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Time-dependent fluctuations in spin systems in d dimensions
Physica 65 (1973) 505-521
0 North-Holland Publishing Co.
TIME-DEPENDENT
FLUCTUATIONS
IN SPIN SYSTEMS
IN d DIMENSIONS P. C. HOHENBERG Bell Laboratories, Murray Hill, New Jersey 07974, USA and M. DE LEENER
and P. RESIBOIS
Faculte’ des Sciences, Universitt Libre de Bruxelles, Bruxelles, Belgique
Received 24 October 1972
Synopsis The time-dependent spin-correlation funcnon is investigated near T, in d dimensions, by a microscopic kinetic theory in a model with long-range forces. For the isotropic ferromagnet it is shown that there exists at least a large precritical region where, in the absence of a magnetic field and for T > T,, dynamic scaling holds up to d = 6. In the range 4 < d < 6, however, the characteristic frequency does not scale from above to below T,, since there are two distinct scaling lengths below T,. For d > 6 the results agree with the conventional (Van Hove) theory. Conjectures are made about the true critical behavior, and a detailed comparison is presented between the kinetic theory and the phenomenological theory of dynamic scaling. For the antiferromagnet the conventional theory is recovered for d 2 4.
1. Introduction. In this paper we wish to generalize the kinetic theory of Rtsibois, De Leener, and Dewellm3) and the phenomenological theory of Halperin and Hohenberg4) to a study of the spin-correlation function*
for arbitrary dimension d. The motivation for our investigation is based on the recent interest in the role of dimensionality in critical phenomena. In particular, Wilson and Fishe?) have argued that the static properties are described by the classical theory for d > 4, and have calculated the nonclassical exponents for small deviations from d = 4. In discussing the dynamics it is important to note that although the static properties of a system may be known exactly, the dynamics may not be soluble, e.g., the one-dimensional superconductor6). In this paper we * Unless otherwise noted the notation is the same as in refs. l-3. 505
506
P. C. HOHENBERG,
M. DE LEENER
AND P. RfiSIBOIS
study a model with long-range forces’-3), in which the static exponents are classical for all ~2,and show that the situation for dynamics is quite complicated, and dependent on the details of the system. In particular, for the isotropic ferromagnet in zero field and above T,, we show that there exists at least a large precritical region where “nonclassical” dynamic scaling4*8) prevails up to d = 6, and the conventional (Van Hove) theory’) only holds for d > 6. The arguments presented here follow closely those of refs. 1-3, in examining the diagrammatic expansion for the kernel of the kinetic equation satisfied by J’g8(t). The derivation is generalized somewhat by re-expressing the results for finite but large z (z is the number of neighbors), stating explicitly the order of magnitude of the neglected terms. It is shown that for T > T, and H = 0 the characteristic frequency CO,(X)of rQ(t) can be determined self-consistently to all orders in the diagrammatic expansion. Since this expansion has no small parameter the derivation is of course by no means rigorous, but we have no reason to believe that the diagrammatic series does not converge. A well-known limitation of the long-range force model is that expansions in l/z do not apply to the immediate vicinity of the critical point for d < 4. For static properties, the size of the critical region has been discussed by a number of authors9), but there also exists a critical region inside which the present dynamical theory cannot be applied. When the characteristic frequency reaches a critical value depending on z, the correction terms to the kinetic equation of refs. l-3 become important, and the present derivation breaks down. Nevertheless the close agreement (at least for d < 4) between our results and those of the phenomenological theory4) (which apply presumably to the true critical region), leads us to conjecture that the results of the kinetic theory can be extended to the immediate vicinity of T,, by merely inserting the correct values of the static exponents into the final formulas*. As stated above, it is found that for the ferromagnet simple dynamic scaling holds for T above T, (and H = 0) up to d = 6, whereas it becomes invalid already at d = 4, for T below T,. Thus the breakdown of scaling as d increases is much more complicated for dynamics than for statics in the ferromagnet, since in the range 4 < d < 6 it is not possible to find the dynamical exponents by matching the behavior above and below T,. As we show below, the reason for the breakdown of the simple phenomenological theory4) is the existence of two scaling lengths below T,, one for longitudinal fluctuations and the other for transverse fluctuations. In the mean-field region, where the kinetic theory applies, there are two lengths even for d < 4, but in the true critical region static scaling holds for d 5 4, requiring that there be only one correlation length, and the simple phenomenological theory4,8) is presumably valid. For d > 6, the kinetic theory agrees with the conventional theory, both above and below T,. * This statement is not totally unambiguous, since there may be corrections disappear in the large-z limit; see eq. (5.14) below.
of order 7 which
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
507
In contrast to the ferromagnet, the situation in the isotropic antiferromagnet is much simpler: for d I 4, there are two scaling lengths in the mean-field region below T,, but only one in the critical region, where the simple phenomenological theory holds; for d > 4 the conventional theory is correct in the whole range of temperatures. In the uniaxial ferro- or antiferromagnet the phenomenological theory4s*) makes no prediction concerning the exponent of the characteristic frequency, and the conventional theory turns out to be applicable for all d. In sec. 2 the results of refs. l-3 are recalled and generalized slightly to the case of finite but large z. In sec. 3 the characteristic frequency in the ferromagnet is determined for arbitrary dimension d in zero field and for T > T,; the case T < T, is studied in sec. 4. The conventional theory and the phenomenological scaling theory are discussed in sec. 5 for arbitrary dimension d, and compared with the results of kinetic theory in sec. 6 (see table 1). The isotropic antiferromagnet, and the uniaxial ferromagnet are discussed briefly in sec. 7, and those results are summarized in table 2. 2. The model and the previous results in three dimensions. We consider a Heisenberg system with dimension d, lattice parameter a, with an exchangeinteraction [its Fourier transform is denoted by, J(q)] of range r 9 a, so that the number of interacting neighbors z is given by z x (r/a)” 4 1.
(2-l)
In this case, the equilibrium pair-correlation function is of the classical Ornstein-Zernike form for reduced temperatures E = IT - T,j/T, satisfying the condition e $-
e1
=
(a/r)” x z-l,
(2.2)
is taken at its mean-field value, k,T, = &J(O). This
if the transition temperature condition becomes E
+
,+
=
(,/,.)'d/'4-d'
E
Z-2/W-d),
(2.3)
if the transition temperature is suitably renormalized9). Eq. (2.3) indicates that, apart from a shift of T,, the O-Z theory is valid for d 2 4 since Ed + 0 as d + 4, thus recovering the result of Fisher and W&or?). If we define J(q) = J(O)(l
- q2r2 + a**),
qr < 1,
(2.4)
then the static correlation function becomes*
~,(O>= A/V (q2 + x2)1,
qr; xr 6 1,
* When we work above T, ‘we may consider, instead of (l.l), the isotropic correlation tion r&)
=.
(2.5) func-
508
P.C. HOHENBERG,
M. DE LEENER
AND P. R&IBOIS
where A is a constant and x x
r-12.
(2.6)
Let us first discuss the ferromagnet for T > T,, H = 0. In ref. 2, starting from the non-markoffian kinetic equation (2.7) it was shown for d = 3 that in the domain w-4
1,
qr < 1,
x/q = finite,
(2.8)
the time scale of pa(t) is entirely determined by intermediate values of q’ of order correlation function
q (or x). Here pa(t) is the normalized time-dependent
(2.9) and the general definition of the kernel G,(tl{pq,)) as an infinite diagrammatic expansion can be found in ref. 2 (we have suppressed the dependence of the kernel c,, on the static correlation functions). In order to allow the reader to follow our dimensional analysis more easily we display explicitly the simplest approximation to (?a in the appendix. From an examination of the full series for eq it was shown in ref. 2 that in the ferromagnet in three dimensions there exists a self-consistent solution of eq. (2.7) in which Pa has a characteristic frequency given by &x)
= (kJ,lfi)
(a/r)3’2 (qr)s’2f(&),
(2.10)
wheref(x) is a dimensionless function. Moreover, it was shown in ref. 2 that the leading correction to eq. (2.10), which comes from q’ values much larger than q (q’r z 1) is characterized by the frequency w!(x) z (k,T,/fi) (a/r)3’2 (qr)2 (q2 + x’) r2 4 w:(x).
(2.11)
Of course, eq. (2.7) is not generally valid but only in the “Weiss limit” where z becomes infinite. This means that, for finite z, there are corrections which we now evaluate, following the discussion of RDLl). 2.1. Corrections due to the imaginary part of p, (see in ref. 1 RDL III, eqs. (5.4) to (5.9) and RDL IV, appendix C). It is readily seen that these corrections are a consequence of the non-commutativity of the quantum-mechanical operators at different times; they are of order (2.12) for small q and x, they are thus always negligible.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
509
2.2. Dynamical corrections. In the derivation of eq. (2.7), we have neglected terms of order l/z in the analysis of the Heisenberg operator S”(t) = exp (iHt) S” exp (-iHt),
(2.13)
appearing in (1.1). This approximation was required in order to obtain an explicit expression for the kernel ea. Yet, by standard projection-operator techniques, it can be readily verified that FJt) obeys the following exact (but formal) kinetic equationlo) (2.14) where, however, no simple explicit form can be given to the exact kernel G,(t). We have for all q:
where G”’ denotes the first correction, of order l/z compared to e,r. Moreover, on the sile basis of the conserved nature of the spin operator S:(r) we know that, for all z, in the ferromagnet
WI G 0 Wr2),
q+
0.
(2.16)
Note that, close to T,, this estimate is probably pessimistic because it only takes into account the “kinematic slowing down” and completely neglects the “dynamical slowing down” which gives rise to the supplementary q3 dependence in our explicit leading term eQ. Combining (2.15) and (2.16), we get for the characteristic frequency wp’(x) associated with Gr’: u$‘(x) I (k,TJk)
(qr)2 (a/r)9’2.
(2.17)
Comparing (2.17) with (2.10), we see that, in the limit (2.8), the effect of Ga” will be negligible provided that (for x % q): (2.18) or, by (2.6) E$&EgXZ
-4
.
(2.19)
2.3. Static corrections. In the above argument, we have not taken into account the l/z corrections which appear in the static properties [they of course come into the canonical average implied by (l.l)]. As already mentioned, these
510
P. C. HOHENBERG,
M. DE LEENER
AND P. Rl%IBOIS
are only negligible in three dimensions if &~&2~.z
-2
(2.20)
.
Note that this limitation is stronger than the dynamical condition (2.19). In general, let us define (2.21)
cc = max (e2, 4, such that our kinetic theory is valid in the “precritical” 8 9 E,.
or “mean-field” region
3. Generalization to arbitrary dimensions. The derivation of (2.14) and its analysis as given in ref. 2, may be repeated for arbitrary dimension. Eq. (2.10) becomes simply* (3.1) whereas the corrections due to q’ values much larger than q are of order
w:(x) =
(k,TJA)
(a/r)fd (qr)2 (q2 + x2) r2.
(3.2)
It is thus seen that for 3 < d < 6, since wr % WI’,the dynamical scaling property of eq. (2.7) remains valid in the same sense as it was shown to hold for d = 3. It is only at d = 6 that co:(q) becomes comparable to co:(q); in that case, the small intermediate wave numbers do not entirely determine the time dependence of pa(t) and we have the “conventional theory” w:(q) z of (q) ~5 q2 (q2 + x2) (see sec. 5). Of course, we have also to reconsider the limit of validity of our eq. (3.1). We may generalize the arguments leading to (2.12), (2.17) and (2.18), and we find, in analogy to (2.19) that our evaluation of the dynamics is only valid so long as F $ .Q z (z-1)4’@-?
(3.3)
Note that, for d 2 4, this is the only condition [and probably too restrictive as already pointed out after (2.16)] because the static properties become classical (E? = 0). For d = 3, it was conjectured in refs. 1 and 2 that in the true critical region (E < ac), eq. (2.10) would remain essentially unchanged, (except for small corrections of order ~7,and a different dependence on the range parameter r) if the cor* Since our arguments are entirely dimensional, it is values, and we shall consider d formally as a continuous our discussion to d’> 3, however, due to the difficulty of tions in a Heisenberg system with long-range forces for d
not necessary to restrict d to integer variable (see ref. 5). We shall restrict a correct description of static correla= 1 or d = 2. See, however, ref. 21.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
511
rect correlation length x-l and exponent v are used. This conjecture is supported by the close agreement with experiment”), with the phenomenological theory4**) (see below), and with the semi-microscopic mode-mode theory12). Before considering the situation for d > 3 in the true critical region, we turn to the case T < T,, where real spin waves can propagate. 4. Kinetic equation below T,. For simplicity we still limit our discussion to the case of zero field (in ref. 13 it is shown that the finite-field case is analogous to the region H = 0, T < TJ. Our starting point is the approximate kinetic equation which describes the dynamical behavior of the spin-correlation function pgP(t) = r:‘(t)/ri’ (t = 0) below Tc3). This equation reads
where mzB(R) denotes the “free spin-wave frequency”
w;-(R) = -w;+(R),
w;(R)
= 0,
(4.2)
R denotes the magnetization,
and the non-markoffian kernel cfl has a structure similar to &q(a simple approximation is displayed in the appendix for a/? = + -). Note that wifi and CC?lP depend on x, the inverse correlation length below T,, and also on the magnetization R. The validity of eq. (4.1) can easily be shown to be restricted by E p
max (.e2, E3) = Ed,
(4.3)
precisely as in the paramagnetic case [eqs. (2.3) and (3.3)]. Let us examine the main features of the solution of eq. (4.1). As discussed in detail in ref. 3, sec. 3 for the case d = 3, eq. (4.1) shows no scaling behavior in the precritical region (E 9 8,). More precisely, writing formally Q:(X) = Re c&t) + i Im W:(X),
(4.4)
where Re U.&X)denotes the characteristic oscillation frequency of pc -, while Im u&x) is the damping frequency, it was found, using the mean-field exponents [R cc E* cc x1-1,that* for
co&)w
4< x,
E B EC,
(+)(qr)‘(m)[l
d=3:
+i($)(s>
+**.I;
* In the following equations we often suppress factors of order unity.
(4.5a)
512
P.C. HOHENBERG,
for
4% x,
M. DE LEENER
E>> E,,
AND P. Rl%IBOIS
d=3:
which indicates that the characteristic simple scaling form4)
(complex) frequency does not have the
(4.6) in the precritical region, for d = 3. As mentioned in ref. 3, however, this breakdown of scaling in the precritical region can be remedied in the true critical region by re-expressing the quantities in eq. (4.5a) in terms of the correct critical exponents R cc E’, xr cc E’ z R2 (neglecting once again the small exponent q), and we find* for
4 $ x,
cog25 (k,T,/fi) for
4% x,
E< &,I (qu)2
(xu)’ [I
+ i (q/x)3 + .--I;
(4.7a)
E < &,I
W&S) z (k,T,/fi) (q#12 [i + (x/q)* + ***I.
(4.7b)
Note that in these equations we have not displayed the r dependence because it is not known. At T, we find the matching condition
where we have taken the limit z + cc first. Another interpretation of eq. (4.5) is in terms of the perpendicular correlation length (x1)-l discussed in sec. 5.2 below. In the mean-field region we have [c$ eq. (5.17)] (4.9) whereas the quantity x of eq. (4.5) is the inverse of the parallel correlation range, given by xl1 0~
R,
(4.10)
* The unusual (q/x)3 dependence for the damping in eq. (4.7a) is due to our approximation for 6:s. It has been shown recently that inclusion of the spin-wave-spin-wave interaction term (which in the ordered region is of order zm2) in 6,afi leads to the usual result (q/x)? (see ref. 14).
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
513
in mean-field theory. Then eq. (4.5) may be written as for
4 4 XI, XII,
E$- EC:
0: (XI, x11)= (kiJClfi) (4rY (xlrY [l + i (a/r)” (4/x,) (4/Q’ for
4% XI,XlI,
mi (XL, xl,) x (kJ,/@
+
***I;WW
&B &,I (qr)5’2 (a/r)3’2 [i + (r/u)3’2 (x,/q)+ + -*-I.
(4.11b)
Thus the breakdown of the scaling “ansatz” (4.6) in the precritical region is caused by the existence of two different scaling lengths, for parallel and perpendicular fluctuations, respectively. In this case, however, we have a “two-length scaling”, (4.12) as can be seen from an examination of eqs. (4.11). In the true critical region, there is only one scaling length for d = 3 (according to “dimensional” scaling), so that (4.13)
XI = XII,
and eq. (4.12) reduces to eq. (4.6). More generally, for arbitrary d, eqs. (4.9) and (4.10) for xL and xl1 are valid in the mean-field region E B cc, and the kinetic theory yields for
4 Q XI,XII,
&$ EC:
d (XI, xi,) = bWclfi) (qr)’ (xlr)(d-2)‘2[1+ i (q/xJds2
(q/x11)2 (a/r)” + -a*]
(4.14a)
f2.1: (xL,
xl,) cz (k,T,/fi)
(qr)f’d+2’ (a/r)?
[i + **.I.
(4.14b)
These expressions are still of the general form (4.12) with r = 3 (d + 2). For d 5 4 the preceding arguments may be repeated, and we again get (4.6) in the true critical region. For d > 4 on the other hand, the static exponents do not change as we go into the true critical region, so it is likely that eq. (4.12) will also remain unchanged, and the simple homogeneity condition (4.6) will be violated. Of course the contribution c# coming from large intermediate momenta must also be considered, and for d 2 6 it will dominate w:, as discussed below.
514
P.C. HOHENBERG,
M. DE LEENER
AND P. Rl%IBOIS
5. Phenomenological theories. Let us generalize the phenomenological
theories
to arbitrary dimension. 5.1. Conventional theory. The conventional theory7) follows from the assumption that transport coefficients are regular at T,, and that apart from kinematic factors, which follow from conservation laws, the critical slowing down comes entirely from the diverging susceptibility. Thus, for a purely damped motion the characteristic frequency is 20) w%(x)cc ix-‘q2 x i (x2 + q2) q2, in the ferromagnet down), and
(5.1)
where the order parameter is conserved (kinematic slowing
0.((x) cc ix-l Z i (x2 + q2),
(5.2)
in the antiferromagnet where the order parameter is not conserved. [We have neglected the exponent q in eqs. (5.1) and (5.2).] When the motion is oscillatory the characteristic frequency is proportional to x-*, and we have15) co’&) cc RqZ w x,,q*,
q@ It,
(5.3)
for the ferromagnet and
4X4 cc Rq =
xllq,
96 x,
(5.4)
for the antiferromagnet, where we have used eq. (4.10) for xl1 . Note that the conventional theory does not provide a simple estimate for the damping of this oscillatory mode. 5.2. Dynamic scaling theory. It is assumed4) that the correlation function has a unique (complex) characteristic frequency of the form
and the unknown exponent 5 is determined, where possible, by matching to the limiting form of eq. (5.5) in a particular region. In the version of the theory formulated by Halperin and one of the authors4), the matching is to the spin waves below T,, which in the ferromagnet have the form15) ilq2,
(5.6)
1 = es/R.
(5.7)
wq x
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
515
In the critical region, the exponent of es was found4) by using static-scaling. Indeed, the perpendicular q-dependent susceptibility has the form15)
xdd = R21esq2~
(5.8)
from which it follows that the correlation function C,(x) dces not decay exponentially at large distances, but rather according to a power law. In that case we define15*16) a perpendicular correlation range XT’, by the relation [cj eq. (2.4) of ref. 41
C,(x) = R* (ll~,xY’,
W)
x-rm
where C,(x) is obtained by Fourier transforming eq. (5.8), yielding a power law with exponent p = d - 2. From eqs. (5.8) and (5.9) it follows that
The parallel correlation function, on the other hand, has a correlation length xi1 of which the exponent Y,, is in general different from the exponent vI of x; ‘. In the true critical region, we assume (according to ’ dimensional scaling”) that Ye = vII = v, whence eq. (5.10) implies v(d-2) es w E
(5.11)
It follows that in the critical region the spin-wave stiffness 3, [eq. (5.7)] goes as (5.12) and comparing eqs. (5.6), (5.12) and (4.6) we get
5 = d-p/v. According to dimensional scaling [dv = /I(6 + 1) = 2 - 01 = y + 28 = + 2/?] this result for 5‘is equal to 5=$(d+2-11).
(5.13) v
(2 - 7)
(5.14)
WagneP) has given an alternative method of estimating the exponent 5, based on matching to the (real) Larmor frequency u)L = g,ugH, in the presence of a magnetic field. The scaling exponent of His .sndw xBd”‘,so
5‘ =
gqv,
which agrees with eq. (5.14) using dimensional scaling.
(5.15)
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P. C. HOHENBERG,
M. DE LEENER AND P. R&IBOIS
In the mean-field region we have
es0~ R2,
(5.16)
whence eq. (5.10) implies (5.17) and lJ1 = (d - 2)-l,
(5.18)
(since p = $), whereas Y,, = 2. l The spin-wave frequency [eq. (5.6)], goes as % w Rq2 M xY-~)‘~ q2 w x,,q’.
(5.19)
6. Comparison of kinetic and phenomenological theories. Let us summarize the situation in the Heisenberg ferromagnet. In the mean-field region (E >> E,), there are two scaling lengths, x;’ and xil, and the characteristic frequency does not obey the simple scaling relation (4.6). Instead, a “two-length” scaling relation (4.12) applies for d I 6, with an exponent [ = 3 (d + 2). In the domain T > T,, H = 0, where the correlation function is isotropic, the length x;l drops out, and “one-length” scaling is recovered. When d > 6 the frequency u$ of the conventional theory dominates the above scaling frequency, and the critical dynamics follows the conventional theory both above and below T, . These results have been verified order by order in perturbation theory in zero field, using the kinetic theory of refs. l-3. In the true critical region (E < a,), the kinetic theory does not strictly apply, but we conjecture that the dynamical behavior can be inferred by changing only the static critical exponents. For d < 4, according to (static) dimensional scaling there is only one scaling length, i.e., vI = vII , and eq. (4.12) reduces to the simple scaling hypothesis4g8) (4.6) with exponent 5 = 3 (d + 2 - 7) = /38/v. For 4 < d < 6, the two lengths x;’ and xi1 remain different in the true critical region, since the static behavior follows mean-field theory5). We are then left with two-length scaling, which again reduces to simple one-length scaling for T > T,, H = 0. In this case it is not possible to infer the behavior above T, by matching to the spin waves below T,. It is interesting to note that Polyakovr7) has suggested that this situation obtains even for d = 3 in the true critical region, thus violating the dynamic scaling “ansatz” (4.6). Clearly, the present work cannot make a definitive statement on this problem, but the arguments given above strongly favor the validity of the matching condition for d = 3, in contradiction to Polyakov’s suggestion. For d 2 6, the conventional theory is again expected to give the correct answer very close to T, . The results for the ferromagnet are summarized in table I.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
517
TABLEI Dynamic scaling exponent [ for the isotropic ferromagnet Region
4
ds4
dr6
1) Mean-field region (E % e,): a) T-c T,, H= 0 or Tarb, H # 0
2 lengths : C=f(d+2)
2 lengths : C=&(d+2)
conventional
theory
b)T>
1 length: 5 = 4 (d + 2)
1 length: conventional i = + (d + 2)
theory
1 length: C=+(d+2-7)
2 lengths:
conventional
theory
1 length:
conventional 1 length: C = f (d + 2)
theory
T,,H=O
2) True critical region (E Q E,): a) T-c T,, H= 0 or Tarb, H# 0 b)T>
T,,H=O
5=3(d+2--rl)
C = t (d + 2)
7. The isotropic antiferromagnet and the uniaxial ferromagnet.
7.1. The anti-
ferromagnet. The analysis of sets. 2 and 3 may be repeated for the isotropic antiferromagnet (see ref. 3, sec. 4) and the results are, for T > T,, w&)
= (kBTCl@ (a/r)3d (qr)*df(4q),
CO:(X)= (k,T,/h) (a/r)‘d (q2 + x2) r* , = (k,T,/fi) (a/r)3d’2.
we’
(7.1) (7.2) (7.3)
It follows that the limit of validity of our dynamical theory is [cJ eq. (2.18), and also the remarks after eq. (2.16)] ES = (z-1)4/d,
(7.4)
and that the conventional theory [eq. (5.2)] becomes valid at d = 4, where UP z 09. For T c T,, the kinetic theory yields q-+ It:
O&X) z (k,T,/h)
q9
O&C) x (k,T,/ti) (qr)‘d (1 + *me),
x:
(xLr)3(d-2’ (qr) (1 + . . *),
(7.5a) (7.5b)
in the precritical region, E g E,. The result may be extended to the true critical region, in analogy to the procedure leading to eq. (4.7a), and we find qQ x:
CJ~(X)x (k,T,/ti) (xa)‘(d-2) (qa) (1 + **a),
for d 5 4, and the conventional result of eq. (5.4) for d 2 4.
(7.6)
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P.C. HOHENBERG,
M. DE LEENER
AND P. RBSIBOIS
The phenomenological theory may be generalized to arbitrary d, by noting that according to hydrodynamics1 “) Co,
x cq(1
+ **.),
(7.7)
c2 = @sIXT,
(7.8)
where XT is the total susceptibility (assumed to go to a constant at T,), and es is related to x1 through the perpendicular staggered susceptibility (7.9) yielding as in eq. (5. IO) (7.10) (No is the staggered magnetization). Since the spin-wave velocity c involves only es, we have, from eqs. (7.8) and (7.10): c a #d--2) _L *
(7.11)
In the scaling region there is only one correlation length (x1 = x,,) and the frequency (7.7) has the simple scaling form (4.6) with (7.12)
5 = W,
which agrees with Wagner’s*) estimate. In the mean-field region, we again have two-length scaling, eq. (4.12), with C given by eq. (7.12). The main difference with the ferromagnet is that the conventional theory, eq. (5.4), becomes valid precisely at d = 4, since the scaling frequency is just q-G x:
d@>x
%I,
d = 4,
(7.13)
which is equal to (7.7) because of (7.12). Thus the anomalies found for 4 < d c 6 in the ferromagnet are not present in the antiferromagnetic case. 7.2. The uniaxial ferromagnet. In this case it is possible to show, using the kinetic theoryl*), or mode-mode couplinglg), that the conventional theory* is valid for all d, up to terms of order ~7,which are zero in our model. In particular, * Let us stress that the general definition (5.1) we have adopted for the conventional theory does not mean that the characteristic frequency may be calculated from the short-time moment expansion20). This remark is particularly important for the uniaxial ferromagnet where the dependence of w:‘(x) on the anisotropy parameter x_t obtained from kinetic theory’*), is completely different from that deduced from the moment method.
FLUCTUATIONS
519
IN SPIN SYSTEMS IN d DIMENSIONS
we find for the longitudinal characteristic frequency above T, oq(2t) w q2$
(x) x x452(x/q),
(7.14)
while the transverse characteristic frequency W: - (x) does not behave critically. The phenomenological theory does not make any predictions for the value of the exponent 5, since there are no propagating modes below T,.The results of the present section are summarized in table II.
TABLEII Dynamic scaling exponent 5 for the isotropic antiferromagnet Region
and uniaxial ferromagnet
d-c4
d>4
Antiferromagnet 1) Mean-field region: E % E, a) T c T,
2 lengths: 5 = 3d 1 length: 5 = 3d
b) T > T, 2) True critical region: E Q E,
conventional
theory
conventional
theory
1 length: ,Z = &d
conventional
theory
c=2
1;=2
Uniaxial ferromagneta Conventional theory One-length scaling a In discussing
the uniaxial case we neglect corrections to the exponent [ which are of order q.
Acknowledgements. One of us (PCH) wishes to thank B.I. Halperin for informative discussions, and the members of the Theoretical Physics Department (TUM) in Munich for their hospitality during the period when this work was initiated.
APPENDIX
For the convenience of the reader, we reproduce here the simplest approximation for the kernels f?@and c: - ; we have”) in the limit (2.9):
~tz,<~) = -y2
(q2 + x2)
s dq,
(A.11
520
P. C. HOHENBERG,
M. DE LEENER
AND P. Rl?SIBOIS
where y2 = PJ (O)‘/O ;
(A.21
(4 is the volume of the Brillouin zone). This formula is valid in any dimension d > 2.
Similarly3)
+
s
q2 dq,
K4 + ‘l!’ - 43 41
1;,:-(z)
I;;‘-,,(z)
.
(A.3)
1
Note that in the estimates made for q + x, we may take p:(z) = 1, while where q 9 x, we have of course pi -(t) = F,(T). Remark also that if we formally set x = 0, the two kernels GUand c?: - become identical. Note added in proof. After this paper was written, we became aware of additional references which bear on the present topic. Early work by Villain22) suggested that the conventional theory would become valid in the ferromagnet for d r 6. This conclusion was confirmed by Kawasakiz3), on the basis of an approximate modemode coupling argument. In addition, Kawasaki considered the antiferromagnet, and found the condition d 2 4. More recently, Van Leeuwen and Guntonz4) have treated the spherical model, which resembles our model quite closely, and have found results similar to ours for the ferromagnet [see their eqs. (5.5) and the discussion after eq. (5.6)]. An advantage of the spherical model is that there is no condition analogous to our eq. (3.3). Another recent development is a microscopic calculation25) of the critical frequency for a uniaxial system, in which the conventional theory is violated to order 17for d < 4.
REFERENCES 1) Rbsibois, P. and De Leener, M., Phys. Rev. 178 (1969) 806, 819. These two papers are referred to in what follows as RDL III and RDL IV, respectively. 2) De Leener, M., Physica 56 (1971) 62. 3) RCsibois, P. and Dewel, G., Ann. Physics 69 (1972) 299. 4) Halperin, B.I. and Hohenberg, P.C., Phys. Rev. 177 (1969) 952. 5) Wilson, K.G. and Fisher, M.E., Phys. Rev. Letters 28 (1972) 240; Wilson, K.G., ibid. 28 (1972) 548. 6) Tucker, J.R. and Halperin, B.I., Phys. Rev. B3 (1971) 3768; Gruenberg, L.W. and Gunther, L., Phys. Letters 38A (1972) 463. 7) Van Hove, L., Phys. Rev. 93 (1954) 1374.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
’
521
8) Wagner, H., Phys. Letters 33A (1970) 58. 9) G&burg, V.L., Fiz. Tver. Tela 2 (1960) 2031 [English translation: Soviet Physics - Solid State 2 (1960) 18241. Vaks, V.G., Larkin, A.I. and Pikin, S.A., Zh. eksper. teor. Fiz. 53 (1967) 281 [English translation: Soviet Physics - JETP 26 (1968) 1881. Thouless, D. J., Phys. Rev. 181 (1969) 954. See also Hohenberg, P.C., Fluctuations in Superconductors, Asilomar, Stanford Research Inst. (Cal., 1968). 10) Zwanzig, R., Phys. Rev. 124 (1961) 983. 11) See, for example, Minkiewicz, V., Intern. J. Magnetism 1 (1971) 149. Lau, H. Y., Corliss, L. M., Delapalme, A., Hastings, J. M., Nathans, R. and Tucciarone, A., Phys. Rev. Letters 23 (1969) 1225. Rtsibois, P. and Piette, C., Phys. Rev. Letters 24 (1970) 514. 12) Kawasaki, K., Progr. theor. Phys. 39 (1968) 2, 285; 40 (1968) 11, 706, 930; Ann. Physics 61 (1970) 1. Kadanoff, L.P. and Swift, J., Phys. Rev. 166 (1968) 89. 13) Dewel, G., to be published. 14) Dewel, G., Resibois, P. and De Leener, M., Phys. Letters 39A (1972) 33. 15) Halperin, B.I. and Hohenberg, P.C., Phys. Rev. 188 (1969) 898. 16) Josephson, B.D., Phys. Letters 21 (1966) 608. 17) Polyakov, A.M., Zh. eksper. teor. Fiz. 57 (1969) 2144 [Soviet Physics - JETP 30 (1970) 11641. 18) Piette, C., M&moire de Licence, Universite Libre de Bruxelles, 1969 (unpublished). 19) Kawasaki, K., Progr. theor. Phys. 39 (1968) 285; 40 (1968) 706. 20) See, for example, Marshall, W., Nat. Bur. Stand. Misc. Publ. 273 (1966) 135. 21) Blume, M., in Proceedings of the 1972 Grenoble International Conference on Neutron Scattering, IAEA (Vienna) to be published. 22) Villain, J., J. Physique 29 (1968) 321. 23) Kawasaki, K., Prog. theor. Phys. 40 (1968) 11; footnote: p. 35. 24) Van Leeuwen, J. M. J. and Gunton, J.D., Phys. Rev. B6 (1972) 231. 25) Halperin, B. I., Hohenberg, P. C. and Ma, S., Phys. Rev. Letters 29 (1972) 1548.
0 North-Holland Publishing Co.
TIME-DEPENDENT
FLUCTUATIONS
IN SPIN SYSTEMS
IN d DIMENSIONS P. C. HOHENBERG Bell Laboratories, Murray Hill, New Jersey 07974, USA and M. DE LEENER
and P. RESIBOIS
Faculte’ des Sciences, Universitt Libre de Bruxelles, Bruxelles, Belgique
Received 24 October 1972
Synopsis The time-dependent spin-correlation funcnon is investigated near T, in d dimensions, by a microscopic kinetic theory in a model with long-range forces. For the isotropic ferromagnet it is shown that there exists at least a large precritical region where, in the absence of a magnetic field and for T > T,, dynamic scaling holds up to d = 6. In the range 4 < d < 6, however, the characteristic frequency does not scale from above to below T,, since there are two distinct scaling lengths below T,. For d > 6 the results agree with the conventional (Van Hove) theory. Conjectures are made about the true critical behavior, and a detailed comparison is presented between the kinetic theory and the phenomenological theory of dynamic scaling. For the antiferromagnet the conventional theory is recovered for d 2 4.
1. Introduction. In this paper we wish to generalize the kinetic theory of Rtsibois, De Leener, and Dewellm3) and the phenomenological theory of Halperin and Hohenberg4) to a study of the spin-correlation function*
for arbitrary dimension d. The motivation for our investigation is based on the recent interest in the role of dimensionality in critical phenomena. In particular, Wilson and Fishe?) have argued that the static properties are described by the classical theory for d > 4, and have calculated the nonclassical exponents for small deviations from d = 4. In discussing the dynamics it is important to note that although the static properties of a system may be known exactly, the dynamics may not be soluble, e.g., the one-dimensional superconductor6). In this paper we * Unless otherwise noted the notation is the same as in refs. l-3. 505
506
P. C. HOHENBERG,
M. DE LEENER
AND P. RfiSIBOIS
study a model with long-range forces’-3), in which the static exponents are classical for all ~2,and show that the situation for dynamics is quite complicated, and dependent on the details of the system. In particular, for the isotropic ferromagnet in zero field and above T,, we show that there exists at least a large precritical region where “nonclassical” dynamic scaling4*8) prevails up to d = 6, and the conventional (Van Hove) theory’) only holds for d > 6. The arguments presented here follow closely those of refs. 1-3, in examining the diagrammatic expansion for the kernel of the kinetic equation satisfied by J’g8(t). The derivation is generalized somewhat by re-expressing the results for finite but large z (z is the number of neighbors), stating explicitly the order of magnitude of the neglected terms. It is shown that for T > T, and H = 0 the characteristic frequency CO,(X)of rQ(t) can be determined self-consistently to all orders in the diagrammatic expansion. Since this expansion has no small parameter the derivation is of course by no means rigorous, but we have no reason to believe that the diagrammatic series does not converge. A well-known limitation of the long-range force model is that expansions in l/z do not apply to the immediate vicinity of the critical point for d < 4. For static properties, the size of the critical region has been discussed by a number of authors9), but there also exists a critical region inside which the present dynamical theory cannot be applied. When the characteristic frequency reaches a critical value depending on z, the correction terms to the kinetic equation of refs. l-3 become important, and the present derivation breaks down. Nevertheless the close agreement (at least for d < 4) between our results and those of the phenomenological theory4) (which apply presumably to the true critical region), leads us to conjecture that the results of the kinetic theory can be extended to the immediate vicinity of T,, by merely inserting the correct values of the static exponents into the final formulas*. As stated above, it is found that for the ferromagnet simple dynamic scaling holds for T above T, (and H = 0) up to d = 6, whereas it becomes invalid already at d = 4, for T below T,. Thus the breakdown of scaling as d increases is much more complicated for dynamics than for statics in the ferromagnet, since in the range 4 < d < 6 it is not possible to find the dynamical exponents by matching the behavior above and below T,. As we show below, the reason for the breakdown of the simple phenomenological theory4) is the existence of two scaling lengths below T,, one for longitudinal fluctuations and the other for transverse fluctuations. In the mean-field region, where the kinetic theory applies, there are two lengths even for d < 4, but in the true critical region static scaling holds for d 5 4, requiring that there be only one correlation length, and the simple phenomenological theory4,8) is presumably valid. For d > 6, the kinetic theory agrees with the conventional theory, both above and below T,. * This statement is not totally unambiguous, since there may be corrections disappear in the large-z limit; see eq. (5.14) below.
of order 7 which
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
507
In contrast to the ferromagnet, the situation in the isotropic antiferromagnet is much simpler: for d I 4, there are two scaling lengths in the mean-field region below T,, but only one in the critical region, where the simple phenomenological theory holds; for d > 4 the conventional theory is correct in the whole range of temperatures. In the uniaxial ferro- or antiferromagnet the phenomenological theory4s*) makes no prediction concerning the exponent of the characteristic frequency, and the conventional theory turns out to be applicable for all d. In sec. 2 the results of refs. l-3 are recalled and generalized slightly to the case of finite but large z. In sec. 3 the characteristic frequency in the ferromagnet is determined for arbitrary dimension d in zero field and for T > T,; the case T < T, is studied in sec. 4. The conventional theory and the phenomenological scaling theory are discussed in sec. 5 for arbitrary dimension d, and compared with the results of kinetic theory in sec. 6 (see table 1). The isotropic antiferromagnet, and the uniaxial ferromagnet are discussed briefly in sec. 7, and those results are summarized in table 2. 2. The model and the previous results in three dimensions. We consider a Heisenberg system with dimension d, lattice parameter a, with an exchangeinteraction [its Fourier transform is denoted by, J(q)] of range r 9 a, so that the number of interacting neighbors z is given by z x (r/a)” 4 1.
(2-l)
In this case, the equilibrium pair-correlation function is of the classical Ornstein-Zernike form for reduced temperatures E = IT - T,j/T, satisfying the condition e $-
e1
=
(a/r)” x z-l,
(2.2)
is taken at its mean-field value, k,T, = &J(O). This
if the transition temperature condition becomes E
+
,+
=
(,/,.)'d/'4-d'
E
Z-2/W-d),
(2.3)
if the transition temperature is suitably renormalized9). Eq. (2.3) indicates that, apart from a shift of T,, the O-Z theory is valid for d 2 4 since Ed + 0 as d + 4, thus recovering the result of Fisher and W&or?). If we define J(q) = J(O)(l
- q2r2 + a**),
qr < 1,
(2.4)
then the static correlation function becomes*
~,(O>= A/V (q2 + x2)1,
qr; xr 6 1,
* When we work above T, ‘we may consider, instead of (l.l), the isotropic correlation tion r&)
=
(2.5) func-
508
P.C. HOHENBERG,
M. DE LEENER
AND P. R&IBOIS
where A is a constant and x x
r-12.
(2.6)
Let us first discuss the ferromagnet for T > T,, H = 0. In ref. 2, starting from the non-markoffian kinetic equation (2.7) it was shown for d = 3 that in the domain w-4
1,
qr < 1,
x/q = finite,
(2.8)
the time scale of pa(t) is entirely determined by intermediate values of q’ of order correlation function
q (or x). Here pa(t) is the normalized time-dependent
(2.9) and the general definition of the kernel G,(tl{pq,)) as an infinite diagrammatic expansion can be found in ref. 2 (we have suppressed the dependence of the kernel c,, on the static correlation functions). In order to allow the reader to follow our dimensional analysis more easily we display explicitly the simplest approximation to (?a in the appendix. From an examination of the full series for eq it was shown in ref. 2 that in the ferromagnet in three dimensions there exists a self-consistent solution of eq. (2.7) in which Pa has a characteristic frequency given by &x)
= (kJ,lfi)
(a/r)3’2 (qr)s’2f(&),
(2.10)
wheref(x) is a dimensionless function. Moreover, it was shown in ref. 2 that the leading correction to eq. (2.10), which comes from q’ values much larger than q (q’r z 1) is characterized by the frequency w!(x) z (k,T,/fi) (a/r)3’2 (qr)2 (q2 + x’) r2 4 w:(x).
(2.11)
Of course, eq. (2.7) is not generally valid but only in the “Weiss limit” where z becomes infinite. This means that, for finite z, there are corrections which we now evaluate, following the discussion of RDLl). 2.1. Corrections due to the imaginary part of p, (see in ref. 1 RDL III, eqs. (5.4) to (5.9) and RDL IV, appendix C). It is readily seen that these corrections are a consequence of the non-commutativity of the quantum-mechanical operators at different times; they are of order (2.12) for small q and x, they are thus always negligible.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
509
2.2. Dynamical corrections. In the derivation of eq. (2.7), we have neglected terms of order l/z in the analysis of the Heisenberg operator S”(t) = exp (iHt) S” exp (-iHt),
(2.13)
appearing in (1.1). This approximation was required in order to obtain an explicit expression for the kernel ea. Yet, by standard projection-operator techniques, it can be readily verified that FJt) obeys the following exact (but formal) kinetic equationlo) (2.14) where, however, no simple explicit form can be given to the exact kernel G,(t). We have for all q:
where G”’ denotes the first correction, of order l/z compared to e,r. Moreover, on the sile basis of the conserved nature of the spin operator S:(r) we know that, for all z, in the ferromagnet
WI G 0 Wr2),
q+
0.
(2.16)
Note that, close to T,, this estimate is probably pessimistic because it only takes into account the “kinematic slowing down” and completely neglects the “dynamical slowing down” which gives rise to the supplementary q3 dependence in our explicit leading term eQ. Combining (2.15) and (2.16), we get for the characteristic frequency wp’(x) associated with Gr’: u$‘(x) I (k,TJk)
(qr)2 (a/r)9’2.
(2.17)
Comparing (2.17) with (2.10), we see that, in the limit (2.8), the effect of Ga” will be negligible provided that (for x % q): (2.18) or, by (2.6) E$&EgXZ
-4
.
(2.19)
2.3. Static corrections. In the above argument, we have not taken into account the l/z corrections which appear in the static properties [they of course come into the canonical average implied by (l.l)]. As already mentioned, these
510
P. C. HOHENBERG,
M. DE LEENER
AND P. Rl%IBOIS
are only negligible in three dimensions if &~&2~.z
-2
(2.20)
.
Note that this limitation is stronger than the dynamical condition (2.19). In general, let us define (2.21)
cc = max (e2, 4, such that our kinetic theory is valid in the “precritical” 8 9 E,.
or “mean-field” region
3. Generalization to arbitrary dimensions. The derivation of (2.14) and its analysis as given in ref. 2, may be repeated for arbitrary dimension. Eq. (2.10) becomes simply* (3.1) whereas the corrections due to q’ values much larger than q are of order
w:(x) =
(k,TJA)
(a/r)fd (qr)2 (q2 + x2) r2.
(3.2)
It is thus seen that for 3 < d < 6, since wr % WI’,the dynamical scaling property of eq. (2.7) remains valid in the same sense as it was shown to hold for d = 3. It is only at d = 6 that co:(q) becomes comparable to co:(q); in that case, the small intermediate wave numbers do not entirely determine the time dependence of pa(t) and we have the “conventional theory” w:(q) z of (q) ~5 q2 (q2 + x2) (see sec. 5). Of course, we have also to reconsider the limit of validity of our eq. (3.1). We may generalize the arguments leading to (2.12), (2.17) and (2.18), and we find, in analogy to (2.19) that our evaluation of the dynamics is only valid so long as F $ .Q z (z-1)4’@-?
(3.3)
Note that, for d 2 4, this is the only condition [and probably too restrictive as already pointed out after (2.16)] because the static properties become classical (E? = 0). For d = 3, it was conjectured in refs. 1 and 2 that in the true critical region (E < ac), eq. (2.10) would remain essentially unchanged, (except for small corrections of order ~7,and a different dependence on the range parameter r) if the cor* Since our arguments are entirely dimensional, it is values, and we shall consider d formally as a continuous our discussion to d’> 3, however, due to the difficulty of tions in a Heisenberg system with long-range forces for d
not necessary to restrict d to integer variable (see ref. 5). We shall restrict a correct description of static correla= 1 or d = 2. See, however, ref. 21.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
511
rect correlation length x-l and exponent v are used. This conjecture is supported by the close agreement with experiment”), with the phenomenological theory4**) (see below), and with the semi-microscopic mode-mode theory12). Before considering the situation for d > 3 in the true critical region, we turn to the case T < T,, where real spin waves can propagate. 4. Kinetic equation below T,. For simplicity we still limit our discussion to the case of zero field (in ref. 13 it is shown that the finite-field case is analogous to the region H = 0, T < TJ. Our starting point is the approximate kinetic equation which describes the dynamical behavior of the spin-correlation function pgP(t) = r:‘(t)/ri’ (t = 0) below Tc3). This equation reads
where mzB(R) denotes the “free spin-wave frequency”
w;-(R) = -w;+(R),
w;(R)
= 0,
(4.2)
R denotes the magnetization,
and the non-markoffian kernel cfl has a structure similar to &q(a simple approximation is displayed in the appendix for a/? = + -). Note that wifi and CC?lP depend on x, the inverse correlation length below T,, and also on the magnetization R. The validity of eq. (4.1) can easily be shown to be restricted by E p
max (.e2, E3) = Ed,
(4.3)
precisely as in the paramagnetic case [eqs. (2.3) and (3.3)]. Let us examine the main features of the solution of eq. (4.1). As discussed in detail in ref. 3, sec. 3 for the case d = 3, eq. (4.1) shows no scaling behavior in the precritical region (E 9 8,). More precisely, writing formally Q:(X) = Re c&t) + i Im W:(X),
(4.4)
where Re U.&X)denotes the characteristic oscillation frequency of pc -, while Im u&x) is the damping frequency, it was found, using the mean-field exponents [R cc E* cc x1-1,that* for
co&)w
4< x,
E B EC,
(+)(qr)‘(m)[l
d=3:
+i($)(s>
+**.I;
* In the following equations we often suppress factors of order unity.
(4.5a)
512
P.C. HOHENBERG,
for
4% x,
M. DE LEENER
E>> E,,
AND P. Rl%IBOIS
d=3:
which indicates that the characteristic simple scaling form4)
(complex) frequency does not have the
(4.6) in the precritical region, for d = 3. As mentioned in ref. 3, however, this breakdown of scaling in the precritical region can be remedied in the true critical region by re-expressing the quantities in eq. (4.5a) in terms of the correct critical exponents R cc E’, xr cc E’ z R2 (neglecting once again the small exponent q), and we find* for
4 $ x,
cog25 (k,T,/fi) for
4% x,
E< &,I (qu)2
(xu)’ [I
+ i (q/x)3 + .--I;
(4.7a)
E < &,I
W&S) z (k,T,/fi) (q#12 [i + (x/q)* + ***I.
(4.7b)
Note that in these equations we have not displayed the r dependence because it is not known. At T, we find the matching condition
where we have taken the limit z + cc first. Another interpretation of eq. (4.5) is in terms of the perpendicular correlation length (x1)-l discussed in sec. 5.2 below. In the mean-field region we have [c$ eq. (5.17)] (4.9) whereas the quantity x of eq. (4.5) is the inverse of the parallel correlation range, given by xl1 0~
R,
(4.10)
* The unusual (q/x)3 dependence for the damping in eq. (4.7a) is due to our approximation for 6:s. It has been shown recently that inclusion of the spin-wave-spin-wave interaction term (which in the ordered region is of order zm2) in 6,afi leads to the usual result (q/x)? (see ref. 14).
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
513
in mean-field theory. Then eq. (4.5) may be written as for
4 4 XI, XII,
E$- EC:
0: (XI, x11)= (kiJClfi) (4rY (xlrY [l + i (a/r)” (4/x,) (4/Q’ for
4% XI,XlI,
mi (XL, xl,) x (kJ,/@
+
***I;WW
&B &,I (qr)5’2 (a/r)3’2 [i + (r/u)3’2 (x,/q)+ + -*-I.
(4.11b)
Thus the breakdown of the scaling “ansatz” (4.6) in the precritical region is caused by the existence of two different scaling lengths, for parallel and perpendicular fluctuations, respectively. In this case, however, we have a “two-length scaling”, (4.12) as can be seen from an examination of eqs. (4.11). In the true critical region, there is only one scaling length for d = 3 (according to “dimensional” scaling), so that (4.13)
XI = XII,
and eq. (4.12) reduces to eq. (4.6). More generally, for arbitrary d, eqs. (4.9) and (4.10) for xL and xl1 are valid in the mean-field region E B cc, and the kinetic theory yields for
4 Q XI,XII,
&$ EC:
d (XI, xi,) = bWclfi) (qr)’ (xlr)(d-2)‘2[1+ i (q/xJds2
(q/x11)2 (a/r)” + -a*]
(4.14a)
f2.1: (xL,
xl,) cz (k,T,/fi)
(qr)f’d+2’ (a/r)?
[i + **.I.
(4.14b)
These expressions are still of the general form (4.12) with r = 3 (d + 2). For d 5 4 the preceding arguments may be repeated, and we again get (4.6) in the true critical region. For d > 4 on the other hand, the static exponents do not change as we go into the true critical region, so it is likely that eq. (4.12) will also remain unchanged, and the simple homogeneity condition (4.6) will be violated. Of course the contribution c# coming from large intermediate momenta must also be considered, and for d 2 6 it will dominate w:, as discussed below.
514
P.C. HOHENBERG,
M. DE LEENER
AND P. Rl%IBOIS
5. Phenomenological theories. Let us generalize the phenomenological
theories
to arbitrary dimension. 5.1. Conventional theory. The conventional theory7) follows from the assumption that transport coefficients are regular at T,, and that apart from kinematic factors, which follow from conservation laws, the critical slowing down comes entirely from the diverging susceptibility. Thus, for a purely damped motion the characteristic frequency is 20) w%(x)cc ix-‘q2 x i (x2 + q2) q2, in the ferromagnet down), and
(5.1)
where the order parameter is conserved (kinematic slowing
0.((x) cc ix-l Z i (x2 + q2),
(5.2)
in the antiferromagnet where the order parameter is not conserved. [We have neglected the exponent q in eqs. (5.1) and (5.2).] When the motion is oscillatory the characteristic frequency is proportional to x-*, and we have15) co’&) cc RqZ w x,,q*,
q@ It,
(5.3)
for the ferromagnet and
4X4 cc Rq =
xllq,
96 x,
(5.4)
for the antiferromagnet, where we have used eq. (4.10) for xl1 . Note that the conventional theory does not provide a simple estimate for the damping of this oscillatory mode. 5.2. Dynamic scaling theory. It is assumed4) that the correlation function has a unique (complex) characteristic frequency of the form
and the unknown exponent 5 is determined, where possible, by matching to the limiting form of eq. (5.5) in a particular region. In the version of the theory formulated by Halperin and one of the authors4), the matching is to the spin waves below T,, which in the ferromagnet have the form15) ilq2,
(5.6)
1 = es/R.
(5.7)
wq x
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
515
In the critical region, the exponent of es was found4) by using static-scaling. Indeed, the perpendicular q-dependent susceptibility has the form15)
xdd = R21esq2~
(5.8)
from which it follows that the correlation function C,(x) dces not decay exponentially at large distances, but rather according to a power law. In that case we define15*16) a perpendicular correlation range XT’, by the relation [cj eq. (2.4) of ref. 41
C,(x) = R* (ll~,xY’,
W)
x-rm
where C,(x) is obtained by Fourier transforming eq. (5.8), yielding a power law with exponent p = d - 2. From eqs. (5.8) and (5.9) it follows that
The parallel correlation function, on the other hand, has a correlation length xi1 of which the exponent Y,, is in general different from the exponent vI of x; ‘. In the true critical region, we assume (according to ’ dimensional scaling”) that Ye = vII = v, whence eq. (5.10) implies v(d-2) es w E
(5.11)
It follows that in the critical region the spin-wave stiffness 3, [eq. (5.7)] goes as (5.12) and comparing eqs. (5.6), (5.12) and (4.6) we get
5 = d-p/v. According to dimensional scaling [dv = /I(6 + 1) = 2 - 01 = y + 28 = + 2/?] this result for 5‘is equal to 5=$(d+2-11).
(5.13) v
(2 - 7)
(5.14)
WagneP) has given an alternative method of estimating the exponent 5, based on matching to the (real) Larmor frequency u)L = g,ugH, in the presence of a magnetic field. The scaling exponent of His .sndw xBd”‘,so
5‘ =
gqv,
which agrees with eq. (5.14) using dimensional scaling.
(5.15)
516
P. C. HOHENBERG,
M. DE LEENER AND P. R&IBOIS
In the mean-field region we have
es0~ R2,
(5.16)
whence eq. (5.10) implies (5.17) and lJ1 = (d - 2)-l,
(5.18)
(since p = $), whereas Y,, = 2. l The spin-wave frequency [eq. (5.6)], goes as % w Rq2 M xY-~)‘~ q2 w x,,q’.
(5.19)
6. Comparison of kinetic and phenomenological theories. Let us summarize the situation in the Heisenberg ferromagnet. In the mean-field region (E >> E,), there are two scaling lengths, x;’ and xil, and the characteristic frequency does not obey the simple scaling relation (4.6). Instead, a “two-length” scaling relation (4.12) applies for d I 6, with an exponent [ = 3 (d + 2). In the domain T > T,, H = 0, where the correlation function is isotropic, the length x;l drops out, and “one-length” scaling is recovered. When d > 6 the frequency u$ of the conventional theory dominates the above scaling frequency, and the critical dynamics follows the conventional theory both above and below T, . These results have been verified order by order in perturbation theory in zero field, using the kinetic theory of refs. l-3. In the true critical region (E < a,), the kinetic theory does not strictly apply, but we conjecture that the dynamical behavior can be inferred by changing only the static critical exponents. For d < 4, according to (static) dimensional scaling there is only one scaling length, i.e., vI = vII , and eq. (4.12) reduces to the simple scaling hypothesis4g8) (4.6) with exponent 5 = 3 (d + 2 - 7) = /38/v. For 4 < d < 6, the two lengths x;’ and xi1 remain different in the true critical region, since the static behavior follows mean-field theory5). We are then left with two-length scaling, which again reduces to simple one-length scaling for T > T,, H = 0. In this case it is not possible to infer the behavior above T, by matching to the spin waves below T,. It is interesting to note that Polyakovr7) has suggested that this situation obtains even for d = 3 in the true critical region, thus violating the dynamic scaling “ansatz” (4.6). Clearly, the present work cannot make a definitive statement on this problem, but the arguments given above strongly favor the validity of the matching condition for d = 3, in contradiction to Polyakov’s suggestion. For d 2 6, the conventional theory is again expected to give the correct answer very close to T, . The results for the ferromagnet are summarized in table I.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
517
TABLEI Dynamic scaling exponent [ for the isotropic ferromagnet Region
4
ds4
dr6
1) Mean-field region (E % e,): a) T-c T,, H= 0 or Tarb, H # 0
2 lengths : C=f(d+2)
2 lengths : C=&(d+2)
conventional
theory
b)T>
1 length: 5 = 4 (d + 2)
1 length: conventional i = + (d + 2)
theory
1 length: C=+(d+2-7)
2 lengths:
conventional
theory
1 length:
conventional 1 length: C = f (d + 2)
theory
T,,H=O
2) True critical region (E Q E,): a) T-c T,, H= 0 or Tarb, H# 0 b)T>
T,,H=O
5=3(d+2--rl)
C = t (d + 2)
7. The isotropic antiferromagnet and the uniaxial ferromagnet.
7.1. The anti-
ferromagnet. The analysis of sets. 2 and 3 may be repeated for the isotropic antiferromagnet (see ref. 3, sec. 4) and the results are, for T > T,, w&)
= (kBTCl@ (a/r)3d (qr)*df(4q),
CO:(X)= (k,T,/h) (a/r)‘d (q2 + x2) r* , = (k,T,/fi) (a/r)3d’2.
we’
(7.1) (7.2) (7.3)
It follows that the limit of validity of our dynamical theory is [cJ eq. (2.18), and also the remarks after eq. (2.16)] ES = (z-1)4/d,
(7.4)
and that the conventional theory [eq. (5.2)] becomes valid at d = 4, where UP z 09. For T c T,, the kinetic theory yields q-+ It:
O&X) z (k,T,/h)
q9
O&C) x (k,T,/ti) (qr)‘d (1 + *me),
x:
(xLr)3(d-2’ (qr) (1 + . . *),
(7.5a) (7.5b)
in the precritical region, E g E,. The result may be extended to the true critical region, in analogy to the procedure leading to eq. (4.7a), and we find qQ x:
CJ~(X)x (k,T,/ti) (xa)‘(d-2) (qa) (1 + **a),
for d 5 4, and the conventional result of eq. (5.4) for d 2 4.
(7.6)
518
P.C. HOHENBERG,
M. DE LEENER
AND P. RBSIBOIS
The phenomenological theory may be generalized to arbitrary d, by noting that according to hydrodynamics1 “) Co,
x cq(1
+ **.),
(7.7)
c2 = @sIXT,
(7.8)
where XT is the total susceptibility (assumed to go to a constant at T,), and es is related to x1 through the perpendicular staggered susceptibility (7.9) yielding as in eq. (5. IO) (7.10) (No is the staggered magnetization). Since the spin-wave velocity c involves only es, we have, from eqs. (7.8) and (7.10): c a #d--2) _L *
(7.11)
In the scaling region there is only one correlation length (x1 = x,,) and the frequency (7.7) has the simple scaling form (4.6) with (7.12)
5 = W,
which agrees with Wagner’s*) estimate. In the mean-field region, we again have two-length scaling, eq. (4.12), with C given by eq. (7.12). The main difference with the ferromagnet is that the conventional theory, eq. (5.4), becomes valid precisely at d = 4, since the scaling frequency is just q-G x:
d@>x
%I,
d = 4,
(7.13)
which is equal to (7.7) because of (7.12). Thus the anomalies found for 4 < d c 6 in the ferromagnet are not present in the antiferromagnetic case. 7.2. The uniaxial ferromagnet. In this case it is possible to show, using the kinetic theoryl*), or mode-mode couplinglg), that the conventional theory* is valid for all d, up to terms of order ~7,which are zero in our model. In particular, * Let us stress that the general definition (5.1) we have adopted for the conventional theory does not mean that the characteristic frequency may be calculated from the short-time moment expansion20). This remark is particularly important for the uniaxial ferromagnet where the dependence of w:‘(x) on the anisotropy parameter x_t obtained from kinetic theory’*), is completely different from that deduced from the moment method.
FLUCTUATIONS
519
IN SPIN SYSTEMS IN d DIMENSIONS
we find for the longitudinal characteristic frequency above T, oq(2t) w q2$
(x) x x452(x/q),
(7.14)
while the transverse characteristic frequency W: - (x) does not behave critically. The phenomenological theory does not make any predictions for the value of the exponent 5, since there are no propagating modes below T,.The results of the present section are summarized in table II.
TABLEII Dynamic scaling exponent 5 for the isotropic antiferromagnet Region
and uniaxial ferromagnet
d-c4
d>4
Antiferromagnet 1) Mean-field region: E % E, a) T c T,
2 lengths: 5 = 3d 1 length: 5 = 3d
b) T > T, 2) True critical region: E Q E,
conventional
theory
conventional
theory
1 length: ,Z = &d
conventional
theory
c=2
1;=2
Uniaxial ferromagneta Conventional theory One-length scaling a In discussing
the uniaxial case we neglect corrections to the exponent [ which are of order q.
Acknowledgements. One of us (PCH) wishes to thank B.I. Halperin for informative discussions, and the members of the Theoretical Physics Department (TUM) in Munich for their hospitality during the period when this work was initiated.
APPENDIX
For the convenience of the reader, we reproduce here the simplest approximation for the kernels f?@and c: - ; we have”) in the limit (2.9):
~tz,<~) = -y2
(q2 + x2)
s dq,
(A.11
520
P. C. HOHENBERG,
M. DE LEENER
AND P. Rl?SIBOIS
where y2 = PJ (O)‘/O ;
(A.21
(4 is the volume of the Brillouin zone). This formula is valid in any dimension d > 2.
Similarly3)
+
s
q2 dq,
K4 + ‘l!’ - 43 41
1;,:-(z)
I;;‘-,,(z)
.
(A.3)
1
Note that in the estimates made for q + x, we may take p:(z) = 1, while where q 9 x, we have of course pi -(t) = F,(T). Remark also that if we formally set x = 0, the two kernels GUand c?: - become identical. Note added in proof. After this paper was written, we became aware of additional references which bear on the present topic. Early work by Villain22) suggested that the conventional theory would become valid in the ferromagnet for d r 6. This conclusion was confirmed by Kawasakiz3), on the basis of an approximate modemode coupling argument. In addition, Kawasaki considered the antiferromagnet, and found the condition d 2 4. More recently, Van Leeuwen and Guntonz4) have treated the spherical model, which resembles our model quite closely, and have found results similar to ours for the ferromagnet [see their eqs. (5.5) and the discussion after eq. (5.6)]. An advantage of the spherical model is that there is no condition analogous to our eq. (3.3). Another recent development is a microscopic calculation25) of the critical frequency for a uniaxial system, in which the conventional theory is violated to order 17for d < 4.
REFERENCES 1) Rbsibois, P. and De Leener, M., Phys. Rev. 178 (1969) 806, 819. These two papers are referred to in what follows as RDL III and RDL IV, respectively. 2) De Leener, M., Physica 56 (1971) 62. 3) RCsibois, P. and Dewel, G., Ann. Physics 69 (1972) 299. 4) Halperin, B.I. and Hohenberg, P.C., Phys. Rev. 177 (1969) 952. 5) Wilson, K.G. and Fisher, M.E., Phys. Rev. Letters 28 (1972) 240; Wilson, K.G., ibid. 28 (1972) 548. 6) Tucker, J.R. and Halperin, B.I., Phys. Rev. B3 (1971) 3768; Gruenberg, L.W. and Gunther, L., Phys. Letters 38A (1972) 463. 7) Van Hove, L., Phys. Rev. 93 (1954) 1374.
FLUCTUATIONS
IN SPIN SYSTEMS IN d DIMENSIONS
’
521
8) Wagner, H., Phys. Letters 33A (1970) 58. 9) G&burg, V.L., Fiz. Tver. Tela 2 (1960) 2031 [English translation: Soviet Physics - Solid State 2 (1960) 18241. Vaks, V.G., Larkin, A.I. and Pikin, S.A., Zh. eksper. teor. Fiz. 53 (1967) 281 [English translation: Soviet Physics - JETP 26 (1968) 1881. Thouless, D. J., Phys. Rev. 181 (1969) 954. See also Hohenberg, P.C., Fluctuations in Superconductors, Asilomar, Stanford Research Inst. (Cal., 1968). 10) Zwanzig, R., Phys. Rev. 124 (1961) 983. 11) See, for example, Minkiewicz, V., Intern. J. Magnetism 1 (1971) 149. Lau, H. Y., Corliss, L. M., Delapalme, A., Hastings, J. M., Nathans, R. and Tucciarone, A., Phys. Rev. Letters 23 (1969) 1225. Rtsibois, P. and Piette, C., Phys. Rev. Letters 24 (1970) 514. 12) Kawasaki, K., Progr. theor. Phys. 39 (1968) 2, 285; 40 (1968) 11, 706, 930; Ann. Physics 61 (1970) 1. Kadanoff, L.P. and Swift, J., Phys. Rev. 166 (1968) 89. 13) Dewel, G., to be published. 14) Dewel, G., Resibois, P. and De Leener, M., Phys. Letters 39A (1972) 33. 15) Halperin, B.I. and Hohenberg, P.C., Phys. Rev. 188 (1969) 898. 16) Josephson, B.D., Phys. Letters 21 (1966) 608. 17) Polyakov, A.M., Zh. eksper. teor. Fiz. 57 (1969) 2144 [Soviet Physics - JETP 30 (1970) 11641. 18) Piette, C., M&moire de Licence, Universite Libre de Bruxelles, 1969 (unpublished). 19) Kawasaki, K., Progr. theor. Phys. 39 (1968) 285; 40 (1968) 706. 20) See, for example, Marshall, W., Nat. Bur. Stand. Misc. Publ. 273 (1966) 135. 21) Blume, M., in Proceedings of the 1972 Grenoble International Conference on Neutron Scattering, IAEA (Vienna) to be published. 22) Villain, J., J. Physique 29 (1968) 321. 23) Kawasaki, K., Prog. theor. Phys. 40 (1968) 11; footnote: p. 35. 24) Van Leeuwen, J. M. J. and Gunton, J.D., Phys. Rev. B6 (1972) 231. 25) Halperin, B. I., Hohenberg, P. C. and Ma, S., Phys. Rev. Letters 29 (1972) 1548.