculations can be carried out if we note that 'li = 0 (e) near the fixed point. We first retain the next order contribution in e in solving (3·21) and then calculate Qq(l;;) from*> Qq(l;;) =Z(p)Zv (p}ps

-

tQ~,k,q-k[z(k) Y~ seek a kind of optimum starting point which might be called "arialytic maximum coarse graining" where one applies as much coarse graining as possible to .a microscopic model without introducing any critical singularity. The Wilson Hamiltonian provides such a starting point for static critical *> -Actual analytic and numerical computations of Q0 (C) are described in Ma and Mazenko.'>

1676

K. Kawasaki

phenomena, and we believe that a stochastic model of the kind treated here .is a ' dynamic analogue of the Wilson Hamiltonian. A similar view was also expressed by Ma and Mazenko, 7> but we feel that the point is sufficiently important worth re-stating it here. In this paper we have left out all the variables other than the order parameter density, but in dynamics thesl} other variables such as the energy density cannot be entirely ignored and often play a ~rucial role. Some such examples will be discussed in Part II of this series where these extra variables· are treated as separate field variables. Alternative way of dealing with these variables would be to r.egard them as composite fields 3> made up of products of the local order parameter and its spatial gradients, and a field-theoretical formulation seems to be well suited for this purpose. Finally, we express a hope that the present RGE approach may be successfully combined with the existing microscopic attempts at critical dynamics15> to obtain a better understanding of critical dynamics on a microscopic level.

Acknowledgements The author would like to thank J. D. Gunton for numerous discussions on the subject of this work. Particular thanks are due to the authors of Ref. 7) and 8) for sending their preprints to this author.

Appendix A

--Renormalizability condition-Since the only new interaction that arises in dynamics is the mode coupling term (3 · 6b), we will consider the renormalizability of the "Hamiltonian" (3 · 5), where H is now Gaussian. Let us consider a term in the perturbation .series described by a diagram p containing E external lines, I internal lines, L loops and n three point vertices (here the counter term is not considered), where each vertex carries one factor of wave number [see (3 · 8}]. The degree of superficial (ultraviolet) divergence of the diagram is3>· 14>

(](p) =n-4I+Ld.

(A·1)

Eliminating I and L from the geometrical relations

L=I- (n-1), 3n=E+2I, we obtain

(A·2) with

Renormalization Group Equations m Critical Dynamics. I

p=;

-5.

1677 (A·3)

Another expression for (J (P) is

tJ(p) =d-i(d-1)E+ipl.

(A·4)

Here we find that for the dimensionality d which satisfies

(A·5) there are only finite number of superficially divergent terms. In particular, for six dimensions the logarithmically divergent second order self-energy term is the only primitive divergence.

Appendix B --Calculation of ZL-Here we briefly describe a perturbational calculation of ZL which was denoted as Z in § 3 in powers of v. Here we use an alternative expression for Gq(r') as follows:

(B·1) or

(B·2) The normalization condition (2 ·16) can then be written as

(B·3) Let us now write

.5-C as . (B·4)

where ${' 1s given by (3·6b) and

.5-Ct = - z-l :E r iiqflaqfl .

(B·5)

q{J

Although (B · 3) allows us to determine Z, we- must show that Z should not depend upon fL explicitly. To see this we express all the wave numbers in unit of fL like k=f-Lx,

q-k=f-L(l-x),

where I is the unit vector in the direction of q. criticality, we can write

(B·6) Since 'Xk =k- 2 and rk =k4 at

(B·7) where CW no longer depends on fL. The perturbation expansion of (B· 3) in ${' can be seen in fact to be an expansion in powers of cr) 2F 1 - 2f-Ld, where f-Ld comes

1678

K.· Kawasaki

from the sum over a wave vector in intermediate states, and F 1 is the unperturbed inverse propagator. Using (B · 7) _and F 1 "-' z- 1/1 4 the expansion parameter behaves 1/1° which is independent as of /L- Thus, (B.~) regarded as an equation for does not contain /L explicitly and we have Z=Z(v). In particular, the second order calculation reproduces the result of § 3 up to this order:'

z-

z

(B·S) References

1) 2)

K. G. Wilson imd J. Kogut, Physics Report 12C (1974), 75. R. Brout, Physics Report 10C (1974), 1.

G. Parisi, Cargese Summer School Lechires (1973). 3) ,E. Brezin, J. C. Le Guillou and J. Zinn-Justin, Field Thefretical Approach to Critical Phenomena, Saclay P;eprint DPh-T/74/100 (1974). 4) Th. Niemeijer and ]. M. ]. van Leeuwen, in Phase Transitions and Critical Phenomena, ed. C. Domb and M. S. Green, Vol. 6 (Academic Press, New York and London) (to. be published). 5) B. I. Halperin, P. C. Hohenberg and s.·K. Ma, Phys. Rev·. Letters 29 (1972), 1548; Phys. Rev. B10 (1974), 139. Further references are listed in Ref. 8). 6) B. I. Halperin, P. C. Hohenberg and E. D. Siggia,.'Phys. Rev. Letters 32 (1974), 1289. 7) S. K. Ma and G. F. Mazenko, Phys. Rev. Letters 33 (1974), 1384 and.to be published. 8) C. de Dominicis, Sacla,y Preprint DPh-T/74/110 (1974). C. de Dominici~, E. Brezin and ]. Zinn-Justin, Saclay Preprint DPh-T/75/17 and DPhT /75/58 (1975). 9)· K. Kawasaki and J. D. Gunton, in Progress in Liquid Physics, ed. C. A._Crqxton, (Wiley, to be published). 10) ]. Villain, ]. de phys. 29 (1968), 321, 687. K. Kawasaki, Prog. Theor. Phys. 40 (1968), 11, 706. 11) K. Kawasaki, Prog. Theor. Phys. 52 (1974), 1527. 12) K. Kawasaki, Ann. of Phys. 61 (1970), 1, and in Proceedings of the International Summer School "Enrico Fermi" Course LI, ed., M. S. Green, (Academic Press, New York a11d London, 1971). 12a) B. I. Halperin and P. C. Hohenberg, Phys. Rev. 177 (1969), 952. 13) J. D. Gunton and K. Kawasaki, J. Phys. AS (1975), L9. 14) N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Interscience, New York; 1959).. 15) A M. Polyakov, Soviet Phys.-JETP 30 (1970), 1164. J. A. Hertz, Int. J. Magnetism 1 (1970), 253, 307, 313. S. V. Maleev, Soviet Phys.-JETP 38 (1974), 613.