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Dynamic scaling laws for Heisenberg spin systems with long-range interactions
Physica 56 (197 1) 62-87 o North-Holland Publishing Co.
DYNAMIC SPIN
SCALING
SYSTEMS
WITH
LAWS
FOR
LONG-RANGE
HEISENBERG INTERACTIONS
M. DE LEENER Service de Chimie-Physique
II, Universitk Libre de Bruxelles, Belgique
Received 2 April 1971
synopsis The dynamic scaling laws proposed by Halperin and Hohenberg are shown to be correct above Tc for Heisenberg magnets with long-range interactions. We derive the asymptotic kinetic equations describing the critical behaviour of the spin correlation functions.
1. Intro&&ion.
During the last few years, considerable progress has been made towards the understanding of the dynamic properties of ferroand antiferromagnets near their critical point, following extensive experimental and theoretical studies of the spin autocorrelation function (a.f.), which we define as
r,(t)
’
= ~
where the brackets denote an average over the canonical equilibrium ensemble; in the Heisenberg model, S,(t) is the q-wavenumber Fourier transform
of the localized
spin operator
Heisenberg picture. A series of recent neutron-scattering transform of the a.f.,
pq(o) =
rdt r,(t)
Sa at the lattice
experimentsi)
site a, in the
measuring the Fourier
(2)
eiot,
0
showed remarkable agreement with the predictions of the dynamic scaling assumptions (DSA) proposed by Halperin and Hohenbergs) and studied by Kawasaki, Mori and Okamotoa). Near the critical point, where the macroscopic equilibrium properties are expected to be dominated by the existence of a unique correlation length ~-1 which becomes infinite at the critical temperature Tc,the DSA state that: 62
DYNAMIC
SCALING
LAWS
(a) when the wavenumber behaviour is expected, fh(o)
FOR
HEISENBERG
SPIN
SYSTEMS
63
q tends
towards any value qc where critical approaches a function
where k = Iq - qc] (for ferromagnets, qc = 0; for antiferromagnets, it has two values, qc = 0 and q*, the latter wavenumber characterizing the staggered spin ordering). (b) the width on(k) is a homogeneous function of K and k: on(k) = kaf(K/k).
(4
Halperin and Hohenberg subsequently predicted the value of a (Q for ferromagnets and 8 for antiferromagnets) as well as the limiting behaviour of the functions f(x) when x + 0 (transition region) and x --f 00 (hydrodynamic region). However powerful the DSA scheme turns out to be, kinetic theories, applying to well-defined microscopic models the techniques of irreversible statistical mechanics, are necessary to test the DSA and to go beyond them, i.e. to determine the unknown functions C and f. Various authors have worked along these line!+). In particular, Resibois and the authors) have studied the isotropic Heisenberg model in the Weiss limitv), where the number z of neighbours interacting with a given spin tends to infinity. The pertinence of this model is most questionable when one deals with real systems where the interactions are short-ranged; moreover, it is known to lead to wrong equilibrium critical indicess). Nevertheless, the simplicity of the model has allowed us to develop an exact theory of its dynamical properties, which turns out to agree with all essential features of the experimental results. We have shown that, at any temperature in the paramagnetic region (T 2 T,), the a.f. f &) obeys the following non-markoffian kinetic equation : W’,(t)
= jdt’(;,(t’,{Q})
f,(t
- t’),
(5)
in which the kernel exhibits a nonlinear dependence on rQf itself and is given by an infinite series of terms which can be associated with graphs. We then studied the properties of the kinetic equation (5) at the critical temperature (T = T,), limiting ourselves to the lowest-order approximation to the kernel. We showed that, in the short-wavenumber and long-time limit, the a.f. P,(t) agrees completely with the predictions of Halperin and Hohenberg’s DSA. Furthermore, we derived the (approximate) equation describing the asymptotic behaviour of the a.f. and concluded that the essential feature of the critical region is that Z’,(t) never reaches a diffusion regime, but decays to zero through damped oscillations. Exploring further the predictions of
M. DE LEENER
64
this kinetic theory, RCsibois and Piettee) have recently extended our work away from the critical temperature (K f 0) and computed the function f(x) of eq. (4); th e agreement with experiment The generality of these results is, however,
is quite satisfactory. limited by the (second-order)
approximation made on the kernel of the kinetic equation (5). In this paper, we remedy this weakness and generalize our results to all orders in the expansion of G,. To this end, we first show that our rules for evaluating
GQ
can be greatly simplified and expressed completely in wavenumber variables instead of the initial spin coordinates; this is the content of section 2. Section 3 is devoted to a detailed study of the ferromagnetic case near the critical point: we derive the asymptotic kinetic equation describing the critical behaviour of the a.f.,show that the corrections are negligible and make the connection with the DSA. The case of antiferromagnets is briefly considered in section 4. Finally, section 5 ends this paper with a short discussion. 2. Simplified
rules for the kernel Gq. For the sake of clarity, we shall here
repeat the rules for evaluating the various contributions to Gab, the Fourier transform (F.t.) of G,t. The reader is, however, urged to refer to RDL III for further explanations. The graphs for Gab(t) are made of dotted lines and plain arrowed lines, connected through the vertices given in fig. 1. Each vertex is labelled with a time between 0 and t, the m times increasing from right to left (tm. = t > t,-1 > . . . > tz > tl = 0). The graphs must start (at time 0) and end (at time t) with a single dotted line; since the vertices “create” or “destroy” one line, the numbers of vertices must be even. Except for the first and last dotted lines, which are labelled with the spin indices b and a, respectively, all lines carry two spin indices (one near each end) which may be identical or different. Only a limited class of graphs should be retained, those which (1) are irreducible, i.e. contain no intermediate state (between two subsequent times) involving only one line; (2) contain no self-energy insertion, i.e. no part starting or ending with a single line (note that any graph, as a whole, has such a self-energy structure). The following rules should be followed to evaluate the contribution of any graph for Gab: t In what follows, we shall always define the Fourier transform of any A ab = A(a - b) as A,
=
2 Ama &-W--o), b
Aae = ;
C A, &*(a--b), 4 where N is the number of spins in the system.
funCtiOn
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
65
k
A-
(1)
c l
= ‘kl
Iid
k
(2’4 “;‘;:
CM9 (40) (4b)
k
=2 ‘k, MI
= $1
$1
k _!_ = @ kl Iii1 >I = 2@k,(Jkm-J,m )Mm
Fig. 1. The vertices and their contribution in coordinate space. Inverting the direction of the arrows changes the sign of the contribution.
Rules A: (1) To everydottedline, except the first and last ones (labelled b and a), starting with spin 1 at time tl and ending with spin k at time ts, associate a factor 4MkMlfkl(t2 - tl), where r,,(t) is the F.t. of T,(t) and the variable Mk attached to every spin k in the graph can take the values +fr, 0, -4. Similarly, to every plain line, associate a factor? 24&, OT+1’&,,oh(t2 if k f
-
h) t
1, or
&*,&k(t2
-
h)
if k = 1, where the displacement as follows : rfpkf(Md
= f(Mk
f
f-lrq+l~
*
@‘k acts on any function
+,a).
(2) To each vertex, associate the operator r& is defined as 17ztr=
opearator
of Mk
(6) the contribution
indicated
in fig. 1. There,
rl+lrQr,
(7)
is the Heisenberg coupling between spins k and I, with F.t. Jq, /3= (kT)-l and @kE is the equilibrium correlation function in the Weiss limit, the F.t. of which is 7)
Jkl
4~ =
Jd(1 - 8BJd.
(8)
(3) Order all operators and M-dependent factors as they appear in the graph. Multiply on the left by a factor M, and on the right by Mb and take the trace (i.e. the sum Z;M,=++) over all spins appearing in the graph. t Everywhere in this paper, a,,, 6Z,Y =
1 0
x=y x #Y.
represents the Kronecker delta function:
M. DE LEENER
66
(4) Multiply in front by 4(3)”
(-)”
; dtzn_rij”& 0
2n
-2
..-OTdta.
0
for a graph of order 2n (with 2n vertices) cluding a and 13). (5) Sum over all m -
involving
m distinct
spins (in-
2 dummy spin indices.
To draw general conclusions about the kinetic equation (.5), we are confronted with two technical difficulties : (1) Rules A are rather awkward, in particular when they involve numerous traces CMl_+$ over the variables Mi. (2) They permit us to calculate Gab in spin coordinate space and the result has to be Fourier transformed afterwards. Yet, as can be verified on the fourth-order contribution Gk4’t, the correct short-wavenumber behaviour is not observed for each individual graph, but only for the sum of all of them at every order. To overcome these difficulties, it is thus desirable to establish compact rules to compute G, directly in wavenumber space, such that the contributions of different graphs are grouped correctly: this turns out to be fairly easy once it is realized that no error is made when one calculates Gab from rules A without paying attention to the special cases where some of the spins a, k, 1, ..,, b are identical. This implies two proofs. First, we must show that, although the graphs for G,, are different from those for Gab (a # b), the correct G aa is obtained by setting b = a in the expression for Gab. We first notice that the F.t. of Gab, #, = c Gab eiq’(b-a),
(9)
all b
could only differ from the exact kernel G, = G,, +
C
Gab eiqatbma),
(10)
b(#a)
by a (q-independent) sum rule r@(t)
=
constant
term.
But the a.f. satisfies
1,
the well-known
(‘1)
for all times t, arising from the conservation of the total magnetization in the Heisenberg spin system; from the kinetic equation (5), we then deduce the sum rule for G,: G,=,,(t) = 0. To ensure that 0,
(12) is identical
t See RDL IV, section IV.
to G,, it is thus sufficient
to check that 8,
DYNAMIC
SCALING FOR HEISENBERG
SPIN SYSTEMS
67
goes to zero in the limit q -+ 0 for all times. This will be verified a poster&i, at the end of this section, to all orders 2n of the expansion. Secondly, we have to prove that, although the contribution of any line with two spin indices (K # I) differs from that with a single index (K = Z), the correct contribution for any graph with a line (F2Fz) can be obtained evaluating it first for the general case (KZ)and setting k = I afterwards.
by
This can be easily verified from rules A(l), A(3) and A(4). As an example, the contribution of any graph with a plain line (kk) ends with the following structure : . . . &fJkkf(~k
(13)
= 0, (wn#k~)~
Inserting a factor 1 =
we
c MI=*+
(‘4)
q+*‘8M,,o,
obtain = 0, {Mmzk,
1)) ik=Z,
(15)
which is precisely the contribution of the same graph with a line (kl), when k is afterwards set equal to 1. We have thus shown that, to calculate the kernel G,, it is sufficient to evaluate the contribution of every graph for G,a taking all spins a, k, I, . . . , b to be different and to sum the result over the dummy spin indices without any exclusion. The result is then Fourier-transformed following eq. (9) or equivalently :
Gg=
-&-z c Gab e a
b
iq.(b-a)
.
(16)
This theorem has the important consequence that the trace over the M variables can easily be taken. Indeed, if all spins in a graph are different, every spin k only appears (once or twice) near a single vertex. Moreover, the variables Mk and displacement operators $‘” appearing in the line contributions [see rules A( I)] can be factored out and associated with the vertex : at every end of a dotted line, we have a factor Mkt ; a plain line contributes a factor 6Mr,Oat itsleft end, and~f1k6nl,,o at its right end. It is then elementary work to combine these factors with the vertex contributions of fig. 1, to multiply by a factor + per spin and to take the trace CMrzft over the two or three spin variables.
7 Note that no special situation arises at the first and last vertices, since rule A(3) tells us to multiply on the left (and right) by M, (and Mb).
M. DE LEENER
68
As an example,
=
vertex
2@kz(C
(3a) leads to the expression
dM,,_&(C Mk
JQ’;)(~)~
=
&@kZ.
Ml
In the same way, we would find for vertex
(‘7)
(3b) the result
-&@kl
(18)
and for vertex @@kz(Jzm
(3c), -
_/km).
(19)
Similar expressions would be obtained for the other vertices of fig. 1. When the trace over the M variables has been taken, it is an easy job to formulate the rules for G, directly in wave-number space. Indeed, the a.f. which appears in the contribution of every line [see rules A(l)] may be expressed in terms of its F.t.: p,,(t)
=
-+
C f*(t)
eiqack--l);
Q
hence, we can associate with every plain or dotted line a factor f a(t) at wavenumber q (to be summed over) and a factor exp(+(or -) iq. k) with the vertex at the left (or right) end of the line. All factors depending upon the coordinates of the spins appearing near a given vertex (i.e. the contribution of the vertex itself, as calculated above, times the exponential factors) can then immediately be summed over these coordinates and the result associated with the vertex t. Moreover, since the lines arriving at the vertex now carry wavenumber indices, the contributions of the vertices with the same general structure (i.e. grouped under the same number in fig. 1) can be added. Coming back to our example (vertices 3a, 3b and 3c), we shall associate a wavenumber qr to the plain line at the right of the vertex, qs to the plain line at the left and q3 to the dotted line. For vertex (3a), we then find the following contribution : K+ F $qjkl eiql*k e-iqa.k e-iq8.l = z
-+ pkl
ei(qa-qd*U-W ei(ql-qa-qd*l
= 3Y <qP+Qah-QnJ
(21)
t Note again that, since we calculate G, through eq. (16), spins a and b play no special role.
DYNAMIC
SCALING LAWS FOR HEISENBERG
where we used the representation J,,,,,
of the Kronecker
SPIN SYSTEMS
delta function:
1 = N 5 ei(ql-qa)*r,
as well as the inversion
(22)
symmetry
of the correlation
function
&l: (23)
+r = &P In the same way, one finds for vertex
(3b) the result (24)
--*N&r,, Q*+G#% and for vertex
(3c),
fN~WL+Qs(6r1
-
(25)
&-_q*) @JCL.
Adding expressions (21), (24) and (25), we are thus led to associate . single graph 3, in wavenumber language, the contribution W,,,
69
q~+dib-as
-
4~2)(1 -
to the
iiB.Jd
the second expression follows from the definition [eq. (8)] of $q after some algebraic manipulations. Note that the total wavenumber is conserved at the vertex, a fact which arises from the translation invariance of the system. Similar calculations can be performed for the other vertices of fig. 1. Except for a factor ftN which will be shortly taken care of, the four resulting contributions are shown in fig. 2( 1). To complete the derivation of our new rules in wavenumber space, there
(a)
(b) q’
s’!
_
,q’
a’
,,*;ll_q” J (1)
( Jqa-q” -J,4
--
(1) (@q’-q”-$q4 (1-pJq42) (t-~Jo~2) “Jq’-q”-Jq”l(l-PJq~~/2)(i-PJql_q,,/2)
(2)
(jql_q”j2_q”2)
(2) (lq’_q”~2_q”2)(K2+,12)
(K2,q”2)(K2+Iq,_q~~,2) Fig. 2. The four vertices in wavenumber language, with their contribution (1) to the general kinetic equation and (2) to the asymptotic equation for ferromagnets. Inverting the direction of the arrows only changes the sign of the contributions.
M. DE
70
LEENEK
remains to determine what factor has to be set in front. From the fact that every vertex involves two plain lines and one dotted line, it follows that any graph involving 2n vertices contains 2n plain lines and (n - 1) dotted lines (excluding both of the dotted lines which start and end the graph). Using rules A( 1) and A(4) as well as eqs. (16) and (20), and multiplying by $N per vertex, we then arrive at the following factor for a graph of order 2n: (-)”
;(L)““(+)“-l(y
_;;(__g
(27)
The number of factors (l/N) of course coincides with that of dummy wavenumbers to be summed over. Finally, the minus sign in eq. (27) can be suppressed by changing the sign of the contribution of the vertices (b) in fig. 2(l), since every graph of order 2n involves the same number n of vertices (a) and (b). Summarizing, we can now give the rules to evaluate the kernel G,(t) of the general kinetic equation (5). Rules B : (1) Connecting dotted and plain arrowed lines through the vertices of fig. 2, draw all distinct graphs of even order 2n, irreducible and containing no self-energy insertion, starting and ending with a dotted line with wavenumber 4. The times attached to the vertices grow from right to left (ts, = t > tsn_i > . . . >ts > tr = 0). The contribution of any graph is found as follows. (2) To every vertex, associate the expression given in fig. 2( 1); to every (plain or dotted) line with wavenumber q’, between the times t and t’, associate a function r,,(t - t’). (3) Multiply by (1 /N)n. (4) Integrate over the ordered intermediate times: ;dlzn:~-dtz,_r
. . . (it2
0
and sum over the n dummy wavenumbers. When one compares rules A and rules B, the simplicity of the latter ones is evident. As an example, fig. 3 shows all the graphs for the second- and
Fig. 3. Examples
of graphs
in wavenumber language: (a) the only (2) the two graphs for Gh*).
graph
for
Gh2);
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
fourth-order contributions to the kernel, Gi2’ is readily found to be
SPIN
SYSTEMS
GL2) and GL4’. The expression
G12W = - +- 2Q’ (JQ-q’- Jg’)(&Pq’-
71
of
$bg’)
x (1 - tBJg)r,44 r,-,+)
the result given in RDL IV, eq. (A.5).
confirming
For Gr’, we get:
-
J&J(J4~+v-a - Jm7”)
x (4Q’+Q”--9 - kJ*) (1 - MJcl-)bh-4”- dcT)(l- 3PJd x 5 dt3 ;dt&,(t 0
x
-
t3)
rg-g’@-
t2)
0
rg*+g4ts
-
is) ra-g"(t3) r,-(tz).
(29)
From rules B, it is now easy to verify the stated property that the kernel G, goes to zero when 4 + 0 [eq. (12)]. Indeed, any graph for G, must start and end with a dotted line 4. From fig. 2(l), we thus conclude that its contribution must involve a factor
(JQ-d- Jcf)(Ja-v - JfiP~ q2. This completes
the proof of the validity
(30) of our rules B.
3. Dynamic scaling for ferromagnets. The compactness of rules kernel of the kinetic equation (5) will now allow us to study the and short-wavenumber behaviour of the a.f. r,r near the critical shall specifically consider an isotropic Heisenberg ferromagnet, Jg of the exchange interaction will thus possess a simple Taylor
B for the long-time point. We the F.t. expansion
around 4 = 0:
Jq = Jo (1 - dq2+ . ..). where 6 is a positive constant. It can be checked from rules B that the time rq characterizing
(31)
the decay of f (Itends for large 4 towards a finite constant Tn, which should remain well behaved when the critical temperature Tc is approached. On the contrary, Tg is expected to become singular near the critical point, i.e. when 4 --f 0 and
M. DE LEENER
72
T + TC, where the equilibrium correlation function (bs.diverges; in the Weiss limit, to which the whole of this work is devoted, we have [see eqs. (8) and
(31)l:
(32) where JO = 2kT, and K, the inverse of the correlation
K2 = (l/d)C(T -
length vanishes at T, : (33)
T,)/T,l.
The study of the lowest-order approximation to the kinetic equation (5) t leads us to expect that the DSA should be correct near the critical point (i.e., that 7;l = ~~(4) should obey eq. (4) with a = &) and that this non-analytic behaviour arises from the fact that any contribution to the kernel G, involves (1) equilibrium correlation functions 4q, (for the intermediate states with wavenumbers 4’) which diverge when 4’ and K become small; (2) a.f.‘s rqt which decay very slowly at short wavenumbers. We shall thus take, as the starting point of our analysis, the hypothesis that, near the critical point (K and CJ+ 0), the long-time behaviour of G,(t) is dominated by those contributions which arise when the intermediate state wavenumbers q’, q”, . . . are all as small as q and K. This hypothesis will only be verified a fiosteriori through the existence of a solution to the asymptotic equation. Following this idea, we shall number inside the first Brillouin
separate any sum over a dummy zone (B) as follows:
wave-
E = c1 + PI, P’PO q’
(34)
with the condition (35) which can always be satisfied if q and K go to zero. In the “asymptotic” domain I, dominant contributions appear when q’ es q and K ; as long as the ratio
K =
the wavenumbers
I’ = qI/q. f See RDL
IV and reference
corresponds
q’
(and
K)
(37)
t We here exclude pected.
to
(36)
K/q
remains finite*, it is then natural to measure in units of q, by introducing reduced variables
which
are expected
6.
the particular
case where
to the “hydrodynamic
q
region”
and K tend to zero with K/q + ~0, where diffusive behaviour is ex-
We shall come back to this point in section 3c.
DYNAMIC
SCALING
LAWS
If our starting hypothesis
FOR
is correct,
HEISENBERG
SPIN
SYSTEMS
73
the sum over 4’ within domain I will
converge rapidly when 4 and K become small and we shall have:
C1 = q 9’
s s dq’ = y
Q’
dz’ -
q+o
2’iqoln
Nq” dz’, d
(38)
s all spaoe
where ,4 is the volume of the first Brillouin zone. In domain II, the a.f. rat is expected to decay after a finite time T*’ 5 rq,; we shall then estimate the corresponding integral in the limit T&/Q + 0
(4 --f 0>4’ > 40).
(39)
Since qo can be made as small as we wish when q and K tend to zero, the error introduced by this arbitrary cutoff in region II should become negligible in the limit. We shall first derive the asymptotic limit of our kinetic equation (5) by neglecting all contributions from domain II; it will be found that the a.f. r,(t) then reduces to a single function F(qQt; K). The corrections introduced by the larger values of the dummy wavenumbers will then be shown to be negligible. Finally, we shall make the connection with the DSA of Halperin and Hohenberg. a) Derivation of the asymptotic equation. the q, K + 0 and t --f 00 limit of the kinetic equation
Let us thus examine
&Z’,(t) = i dt' : Ga")(t' j{fa$ fq(t - t’), n=l
where, in every contribution to the kernel, we only retain the sums over the dummy wavenumbers in region I, where q’ < qo. We may then expand the vertices contributions of fig. 2( 1) around the origin in wavenumber space; using eqs. (31) and (32), we find for the vertices of type (a): dJo(q”2 Jc~v - Jc tq,,q~+Oj
-
Iq’ -
q”12);
(41)
and for the vertices (b) :
(Jq’-q”- Jq”) (1 -
(1 - +PJe,) @Jqn)(l - $@Jq,_q")
M Jo (f2 - k' - @i2)(K2 + q'2) (K2 + qM2)(K2+ IQ'- q"12).
(42)
To get a hint at what is going to happen, let us first return to the approximate kinetic equation obtained with the second-order kernel Gp) given by eq. (28). W e use the expanded expressions (41) and (42) and make the
M. DE LEENER
74
changes
of variables
(36) and
(37); we get:
(2’2 - 11 - 2’12)2 &l’,(t) = -y2q5(K2 + 1) ds’ (K2 + 2’2)(P + \I - z’(2) s 1
x
dt’r,,$‘)
r,,,_,#‘)
ra(t - t’),
(43)
0
where I is a unit vector parallel to Q and y2 = dJ;/A. Let us introduce T
=
(44) the dimensionless 7'
y@t,
=
variables
(45)
yq’t’
and the function
(46) eq. (43) becomes &l’,(7) = -(K2
+ 1)
7
X
J
dO,,+‘%‘)
(2’2 12 2’12)2 +2’2)(lP +(1- 2’12) J (K2 dz’
&,_.,#
- 2’1’ T’) F,(T -
7’).
0
(47)
We note that the variable q only appears in this equation as a factor in the index of every function I’. Let Fq(7) b e one solution, then F&*(T) is clearly also a solution, whatever the value of the positive factor CI. If eq. (47) possesses a unique solution, then r@(T) and r&T) must be identical functions, i.e. they cannot depend on the variable q. Hence, eq. (47) reduces to the following asymptotic equation : &F(T) = - (K2 + 1) 7
(2’2 12 z’py +2’2)(K2 +\I - z’(2) J (K2 dz’
dT’r(z’%‘) r(ll
X
-
Z’jf
T’) r(~
-
T’),
(48)
s 0
for the single function rP (t) M
r(T;
K)
=
r(y@t;
K/Q),
(49)
describing the limiting behaviour of the a.f. I’,(t) near the critical point (i.e. when q and K tend to zero) and for long times (i.e. for t -+ co as q-‘, so that T remains finite).
DYNAMIC SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
75
A brief remark should be made concerning the uniqueness of the solution to eq. (47): this property is not at all evident. Indeed, there clearly is a unique solution to the general kinetic equation (40), with the initial value f n(O) = 1 for any 4 [see eq. (l)], but this initial condition does not necessarily apply to the asymptotic limit of this equation for long times: it is easily verified that the solution of eq. (47) is singular at T = 0, i.e. possesses no Taylor expansion around this point. Note also that the evolution of the asymptotic a.f. F(7) has no causal character: the right-hand side of eq. (48) involves values of r(x) for all x, larger and smaller than 7. We now wish to show that the behaviour of the a.f., described by eq. (49), is a property of the general kinetic equation (40) with the complete kernel. To this aim, it is important to note that the proof given above relies on the single fact that the changes of variables (36), (37) and (45) have made the asymptotic equation (48) dimensionless. Extending the proof to all orders of approximation to the kernel G, thus amounts to showing that this nondimensionality is general, i.e. that the factors 4 (and JO, 6, . . .) which appear through the small-q-and-K expansion of the vertices contributions and through the changes of variables (36)-(38) are exactly eliminated through the introduction of dimensionless times T as defined by eq. (45). To establish this fact, it is sufficient to note from rules B that any contribution of order 2n to the kernel GP.is represented by a graph with n vertices of type (a) and 1z of type (b) (see fig. 2) and that it involves n summations over dummy wavenumbers and (PZ- 2) integrals over intermediate times. Using eqs. (41) and (42), we thus conclude that, when we switch to reduced wavenumbers as in eqs. (36)-(38), the variable 4 will appear in front in a factor
When we then substitute the resulting contribution to Gr’ into the kinetic equation (40), we ascertain that the introduction of n reduced times as defined by eq. (45) exactly cancels the factor (50). Simultaneously introducing functions F,(T) through eq. (46), we remark again that the variable q only appears as an arbitrary factor in their indices; hence the solution to the asymptotic equation cannot depend on q. We have thus shown that, if it is true that in the q -+ 0, K -+ 0 and t -+ 00 limits the dominant contributions to the kernel G, arise when the dummy wavenumbers q’, q”, . . . are as small as q and K, then the a.f. f,(t) tends asymptotically towards a single function r(yqb; K/q). This function r(~; K) obeys the following dimensionless equation : &T(T)
= [dT’G(7’/r)
r(T
-
where the kernel G is defined as
T’),
M. DE LEENER
76
GT being the asymptotic rules to calculate
expression for the general kernel G,(tJ{f,,}). The easily deduced from rules B and are given in
G(T) are
appendix A. The solution of the simplest approximation (48) to the asymptotic equation (51) has been studied (see RDL IV and ref. 6) and has shown very good agreement with experiment 1). Therefore, although it is beyond our present mathematical ability to establish the convergence of the series exas well as the existence of a solution to the nonlinear pansion for G(T), integrodifferential equation (51), it seems reasonable to admit that this equation has a meaning. This then justifies a posteriori our hypothesis that, in any contribution to the kernel Gg, values of the dummy wavenumbers of the same order of magnitude as q and K dominate near the critical point. Let us, however, insist again on the essential limitation of our results: they only apply to a Heisenberg magnet in the Weiss limit, i.e. with long-range interactions. We shall now examine the terms neglected in the calculation above, i.e. those involving sums over wavenumbers in region II (> 40). We shall show that these corrections all appear in eq. (51) multiplied by a factor qa, with 012 #, and are thus negligible in the asymptotic limit we are considering. b) Behaviour of the neglected terms. As an example, we first consider the correction to the asymptotic form of the kernel contribution Gr) [see eq. (29)], which arises when q’ is in region I(< qo) while q” is in region II (> 40). Keeping q” finite, we expand the vertices contributions for q and q’ + 0 using, in addition to eqs. (41) and (42), the following relations valid near the critical temperature (K --f 0) t :
and
k4z-4’- dw)(1 - iM.Ja)
(54)
We then get the correction 6Gi4’(f) = -&
dq’ dq”dJr,(q’s s s cc70 =-c?o
x WC2+
-Jo IQ -
](l q’j2)
jq -
_&!J
q’12)(q’.VJq”)
(-q*v$qe)
d(K2 + q2) Q(t). (55)
t Here again, we take K = I& to be finite, thus excluding the strict hydrodynamic limit, which will be discussed in section 3c.
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
SPIN
SYSTEMS
77
The function @ is the integral over intermediate times of a product of five a.f.‘s. We wish to evaluate it in the asymptotic limit where q M q’ m K -+ 0, while t --f co as rq cc q-‘, keeping q” (and hence the characteristic time Tag) finite. There will then be a complete separation between these two classes of time scales (7-q w Tq’ > Tqe), which will allow us to apply the usual procedure leading to markoffian limits?; we find in this way: Q(t) = ; dta jdtJq+ 0
- ta)
0
x
rq-q+
M
r,+)
-
t2) rq'+q"-q(t3
r,_,,(t)
-
i2) rq-q"(t3)
r&2)
yb",
(56)
where !Pqn = ;dt, ; d&r,* (t3 - t2) rqe(t3) f q+2). 0 0
(57)
For the small-wavenumber functions rq, and rq-qf, we now use the asymptotic limit (49). Switching to the reduced wavenumbers and times defined by eqs. (36), (37) and (45) we arrive at the following correction to the kernel G(T) of the asymptotic equation [see eqs. (52) and (44)] :
, xs dz
6G@)(7) = q2(W
+
2’2 -
1)
‘I - z”2 r(Z’fT) q1 z’12
K2 + 11 -
x
$
s
dq”(z’.PJ,~)(1.~+,~)
(
-
Z’jf,)
1 - k- !Pqa. Jo >
(58)
This expression is proportional to q2 and thus negligible in the limit q --f 0. We now intend to show that this result is general, more precisely, that any correction to the kernel G appears in the asymptotic equation proportional to qa, where CY 2 8, and hence may be neglected. To achieve this aim, we need to compute (at least) a lower bound to the number P of factors q appearing in the contribution of any graph to the asymptotic equation (5 1). It will of course be helpful to introduce a more detailed diagrammatic technique, specifying whether a particular wavenumber is “small” (< qo) or “large” (> qo). Since we are not interested in denumerating the graphs, nor in calculating their contribution, we may forget about the difference between dotted and (arrowed) plain lines as well as the precise values of wavenumbers. From our central hypothesis that the dominant contributions for any “small” wavenumber q’ come from the region where q’ M q and K, both much smaller than the cutoff qo which plays thus no essential role, we t See for instance, RDL
II, eq. (111.23).
M. DE LEENER
78
(2)
(I)
(3)
(4)
q q
a;2 15) 9
-<
q )_
<+q aZ=l
a3.0
a4.0
a7=-2
a5=0
pgp
q
a,=0
as=3
Fig. 4. Vertices of the representation used in section 3b. The index q represents a wavenumber inside region I (q < 40); Q is in region II (Q > qo). The index ori below any vertex is the power of q it contributes in the asymptotic limit.
conclude that the only possible vertices are those displayed in fig. 4, where q denotes a wavenumber in region I (q -=c qo) and Q is in region II (Q > 40). When we evaluate the contribution of any graph, in addition to the explicit q dependence, a number of factors q will appear when we introduce the reduced wavenumbers defined by eqs. (36) and (37); then, when we switch to the reduced times (45), the total power of q will be correspondingly diminished. What is then the final number of factors q in front of the contribution of an arbitrary graph involving 2n vertices, nt of type i (see fig. 4) ? (1) Eq. (38) indicates that 31 factors q will appear if a graph involves I “small” dummy wavenumbers. This number may be found by choosing the largest of the two results obtained by counting the vertices where a new line q appears, when we read the graph from left to right (i = 1 to 4) and then from right to left (i = 5 to 8) : I = Max(Zr, 1s),
(59)
with Jr = nr + ~3.
12 = n5 + n7.
(60)
(2) When we expand the contribution of any vertex i, for all “small” wavenumbers around zero, and introduce the reduced variables (36) and (37), a power olt of factors q will emerge. Using eqs. (41), (42), (53) and (54), one easily computes the indices ~.f given in fig. 4. The whole graph then contributes the following number of factors q:
A = ; uani = 2nr + ns + 3na - 2nT. i=l
(61)
(3) Rules B tell us that a graph of order 2n leads to 2(n - 1) time integrals. Reduced time variables [see eq. (45)] will only be introduced for those intervals where all a.f.‘s Frrr have a time scale TV’such that T~//T~= O( 1)) i.e. q. Two upper bounds to the where all lines carry “small” wavenumbers number N of such intervals may be found by subtracting from 2(n - 1) the
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
number of vertices where at least one “large” wavenumber right (Nr) or on the left (Ns). We thus get :
79
Q appears on the
N 2 N1 = 2(n -
1) - ne -
n7 - ng
(62)
N I Ns = 2(” -
1) - ?%s- n4 - ?Ze- n7 - ns.
(63)
n3
-
n4
-
and
The introduction of every reduced time variable absorbs a factor 4’; the total power of q lost for the graph as a whole is thus QN. This result applies to the kernel G,; when we substitute it into the kinetic equation (5) and take the asymptotic limit, an additional factor q5 is lost [see eq. (52)]. We thus arrive at the conclusion that
P = 31 + A - #(N + 2),
(64)
where I, A and N obey the relations (59) to (63) and P is the number of factors q appearing in front of any contribution to the kinetic equation in the asymptotic limit, when expressed in terms of reduced time and wavenumber variables. In appendix B, using elementary conservation relations arising from the graphs structure, we prove that P = 0 if (and only if) the graph involves exclusively “small” wavenumbers, and that otherwise P 2 8.This establishes the fact that all corrections to the asymptotic equation (51) appear proportional to (at least) q* and are thus negligible. c) Connection with the dynamic scaling laws. We have thus shown that, when the critical point is approached from “above” (T 2 T,), i.e. in the limit where the wavenumber q, as well as the inverse correlation length K, go to zero, the a.f. r,(t) of a Heisenberg ferromagnet tends asymptotically for long times toward a single function r(yq%t; K/q). The only approximation made is that the number of neighbours z interacting with a given spin has been supposed to be very large. We now wish to make the connection with the dynamic scaling laws of Halperin and Hohenbergs) as described in the introduction [eqs. (3) and (4)]. To this end, we have to compute the width co&(q) of the Fourier transform fq(w) of r,(t) [see eq. (2)]. The definition of this width is of course largely arbitrary and we shall use the same convention as Resibois and Piettesr: wE(q)-l
=
&co = 0) =O~dV,(t).
Using the asymptotic
form (49) of r,(t),
we then immediately
get that, near
t The results below are of course general and could be derived as well with Halperin and Hohenberg’s definition of the width2).
M. DE LEENER
80
the critical
point, 00
with the dimensionless the DSA homogeneity
time defined by eq. (45) ; this result is identical assumption
with
[eq. (4)] :
(-&7) = Pfb+l)~
(67)
with a = 4 and /(K/4) = Y( p,rcT;
‘+))-l*
(68)
It is then evident that the asymptotic of the width o&), i.e. that r,(t)
M r(&;
K/q) = F&(q)
form (45) may be rewritten
t; K/q).
Taking the Fourier transform of this expression immediately find that, near the critical point,
in terms
(69) as defined in eq. (2), we
(70) confirming eq. (3). Hence, we prove that the Heisenberg ferromagnet (with long-range interactions) satisfies the DSA with the value Q for the power a, as suggested by Halperin and Hohenberg and verified experimentallyl). In addition to this result, we established the asymptotic equation (51) satisfied by the a.f., wherefrom we can compute the unknown functions C and f of eqs. (68) and (70) [see RDL IV and ref. 61. We shall here consider only the qualitative behaviour of f(x). For any finite value of x = K/q, the analysis developed in section 3 (a and b) showed that WIG(q) goes to zero near the critical point, as 4%; thus, as long
as x remains finite, so does f(x). On the other hand, the behaviour of f(x) region” above T,) has still to be when x + co (in the “hydrodynamic determined, since our analysis above does not apply there. We have studied the general kinetic equation (5) in this particular asymptotic limit, to all orders of approximation to the kernel G,. It appears that, when x --f co, the dominant contributions to the kernel arise when the wavenumbers q’ of the intermediate states have the same order of magnitude as K. The characteristic decay time of the kernel, although very large since K + 0, is then much smaller than that of the a.f. rq when q +- 0 with q/K --f 0. Exactly as in the high-temperature case t, this complete separation of time scales transforms t See again RDL II, eq. (II. 23).
DYNAMIC
the kinetic
SCALING
LAWS
FOR
HEISENBERG
into its markoffian
equation
SPIN
asymptotic
limit,
SYSTEMS
81
i.e. a diffusion
equation : &r,(t)
= --DqV&.
(70)
We found that the diffusion critical
limit
equations
of the
coefficient
,,hydrodynamic”
D vanishes regime
as K* when
clearly
satisfies
K
-+ 0. This the DSA
(70) and (67) with
CJ,&) Cc Dq2 Cc K’q2
(71)
;
hence, f(x -+ co) K d.
(72)
To establish these results in full generality, the analysis follows exactly the same lines as that given in sections 3 (a and b) for finite values of ~/q. To avoid tedious lengthiness, we shall therefore skip those details and limit ourselves to briefly examining the case of the lowest-order kernel Gi2’. We start again from eq. (28) and make use of the critical expansions (41) and (42). If the dominant contributions to the sum over q’ arise when q’ w K, we may approximate the resulting expression taking q < q’ w K. The time scales of the a.f.‘s 74 > TV’ are:then completely separated and the corresponding kinetic
equation
tends towards its markoffian
limit:
(73) (as before, A is the volume of the first Brillouin zone). To determine how the expression between brackets, the diffusion coefficient, depends on K, we use the fact that near the critical point, the a.f. r,s(t) satisfies the DSA expressions (69) and (67), whence
(74 On the other hand, introducing 5’ =
q’/K
the reduced variable (75)
makes it clear that the integral over q’ is proportional to K3. Altogether we thus find: D cc K~K-$ cx Kt as stated above. This ends our analysis of the critical behaviour of the Heisenberg ferromagnet in the Weiss limit, from which we conclude that the agreement is complete between our results and the predictions of the dynamic scaling assumptions of Halperin and Hohenberg. 4. Dynamic ferromagnets,
scaling for antifewomagnets. As soon as a theory exists for it can be developed for antiferromagnets as well. We have
M. DE LEENER
82
thus applied the whole argumentation of section 3 to a Heisenberg (twosublattice) isotropic antiferromagnet. Since the analysis follows the same lines as for ferromagnets, we shall limit ourselves to stating the most important points without giving the mathematical proofs in detail. It is interesting to notice here the essential difference between ferro- and antiferromagnets from the point of view of critical dynamic behaviour. For ferromagnets, at the transition temperature TC, the characteristic time scale Tq of the a.f. r,(t) is expected to diverge at the single point Q = 0; this is due to the superposition of two effects: (1) the temperature-independent “kinematic slowing-down” arising from the conservation of the total angular momentum [see eq. (1 l)] ; (2) the critical “thermodynamic slowing-down” due to the divergence of the equilibrium a.f. 4q [see eq. (32)]. On the other hand, for antiferromagnets, these two phenomena occur at different points in wavenumber space : the conservation of angular momentum still implies that r,+(t) is a constant but the divergence of 4q now occurs at some point q = q* of the Brillouin zone, characterizing the staggered spin ordering. The exchange interaction Jq can be expanded around this point, where it has its maximum :
Jq = Jq* (1 - 6 Iq - q*12+ . ..I.
(76)
in the Weiss limit, to which the whole of this work is devoted, as follows :
4Q=
+q diverges
J /+2
+
,;*_
(77)
q*j2) '
where the inverse correlation length K vanishes linearly at the Neel ature TN [see eq. (33)]. Thus, critical behaviour of the a.f. r,(t) is expected in both regions q = 0 and q = q*. We have indeed shown that, when T + TN (from the long-time behaviour of r,(t) near those points has the following totic form : rd4 r,-+,*@)
temperaround above), asymp-
= r(Yq’t ; K/q), es r*b’
where the constant
iq -
q*i’ t; ‘&
-
q*j),
(78)
y is given by
ys = 2J;./A
(79)
[compare with eq. (44)]. The functions r and r* obey two coupled equations:
a,r*(T) =iTdr’G*(T’]r,r’)
r*(T
-
T’),
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
where the kernels are nonlinear functionals as in the ferromagnetic
case (appendix
obtainable from well-defined
diagrammatic
SPIN
of the unknowns
B), by an infinite
SYSTEMS
83
and are given, series of terms
rules. This result was established
as in sections 3 (a and b) by showing that, near the critical point, the dominant contributions to the kernel GQ in the general kinetic equation (5) arise when all intermediate wavenumbers are in the vicinity of the points 4 = 0 or q = q*. As is done in section 3c, it is then easy to deduce from (78) that the critical behaviour of the Fourier transform p*(o) is described, in accordance with the dynamical scaling assumptions2), by two (different) functions of the form given in eqs. (3) and (4) with K = q and k = jq - q*I, respectively, and with a = $. But our kinetic theory of course permits us to go beyond these qualitative statements and to determine all functions precisely. In particular, we have examined the functions f(x) and f*( x ) ; we showed that they remain finite fdr all finite values of x and that
(81) in complete accordance for antiferromagnets.
with the predictions
of Halperin
and Hohenberg
5. Discussion. In this paper, we have studied the critical behaviour of the time-dependent autocorrelation function in isotropic ferro- and antiferromagnetic systems described by the Heisenberg model. Our analysis is exact within the limits of validity of our starting point, the general kinetic equation (S), which applies to the paramagnetic region (T 2 T,) and to the Weiss limit 7), where the number z of neighbours interacting with a given spin is taken to be infinite. The extension of the theory below Tc has been studied by Rbsibois and Dewel and will be published soon. The Weiss approximation is a serious limitation when we compare our theory with the experimentsl) and with their remarkably accurate phenomenological description, the dynamic scaling assumptions of Halperin and Hohenbergz). Indeed, the exchange interaction is known to be of very short range (z = 6-12) ; moreover, the Weiss limit leads to wrong equilibrium critical indices*). It may then seem surprising that our results are in accordance with the qualitative predictions of the DSA and, even more, that there seems to be good quantitative accordance with the experimental facts. This problem has already been discussed in reference 5 (RDL IV, section V) which will allow us to be very brief. Concerning the comparison with experiments on systems where z is not large, we believe that the Weiss approximation may well be better for dynamic correlation functions than
a4
M. DE LEENER
for other quantities. Indeed, it is known that the Ornstein-Zernike form of the equilibrium correlations [eq. (32)] is very nearly correct (the critical index 7 measuring the deviation from this approximation is experimentally very small) 8). It is therefore possible that the self-consistent description that we obtained for the dynamics of the correlations would not be deeply modified by the finiteness of z. However, to get agreement with experiments, we must use the measured temperature dependence of the correlation length relation [K-l cc (T Tc)‘, withv = 0.71, instead of the Ornstein-Zernike K -1 [eq. (33)l. The accordance of the qualitative predictions of our theory with the dynamic scaling assumptions leads to the following question: are the DSA correct outside the Weiss limit? Mori and Okamoto3) have suggested that the answer is negative. In fact, this seems plausible, since the static scaling laws themselves are believed to be only approximately correct in threedimensional systems. Although unable to answer this fundamental question, our work justifies the DSA for a specific model; in fact, it goes beyond them and furnishes a detailed quantitative description. The author wishes to thank Professor I. Prigogine Acknowledgments. for his interest in this work and Professor P. RCsibois for many helpful discussions.
APPENDIX
A
The reader will find here the rules permitting to calculate the kernel G(T; K) of the asymptotic equation (51) describing the critical behaviour of the autocorrelation function in ferromagnets. The vocabulary used is defined at the beginning of section 2; note, however, that here the “times” are dimensionless variables as defined by eqs. (36), and “wavenumbers” (37) and (45). Rules C: (1) Connecting dotted and plain arrowed lines through the vertices of fig. 2, draw all distinct graphs of even order 2n, irreducible and containing no self-energy insertion, starting and ending with a dotted line labelled with a wavenumber 1 of unit length and arbitrary direction. The times attached to the vertices grow from right to left
(T2n --
T
>
7212-l
>
..a
>
72
>
71
=
0).
The contribution of any graph is found as follows. (2) To every vertex, associate the expression given in fig. 2 (bottom) ; to every (plain or dotted) line with wavenumber q’, between the times 7 and T’, associate a function Q’)(T T’)].
DYNAMIC
SCALING
(3) Integrate
LAWS
FOR
HEISENBERG
over the ordered intermediate
jdrz,_l~~~s,_z
SPIN
85
SYSTEMS
times:
...Oj&
0
and integrate (over all space) over the n dummy wavenumbers.
APPENDIX
B
We consider an arbitrary graph made up of lines q and Q connected by 2n vertices of the types shown in fig. 4 (nr vertices of type i, with i = 1 to 8) ; the graph starts and ends with a single line q. The numbers 1~~obey two simple conservation laws arising from the fact that the numbers of left and right ends of q (and Q) lines must be equal:
fll+
fi3 +
2?%3 +
n4
126 = =
%? +
a5 +
127,
P.1)
2~~6 -+- ?%3.
w?
Adding these relations, we get: n1-k
fin2 +
123 +
124 =
n5 +
fin6+
127 -k n3=
(B.3)
fl.
Note that (B.3) implies that (62) and (63) may be rewritten as N I Ni = ~21+ ?25+ %j -
2,
(B.4)
N I Ns = nl + ns + ns -
2.
(B.5)
Using these relations, as well as the definitions that
P = 3Z+ A -
(59) to (61), we shall show
$(N + 2)
(B.6)
is equal to zero if (and only if) the graph involves no line Q, and that otherwise P 2 8. Let us start by considering the particular family of “dominant” graphs that have lead us to the asymptotic equation (5 I) ; they only involve “small” wavenumber 1~1= n5 =
lines q and thus only contain
vertices
of types
1 and 5:
n.By mere inspection, we then get:
1 = 11 = 12 = n, N = N1 = Nz = 2n,
A = 2n
and thus
P = 0,
(B-7)
as it should. Passing now to the case of “non-dominant”
graphs, we remark that we
M. DE LEENER
86
necessarily have and
722> 0
n6 >
(‘3.8)
0,
since only the vertices 2 and 6 transform a line 4 into lines Q. We then have three cases to consider separately: a) WI + a3 > 1~5 + n7. Then eqs. (B.l) to (B.5) tell us that (B.9)
N2--l=fi2--6=(fil+%3)-((n5+%7)>0;
hence, using Nr as an upper bound to N, we get from eqs. (59) to (64) : P 2 PI = 3zr+
A -
#(Nr + 2)
= 3(nl+
n3) +
2fil-t fi2 f
=
fi3) -
@5
f(nl +
+
3fi6 -
fi7) f
2n7 -
&(a3 +
@I$
3fi6 f
n5 +
%6)
n7);
Eqs. (B.8) and (B.9) then imply that (13.10)
P 2 5. b) n5 + ~7 > rtl+ ~3. Along the same lines, we get: Nl-
N2
=
fi6 -
fl2 =
(a5 +
'yz7)-
(?Zl +
923) >
0,
(B.11)
whence, using the upper bound N2 to N: p 2 P2 = 312 + A = f(fi5 + fl7) -#1
Q(N2 + 2) +
n3) +
$(3ns + n3
n7) ;
thus (B. 12)
P 2 5. c) n1+ Here,
n3 =
Nr = N2,
a5 +
n7.
n2 =
a6
and P 2 &(3n2 + n3 + ~7). Excepting the particular case of the “dominant” we get from eq. (B.8) the bound
above, (B. 13)
p 2 #> which completes
graphs considered
the proof.
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
87
REFERENCES 1) For a review of these experiments, see for instance: Minkiewicz, V., to appear. 2) Halperin, B. and Hohenberg, P., Phys. Rev. Letters 19 (1967) 700; Phys. Rev. 177 (1969) 952. 3) Kawasaki, K., Progr. theor. Phys. (Kyoto) 39 (1968) 1133, 40 (1958) 11, 706, 930; Mori, H. and Okamoto, H., Progr. theor. Phys. (Kyoto) 40 (1968) 1287, 41 (1969) 1177. 4) Bennett, P. and Martin, P., Phys. Rev. 138 (1965) 607; Kawasaki, K., J. Phys. Chem. Solids 28 (1967) 1277; Progr. theor. Phys. (Kyoto) 39 (1968) 285; Villain, J., J. Phys. (France) 29 (1968) 321, 687. 5) RBsibois, P. and De Leener, M., Phys. Rev. 152 (1966) 305,318, 178 (1969) 805, 8 19 (hereafter referred to as RDL I, II, III and IV, respectively); Phys. Letters 25A (1967) 65. 6) RBsibois, P. and Piette, C., Phys. Rev. Letters 24 (1970) 514. 7) See Brout, R., Phase Transitions, W. A. Benjamin, Inc. (New York, 1965). 8) See, for instance, Fisher, M., Rep. Progr. Phys. 30 (1967) 615. 9) R&ibois, P. and Dewel, G., Ann. Phys., to appear.
DYNAMIC SPIN
SCALING
SYSTEMS
WITH
LAWS
FOR
LONG-RANGE
HEISENBERG INTERACTIONS
M. DE LEENER Service de Chimie-Physique
II, Universitk Libre de Bruxelles, Belgique
Received 2 April 1971
synopsis The dynamic scaling laws proposed by Halperin and Hohenberg are shown to be correct above Tc for Heisenberg magnets with long-range interactions. We derive the asymptotic kinetic equations describing the critical behaviour of the spin correlation functions.
1. Intro&&ion.
During the last few years, considerable progress has been made towards the understanding of the dynamic properties of ferroand antiferromagnets near their critical point, following extensive experimental and theoretical studies of the spin autocorrelation function (a.f.), which we define as
r,(t)
= ~
where the brackets denote an average over the canonical equilibrium ensemble; in the Heisenberg model, S,(t) is the q-wavenumber Fourier transform
of the localized
spin operator
Heisenberg picture. A series of recent neutron-scattering transform of the a.f.,
pq(o) =
rdt r,(t)
Sa at the lattice
experimentsi)
site a, in the
measuring the Fourier
(2)
eiot,
0
showed remarkable agreement with the predictions of the dynamic scaling assumptions (DSA) proposed by Halperin and Hohenbergs) and studied by Kawasaki, Mori and Okamotoa). Near the critical point, where the macroscopic equilibrium properties are expected to be dominated by the existence of a unique correlation length ~-1 which becomes infinite at the critical temperature Tc,the DSA state that: 62
DYNAMIC
SCALING
LAWS
(a) when the wavenumber behaviour is expected, fh(o)
FOR
HEISENBERG
SPIN
SYSTEMS
63
q tends
towards any value qc where critical approaches a function
where k = Iq - qc] (for ferromagnets, qc = 0; for antiferromagnets, it has two values, qc = 0 and q*, the latter wavenumber characterizing the staggered spin ordering). (b) the width on(k) is a homogeneous function of K and k: on(k) = kaf(K/k).
(4
Halperin and Hohenberg subsequently predicted the value of a (Q for ferromagnets and 8 for antiferromagnets) as well as the limiting behaviour of the functions f(x) when x + 0 (transition region) and x --f 00 (hydrodynamic region). However powerful the DSA scheme turns out to be, kinetic theories, applying to well-defined microscopic models the techniques of irreversible statistical mechanics, are necessary to test the DSA and to go beyond them, i.e. to determine the unknown functions C and f. Various authors have worked along these line!+). In particular, Resibois and the authors) have studied the isotropic Heisenberg model in the Weiss limitv), where the number z of neighbours interacting with a given spin tends to infinity. The pertinence of this model is most questionable when one deals with real systems where the interactions are short-ranged; moreover, it is known to lead to wrong equilibrium critical indicess). Nevertheless, the simplicity of the model has allowed us to develop an exact theory of its dynamical properties, which turns out to agree with all essential features of the experimental results. We have shown that, at any temperature in the paramagnetic region (T 2 T,), the a.f. f &) obeys the following non-markoffian kinetic equation : W’,(t)
= jdt’(;,(t’,{Q})
f,(t
- t’),
(5)
in which the kernel exhibits a nonlinear dependence on rQf itself and is given by an infinite series of terms which can be associated with graphs. We then studied the properties of the kinetic equation (5) at the critical temperature (T = T,), limiting ourselves to the lowest-order approximation to the kernel. We showed that, in the short-wavenumber and long-time limit, the a.f. P,(t) agrees completely with the predictions of Halperin and Hohenberg’s DSA. Furthermore, we derived the (approximate) equation describing the asymptotic behaviour of the a.f. and concluded that the essential feature of the critical region is that Z’,(t) never reaches a diffusion regime, but decays to zero through damped oscillations. Exploring further the predictions of
M. DE LEENER
64
this kinetic theory, RCsibois and Piettee) have recently extended our work away from the critical temperature (K f 0) and computed the function f(x) of eq. (4); th e agreement with experiment The generality of these results is, however,
is quite satisfactory. limited by the (second-order)
approximation made on the kernel of the kinetic equation (5). In this paper, we remedy this weakness and generalize our results to all orders in the expansion of G,. To this end, we first show that our rules for evaluating
GQ
can be greatly simplified and expressed completely in wavenumber variables instead of the initial spin coordinates; this is the content of section 2. Section 3 is devoted to a detailed study of the ferromagnetic case near the critical point: we derive the asymptotic kinetic equation describing the critical behaviour of the a.f.,show that the corrections are negligible and make the connection with the DSA. The case of antiferromagnets is briefly considered in section 4. Finally, section 5 ends this paper with a short discussion. 2. Simplified
rules for the kernel Gq. For the sake of clarity, we shall here
repeat the rules for evaluating the various contributions to Gab, the Fourier transform (F.t.) of G,t. The reader is, however, urged to refer to RDL III for further explanations. The graphs for Gab(t) are made of dotted lines and plain arrowed lines, connected through the vertices given in fig. 1. Each vertex is labelled with a time between 0 and t, the m times increasing from right to left (tm. = t > t,-1 > . . . > tz > tl = 0). The graphs must start (at time 0) and end (at time t) with a single dotted line; since the vertices “create” or “destroy” one line, the numbers of vertices must be even. Except for the first and last dotted lines, which are labelled with the spin indices b and a, respectively, all lines carry two spin indices (one near each end) which may be identical or different. Only a limited class of graphs should be retained, those which (1) are irreducible, i.e. contain no intermediate state (between two subsequent times) involving only one line; (2) contain no self-energy insertion, i.e. no part starting or ending with a single line (note that any graph, as a whole, has such a self-energy structure). The following rules should be followed to evaluate the contribution of any graph for Gab: t In what follows, we shall always define the Fourier transform of any A ab = A(a - b) as A,
=
2 Ama &-W--o), b
Aae = ;
C A, &*(a--b), 4 where N is the number of spins in the system.
funCtiOn
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
65
k
A-
(1)
c l
= ‘kl
Iid
k
(2’4 “;‘;:
CM9 (40) (4b)
k
=2 ‘k, MI
= $1
$1
k _!_ = @ kl Iii1 >I = 2@k,(Jkm-J,m )Mm
Fig. 1. The vertices and their contribution in coordinate space. Inverting the direction of the arrows changes the sign of the contribution.
Rules A: (1) To everydottedline, except the first and last ones (labelled b and a), starting with spin 1 at time tl and ending with spin k at time ts, associate a factor 4MkMlfkl(t2 - tl), where r,,(t) is the F.t. of T,(t) and the variable Mk attached to every spin k in the graph can take the values +fr, 0, -4. Similarly, to every plain line, associate a factor? 24&, OT+1’&,,oh(t2 if k f
-
h) t
1, or
&*,&k(t2
-
h)
if k = 1, where the displacement as follows : rfpkf(Md
= f(Mk
f
f-lrq+l~
*
@‘k acts on any function
+,a).
(2) To each vertex, associate the operator r& is defined as 17ztr=
opearator
of Mk
(6) the contribution
indicated
in fig. 1. There,
rl+lrQr,
(7)
is the Heisenberg coupling between spins k and I, with F.t. Jq, /3= (kT)-l and @kE is the equilibrium correlation function in the Weiss limit, the F.t. of which is 7)
Jkl
4~ =
Jd(1 - 8BJd.
(8)
(3) Order all operators and M-dependent factors as they appear in the graph. Multiply on the left by a factor M, and on the right by Mb and take the trace (i.e. the sum Z;M,=++) over all spins appearing in the graph. t Everywhere in this paper, a,,, 6Z,Y =
1 0
x=y x #Y.
represents the Kronecker delta function:
M. DE LEENER
66
(4) Multiply in front by 4(3)”
(-)”
; dtzn_rij”& 0
2n
-2
..-OTdta.
0
for a graph of order 2n (with 2n vertices) cluding a and 13). (5) Sum over all m -
involving
m distinct
spins (in-
2 dummy spin indices.
To draw general conclusions about the kinetic equation (.5), we are confronted with two technical difficulties : (1) Rules A are rather awkward, in particular when they involve numerous traces CMl_+$ over the variables Mi. (2) They permit us to calculate Gab in spin coordinate space and the result has to be Fourier transformed afterwards. Yet, as can be verified on the fourth-order contribution Gk4’t, the correct short-wavenumber behaviour is not observed for each individual graph, but only for the sum of all of them at every order. To overcome these difficulties, it is thus desirable to establish compact rules to compute G, directly in wavenumber space, such that the contributions of different graphs are grouped correctly: this turns out to be fairly easy once it is realized that no error is made when one calculates Gab from rules A without paying attention to the special cases where some of the spins a, k, 1, ..,, b are identical. This implies two proofs. First, we must show that, although the graphs for G,, are different from those for Gab (a # b), the correct G aa is obtained by setting b = a in the expression for Gab. We first notice that the F.t. of Gab, #, = c Gab eiq’(b-a),
(9)
all b
could only differ from the exact kernel G, = G,, +
C
Gab eiqatbma),
(10)
b(#a)
by a (q-independent) sum rule r@(t)
=
constant
term.
But the a.f. satisfies
1,
the well-known
(‘1)
for all times t, arising from the conservation of the total magnetization in the Heisenberg spin system; from the kinetic equation (5), we then deduce the sum rule for G,: G,=,,(t) = 0. To ensure that 0,
(12) is identical
t See RDL IV, section IV.
to G,, it is thus sufficient
to check that 8,
DYNAMIC
SCALING FOR HEISENBERG
SPIN SYSTEMS
67
goes to zero in the limit q -+ 0 for all times. This will be verified a poster&i, at the end of this section, to all orders 2n of the expansion. Secondly, we have to prove that, although the contribution of any line with two spin indices (K # I) differs from that with a single index (K = Z), the correct contribution for any graph with a line (F2Fz) can be obtained evaluating it first for the general case (KZ)and setting k = I afterwards.
by
This can be easily verified from rules A(l), A(3) and A(4). As an example, the contribution of any graph with a plain line (kk) ends with the following structure : . . . &fJkkf(~k
(13)
= 0, (wn#k~)~
Inserting a factor 1 =
we
c MI=*+
(‘4)
q+*‘8M,,o,
obtain = 0, {Mmzk,
1)) ik=Z,
(15)
which is precisely the contribution of the same graph with a line (kl), when k is afterwards set equal to 1. We have thus shown that, to calculate the kernel G,, it is sufficient to evaluate the contribution of every graph for G,a taking all spins a, k, I, . . . , b to be different and to sum the result over the dummy spin indices without any exclusion. The result is then Fourier-transformed following eq. (9) or equivalently :
Gg=
-&-z c Gab e a
b
iq.(b-a)
.
(16)
This theorem has the important consequence that the trace over the M variables can easily be taken. Indeed, if all spins in a graph are different, every spin k only appears (once or twice) near a single vertex. Moreover, the variables Mk and displacement operators $‘” appearing in the line contributions [see rules A( I)] can be factored out and associated with the vertex : at every end of a dotted line, we have a factor Mkt ; a plain line contributes a factor 6Mr,Oat itsleft end, and~f1k6nl,,o at its right end. It is then elementary work to combine these factors with the vertex contributions of fig. 1, to multiply by a factor + per spin and to take the trace CMrzft over the two or three spin variables.
7 Note that no special situation arises at the first and last vertices, since rule A(3) tells us to multiply on the left (and right) by M, (and Mb).
M. DE LEENER
68
As an example,
=
vertex
2@kz(C
(3a) leads to the expression
dM,,_&(C Mk
JQ’;)(~)~
=
&@kZ.
Ml
In the same way, we would find for vertex
(‘7)
(3b) the result
-&@kl
(18)
and for vertex @@kz(Jzm
(3c), -
_/km).
(19)
Similar expressions would be obtained for the other vertices of fig. 1. When the trace over the M variables has been taken, it is an easy job to formulate the rules for G, directly in wave-number space. Indeed, the a.f. which appears in the contribution of every line [see rules A(l)] may be expressed in terms of its F.t.: p,,(t)
=
-+
C f*(t)
eiqack--l);
Q
hence, we can associate with every plain or dotted line a factor f a(t) at wavenumber q (to be summed over) and a factor exp(+(or -) iq. k) with the vertex at the left (or right) end of the line. All factors depending upon the coordinates of the spins appearing near a given vertex (i.e. the contribution of the vertex itself, as calculated above, times the exponential factors) can then immediately be summed over these coordinates and the result associated with the vertex t. Moreover, since the lines arriving at the vertex now carry wavenumber indices, the contributions of the vertices with the same general structure (i.e. grouped under the same number in fig. 1) can be added. Coming back to our example (vertices 3a, 3b and 3c), we shall associate a wavenumber qr to the plain line at the right of the vertex, qs to the plain line at the left and q3 to the dotted line. For vertex (3a), we then find the following contribution : K+ F $qjkl eiql*k e-iqa.k e-iq8.l = z
-+ pkl
ei(qa-qd*U-W ei(ql-qa-qd*l
= 3Y <qP+Qah-QnJ
(21)
t Note again that, since we calculate G, through eq. (16), spins a and b play no special role.
DYNAMIC
SCALING LAWS FOR HEISENBERG
where we used the representation J,,,,,
of the Kronecker
SPIN SYSTEMS
delta function:
1 = N 5 ei(ql-qa)*r,
as well as the inversion
(22)
symmetry
of the correlation
function
&l: (23)
+r = &P In the same way, one finds for vertex
(3b) the result (24)
--*N&r,, Q*+G#% and for vertex
(3c),
fN~WL+Qs(6r1
-
(25)
&-_q*) @JCL.
Adding expressions (21), (24) and (25), we are thus led to associate . single graph 3, in wavenumber language, the contribution W,,,
69
q~+dib-as
-
4~2)(1 -
to the
iiB.Jd
the second expression follows from the definition [eq. (8)] of $q after some algebraic manipulations. Note that the total wavenumber is conserved at the vertex, a fact which arises from the translation invariance of the system. Similar calculations can be performed for the other vertices of fig. 1. Except for a factor ftN which will be shortly taken care of, the four resulting contributions are shown in fig. 2( 1). To complete the derivation of our new rules in wavenumber space, there
(a)
(b) q’
s’!
_
,q’
a’
,,*;ll_q” J (1)
( Jqa-q” -J,4
--
(1) (@q’-q”-$q4 (1-pJq42) (t-~Jo~2) “Jq’-q”-Jq”l(l-PJq~~/2)(i-PJql_q,,/2)
(2)
(jql_q”j2_q”2)
(2) (lq’_q”~2_q”2)(K2+,12)
(K2,q”2)(K2+Iq,_q~~,2) Fig. 2. The four vertices in wavenumber language, with their contribution (1) to the general kinetic equation and (2) to the asymptotic equation for ferromagnets. Inverting the direction of the arrows only changes the sign of the contributions.
M. DE
70
LEENEK
remains to determine what factor has to be set in front. From the fact that every vertex involves two plain lines and one dotted line, it follows that any graph involving 2n vertices contains 2n plain lines and (n - 1) dotted lines (excluding both of the dotted lines which start and end the graph). Using rules A( 1) and A(4) as well as eqs. (16) and (20), and multiplying by $N per vertex, we then arrive at the following factor for a graph of order 2n: (-)”
;(L)““(+)“-l(y
_;;(__g
(27)
The number of factors (l/N) of course coincides with that of dummy wavenumbers to be summed over. Finally, the minus sign in eq. (27) can be suppressed by changing the sign of the contribution of the vertices (b) in fig. 2(l), since every graph of order 2n involves the same number n of vertices (a) and (b). Summarizing, we can now give the rules to evaluate the kernel G,(t) of the general kinetic equation (5). Rules B : (1) Connecting dotted and plain arrowed lines through the vertices of fig. 2, draw all distinct graphs of even order 2n, irreducible and containing no self-energy insertion, starting and ending with a dotted line with wavenumber 4. The times attached to the vertices grow from right to left (ts, = t > tsn_i > . . . >ts > tr = 0). The contribution of any graph is found as follows. (2) To every vertex, associate the expression given in fig. 2( 1); to every (plain or dotted) line with wavenumber q’, between the times t and t’, associate a function r,,(t - t’). (3) Multiply by (1 /N)n. (4) Integrate over the ordered intermediate times: ;dlzn:~-dtz,_r
. . . (it2
0
and sum over the n dummy wavenumbers. When one compares rules A and rules B, the simplicity of the latter ones is evident. As an example, fig. 3 shows all the graphs for the second- and
Fig. 3. Examples
of graphs
in wavenumber language: (a) the only (2) the two graphs for Gh*).
graph
for
Gh2);
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
fourth-order contributions to the kernel, Gi2’ is readily found to be
SPIN
SYSTEMS
GL2) and GL4’. The expression
G12W = - +- 2Q’ (JQ-q’- Jg’)(&Pq’-
71
of
$bg’)
x (1 - tBJg)r,44 r,-,+)
the result given in RDL IV, eq. (A.5).
confirming
For Gr’, we get:
-
J&J(J4~+v-a - Jm7”)
x (4Q’+Q”--9 - kJ*) (1 - MJcl-)bh-4”- dcT)(l- 3PJd x 5 dt3 ;dt&,(t 0
x
-
t3)
rg-g’@-
t2)
0
rg*+g4ts
-
is) ra-g"(t3) r,-(tz).
(29)
From rules B, it is now easy to verify the stated property that the kernel G, goes to zero when 4 + 0 [eq. (12)]. Indeed, any graph for G, must start and end with a dotted line 4. From fig. 2(l), we thus conclude that its contribution must involve a factor
(JQ-d- Jcf)(Ja-v - JfiP~ q2. This completes
the proof of the validity
(30) of our rules B.
3. Dynamic scaling for ferromagnets. The compactness of rules kernel of the kinetic equation (5) will now allow us to study the and short-wavenumber behaviour of the a.f. r,r near the critical shall specifically consider an isotropic Heisenberg ferromagnet, Jg of the exchange interaction will thus possess a simple Taylor
B for the long-time point. We the F.t. expansion
around 4 = 0:
Jq = Jo (1 - dq2+ . ..). where 6 is a positive constant. It can be checked from rules B that the time rq characterizing
(31)
the decay of f (Itends for large 4 towards a finite constant Tn, which should remain well behaved when the critical temperature Tc is approached. On the contrary, Tg is expected to become singular near the critical point, i.e. when 4 --f 0 and
M. DE LEENER
72
T + TC, where the equilibrium correlation function (bs.diverges; in the Weiss limit, to which the whole of this work is devoted, we have [see eqs. (8) and
(31)l:
(32) where JO = 2kT, and K, the inverse of the correlation
K2 = (l/d)C(T -
length vanishes at T, : (33)
T,)/T,l.
The study of the lowest-order approximation to the kinetic equation (5) t leads us to expect that the DSA should be correct near the critical point (i.e., that 7;l = ~~(4) should obey eq. (4) with a = &) and that this non-analytic behaviour arises from the fact that any contribution to the kernel G, involves (1) equilibrium correlation functions 4q, (for the intermediate states with wavenumbers 4’) which diverge when 4’ and K become small; (2) a.f.‘s rqt which decay very slowly at short wavenumbers. We shall thus take, as the starting point of our analysis, the hypothesis that, near the critical point (K and CJ+ 0), the long-time behaviour of G,(t) is dominated by those contributions which arise when the intermediate state wavenumbers q’, q”, . . . are all as small as q and K. This hypothesis will only be verified a fiosteriori through the existence of a solution to the asymptotic equation. Following this idea, we shall number inside the first Brillouin
separate any sum over a dummy zone (B) as follows:
wave-
E = c1 + PI, P’
(34)
with the condition (35) which can always be satisfied if q and K go to zero. In the “asymptotic” domain I, dominant contributions appear when q’ es q and K ; as long as the ratio
K =
the wavenumbers
I’ = qI/q. f See RDL
IV and reference
corresponds
q’
(and
K)
(37)
t We here exclude pected.
to
(36)
K/q
remains finite*, it is then natural to measure in units of q, by introducing reduced variables
which
are expected
6.
the particular
case where
to the “hydrodynamic
q
region”
and K tend to zero with K/q + ~0, where diffusive behaviour is ex-
We shall come back to this point in section 3c.
DYNAMIC
SCALING
LAWS
If our starting hypothesis
FOR
is correct,
HEISENBERG
SPIN
SYSTEMS
73
the sum over 4’ within domain I will
converge rapidly when 4 and K become small and we shall have:
C1 = q 9’
s s dq’ = y
Q’
dz’ -
q+o
2’iqoln
Nq” dz’, d
(38)
s all spaoe
where ,4 is the volume of the first Brillouin zone. In domain II, the a.f. rat is expected to decay after a finite time T*’ 5 rq,; we shall then estimate the corresponding integral in the limit T&/Q + 0
(4 --f 0>4’ > 40).
(39)
Since qo can be made as small as we wish when q and K tend to zero, the error introduced by this arbitrary cutoff in region II should become negligible in the limit. We shall first derive the asymptotic limit of our kinetic equation (5) by neglecting all contributions from domain II; it will be found that the a.f. r,(t) then reduces to a single function F(qQt; K). The corrections introduced by the larger values of the dummy wavenumbers will then be shown to be negligible. Finally, we shall make the connection with the DSA of Halperin and Hohenberg. a) Derivation of the asymptotic equation. the q, K + 0 and t --f 00 limit of the kinetic equation
Let us thus examine
&Z’,(t) = i dt' : Ga")(t' j{fa$ fq(t - t’), n=l
where, in every contribution to the kernel, we only retain the sums over the dummy wavenumbers in region I, where q’ < qo. We may then expand the vertices contributions of fig. 2( 1) around the origin in wavenumber space; using eqs. (31) and (32), we find for the vertices of type (a): dJo(q”2 Jc~v - Jc tq,,q~+Oj
-
Iq’ -
q”12);
(41)
and for the vertices (b) :
(Jq’-q”- Jq”) (1 -
(1 - +PJe,) @Jqn)(l - $@Jq,_q")
M Jo (f2 - k' - @i2)(K2 + q'2) (K2 + qM2)(K2+ IQ'- q"12).
(42)
To get a hint at what is going to happen, let us first return to the approximate kinetic equation obtained with the second-order kernel Gp) given by eq. (28). W e use the expanded expressions (41) and (42) and make the
M. DE LEENER
74
changes
of variables
(36) and
(37); we get:
(2’2 - 11 - 2’12)2 &l’,(t) = -y2q5(K2 + 1) ds’ (K2 + 2’2)(P + \I - z’(2) s 1
x
dt’r,,$‘)
r,,,_,#‘)
ra(t - t’),
(43)
0
where I is a unit vector parallel to Q and y2 = dJ;/A. Let us introduce T
=
(44) the dimensionless 7'
y@t,
=
variables
(45)
yq’t’
and the function
(46) eq. (43) becomes &l’,(7) = -(K2
+ 1)
7
X
J
dO,,+‘%‘)
(2’2 12 2’12)2 +2’2)(lP +(1- 2’12) J (K2 dz’
&,_.,#
- 2’1’ T’) F,(T -
7’).
0
(47)
We note that the variable q only appears in this equation as a factor in the index of every function I’. Let Fq(7) b e one solution, then F&*(T) is clearly also a solution, whatever the value of the positive factor CI. If eq. (47) possesses a unique solution, then r@(T) and r&T) must be identical functions, i.e. they cannot depend on the variable q. Hence, eq. (47) reduces to the following asymptotic equation : &F(T) = - (K2 + 1) 7
(2’2 12 z’py +2’2)(K2 +\I - z’(2) J (K2 dz’
dT’r(z’%‘) r(ll
X
-
Z’jf
T’) r(~
-
T’),
(48)
s 0
for the single function rP (t) M
r(T;
K)
=
r(y@t;
K/Q),
(49)
describing the limiting behaviour of the a.f. I’,(t) near the critical point (i.e. when q and K tend to zero) and for long times (i.e. for t -+ co as q-‘, so that T remains finite).
DYNAMIC SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
75
A brief remark should be made concerning the uniqueness of the solution to eq. (47): this property is not at all evident. Indeed, there clearly is a unique solution to the general kinetic equation (40), with the initial value f n(O) = 1 for any 4 [see eq. (l)], but this initial condition does not necessarily apply to the asymptotic limit of this equation for long times: it is easily verified that the solution of eq. (47) is singular at T = 0, i.e. possesses no Taylor expansion around this point. Note also that the evolution of the asymptotic a.f. F(7) has no causal character: the right-hand side of eq. (48) involves values of r(x) for all x, larger and smaller than 7. We now wish to show that the behaviour of the a.f., described by eq. (49), is a property of the general kinetic equation (40) with the complete kernel. To this aim, it is important to note that the proof given above relies on the single fact that the changes of variables (36), (37) and (45) have made the asymptotic equation (48) dimensionless. Extending the proof to all orders of approximation to the kernel G, thus amounts to showing that this nondimensionality is general, i.e. that the factors 4 (and JO, 6, . . .) which appear through the small-q-and-K expansion of the vertices contributions and through the changes of variables (36)-(38) are exactly eliminated through the introduction of dimensionless times T as defined by eq. (45). To establish this fact, it is sufficient to note from rules B that any contribution of order 2n to the kernel GP.is represented by a graph with n vertices of type (a) and 1z of type (b) (see fig. 2) and that it involves n summations over dummy wavenumbers and (PZ- 2) integrals over intermediate times. Using eqs. (41) and (42), we thus conclude that, when we switch to reduced wavenumbers as in eqs. (36)-(38), the variable 4 will appear in front in a factor
When we then substitute the resulting contribution to Gr’ into the kinetic equation (40), we ascertain that the introduction of n reduced times as defined by eq. (45) exactly cancels the factor (50). Simultaneously introducing functions F,(T) through eq. (46), we remark again that the variable q only appears as an arbitrary factor in their indices; hence the solution to the asymptotic equation cannot depend on q. We have thus shown that, if it is true that in the q -+ 0, K -+ 0 and t -+ 00 limits the dominant contributions to the kernel G, arise when the dummy wavenumbers q’, q”, . . . are as small as q and K, then the a.f. f,(t) tends asymptotically towards a single function r(yqb; K/q). This function r(~; K) obeys the following dimensionless equation : &T(T)
= [dT’G(7’/r)
r(T
-
where the kernel G is defined as
T’),
M. DE LEENER
76
GT being the asymptotic rules to calculate
expression for the general kernel G,(tJ{f,,}). The easily deduced from rules B and are given in
G(T) are
appendix A. The solution of the simplest approximation (48) to the asymptotic equation (51) has been studied (see RDL IV and ref. 6) and has shown very good agreement with experiment 1). Therefore, although it is beyond our present mathematical ability to establish the convergence of the series exas well as the existence of a solution to the nonlinear pansion for G(T), integrodifferential equation (51), it seems reasonable to admit that this equation has a meaning. This then justifies a posteriori our hypothesis that, in any contribution to the kernel Gg, values of the dummy wavenumbers of the same order of magnitude as q and K dominate near the critical point. Let us, however, insist again on the essential limitation of our results: they only apply to a Heisenberg magnet in the Weiss limit, i.e. with long-range interactions. We shall now examine the terms neglected in the calculation above, i.e. those involving sums over wavenumbers in region II (> 40). We shall show that these corrections all appear in eq. (51) multiplied by a factor qa, with 012 #, and are thus negligible in the asymptotic limit we are considering. b) Behaviour of the neglected terms. As an example, we first consider the correction to the asymptotic form of the kernel contribution Gr) [see eq. (29)], which arises when q’ is in region I(< qo) while q” is in region II (> 40). Keeping q” finite, we expand the vertices contributions for q and q’ + 0 using, in addition to eqs. (41) and (42), the following relations valid near the critical temperature (K --f 0) t :
and
k4z-4’- dw)(1 - iM.Ja)
(54)
We then get the correction 6Gi4’(f) = -&
dq’ dq”dJr,(q’s s s cc70 =-c?o
x WC2+
-Jo IQ -
](l q’j2)
jq -
_&!J
q’12)(q’.VJq”)
(-q*v$qe)
d(K2 + q2) Q(t). (55)
t Here again, we take K = I& to be finite, thus excluding the strict hydrodynamic limit, which will be discussed in section 3c.
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
SPIN
SYSTEMS
77
The function @ is the integral over intermediate times of a product of five a.f.‘s. We wish to evaluate it in the asymptotic limit where q M q’ m K -+ 0, while t --f co as rq cc q-‘, keeping q” (and hence the characteristic time Tag) finite. There will then be a complete separation between these two classes of time scales (7-q w Tq’ > Tqe), which will allow us to apply the usual procedure leading to markoffian limits?; we find in this way: Q(t) = ; dta jdtJq+ 0
- ta)
0
x
rq-q+
M
r,+)
-
t2) rq'+q"-q(t3
r,_,,(t)
-
i2) rq-q"(t3)
r&2)
yb",
(56)
where !Pqn = ;dt, ; d&r,* (t3 - t2) rqe(t3) f q+2). 0 0
(57)
For the small-wavenumber functions rq, and rq-qf, we now use the asymptotic limit (49). Switching to the reduced wavenumbers and times defined by eqs. (36), (37) and (45) we arrive at the following correction to the kernel G(T) of the asymptotic equation [see eqs. (52) and (44)] :
, xs dz
6G@)(7) = q2(W
+
2’2 -
1)
‘I - z”2 r(Z’fT) q1 z’12
K2 + 11 -
x
$
s
dq”(z’.PJ,~)(1.~+,~)
(
-
Z’jf,)
1 - k- !Pqa. Jo >
(58)
This expression is proportional to q2 and thus negligible in the limit q --f 0. We now intend to show that this result is general, more precisely, that any correction to the kernel G appears in the asymptotic equation proportional to qa, where CY 2 8, and hence may be neglected. To achieve this aim, we need to compute (at least) a lower bound to the number P of factors q appearing in the contribution of any graph to the asymptotic equation (5 1). It will of course be helpful to introduce a more detailed diagrammatic technique, specifying whether a particular wavenumber is “small” (< qo) or “large” (> qo). Since we are not interested in denumerating the graphs, nor in calculating their contribution, we may forget about the difference between dotted and (arrowed) plain lines as well as the precise values of wavenumbers. From our central hypothesis that the dominant contributions for any “small” wavenumber q’ come from the region where q’ M q and K, both much smaller than the cutoff qo which plays thus no essential role, we t See for instance, RDL
II, eq. (111.23).
M. DE LEENER
78
(2)
(I)
(3)
(4)
q q
a;2 15) 9
-<
q )_
<+q aZ=l
a3.0
a4.0
a7=-2
a5=0
pgp
q
a,=0
as=3
Fig. 4. Vertices of the representation used in section 3b. The index q represents a wavenumber inside region I (q < 40); Q is in region II (Q > qo). The index ori below any vertex is the power of q it contributes in the asymptotic limit.
conclude that the only possible vertices are those displayed in fig. 4, where q denotes a wavenumber in region I (q -=c qo) and Q is in region II (Q > 40). When we evaluate the contribution of any graph, in addition to the explicit q dependence, a number of factors q will appear when we introduce the reduced wavenumbers defined by eqs. (36) and (37); then, when we switch to the reduced times (45), the total power of q will be correspondingly diminished. What is then the final number of factors q in front of the contribution of an arbitrary graph involving 2n vertices, nt of type i (see fig. 4) ? (1) Eq. (38) indicates that 31 factors q will appear if a graph involves I “small” dummy wavenumbers. This number may be found by choosing the largest of the two results obtained by counting the vertices where a new line q appears, when we read the graph from left to right (i = 1 to 4) and then from right to left (i = 5 to 8) : I = Max(Zr, 1s),
(59)
with Jr = nr + ~3.
12 = n5 + n7.
(60)
(2) When we expand the contribution of any vertex i, for all “small” wavenumbers around zero, and introduce the reduced variables (36) and (37), a power olt of factors q will emerge. Using eqs. (41), (42), (53) and (54), one easily computes the indices ~.f given in fig. 4. The whole graph then contributes the following number of factors q:
A = ; uani = 2nr + ns + 3na - 2nT. i=l
(61)
(3) Rules B tell us that a graph of order 2n leads to 2(n - 1) time integrals. Reduced time variables [see eq. (45)] will only be introduced for those intervals where all a.f.‘s Frrr have a time scale TV’such that T~//T~= O( 1)) i.e. q. Two upper bounds to the where all lines carry “small” wavenumbers number N of such intervals may be found by subtracting from 2(n - 1) the
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
number of vertices where at least one “large” wavenumber right (Nr) or on the left (Ns). We thus get :
79
Q appears on the
N 2 N1 = 2(n -
1) - ne -
n7 - ng
(62)
N I Ns = 2(” -
1) - ?%s- n4 - ?Ze- n7 - ns.
(63)
n3
-
n4
-
and
The introduction of every reduced time variable absorbs a factor 4’; the total power of q lost for the graph as a whole is thus QN. This result applies to the kernel G,; when we substitute it into the kinetic equation (5) and take the asymptotic limit, an additional factor q5 is lost [see eq. (52)]. We thus arrive at the conclusion that
P = 31 + A - #(N + 2),
(64)
where I, A and N obey the relations (59) to (63) and P is the number of factors q appearing in front of any contribution to the kinetic equation in the asymptotic limit, when expressed in terms of reduced time and wavenumber variables. In appendix B, using elementary conservation relations arising from the graphs structure, we prove that P = 0 if (and only if) the graph involves exclusively “small” wavenumbers, and that otherwise P 2 8.This establishes the fact that all corrections to the asymptotic equation (51) appear proportional to (at least) q* and are thus negligible. c) Connection with the dynamic scaling laws. We have thus shown that, when the critical point is approached from “above” (T 2 T,), i.e. in the limit where the wavenumber q, as well as the inverse correlation length K, go to zero, the a.f. r,(t) of a Heisenberg ferromagnet tends asymptotically for long times toward a single function r(yq%t; K/q). The only approximation made is that the number of neighbours z interacting with a given spin has been supposed to be very large. We now wish to make the connection with the dynamic scaling laws of Halperin and Hohenbergs) as described in the introduction [eqs. (3) and (4)]. To this end, we have to compute the width co&(q) of the Fourier transform fq(w) of r,(t) [see eq. (2)]. The definition of this width is of course largely arbitrary and we shall use the same convention as Resibois and Piettesr: wE(q)-l
=
&co = 0) =O~dV,(t).
Using the asymptotic
form (49) of r,(t),
we then immediately
get that, near
t The results below are of course general and could be derived as well with Halperin and Hohenberg’s definition of the width2).
M. DE LEENER
80
the critical
point, 00
with the dimensionless the DSA homogeneity
time defined by eq. (45) ; this result is identical assumption
with
[eq. (4)] :
(-&7) = Pfb+l)~
(67)
with a = 4 and /(K/4) = Y( p,rcT;
‘+))-l*
(68)
It is then evident that the asymptotic of the width o&), i.e. that r,(t)
M r(&;
K/q) = F&(q)
form (45) may be rewritten
t; K/q).
Taking the Fourier transform of this expression immediately find that, near the critical point,
in terms
(69) as defined in eq. (2), we
(70) confirming eq. (3). Hence, we prove that the Heisenberg ferromagnet (with long-range interactions) satisfies the DSA with the value Q for the power a, as suggested by Halperin and Hohenberg and verified experimentallyl). In addition to this result, we established the asymptotic equation (51) satisfied by the a.f., wherefrom we can compute the unknown functions C and f of eqs. (68) and (70) [see RDL IV and ref. 61. We shall here consider only the qualitative behaviour of f(x). For any finite value of x = K/q, the analysis developed in section 3 (a and b) showed that WIG(q) goes to zero near the critical point, as 4%; thus, as long
as x remains finite, so does f(x). On the other hand, the behaviour of f(x) region” above T,) has still to be when x + co (in the “hydrodynamic determined, since our analysis above does not apply there. We have studied the general kinetic equation (5) in this particular asymptotic limit, to all orders of approximation to the kernel G,. It appears that, when x --f co, the dominant contributions to the kernel arise when the wavenumbers q’ of the intermediate states have the same order of magnitude as K. The characteristic decay time of the kernel, although very large since K + 0, is then much smaller than that of the a.f. rq when q +- 0 with q/K --f 0. Exactly as in the high-temperature case t, this complete separation of time scales transforms t See again RDL II, eq. (II. 23).
DYNAMIC
the kinetic
SCALING
LAWS
FOR
HEISENBERG
into its markoffian
equation
SPIN
asymptotic
limit,
SYSTEMS
81
i.e. a diffusion
equation : &r,(t)
= --DqV&.
(70)
We found that the diffusion critical
limit
equations
of the
coefficient
,,hydrodynamic”
D vanishes regime
as K* when
clearly
satisfies
K
-+ 0. This the DSA
(70) and (67) with
CJ,&) Cc Dq2 Cc K’q2
(71)
;
hence, f(x -+ co) K d.
(72)
To establish these results in full generality, the analysis follows exactly the same lines as that given in sections 3 (a and b) for finite values of ~/q. To avoid tedious lengthiness, we shall therefore skip those details and limit ourselves to briefly examining the case of the lowest-order kernel Gi2’. We start again from eq. (28) and make use of the critical expansions (41) and (42). If the dominant contributions to the sum over q’ arise when q’ w K, we may approximate the resulting expression taking q < q’ w K. The time scales of the a.f.‘s 74 > TV’ are:then completely separated and the corresponding kinetic
equation
tends towards its markoffian
limit:
(73) (as before, A is the volume of the first Brillouin zone). To determine how the expression between brackets, the diffusion coefficient, depends on K, we use the fact that near the critical point, the a.f. r,s(t) satisfies the DSA expressions (69) and (67), whence
(74 On the other hand, introducing 5’ =
q’/K
the reduced variable (75)
makes it clear that the integral over q’ is proportional to K3. Altogether we thus find: D cc K~K-$ cx Kt as stated above. This ends our analysis of the critical behaviour of the Heisenberg ferromagnet in the Weiss limit, from which we conclude that the agreement is complete between our results and the predictions of the dynamic scaling assumptions of Halperin and Hohenberg. 4. Dynamic ferromagnets,
scaling for antifewomagnets. As soon as a theory exists for it can be developed for antiferromagnets as well. We have
M. DE LEENER
82
thus applied the whole argumentation of section 3 to a Heisenberg (twosublattice) isotropic antiferromagnet. Since the analysis follows the same lines as for ferromagnets, we shall limit ourselves to stating the most important points without giving the mathematical proofs in detail. It is interesting to notice here the essential difference between ferro- and antiferromagnets from the point of view of critical dynamic behaviour. For ferromagnets, at the transition temperature TC, the characteristic time scale Tq of the a.f. r,(t) is expected to diverge at the single point Q = 0; this is due to the superposition of two effects: (1) the temperature-independent “kinematic slowing-down” arising from the conservation of the total angular momentum [see eq. (1 l)] ; (2) the critical “thermodynamic slowing-down” due to the divergence of the equilibrium a.f. 4q [see eq. (32)]. On the other hand, for antiferromagnets, these two phenomena occur at different points in wavenumber space : the conservation of angular momentum still implies that r,+(t) is a constant but the divergence of 4q now occurs at some point q = q* of the Brillouin zone, characterizing the staggered spin ordering. The exchange interaction Jq can be expanded around this point, where it has its maximum :
Jq = Jq* (1 - 6 Iq - q*12+ . ..I.
(76)
in the Weiss limit, to which the whole of this work is devoted, as follows :
4Q=
+q diverges
J /+2
+
,;*_
(77)
q*j2) '
where the inverse correlation length K vanishes linearly at the Neel ature TN [see eq. (33)]. Thus, critical behaviour of the a.f. r,(t) is expected in both regions q = 0 and q = q*. We have indeed shown that, when T + TN (from the long-time behaviour of r,(t) near those points has the following totic form : rd4 r,-+,*@)
temperaround above), asymp-
= r(Yq’t ; K/q), es r*b’
where the constant
iq -
q*i’ t; ‘&
-
q*j),
(78)
y is given by
ys = 2J;./A
(79)
[compare with eq. (44)]. The functions r and r* obey two coupled equations:
a,r*(T) =iTdr’G*(T’]r,r’)
r*(T
-
T’),
DYNAMIC
SCALING
LAWS
FOR
HEISENBERG
where the kernels are nonlinear functionals as in the ferromagnetic
case (appendix
obtainable from well-defined
diagrammatic
SPIN
of the unknowns
B), by an infinite
SYSTEMS
83
and are given, series of terms
rules. This result was established
as in sections 3 (a and b) by showing that, near the critical point, the dominant contributions to the kernel GQ in the general kinetic equation (5) arise when all intermediate wavenumbers are in the vicinity of the points 4 = 0 or q = q*. As is done in section 3c, it is then easy to deduce from (78) that the critical behaviour of the Fourier transform p*(o) is described, in accordance with the dynamical scaling assumptions2), by two (different) functions of the form given in eqs. (3) and (4) with K = q and k = jq - q*I, respectively, and with a = $. But our kinetic theory of course permits us to go beyond these qualitative statements and to determine all functions precisely. In particular, we have examined the functions f(x) and f*( x ) ; we showed that they remain finite fdr all finite values of x and that
(81) in complete accordance for antiferromagnets.
with the predictions
of Halperin
and Hohenberg
5. Discussion. In this paper, we have studied the critical behaviour of the time-dependent autocorrelation function in isotropic ferro- and antiferromagnetic systems described by the Heisenberg model. Our analysis is exact within the limits of validity of our starting point, the general kinetic equation (S), which applies to the paramagnetic region (T 2 T,) and to the Weiss limit 7), where the number z of neighbours interacting with a given spin is taken to be infinite. The extension of the theory below Tc has been studied by Rbsibois and Dewel and will be published soon. The Weiss approximation is a serious limitation when we compare our theory with the experimentsl) and with their remarkably accurate phenomenological description, the dynamic scaling assumptions of Halperin and Hohenbergz). Indeed, the exchange interaction is known to be of very short range (z = 6-12) ; moreover, the Weiss limit leads to wrong equilibrium critical indices*). It may then seem surprising that our results are in accordance with the qualitative predictions of the DSA and, even more, that there seems to be good quantitative accordance with the experimental facts. This problem has already been discussed in reference 5 (RDL IV, section V) which will allow us to be very brief. Concerning the comparison with experiments on systems where z is not large, we believe that the Weiss approximation may well be better for dynamic correlation functions than
a4
M. DE LEENER
for other quantities. Indeed, it is known that the Ornstein-Zernike form of the equilibrium correlations [eq. (32)] is very nearly correct (the critical index 7 measuring the deviation from this approximation is experimentally very small) 8). It is therefore possible that the self-consistent description that we obtained for the dynamics of the correlations would not be deeply modified by the finiteness of z. However, to get agreement with experiments, we must use the measured temperature dependence of the correlation length relation [K-l cc (T Tc)‘, withv = 0.71, instead of the Ornstein-Zernike K -1 [eq. (33)l. The accordance of the qualitative predictions of our theory with the dynamic scaling assumptions leads to the following question: are the DSA correct outside the Weiss limit? Mori and Okamoto3) have suggested that the answer is negative. In fact, this seems plausible, since the static scaling laws themselves are believed to be only approximately correct in threedimensional systems. Although unable to answer this fundamental question, our work justifies the DSA for a specific model; in fact, it goes beyond them and furnishes a detailed quantitative description. The author wishes to thank Professor I. Prigogine Acknowledgments. for his interest in this work and Professor P. RCsibois for many helpful discussions.
APPENDIX
A
The reader will find here the rules permitting to calculate the kernel G(T; K) of the asymptotic equation (51) describing the critical behaviour of the autocorrelation function in ferromagnets. The vocabulary used is defined at the beginning of section 2; note, however, that here the “times” are dimensionless variables as defined by eqs. (36), and “wavenumbers” (37) and (45). Rules C: (1) Connecting dotted and plain arrowed lines through the vertices of fig. 2, draw all distinct graphs of even order 2n, irreducible and containing no self-energy insertion, starting and ending with a dotted line labelled with a wavenumber 1 of unit length and arbitrary direction. The times attached to the vertices grow from right to left
(T2n --
T
>
7212-l
>
..a
>
72
>
71
=
0).
The contribution of any graph is found as follows. (2) To every vertex, associate the expression given in fig. 2 (bottom) ; to every (plain or dotted) line with wavenumber q’, between the times 7 and T’, associate a function Q’)(T T’)].
DYNAMIC
SCALING
(3) Integrate
LAWS
FOR
HEISENBERG
over the ordered intermediate
jdrz,_l~~~s,_z
SPIN
85
SYSTEMS
times:
...Oj&
0
and integrate (over all space) over the n dummy wavenumbers.
APPENDIX
B
We consider an arbitrary graph made up of lines q and Q connected by 2n vertices of the types shown in fig. 4 (nr vertices of type i, with i = 1 to 8) ; the graph starts and ends with a single line q. The numbers 1~~obey two simple conservation laws arising from the fact that the numbers of left and right ends of q (and Q) lines must be equal:
fll+
fi3 +
2?%3 +
n4
126 = =
%? +
a5 +
127,
P.1)
2~~6 -+- ?%3.
w?
Adding these relations, we get: n1-k
fin2 +
123 +
124 =
n5 +
fin6+
127 -k n3=
(B.3)
fl.
Note that (B.3) implies that (62) and (63) may be rewritten as N I Ni = ~21+ ?25+ %j -
2,
(B.4)
N I Ns = nl + ns + ns -
2.
(B.5)
Using these relations, as well as the definitions that
P = 3Z+ A -
(59) to (61), we shall show
$(N + 2)
(B.6)
is equal to zero if (and only if) the graph involves no line Q, and that otherwise P 2 8. Let us start by considering the particular family of “dominant” graphs that have lead us to the asymptotic equation (5 I) ; they only involve “small” wavenumber 1~1= n5 =
lines q and thus only contain
vertices
of types
1 and 5:
n.By mere inspection, we then get:
1 = 11 = 12 = n, N = N1 = Nz = 2n,
A = 2n
and thus
P = 0,
(B-7)
as it should. Passing now to the case of “non-dominant”
graphs, we remark that we
M. DE LEENER
86
necessarily have and
722> 0
n6 >
(‘3.8)
0,
since only the vertices 2 and 6 transform a line 4 into lines Q. We then have three cases to consider separately: a) WI + a3 > 1~5 + n7. Then eqs. (B.l) to (B.5) tell us that (B.9)
N2--l=fi2--6=(fil+%3)-((n5+%7)>0;
hence, using Nr as an upper bound to N, we get from eqs. (59) to (64) : P 2 PI = 3zr+
A -
#(Nr + 2)
= 3(nl+
n3) +
2fil-t fi2 f
=
fi3) -
@5
f(nl +
+
3fi6 -
fi7) f
2n7 -
&(a3 +
@I$
3fi6 f
n5 +
%6)
n7);
Eqs. (B.8) and (B.9) then imply that (13.10)
P 2 5. b) n5 + ~7 > rtl+ ~3. Along the same lines, we get: Nl-
N2
=
fi6 -
fl2 =
(a5 +
'yz7)-
(?Zl +
923) >
0,
(B.11)
whence, using the upper bound N2 to N: p 2 P2 = 312 + A = f(fi5 + fl7) -#1
Q(N2 + 2) +
n3) +
$(3ns + n3
n7) ;
thus (B. 12)
P 2 5. c) n1+ Here,
n3 =
Nr = N2,
a5 +
n7.
n2 =
a6
and P 2 &(3n2 + n3 + ~7). Excepting the particular case of the “dominant” we get from eq. (B.8) the bound
above, (B. 13)
p 2 #> which completes
graphs considered
the proof.
DYNAMIC
SCALING LAWS FOR HEISENBERG
SPIN SYSTEMS
87
REFERENCES 1) For a review of these experiments, see for instance: Minkiewicz, V., to appear. 2) Halperin, B. and Hohenberg, P., Phys. Rev. Letters 19 (1967) 700; Phys. Rev. 177 (1969) 952. 3) Kawasaki, K., Progr. theor. Phys. (Kyoto) 39 (1968) 1133, 40 (1958) 11, 706, 930; Mori, H. and Okamoto, H., Progr. theor. Phys. (Kyoto) 40 (1968) 1287, 41 (1969) 1177. 4) Bennett, P. and Martin, P., Phys. Rev. 138 (1965) 607; Kawasaki, K., J. Phys. Chem. Solids 28 (1967) 1277; Progr. theor. Phys. (Kyoto) 39 (1968) 285; Villain, J., J. Phys. (France) 29 (1968) 321, 687. 5) RBsibois, P. and De Leener, M., Phys. Rev. 152 (1966) 305,318, 178 (1969) 805, 8 19 (hereafter referred to as RDL I, II, III and IV, respectively); Phys. Letters 25A (1967) 65. 6) RBsibois, P. and Piette, C., Phys. Rev. Letters 24 (1970) 514. 7) See Brout, R., Phase Transitions, W. A. Benjamin, Inc. (New York, 1965). 8) See, for instance, Fisher, M., Rep. Progr. Phys. 30 (1967) 615. 9) R&ibois, P. and Dewel, G., Ann. Phys., to appear.