Nonlinear dynamic renormalization-group analysis above and below the lambda transition in 4He

Physica 109 & I10B (1982) 1549-1556 North-Holland Publishing Company

1549

NONLINEAR DYNAMIC RENORMALIZATION-GROUP ANALYSIS A B O V E A N D B E L O W T H E L A M B D A T R A N S I T I O N IN 4He

V. D O H M lnstitut fiir Festk6rperforschung der Kernforschungsanlage Jiilich, D-5170 Jiilich, Fed. Rep. Germany

and R. F O L K * lnstitut fiir Theoretische Physik, Universitdit Linz, A-4045 Linz, Austria t and Institut fiir Festk6rperforschung der Kernforschungsanlage Jiilich, D-5170 Jiilich, Fed. Rep. Germany.

Recent developments in the theory of the dynamics of liquid 4He near the lambda transition are reviewed. A nonlinear renormalization-group treatment is presented which provides a quantitative description of the dynamics in the entire region between noncritical background and criticality. The previous discrepancies between asymptotic scaling theories and various experiments above and below Ta are explained in terms of crossover effects. This crossover is governed (i) close to Ta by the effect of the weak-scaling fixed point, and (ii) farther away from T~ by a strong decrease of the effective dynamic coupling. New results are presented which are based on the complete flow equations for model F in two-loop order. They tentatively confirm the conclusions of an earlier analysis based on model E and invalidate part of a recent analysis treating model F in an unsystematic way. Excellent agreement is found with new accurate measurements of second-sound damping over four decades in relative temperature.

1. Introduction: the problems In the early d y n a m i c - s c a l i n g picture [1, 2] the o b s e r v a b l e critical d y n a m i c s of liquid 4He n e a r t h e l a m b d a t r a n s i t i o n was i n t e r p r e t e d as an asymptotic b e h a v i o r c h a r a c t e r i z e d by t h e usual p o w e r laws, a p a r t f r o m c o r r e c t i o n s d u e to the (almost l o g a r i t h m i c ) t e m p e r a t u r e d e p e n d e n c e of the specific heat. A l t h o u g h r e n o r m a l i z a t i o n g r o u p ( R G ) a n d m o d e - c o u p l i n g t h e o r i e s [3-5] s e e m e d to confirm this p i c t u r e t h e r e e x i s t e d a n u m b e r of s e r i o u s p r o b l e m s . (i) T h e r m a l c o n d u c t i v i t y m e a s u r e m e n t s [6] v e r y close to T~ y i e l d e d an effective e x p o n e n t which was a b o u t 20% l a r g e r than that p r e d i c t e d by t h e scaling t h e o r y .

(ii) T h e w e a k t e m p e r a t u r e d e p e n d e n c e of the linewidth of the light-scattering s p e c t r u m a b o v e a n d b e l o w T~ [7-10] was in gross c o n t r a d i c t i o n with t h e scaling p r e d i c t i o n [1-3] a n d with a q u a n t i t a t i v e m o d e - c o u p l i n g c a l c u l a t i o n [4]. A l s o , the m a g n i t u d e of t h e t r a n s p o r t coefficients disa g r e e d d r a s t i c a l l y with light s c a t t e r i n g d a t a [ 7 -

10l. (iii) T h e a s y m p t o t i c ratio w* b e t w e e n the order parameter and the entropy relaxation rate in 4 - e d i m e n s i o n s was f o u n d to b e w * = 1-1.41e [3]. This c o u l d not b e p r o p e r l y e x t r a p o l a t e d to e = 1 to yield a r e l i a b l e e s t i m a t e in t h r e e d i m e n s i o n s for w* a n d for the universal a m p l i t u d e s a n d scaling functions d e p e n d i n g on

* Supported in part by the Fonds zur F6rderung der wissenschaftlichen Forschung. + Permanent address.

0378-4363/82/0000-0000/$02.75 O 1982 N o r t h - H o l l a n d

W*.

(iv) T h e m o s t striking d i s c r e p a n c y b e t w e e n t h e o r y [4, 11] a n d e x p e r i m e n t [12] for t h e amp l i t u d e of s e c o n d - s o u n d d a m p i n g was r e c e n t l y

~,'. D o h n l a m t R. b}dk / l ) v n a t n i c ~malvsts near the lanibda tran~iticm tit ~Hc

1550

reduced by new experiments [131. Nevertheless. n o reliable theoretical prediction for this amplitude was awfilable. (v) Very recently, the publication of new thermal conductivity data [14-1(q revealed an unexpectedly large systematic uncertainty of the earlier [6] experiments due to an unexplained cell-size d e p e n d e n c e . At present it is not clear to what accuracy the new data represent the truc critical thermal conductivity of bulk ;He [15 I. It is the purpose of this paper to review how these problems, except for problcm (v), wcrc resolved recently, apart from refinements. Nov,' preliminary results based on the complete llow equations of the asymmetric spin model (F) [31 in two-loop order will also be presented. The results tentatively confirm earlier conclusions [17-19] obtained on the basis of the symmetric model E [3] and invalidate part of a very recent analysis [14, 15] employing model F in an unsystematic approximation. Excellent agreement of our theory with new accurate s e c o n d - s o u n d d a m p i n g data [20] will also be reported.

2. A s y m p t o t i c

theory

The asymptotic critical dynamics of the lambda transition can be described by the wellknown model [ of Halperin. Hohenberg and $iggia Pl which consists of coupled l.angevin e q u a t i o n s for the o r d e r p a r a m e t e r & ( s , t) a n d e n t r o p y ##l(X, t).

the ralio~ / '<4 ,, "

w,, ::

A,~

,k']

l~

mid

/'.,A.

(

2.-1 )

whose (dimensionless) lixed-point values will hc denoted by w and l i\n important ,+,.p towards it proper understanding of the dvlmmio, within model E in three dimensions was the study t~v Dc l)ominicis and Peliti (DP) I21 t of n - d e p e n d e n t gcneralizatilms of this model in d ~t s dimensions tip [0 ((,f ') (/1 is the n u m b e r ~>1 c o m p o n e n t s ~f the order parameter). This led I~ the discovery of a b o u n d a r y in the n - d plane which separates :1 scaling region with w -ii from a "'weak-scaling' region with w it I'~,r the SSS-model [22] this bcmndary emerges it{mt ~ :: ~L n 3/2 [23.24]. with a linear cxtrapolalh*u [21.25] to thc \icinity of the physical point n 2. d 3 (s 1 ) (dashed line in li~. l). This implies
I21l a,~d

~I the ordcr ~I ().I [2~'~.271
,' I. n 2. ' l h c cxistcm_'c of the ncxv boundar~ explained I271 ihc p~>~, c~H/\crgcncc oi Ihc traditional strict ~-cxpam, itm I3.5,211 l'lu n and appropriate self-consistent cxlrcipolatioil l~rocc'dtircs I>-2Sl could be ilst:d to relllO,~t_" this dilticultv. Suhsequcntly. the clfeci of the small

w¸

(.

J

8H ,i, = -_t <, a0* ,~.,

t,l

.

8H

(2.

~,m,#, m~7 + o,,,,

: A,,V-• &~SH+ 2g,, lm~[ &".~.~ )

+ 0,,, .

i)

W

: L~

(2.2) 01

4

3

~--',V,, ' ',, :}. (2.3)

dlm~nsi0r Fig. I. I,mear c',trapolali~m (da,,hcd Imc) lk,r lhc ,,mhiliI,,

with kinetic coefficients /',. A,, and ;J dynamic coupling go. C o n v e n i e n t dynamic parameters illc

boundar,~ ~q lilt_'biY,S-m~,dcl ~,ichliug ;i hordcrlim: dimem>hm d,~s~ 2.,'42for model [i (n 2) ()m mlaI~.,,is I,,cclion f~l ,,f the avMlablc lhcrmal conchlclb, ii.}¢ data xuggcq,, lll;,~ ll~c bounc:[;.iI-\ /icx ,,C~tllC,.v[Icrc irl lhu ,;h~ided r¢~ion l:~r ih<~ dottc,.] Cl.lf\C "~cc v.cciiOll tl

H

f d"xO-,,la.I ~ + ~lv~F +

.,,l#.l ~

V. Dohm and R. Folk / Dynamic analysis near the lambda transition in 4He

w* on the light-scattering spectrum [29] and later on second-sound damping [30] were investigated in the asymptotic critical region. Since corrections to scaling had not been taken into account in these studies, it was not yet possible to carry out a conclusive comparison with experiment.

3. Nonasymptotic theory It has already been noted by D P [21] that the small transient exponent tow largely shrinks the asymptotic scaling region and that it may account for some of the discrepancies between scaling theory and experiments. A quantitative analysis of thermal conductivity and light-scattering data, including corrections to scaling, was first carried out by Ferrell and Bhattacharjee (FB) [31-34]. Equally important, these authors also opened up a more general point of view by incorporating in their analysis the "precritical" behavior far from criticality, which they described by means of a "high-temperature" expansion in powers of the inverse correlation length. Thus, there existed two separate approaches which were appropriate for the limiting cases far from and close to criticality. A complete description which accounted both for these limiting cases and for the crossover between criticality and background above and below Ta was introducted by the present authors in terms of a nonlinear dynamic renormalizationgroup treatment [17-19]. This description is based on the field-theoretic renormalization group [21, 35, 36]. Independent suggestions, somewhat similar in spirit, were also made by other authors [37, 38], although at that time they did not recognize the necessity of a fully nonlinear treatment. We first sketch the nonlinear analysis for the example of the thermal conductivity A whose scaling form [3] reads A =°" l x~x ~1; /O2S +

Cp1/2 ,

(3.1)

1551

where ~:+ is the correlation length and Cp is the specific heat. The amplitude RA is of interest for a test of the theory since it depends rather sensitively on the fixed point value w* according to [3, 21, 27] R,~ : (2"rr2w*f*)-'/2(1

- f*/4).

(3.2)

Within the field-theoretic R G approach the departures from the asymptotic form of A, eqs. (3.1) and (3.2), can be described in a compact way by generalizing R~ to an effective amplitude [17, 18, 38] R~'~ = [ 2 ~ 2 w ( O f ( e ) ]

1/211 - -

f(g)/4].

(3.3)

Here w(#) and f(¢) are effective parameters determined by the flow equations (of model E) dw

(~-{= flw(w,f),

dr_

t¢'~ - ~r(w,f),

(3.4)

with /3-functions that are presently known in two-loop order [21,27] (second order in the effective coupling f). The w-dependence in (3.4) is nonlinear and is known exactly within the loop-expansion. Rather than studying the asymptotic critical dynamics corresponding to the ( ~ 0 limit with w* =- w(0), f* - f(0), we have fully exploited the complete information about the nonasymptotic dynamics contained in (3.4). This can be represented in terms of a flow diagram (fig. 2) [18]. The 4He dynamics at a particular pressure is represented by a particular trajectory which can be identified via a comparison of R]" with experimental data (in practice by a fit with two adjustable parameters w(G) and f ( G ) at some convenient initial value G). For the present example, ( may be identified as t=

t~.

t = (r-

T,)IT,.

(3.5)

with u = 2/3. The necessity for a complete integration of the nonlinear flow equation comes from the fact that tow <~ 1. This implies that lead-

1552

V. l)ohm and R. l'blk I l)ynamic analysis near the larnbda transition in ~Hc 06

d:d'=3

i

{, AHLERS, MEASURED DATA (EELL b) -~ AHK, GENERATED DATA

/ I

//

2

/

05-

'2

f

R~ff

-

-

---

f

BEST FIT,MODEL AHK F I T •

E

' i

/

O&

-,,~o (; 3 ,f

0

';

I 0

:

OZL !0

',. 05

--.~.

1

e,

10 ~'

10 '+

!~,,. ~

10 •

W 1+W

Fig. 2. Flow d i a g r a m for the d y n a m i c p a r a m e t e r s of m o d e l R e l e v a n t for the superfluid t r a n s i t i o n in three d i m e n s i o n s the s m a l l - f d o m a i n . The t r a j e c t o r y r e p r e s e n t i n g aHc s a t u r a t e d v a p o r p r e s s u r e is close to thai leaving the vicinity the f 0 axis n e a r w / ( l + w) 0.4 (from ref. 1N).

l{. is a! el

ing (~- (<°*) and subleading (--l :'°,,. t ~'",,,...) cot-rection terms are almost equally important a n d that therefore the standard concept [21, 31,35, 36, 38] of expanding the [3-functions around the fixed point becomes inapplicable. Furthermore, the strong variation of f, as shown in the flow diagram and in fig. 4 below, can he properly described only by means of a nonlinear treatment. An appealing feature of this approach is that ii easily allows us to make quantitative predictions about the nonasymptotic, i.e. nonuniversal ciitical dynamics of all other physical quantities. without further adjustable parameters. T h e r e is n o nonuniversal p a r a m e t e r left (within model E) once w(l) and f ( l ) have been identified (via R~ 'T, for example). W e illustrate this point in section 5 for the dynamic structure factor below Ta, after a n analysis of Re," abow, Ta.

4. Thermal conductivity The thermal conductivity data at saturated vapor pressure for difl'erent He cells J¢~, 14-16] are shown in fig. 3(a and b). (They differ for I < ll) a due to an unexplained cell-size dependence.) Also shown are "'generated d a t a " Jl4. 151

• AHLERS AND BEHRINGER (CELL A) 05#--BEST FIT , ,. . . . AHK FIT i MODEL E R)elf i

04

03-

0 2i

,

~0 ~

~o ~

lo 3

1'0

Fig, 3. (a, h) T h e r m a l conductiviI,, d a t a a c c o r d i n g to lig. 2 el ref. 14. D a s h e d curves from refs. 14 and 15: full c u r v e s ave o u r least s q u a r e lit usmg precisely thc p r o c e d u r e d e s c r i b e d m rcf. 15. The d a t a and c u r v e s arc s h o w n only for the r a n g e el the fits (l(i ", t• I(I 2)

whose physical significance is obscure (we include them here for reasons of comparison with A H K only). The full curves represent R~ t~ according to (3.3)-(3.5) with the two p a r a m e t e r s (initial conditions) w(1 ) and f ( l ) adjusted st) as to yield a least square fit to the data in the range l(i " < t < I l l :. We see that model E provides a reasonably accurate fit to the data, as found earlier [17-19] for cell D data [39]. Bv contrast. A H K claimed [14, 15] that model E shows significant deviations from the data. as exhibited by the dashed curves in fig. 3(a, b). Since wc have used precisely the s a m e fitting procedure as described bv A H K [15] we must attribute this discrepancy to an error of A H K . (With this procedure we could reproduce the full curves in

1553

v. Dohm and R. Folk / Dynamic analysis near the lambda transition in 4He

fig. 2a of [14] and in fig. 17 of [15].) We have found a similar disagreement with the dashed curves in fig. 15 of [15] and in figs. 4 and 5 of Ahlers [40]. Therefore we consider as incorrect the conclusions of A H K concerning the accuracy of model E. The temperature dependence of R]~ is explained [17-19] in terms of two crossover effects induced by the parameter flows w(#) and f(#) in fig. 4. (i) w(#) decays towards w(O)= w * ~ 1 with a slow transient due to the effect of the weak-scaling fixed point. This implies an increase of R]" for t < 10 -4.

(ii) f(#) decreases for # > 10-2 towards a small value f(1),~ 1. This implies an increase of R ] n farther away from T,. The competition between both effects is responsible for the minimum of R ] ~ near t = 10 -3"5. The latter point (ii) has subsequently been emphasized by Ahlers, Hohenberg and Kornblit [14, 15].

5. D y n a m i c structure factor

A nontrivial quantitative test of our theory is provided by the dynamic structure factor below T,. In the hydrodynamic region, at low frequencies w, it has the form S(w) = const.

(~,_~2, Ko)2r

+

where /2 = c2k is the second-sound frequency. Near Ta the asymptotic scaling form of F2 = ½D2k2 reads [5] /)2 = 2R2c2~- ,

where ~:_ is the correlation length. The universal amplitude [3-5, 11,30] R 2 ( w * , f * ) can be generalized to the nonasymptotic regime according to [17]

and can be calculated without adjustable parameters if w(#) and f(#) are already identified (via R~). Within the expected accuracy of model E the resulting D2 (dashed curve in fig. 5) [17] compares reasonably well with experiments [13, 20,41]. A somewhat different prediction in the precritical region based on the high-temperature expansion has subsequently been made by Ferrell and Bhattacharjee [42]. Our procedure can also be applied at higher pressure in order to compare with light-scattering data. The result is shown in fig. 6. Similarly, good agreement is found with other light-scattering data below and above T, [18, 19]. In fig. 7 the separate contributions F, and F~ to F2 = F~ + F~ are also compared with experiment. The agreement is encouraging. The departure from 16 r~

,

kk\~ ~

K

F 1.0 f

" I]HANSONANDPELAM

f(l} I •

"

o,i

N

:'~

T

g

w(t) -5

-5

, , • ROBINSON AND CROOKS x AHLERS

i %_-

'o

!

(5.3)

R~" = R2(w(#), f ( # ) )

(5.1)

(..02) 2 -I"- 4/-'2to 2 '

(5.2)

-~.

-3

-2

-1

-g

0

tOglo it)

-3 10g10 Ill

-2

-1

Fig. 4. Parameter flow corresponding to the full curve in fig.

Fig. 5. Second-sound damping D2. Dashed curve corresponds to the parameter flow of fig. 4 (model E), full curve from

3(a).

model F (section 6).

1554

~/r. D o h m a n d R. Folk I D y n a m i c analysis near the l a m b d a transition in +He

10~ • TARVIN,VIOAL,GREYTAK

I-2 2E

•

P=231 BAR T
i

2P

•

+

•

o

•

O6 Z

t

(MHz)~ . •

¢)

~04i

o

L

10

I

IT -Tx(P)l

r,

o

o

o

<,

02:

/

100

;

01

i o + TARVIN,V[DAL,GREYTAK P=231BAR I 0! 10 5 2 l,(P) I {inK)

0 0t

(inK)

Fig. (~. Halfwidth for the dynamic structure factor below 7] M . Dashed curve is the scaling prediction 141. Our ihcorv

I,ig 7. l r a n s p o r ! coeflicicnls, cq. (5 1), in the hvdrod~nanlic rcgion below 7] I<~i ()ur thcor;, vickts lhe ,olid cur'+c', (l:rom r01L It) )

yields the solid curve. The theory is applicable to the hydrodynamic region. (From ref. lC,L} the scaling behavior [41 (dashed curve in fig, ~) is thus explained in terms of the decreasing /(+') farther away from TA [18, 191. Since R+ +-+f ( i ) ~/:, this decrease essentially c o m p e n s a t e s the temperature d e p e n d e n c e of c2,£ in D:, eq. (5.2). Thus, the long-standing problem (it) m e n t i o n e d in the introduction is resolved, apart from refinements. The more complicated critical (k~ ~ 1) region below Ta has also been studied [43], and comparison with experiment by means of the nonlinear analysis is in progress. A b o v e +I'~ it is straightforward to generalize the asymptotic expression [27] for the dynamic structure factor lo the n o n a s y m p t o t i c regime [44]. A similar study employing model E was carried out recently by H o h e n b e r g and Sarkar [45[. Their qualitative conclusions agree with those of Ferrell and Bh attacharjee [31-341 . Some diltere n ces concerning the shape of the spectrum were found but the experimental situation above T~ [7-10] is not yet sufficiently accurate for a conclusive c o m p a r i s o n with the theory.

be c o m p l e m e n t e d by additional terms y,m[@[" and h,m in (2.3) and that / +. is to be taken complex. This is model F [31 which has the same asymptotic behavior as model E. T h e additional n o n a s y m p t o t i c contributions were expected to yield corrections to our model K analysis, nt about 15% [171. In view of the more fundan3cntal problems to bc resolved, however, it did not seem urgent to study such corrections. In fact. it did not even seem /~,asihh" to carr~ oul a model F analysis since t h e / J - f u n c t i o n s entering the flow equations were not known completely in twoloop order, as noted earlier [IM I. Nevertheless. A H K did carry out such an analysis [14. 151 employing an approximation suggested 1~ H o h e n b e r g ci at. 13vq. This approximalion includes in the (complex) /3-fut~cliotl /3,, tmc-lt~ol~ /, i T f l:. and y ,

tcrnls

and two-loop ternl~

[ml #ief,lecls other two-loop terms, i 7 ) I :. y tL izf":, and y+. Since l(t ) is of I)(1) close to 77, the terms y-~/' and iT/"~-~ inc c o n l p a r a b l c to v: ~illcl i7/l':, respectively. Thus, the A I - | K analvsi~ ix based

Oll

all

unsystonlatic

sequently, lhe conclusive.

approxinmlion.

corresponding

restllis

M o d e l F is ~i special ca~,t_" ill' ~1 illorc

6. Model F analysis It is well known [31 that in the n o n a s y m p t o t i c region the model E equations (2.1)-(2.3) have to

t".

bicritical

model

of

Janshcn

mid

<)no

('Oll

{ilC

ilol

gt_'llcr;i] of

Ibm

present authors 14+ 1, part of which has bccn i n v e s l i g a t e d p r e v i o u s l y in lw, o - l o o p order. Fhi~

two-loop sttidy h a s b e e n c o n l p l c t e d

\cry

recciltb,

v. Dohm and R. Folk / Dynamic analysis near the lambda transition in 4He [47] as far as model F contributions are concerned. W e have used the c o r r e s p o n d i n g two-loop flow equations to carry out a model F analysis. It yields the following preliminary results. (a) T h e t w o - l o o p terms neglected by A H K have a considerable effect on RE" in the regime close to T,. This is illustrated in fig. 8 (compare fig. lb of [14] and the corresponding figures in [15, 40]). (b) O u r analysis tentatively implies that the borderline dimension d* below which dynamic scaling breaks down (fig. 1) is close to 3 (based on the cell A data) or slightly larger than 3 (based on the cell D data) [48], in a g r e e m e n t with o u r earlier model E analysis [17-19]. It should be noted, of course, that this estimate of d~* should be considered as tentative because of the neglect of higher-loop contributions to (3.3) and (3.4). (c) Excellent a g r e e m e n t with accurate secondsound d a m p i n g m e a s u r e m e n t s [20] is f o u n d over four decades in relative t e m p e r a t u r e (full curve in fig. 5), without having adjusted a three-loop term [15] a b o v e T~. A c c o r d i n g to these points and to our results discussed in section 4 above, we conclude that

• AHLERS AND BEHRINQER - - - AHK MODEL F

ef ' ~ 06

/

R;~

02 ,

-

-5

_

,

,

-4

-3

TI -2

-I

IOglo 1"

Fig. 8. Fits of the effective amplitude R,~" for cell A using data in the limited range 0.003 < t <0.01, between the vertical arrows. Dashed curve from fig. lb of ref. 14 or fig. 16b of ref. 15 which is based on an unsystematic approximation for model F. Full curve from model F in two-loop order. The large difference for t < 10 4 is due to the two-loop terms neglected by AHK. The precise agreement of the full curve with the data indicates d*~3, provided that the data represent the true thermal conductivity of bulk 4He.

1555

essential parts of the work of A H K [14, 15] are not valid. Finally, we briefly c o m m e n t on fig. 1 in view of our new results. T h e y suggest that the stability b o u n d a r y of the SSS model lies s o m e w h e r e in the shaded region, thus being close to the physical point n = 2, d = 3. Even if the dimension d*, where this b o u n d a r y meets the n = 2 horizontal line, were considerably smaller than 3, this would not imply a small effect of the weak-scaling fixed point on the data, as claimed recently [14]. T h e relevant quantity determining this effect is the distance of the stability boundary in the n - d plane from the point n = 2, d = 3 rather than the difference 3 - d*. T h e distance is small also for a b o u n d a r y like the d o t t e d curve in fig. I c o r r e s p o n d i n g to the A H K approximation. M o r e details will be given elsewhere [49].

7. Conclusions It can now be considered as almost established that the dynamic-scaling predictions [1-3], alt h o u g h successful in a variety of o t h e r phase transitions [5], are actually not valid just for the lambda transition for which this theory had been p r o p o s e d originally [1], except, perhaps, in an experimentally inaccessible regime. Large crossover effects of genuine dynamic origin, as described in this brief review, govern the observable critical dynamics. Owing to the powerful field-theoretic r e n o r m a l i z a t i o n - g r o u p approach, however, this crossover is n o w well u n d e r control, apart f r o m details, and can be treated quantitatively with a few nonuniversal p a r a m e t e r s taken from experiment. T h e situation on the experimental side, on the o t h e r hand, is less satisfactory. Additional efforts towards m o r e accurate m e a s u r e m e n t s are certainly desirable in order to m a k e possible a quantitative test of the theory of critical dynamics.

Acknowledgements W e wish to thank G. Ahlers, J.K. Bhattacharjee, M.J. Crooks, R . A . Ferrell, P.C. H o h e n b e r g ,

V. Dohm and R, Folk / Dynamic analysis' near the lamhda transition in aHe

1550

A. Kornblit, and B,J. Robinson for sending us their results prior to publication, and K. Binder for useful comments.

References

1241 [251 [21q [271 [2• l [291 [301

[I] R.A. Ferrel[, N. Menyhard, H. Schmidt, F. Schwabl and P. Szepfalusy, Ann. Phys. (N.Y.) 47 (1968) 565. [21 B.I. Halperin and P.C. Hohenberg, Phys. Re,~. 177

(i~69) 952. 13] B.I. Halperin, P.C. Hohenberg and E.D. Siggia. Phys. Rev. B13 (1976) 1299: erratum B21 (198(}) 2044. [41 P.C. Hohenberg. E.D. Siggia and B.I. Halperin. I'hvs. Rev. BI4 (1~,76) 2865. 151 P.C. H o h e n b e r g and B.I. Halperin. Rev. Mod. Phys. !t, (1977) 435. [6] (3. Ahlers, in: Proceedings Twelfth hlternational ('onference on Low T e m p e r a t u r e Physics, E. Kanda. ed. (Academic Press, Japan, It?71) p. 21: and in: The Physics of Liquid and Solid Helium, Part 1, K.H. B e n n e m a n n and J.B. Ketterson. eds. (Wiley, New York, 1076)~ see also M. Archibald, J.M. Mochel and L. Weaver. Phys. Rev. Lett. 21 (1968) 1165. [7] G. Winterling, F.S. Holmes and T.J. Greytak, Phys. Rev. l,ett. 30 119731 427. G. Winterling, J. Miller and T J . Greytak, Phys. Lett. 48A (1974) 343. [8] W.F. Vinen and D.L. Hurd. Advan. Phys. 27 (1978) 533. [9] J.A. Tarvin, F. Vidal and T.J. Greytak, Phys. Rev. BI5 (1977) 4193. [10] T.J. Greytak, in: Proceedings of the International Conference on Dynamic Critical P h e n o m e n a , C.P. Enz, ed. (Springer, Berlin, 1979). [11] E,D. Siggia, Phys. Rev. B131197613218. [121 J.A. Tyson, Phys. Rev. Left. 21 (19681 1235. [13] G. Ahlers. Phys. Rev. Lett. 43 (1979) 1417. [141 G. Ahlers, P.C. H o h e n b e r g and A. Kornblit, Phys. Rev. Lett. 46 (1981) 493; erratum 46 11981) 686. [15] G. Ahlers, P.C. H o h e n b e r g and A. Kornblit, preprint (May 1981). [16] G. Ahlers and R.P. Behringer 119771 as quoted m refs. 14 and 15. [17] V. D o h m and R. Folk, Phys. Rev. Lett. 46 (19811 34~,L [18] V. D o h m , R. Folk, Z. Phys. B40 (19801 79. [19] V. D o h m and R. Folk, Z. Phys. B41 (1981)25l. [20] M.J. Crooks and B.J. Robinson, Physica 107B (1981) 339: and private communication. [21] (7`. de Dominicis and L. Peliti, Phys. Rev. Lett. 38 (1977) 505; Phys. Rev. BIg (197g) 353. [221 L. Sasvfiri, F. Schwabl and P. Sz~pfalusy, Physica 81A (1975) HIS. [23] L. Sasvfiri and P. Sz6pfalusy, Physica 87A (1977) 1.

[31] [321

[33]

1341 {351

H.K. Janssen, Z. Phys. 1320 (1~,~771 187. V. D o h m and R.A. FerrelI, Phys. I,elt. 67A (It#7Sl ',ST V. Dohm, Z. I'hys. t331 (b,~7S) 327 V. I)ohm, Z. Phys. B33 (197<,~) 70. R.A. Ferrell and ,I.K. Bhattacharjce. J. l o w I'cmp. Phys. 36 (197~) 165. R A . Ferrell, V. I)ohm and .I.K. Bhattacharjcc, Phys Rev. l.ctt. 41 (19781 181,% V. D o h m and R. Folk, / Phys. B35 (l~,~7t~) 277: addendum B39 (198(I) 94. R.A. Ferrell a n d . l . K Bhattacharjcc, Phys. Rcv. l.ctt. 42 (1979) 163& R.A. Ferrell and.I.K. Bhattacharjee, m: Proceedings ol the International Conference on Dynamic Critical Phenomena, ('.P. Enz, ed. (Springer, Berlin, 19791. R.A. Ferrell a n d . l K . Bhattacharjec, in: Proceedings of Second U S - U S S R Symposium on the Scattering of I,ight by C o n d e n s e d Matter, H.Z. ( ' u m m i n s , ed. (Plenum, New York, It~79). R.A. Ferrell and ,I.K. Bhattacharjee. Phys. Rcv. B20 (1979) 3e,90. R. Bausch, H.K. Janssen and H. Wagner. Z. Phys. B24

(b~7(,) 113. [36] H.K..lanssen, in: P r o c e e d m g s o f the International Conference on Dynamic Critical Phenomena. C.P. En/, c d (Springer, Berlin, 1'07~,~). [371 G. Ahlers, P.C. Hohenberg and A. Kornblil, as quoted in G. Ahlers, Rev. Mod. Phys. 52 (19801 489. [381 P.C. Hohenberg, B.I. Halperm and D.R. Nelson, Phys Rev. B22 (19801 2373. [3~J] We should like to thank (i. Ahlers and P.C. Hohenberg for providing us with the data in an accurate, numerical form in April 1981. 140] G. Ahlers, in: Proceedings of the ItJgll Carg6se S u m m e r Institute on Phase Transitions, M. I,evy, ,1.('. I.e Guillou and J. Zinn-Justin, eel. (Plenum, New York, 1981). [41] W.B. Hanson and J.R. Pellam, Phys. Re,,,. '~5 (1954)321. [42] R.A. Ferrell and J.K. Bhattacharjee, Phys. Rev. B24 (1981) 5071; Physica 107B (198l) 341. [431 (7". De Dominicis and V. D o h m (I979), unpublished. [441 See section III.A of r e f IS [451 P.C. H o h e n b e r g and S. Sarkar, Phys. Re','. B [1981). [46] V. D o h m and H . K . . l a n s s e n , Phys. Rev. l,ett. 30 (I9771 946: J. Appl. Phys. 4 t) (1~781 1347 V. Dohm, Report of the KFA J6lich no. 1578 (1tl791, available from Zentralbibliothek der K F A Jiilich, 17)5170 Jiilich, FRG. [471 V. Dohm, unpublished. [481 Both m e t h o d s of varying the model dimension (refs 17-19) or adjusting a three-loop term (refs. 14, 15) yield essentially the same answer, contrary to what is implied by A H K . T h e behavior of R]" in the high-t limit [14, 15] is irrelevant for a determination of d*. I49] V. D o h m and R. Folk, Z. Phys. B45 (1981) 129.