GOLDSTONE SINGULARITIES IN ISOTROPIC FERROMAGNETS

HAL Id: jpa-00228953

https://hal.archives-ouvertes.fr/jpa-00228953

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

GOLDSTONE SINGULARITIES IN ISOTROPIC

FERROMAGNETS

H.-O. Heuer

To cite this version:

JOURNAL DE PHYSIQUE

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

GOLDSTONE SINGULARITIES IN ISOTROPIC FERROMAGNETS

H.-0. Heuer

Institut fiir Theoretische Physik 111, Ruhr- UniversitZt Bochum, 0-4630 Bochum, F.R. G.

Abstract.

-

The equation of state and the susceptibilities of isotropic magnets are calculated in exponentiated scaling form to 0 ( E ) by repetitive use of the trajectory integral method. The resuIting coexistence-curve singularities do not depend on the dimension for d 2 3 since the Goldstone modes show Fisher-renormalized classical tricritical behaviour.

It is well-established [I, 21 that the coexistence-curve of isotropic ferromagnets is a line of critical points where the transverse susceptibility diverges. It ends at the ordinary critical point, which is a kind of bicritical point where longitudinal and transverse fluctuations become critical. In this paper, I apply the trajectory integral method [3] to calculate the crossover between the ordinary critical behaviour and the coexistence be- haviour of isotropic magnets in 0 (E) (E = 4

-

d)

.

My starting point is the usual Ginzburg-Landau-Wilson- Harniltonian for an %component isotropic spin system:

Separating longitudinal and transverse modes by a shift aq =

Sq

-

M.6 (q)

,

one obtains the Hamilto- nian

with the coupling parameters TL=T

+

1 2 u ~ ~ , TT = T

+

~ u M ~ , w1 = w2 = 4uM, ~1 = u2 = u, UQ = 2u and H = H

-

r M

-

4 u ~ ~ . M is the magnetiza- tion given by the condition (ao) = 0. It is the idea of the trajectory integral method to renormalize the sys- tem into the noncritical region r (e*) = 0 (1) and to match it with Landau theory plus fluctuation correc- tions. Mean field theory shows that this concept faiIs near the coexistence-curve: if one chooses

e*

in such a way that the longitudinal fluctuations are noncrit- ical (TL (e*) = 0 (I)), the transverse fluctuations are still critical (TT (1*) 1:0). These considerations led to a parquet graph summation of the susceptibilities in Nelsons work [I]. In this paper I apply the trajectory integral method throughout. The idea is to continue the renormalization of the system

'f?

(e*) [I, 31 until the critical Goldstone modes are noncritical at some matching point

e.

Before, it is necessary to eliminate

the noncritical longitudinal modes in the renormalized free energy

exp ~ ( u ,

SL;

1*) (3) by integration over a. The Feynman-graph expansion leads t o

AF, is the integrated free energy of the longitudinal modes in

f i

(a,

SI;

e*)

.

Since these modes are non- critical, AF, can be calculated by Landau theory plus leading fluctuation corrections:

The effective Hamiltonian (5)

in (4) describes the critical Goldstone modes coupled to the noncritical longitudinal modes. The coup- ling parameters .i. and & are calculated t o O(ui (e*)

,

w"e*)) as

and

GL in (5, 7, 8) is the longitudinal propagator at

e*

: GL= (TL (e*)

+

P2)

.

Before I work out the new cou- pling parameters p and 6 in terms of the original pa-

C8

-

1562 JOURNAL DE PHYSIQUE rarneters of

H

it is neccesary first t o calculate the su-

ceptibilities and the equation of state from the con- dition ( 0 0 ) = 0 . The usual Feynman graph expansion

would lead to logarithmic terms In ?'. A better way is to make use of ( 2 ) which shows that H is the source term for (T :

H

(t*)

-

T

(e*)

M

(e*)

-

4~

(e*)

~ " e * )

= 1 2 ~

(e*)

M

(e*)

Jq

G~ ( T L

(e*))

+

8~

(P-)

M

(e*)

.B

(i,

6 ) ( 9 )

where

is the energy of the ( n

-

1)-dimensional Goldstone sys- tem which has been calculated in exponentiated scal- ing form [4, 51. It is sensible to introduce the abbrevi- ations

T L

(e*)

:= t ( e * ) + 1 2 ~

(e*)

M~

(e*)

= T L

@*)-to

(U

(e*))

TT

(e*)

:= t ( e * ) + 4 ~

(e*)

M~

(e*)

= TT

(e*)+o

(U

(e*))

where t

(t*)

= T

(e*)

+

A/2u

(e*)

is the renormalized

temperature scaling field [3]. T L

(e*)

1:I-2t

(e*)I

is the

renormalized temperature distance, whereas TT

(e*)

is the renormalized distance from the coexistence-curve. Choosing T L

(t*)

= 1 as the matching condition, the effective temperature

f

:=

P

+

A/2.6 of the Goldstone system follows from ( 7 ) as

t^

= TT

(e*)

+

(1 - TT

(e*))

.8u

(e*)

E

(t^,

O)

.

(12) The effective interaction C ( 8 ) is momentum indepen- dent in

0

( E )

,

given by

TT

(t*)

a = ~ ( r ) . -

T L

(t*)

'

The equation of state ( 9 ) and the susceptibilities which follow from (9) depend on the~ffective temperature

t^

(12) and 6 (13) via the energy E

.

t^

and O itself depend on the equation of state via TT ( e * ) . Thus, the mag- netic properties for the whole M

-

T

-

phase diagram result from the coupled equations (9, 12, 13) [6].

In this paper I present the results for the coexistence-curve only setting the ~ * - c o u ~ l i n ~ to its critical value u = u * . The behaviour a t the coexistence-curve is obtained from the general results (9, 12, 13) in the limit

TT

(t*)

<<

1. Inserting the en- ergy of the Goldstone system [5, 61

E (f, 6 ) = ( n

-

1) K4 _.t

-

_-

( n

-

1) K4 .fl-fft . F ~

(Ct)

4at 4at

(14) in tricritical scaling fields into (12), leads to

9 n - 1

TT

(e*)

2! -.i+

-.

$I-"* . F ~ ( h )

.

n + 8 n + 8

This equation explicitly shows that the critical dis- tance TT

(e*)

is related to the temperature

t^

of the Goldstone system by a Fisher-renormalization [7] since the nonanalytic term with ( Y , = E / ~ dominates

near the coexistence-curve. The physical reason for this Fisher-renormalization is the coupling of the crit- ically fluctuating transverse modes t o the longitudinal modes in the Goldstone regime. The equation of state and the transverse susceptibility n e a the coexistence- curve follow from (9) and (15) as

The longitudinal susceptibility is evaluated in the same way observing that the term

~ / t ^

dominates:

Note that the equation of state and the transverse sus- ceptibility near the coexistence-curve have the very simple structure of a system with temperature

f

and

vanishing interaction 6. This point is verified from (13), which leads to

Working out the matching condition T L (l*) = 1 and TT

(e*)

in (15), one obtains the equat:ion of state near the coexistence-curve

using the usual normalization condit:ions f (-1) = ' 0 and f (0) = 1. The longitudinal suscelptibility is given bv

near the coexistence-curve. Note that the results (20, 21) for the functional form of the equation of state and the susceptibilities near the coexistence-curve are 1 exact for d

2

3 since the specific heat exponent at=-

2 is classical tricritical for d

2

3.

[ I ] Nelson, D. R., Phys. Rev. B 13 (1976) 2222. [2] Lawrie, I. D., J. Phys. A 14 (19131) 2489. [3] Rudnick, J . and Nelson, D. R., I'hys. Rev. B 13

(1976) 2208.

[4] Heuer, H.-0. and Wagner, D., to be published.

[5] Heuer, H.-0. and Wagner, D., J. Phys. Suppl.,

this issue.

[6] Beuer, H.-O., t o be published.