Nonasymptotic critical behaviour of uniaxial systems with strong dipolar interaction

26 January 1987

PHYSICS LETTERS A

Volume 120, number 1

NONASYMPTOTIC CRITICAL BEHAVIOUR OF UNIAXIAL SYSTEMS WITH STRONG DIPOLAR INTERACTION *

R. FOLK and G. MOSER Institut ftir theoretische Physik, Universitiit Linz, A-4040 Linz, Austria Received 24 April 1986; revised manuscript received 24 September 1986; accepted for publication 7 November 1986

Within a field-theoretic renormalization group approach general relations between specific heat, susceptibility and order parameter for uniaxial dipolar ferromagnets or ferroelectrics are derived. An advantage of these relations is that no explicit solution of the flow equation for the temperature dependence of the effective four-point coupling is needed. In this way the crossover from critical behaviour to mean field behaviour can be studied. The theory is applied to LiTbF, and TSCC.

Renormalization group (RG) theory predicts that the asymptotic critical behaviour of systems at their critical dimension is mean field like modified by logarithmic corrections. Ferromagnets or ferroelectrics with uniaxial dipolar interaction are in d=3 at their critical dimension and can be used to test this predictions or RG theory [ l-3 1. The existence of the logarithmic corrections has been shown unambiguously for the first time in ferromagnet LiTbF, by specific-heat measurements [ 41, and afterwards also by measurements of the magnetization [ 51,the equation of state [ 61, and the magnetic susceptibility above T, [ 7,8]. Recently those corrections were also observed in the susceptibility [ 93 and the specific heat [ lo] of the ferromagnetic tris-sarcosine calcium chloride (TSCC). These measurements are usually made for relative temperature distances not smaller than I tI - 10 -3.5 (t= (T- T,)/T,), where rounding effects are not yet important, and thus to a large amount are performed in the crossover region from the background behaviour to the true asymptotic critical behaviour. However, as has been mentioned in refs. [ 8,9] no crossover functions have been calculated so far. In order to go beyond the asymptotic theory, usually a nonuniversal parameter tn is introduced [ 111 in the parametrization of the asymptotic temperature dependence of the corresponding physical quantityf( t) , j(t) +I

tl ~‘lln(t&t)

I x’.

(1)

Here f( t) may be the susceptibility, order parameter, etc., with the universal exponents a, xfand the nonuniversal amplitude A, Since in tfo the corrections to the asymptotic behaviour have been absorbed, this parameter may be different for the various quantities considered. Thus, apart from a test of the universal features, it seems to be difficult to verify the consistency between the different measurements. One of the most important disadvantages of expression (1) is that it does not lead to a crossover-to-meanfield behaviour at sufficiently high t but instead leads to a divergence or a zero at I t I = tfo. To describe this crossover further refined ad hoc corrections have been introduced (see e.g. refs. [ 8,9]). It is the aim of this paper to derive general relations between the specific heat, susceptibility and order parameter valid in the whole cossover region similar to those in systems with short range interaction given by Dohm [ 12-I 41. Those relations are then used in one loop order for the application to LiTbF4 and TSCC. Let us start from the Ginzburg-Landau hamiltonian for uniaxial systems with strong dipolar interaction [ 31, Jf’= *

s

ddk (ro+P2 +g~42~P2)@ook@Jo--k +Uo ddk, . . . ddb s

@Ok,@Ok>@Okj@O

-k,

-kz

-kl

.

(2)

Supported by the Fonds zur Fiirderung der Wissenschaftlichen Forschung.

03759601/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

39

Volume

120, number

1

PHYSICS

Then the specific heat, the susceptibility

s s

ddx (@o(x)@,(O)> -

m,(t) =

and the order parameter

26 January

A

(3a)

(@o > 2 .

(3b)

ddx (@o(x) >

We now introduce

1987

are given by

1(@;(xMi%O) > - (4; > ‘I .

Co(t) =/dds

x0(1) =

LETTERS

(3C)

renormalized

quantities

(in the minimal

subtraction

scheme)

as

with K an arbitrary reference wave number and Ad a dimension-dependent factor, which will be specified later for explicit calculations. There is no additional Z factor for the parameter g since no pole terms exist for the q2/p2 part of the inverse propagator. Apart from analytic contributions in t and cut-off dependent corrections one can identify the following measurable dimensionless quantities with their renormalized counterparts,

c,+ C’ _-c’ C, The renormalized

dlnxo

dlnC’

dlnC,t d In t

=dlnt’

quantities

dlnt

d In m d In m, dlnt =- dlnr

d lnx =dIn

can be found from the solutions

of their RG equation

As usual we have defined [,= (a In Z,‘/~K)~,, i=@, Y and /3,,=~(d~ld~)~~. renormalization of C. Integration by the method of characteristics leads to

C’(r, g, U,

K)

=g-’

B(g,

u)

is related to the additive

exp /

t”(r(f)/K’f’,

:48@(l))

E(f))-J-

exp -

I

X-‘(r, g, U,

K)

=K2f2

X^-‘(r(f)/K2f2,

‘2Xp

f?(f)),

( 5b

)

(SC

)

1

WZ(r,

g,

U, K)

=,$-“2Kf

eXp(

-

{j

$+i(r(l)/K2f2,

c(f))

I

We have made use of the special structure ~2= ulg. The characteristic equations read f~r(f)=W(f))r(f),

40

f$2(f)=Ai

of the perturbation

>

theory for the hamiltonian

(1) by introducing

(6a,b)

PHYSICS LETTERS A

Volume 120, number 1

26 January 1987

withBn=lc( &?I&c)~ which can be related to BUand &,. The flow parameter is arbitratraty and will be chosen as usual, namely r(l)ld12

= 1 ) or

2r(l)llc21* = 1 ,

for the quantities above or below T,, respectively. This choice of 1 relates the temperature t+ above T, to the temperature t_ below T,, which has to be taken for the experimental quantities in (3a)-( 3c) as t+ = -2t_. Defining the effective exponents y

ff(t)=

e

dlnxo’(t) dlnt

’

Gdf)=-

d In C,(t) 9 d In t

and the ratio R(t)=G(fItl)G(t) weobtainind=3

R(t)-1 - o,ff(t)

3

(Z=O)

C-(1, u”)-C’+(l, 5)[2-&(ti)] = 4B(17)-[2r,(~)+fr,(~)]~+(1,ii)-Pna~+(l,ii)laa’

(9)

We want to point out that eqs. (8) and (9) are different from the corresponding relations for short range isotropic systems in the dependence on c+,because of the appearance of the extra g factors in (3a) and 3~). The specific heat, susceptibility and order parameter are connected via ii(t) by these relations. We now apply these relations to the measurements in LiTbF, and TSCC mentioned above. For this purpose we use the result of a one-loop calcultion. As is well known [ 31 the universal exponents and amplitude ratios in unaxial dipolar systems at d= 3 in one loop order are the same as those in isotropic short range systems at d= 4. For a suitable choice of the factor Ad in the renormalization of the fourth-order coupling u (12respectively) of eq. (2) this is also the case for the nonuniversal functions ” appearing on the right-hand side of eqs. (7)-( 9). Although the systems considered are in d= 3 at their critical dimension, the calculation has to be performed at Z# 0, only after the cancelation of the pole terms one can set Z= 0. Since the Z independent part contains terms resulting from the multiplication of the pole terms with “Epowers, those terms depend on the Z expansion and in this way on the geometrical factor Ad [ 141. We chose Ad to be &=2’-’

1-z 1-tr

(1-E”)(l+@) 1-jg

s,_, 2 ’

where S,=2’-drr -“*P’( id). Then the one loop order results for the crossover function f, c’, vf?and the functions c, and pli are the same as for the isotropic short-range case in d= 4 [ 141 (in the minimal subtraction scheme). Explicitly we have in one loop order c,=O,

(,=12&

j&=36ii2,

j=l,

A=1/22?,

c+=-1,

c-=1/2fi-4,

B(@=t.

(10)

One can now use the solution of the flow equation (6b) in ( 7)-( 10) and fit the temperature dependence of the physical quantities in order to fix the nonuniversal initial conditions ii0 = ii( to) at some temperature to in the ” Nonuniversal in the sense that these functions still depend on the RG procedure chosen.

41

Volume

120, number

I

PHYSICS

,;H, ,#/*..

OW+

0

-i

/A”’

A

26 January

1987

/,,’ ’

,,/”

0 01

LETTERS

’

c *;y

_

-i

-1

-i

i

‘a*

1

Fig. 1. Effective flow u‘(t) for LrTbF4 versus temperature calculated from experimental data. The dash-dotted line is calculated from eq. (7) with the susceptibility of ref. [ 71, the dashed line from eq. (8) with the magnetization of ref. [ 51 and the solid line from eq. (9) with the specific heat of ref. [ 41. The open circles are calculated from eq. ( 7) with the data for yen of ref. [ 81.

background. On the other hand one can directly calculate the flow u(t) from the experimental quantities on the left-hand side, without using the solution of (6b). In order to treat the crossover to mean field behaviour, we have to choose this second possibility as explained below. The results for fi( t) obtained from LiTbF, are shown in fig. 1. In the ideal case all three curves shown should coincide and together with the open circles should lie on one curve. The observed small differences may be due to: (i) nonuniversal values of ti( to) in the background region, because we have different LiTbF, crystals, (ii) uncertainties because we have used only the results of one loop order. (iii) effects of other contributions not taken into account in the hamiltonian (2). From the data of ref. [ 81 we see that the crossover to mean field behaviour is reached for t- 0.8. A similar behaviour can also be seen (fig. 2) in the measurements of the susceptibility in TSCC [ 91. This nonotonic behaviour of z7(t) could not have been obtained from one or higher loop solutions of the flow equations. The solutions in one loop areofthetype(b=18u,) c(t)-

20 1 +bln(t,lt)

’

(11)

which diverges at some finite value t> t 0. In two loop order there is no divergence but the solution is monotonically increasing and reaches a finite value U for large t. These features and maintained in higher loop order.

Fig. 2. Effective flow iI( f) for TSCC versus temperature calculated from eq. (7) with the experimental susceptibility of ref. [ 91.

42

Fig. 3. Logarithmic derivative Pen of the polarization in TSCC versus temperature predicted by eq. (8) using the flow a( /) shown in fig. 2.

Volume 120, number 1

PHYSICS LETTERS A

26 January 1987

Neither solution can therefore be used to describe the crossover between critical and mean field behaviour. We can now employ eq. (8) to predict Peff in TSCC without using the solution of eq. (6b) but taking the flow ii(t) found from the susceptibility measurements (see fig. 3). Since we take our expressions in one loop order we have the relation

(12) This relation is also fulfilled in the Larkin-Khmelnitskii theory [ 1,151. We want to point out that no free parameter is involved in eq. (12). Therefore one has no freedom to readjust the parameters used in a parametrization of yeff in order to obtain a specific /.Iefl.This has been done in ref. [ 91 in order to get agreement with the data of ref. [ 161. These data represent a considerable deviation from the mean field value B = 0.5 far in the background, namely Beff- i at t - 0.15. Independent measurements [ 171 which do not use the EPR technique of ref. [ 161 lead to Peff~ 0.5 for t G 0.1 and subsequent measurements [ 181 showed a crossover at t- 1O-‘.6 from /3N 0.5 near T, to B _ 0.33 in the region 0.023 < t < 0.23, which is more compatible with fig. 3. However, more precise measurements will be necessary to check our predictions in detail. There are also measurements of the specific heat [ 19,101 which show that besides critical effects there is a large temperature dependent background. This makes the extraction of the critical contribution strongly dependent on the subtraction procedure. Further experiments will have to be done to obtain the flow fi( t) from those data with satisfactory accuracy.

References [ 1] A.I. Larkin and D.E. Khmelnitskii, Zh. Eksp. Teor. Fiz. 56 (1969) 2087 [ Sov. Phys. JETP 29 (1969) 1123). [ 21 A. Aharony, Phys. Rev. B 8 (1973) 2263. [ 31 E. Brezin and J. Zinn-Justin, Phys. Rev. B 13 (1976) 25 1. [4] G. Ahlers, A. Komblitt and H.J. Guggenheim, Phys. Rev. Lett. 34 (1975) 1227; G. Ahlers, private communication. [5] J.A. Grit%, J.D. Listerand A. Linz, Phys. Rev. Lett. 38 (1977) 251. [6] R. Frowein, J. Kiitzler and W. Assmus, Phys. Rev. Lett. 42 (1979) 739. [ 71 P. Beauvillain, C. Chappert and I. Laursen, J. Phys. C 13 (1980) 1481. [S] R. Frowein, J. KBtzler, B. Schaub and H.G. Schuster, Phys. Rev. B 25 (1982) 4905. [ 91 E. Sandvold and E. Courtens, Phys. Rev. B 27 (1983) 5660. [ 10lM.J. Tello, M.A. Perez-Jubindo, A. Lopez-Echari and C. Socias, Solid State Commun. 50 (1984) 957. [ 111 A. Aharony and B.I. Halperin, Phys. Rev. Lett. 35 (I 975) 1308. [ 121 V. Dohm, Phys. Rev. Lett. 53 (1984) 1379. [ 131 V. Dohm, in: Applications of field theory to statistical mechanics, ed. L. Carrido (Springer, Berlin, 1985). [14]V.Dohm,Z.Phys.B60(1985)61. [ 151 Th. Nattermann, Phys. Stat. Sol. (b) 85 (1978) 29 1. [ 161 W. Windsch, Ferroelectrics 12 (1976) 63. [ 171 A. Levstik, C. Filipic and R. Blinc, Solid State Commun. 18 (1976) 123 1. [ 181 G. Serge and U. Straube, Phys. Stat. Sol. (a) 51 (1979) 117. [ 191 T. Matsuo, M. Mansson and S. Sunner, Acta Chem. Stand. A 33 (1979) 78 1.

43