Critical behavior of uniaxial ferromagnetics with dipolar interactions

Volume 44A, number 5

PHYSICS LETTERS

2 July 1973

CRITICAL BEHAVIOR OF UNIAXIAL FERROMAGNETICS WITH DIPOLAR INTERACTIONS A. AHARONY Baker Laboratory, Cornell University, Ithaca, New York 14850, USA Received 14 May 1973 The critical behavior of uniaxial ferromagnets with exchange and dipolar interaction is studied by exact renormalization group techniques. A crossover occurs from short-range (Ising) to characteristic dipolar behavior, which is described.

Magnetic dipole-dipole interactions exist in all magnetic materials, and should be important in determining the critical behavior of materials with low transition temperatures. The effects of these interactions for Heisenberg like materials have recently been studied near four dimensions, revealing changes both in the critical exponents and in the angular dependence of the correlation functions [1]. Many materials (e.g., uniaxial ferroelectrics) have only an Ising-like or uniaxial symmetiy, for which a different behavior is to be expected. Indeed, Larkin and Khmel’nitskii [2] concluded that for d = 3 the critical behavior deviates only logarithmically from that predicted classically. Larkin and Khmel’nitskii discuss only the threedimensional (d = 3) dipolar In this note(a) we report renormalization groupbehavior. calculations which show that d = 3 represents the boundary between classical (d> 3) and nonclassical (d < 3) behavior (instead of d = 4 as for the pure exchange case), (b) yield the crossover ~ from exchange to dipolar behavior,exponent and (c) lead to anpure expansion in powers of e’ = (3—d) for the critical exponents for d<3. The magnetic (or ferroelectric Hamiltonian considered can be written in the form

u

=—~

E

[J(R-_R’)+(g

R,R’

2A(R_R’)]S~S~. (1)

d

3) be measured by

=

~=

[

4irg~p~/3v~

LJ(R)]

=

~Td~/TC,

(3)

R

which becomes appreciable in systems with small values of T~. For renormaltzation group calculation, we utilize the reduced Hamiltonian

H=

— ~

fu2(q)sqs_q q

U

fff ~ q q’ q” 1q means (2ir)°’J’

(4) d’1q. The effective pair in-

where teraction potential is u 2 h(qZ)2 +g(qZfq)2 (5) 2(q) = r+ As usual [3]q r is proportional to (T— T 0), while the 2 is fixed at unity by a spin rescaling. coefficient of q g and h arise from the Fourier transThe coefficients form of the dipolar interaction (2) and are all —

.

,

proportional to ~ [1] with coefficients of order one. The Gaussian propagator for the graphical expansion then has the form ,

G(q)~=r+q2+g(qZ/q)2_h(qZ)2,

(6)

5#~) which is quite different from the one in the isotropic case [1]. For g, h ~ r, or equivalently for t = (T/T~) 1 ~ i, this reduces to the standard propagator

where J(R) is the short range exchange coupling while, in d dimensions, the dipolar coupling is A(R) = dZ2/R’1~2—l/R’~ where the z axis is taken as the axis of anisotropy. The relative strength of the dipolar terms can (for

—

(2)

for the short range Ising model. We find the usual nontrivial fixed point with exponents [3] (e = 4—d) 2v

1

+ ~ e + ~ ~2

~¼ ~2( 1 + ~e)

,

(7) 313

\olume 44/s. number ~

PHYSICS LETtERS

and g = /1 = (2. This fixed point. however, is strongl~ 2 unst ible ~sith respect 10 the dipolai itifli ~q q) the corresponding crnssover exponent is I ound 4 to be exactly T()i7=

O(7

c~g/d~=(~-~)g. i~=O(u2g1). =

2r +

2g

.,l~

=

(4

d)u

f O~u3g

.

~

$h.’I ~u2g I 2 r)

At T~.the correlation ion I~n necomes /;,q’’1/)~i

~sith~ = ford 3 h~np; :11) /2 I lola Ihe susceptibility LlispiaYs lie expected demagnel /atiori etfccl\

ianiei\ -,

classical behavior of ~ and of l’(q), However, it would he very interesting to attempt to detect the existence

I

.

tor(h). The important parameter in the expansion is thus in i 2 which replaces u On solvinf i I I) is oP the aid of ( 0), one finds that w goes to zero whenever . d .-> n. so that classical behavior is maintained. F os = 3, however. w varies as (1+10) which leads logarithmic factors in the [-dependence. For d the solution is exactly analogous to the usual Wilson Fishier e expansion near d = 4. Explicitly, the dipolar spin-spin correlation tune-

P logarithmic corrections. Series calculations ii / 3 models (e.g.. t’or sI = 7) would also he valnable. he author acknowledges his inieracW ns with . .,. Professor ME. F’ slier, the support of the National . Science Foundation - pa rd’s through the M al erials Science t entem a it oruell 13 nive rsity and a F’ulbrigli I Hays scholarship.

References J Ml-. I mslier and A. ‘\harony. Phvs. Rev. [ett. 30 I 977 S 59A. Aharony and ML. 1’ ether, 10 be published: A. .Almarony, to be published.

lion nas tile I orm

I’(q) ~ ~ II +(~q)2 h 0 (~q2l2+g (q/q)2I 0

t2t

with

121 Al. and208” 1)1’.. Khmcl’nitsku, ‘Feor. 1231 1w Larkin 5611969) ISo” Phy’~ ti Zh. TP Eksp. 29)1969)1 131 KG. Wilson and J. Kogut, Physics Reports, to be published.

2

llnt[’~’

for d > .3

13)

for d= 3

14

for d < .3

( I ~)

-

314

~

a shape-depende iii ~ini1. Experimenis on thrce-dinieiisional sing-like errsmegnets hI and on uniaxial ferroelectrics 17 I indicalc

2r

where -4d is a constant of order unity. I he factor 1 2 results form angular integrals over the propa~c-

t

~

svhe re

iig

1g I 2 ~O(u

du/dI

I

‘)qi°~q

t~c+~~.

Ihus. oil
dr/dI

with

2Juiv

“

141 51

The details of the calculation will be published elsewhere KG, Wilson and Ml’, Fisher, Phvs. Rev. Let). 28(19721 240. 161 (‘A. Catancsc, Al. Skjehrop, Ill:. Meissner andW P Wolf, yale preprint 11972) 171 V. Yamada. C Shirane and A. I inz, Phys. Rev. 1 7” 11969) 848