RENORMALIZED FIELD THEORY FOR THE STATIC CROSSOVER IN DIPOLAR FERROMAGNETS

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RENORMALIZED FIELD THEORY FOR THE

STATIC CROSSOVER IN DIPOLAR

FERROMAGNETS

E. Frey, F. Schwabl

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, dbcembre 1988

RENORMALIZED FIELD THEORY FOR THE STATIC CROSSOVER IN DIPOLAR

FERROMAGNETS

E. Frey and F. Schwabl

Institut fiir Theoretisehe Physik, Physik-Department der Technischen Universitat MCnchen, D-8046 Garching, F. R. G.

Abstract.

-

A field theoretical description for the static crossover in dipolar ferromagnets is presented. New non leading critical exponents for the longitudinal static susceptibility are identified and the existence and magnitude of the dip in

the effective critical exponent of the transverse susceptibility found by matching techniques is scrutinized. It was first shown by Aharony and Fisher [l] that

the short range Heisenberg fixed point (FP) of the renormalization group (RG) is unstable against per- turbations from the long range dipole-dipole interaction leading to a new stable dipolar FP. In subsequent papers the crossover from a critical behavior domi- nated by the short range exchange interaction to the asymptotic dipolar critical behavior was investigated by parquet graph [2] and matching techniques [3].

The latter follow closely the concept of phenomeno- logical crossover theory, which leads to several difficul- ties: (i) the use of two temperature variables makes the analysis quite complicated and (ii) one has to as- sume that the dipole-dipole interaction is sufficiently weak to guarantee that the RG trajectories traverse the region close to the unstable Heisenberg FP. In ferromagnets like EuO and EuS with relatively strong dipolar interaction this assumption may not be cor- rect. Thus the dip in the effective susceptibility exponent yea (r) found by these earlier studies [2] may not be universal. Here we give a field theoretical description of the crossover solely in terms of the true reduced temperature r using a generalized minimal subtraction procedure introduced by Amit and Goldschmidt [4] for bicritical points.

The Hamiltonian for a spin system with both exchange and dipolar interaction [I] is given by

91 92 93

xs: (q3)

sop

(-ql

-

q2

-

qs)

.

(1) Sz (q) (a = 1,2,

...,

n) are the components of the bare spin variable with n equal to the space dimensionality d. We further used the abbreviation

I =

/ g $ P r 0

is the bare reduced temperature and go denotes the bare relative strength of the dipolar interaction.

The dipolar interaction in equation (1) breaks the symmetry of the spin fluctuations transverse and longitudinal to the wave vector q, which is reflected in

where G; (ro, go, q) = (ro

+

q2)

.

The critical behavior is most conveniently studied by the di- mensional regularization and minimal subtraction procedure o f t ' Hooft and Veltmam [5]. In this framework an arbitrary momentum scale p is introduced, which allows for the definition of a dimensionless renormalized coupling constant a by a0 = p E ~ u ~ ~ l ~ , where the factor Sa = 2/ ((47~14

r

($))

is introduced for convenience. To remove fully the singularities it is also necessary to introduce. renormalized parameters and fields according to ro = Z,r, go = Zgg and

L 1/2 L T 1/2 T

sz

(9) =

(z*)

papsy (q)

+

(z*)

Pff,SY (q)

.

In the conventional minimal subtraction scheme the Zfactors are functions of E (d = 4 - E ) and of the renor-

malized quantities, singular as E -4 0, in order to re-

move the singularities in the vertex functions (VF). However in the present case this renormalization pre- scription is inadequate as the critical region is ap- proached for reasons similar to the case of bicritical points [4]. Therefore we adopt the following condi- tions of the

Z

factors: (i) for finite dipolar coupling g the renormalization constants cancel the poles in E

in all VF (ii) in the limit of infinite dipolar coupling g -+ co all VF are finite order by order in a and E .

There is no singular contribution proportional to the dipolar coupling g. This implies that the renormalization constant for g simply is given by 2;' = 2;.

Now we turn to the RG equation for the renormalized two point VF

I'g)aP

(r,g, k, u, ,u)

,

which can be decomposed into a longitudinal and transverse part in the same way as the free propagator. The correspond- ing RG equation is (a = L,

T)

O =

[ap

p-+Cr a,- r-+Cg a , - g-f

( 2 ) ; 2 ) :

where

the free propagator

a

~ ( 1 1 , ~ ) ~ ( u , $ ) = P - l n ~ - l

a~

lo.

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C8

-

1570 JOURNAL DE PHYSIQUE Note that due t o the generalized remmalization pro-

cedure the

p-

and [-functions depend on u as well as

on

9.

w e find p2

The flow equation is solved by

This gives four FP (u*, g*) : Gaussian (0,O) Heisen- berg

(E,

0)

,

Gaussian dipolar (0, m) and dipolar

(%,

m)

,

where only the dipolar FP is infrared stable.

The solution of the RG equation is

r;

(T, g, k, u) =

d r (1)

where p (I) = pl, 1- dg (1)

dl = r ( l ) C r ( l ) , l T =

du (I)

g (1)

Cg

(I) and 1- =

p

(I) with the initial condi- dl

tions T (1) = r, g (1) = g and u (1) = u.

Next we study the behavior of the transverse and longitudinal two point VF in the asymptotic region (I -+ 0). (i) For T =

T,

and by choosing the flow pa- rameter l according t o

b

= l we find

/I (1)

12

where c is a constant, u* = lim u (I) = - E and

Cg

=

1-0 17

lim (1) . From equation (7) we may identify the lead-

1-0

20

ing exponent gT =

-&*

= -a2 and a new non 867

leading critical exponent gL =

-<'*

=

-

l3 E ~ . (ii)

or

289 k = 0 and by choosing

fi

= 1 one gets

p212

where 2. is a constant. Therefrom we find the leading

critical exponents

and y~ = v (2

-

g ~ ) for the transve:rse and the non leading exponent y~ = v (2

-

q ~ ) for the longitudinal susceptibility.

Now we study the crossover of the transverse susceptibility at zero wave vector k = 0. We find for the

a In x-l

effective critical exponent defined by ,yetf =

-

a

In T

with 1 = and u (1) is given by equation (5).

7'

Figure 1 shows yetf (r, 9 ) versus

--

at fixed dipo-

cr

lar couplings for a series of initial d u e s u. For weak dipolar systems (g =

low6)

all curves join and the minimum of the effective susceptibility exponent is a universal property of the system, whereas for stronger dipolar systems (g = it depends on u whether there is a minimum or not. If u

>

u& there is a minimum at the same position as for u = u:, but for u

<

u& the dip in y,e diminishes with decreasing u.

Fig. 1. -Effective susceptibility exponent yes for g =

(solid), g = (poiht-dashed), E = j l and a series of initial values u =

-

2 + with Ic = 1,

...,

8 indicated in the

10 graph.

[I] Aharony, A. and Fisher, M. E., IJhys. Rev. B 8

(1973) 3323.

[2] Nattermann, T. and Trimper, S., J. Phys. C 9

(1976) 2589.

[3] Bruce, A. D., Kosterlitz, J. M. and Nelson, D. R.,

J. Phys. C 9 (1976) 825;

Bruce, A. D., J. Phys.

C

10 (197'7) 419.

[4] Amit, D. J. and Goldschmidt, Y. Y., Ann. Phys. 114 (1978) 356.

[5] t'Hooft, G. and Veltman, M., Nucl. Phys. B 44