TIME DEPENDENT GINZBURG - LANDAU EQUATIONS IN SPIN GLASSES

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TIME DEPENDENT GINZBURG - LANDAU

EQUATIONS IN SPIN GLASSES

B. Shastry, S. Shenoy

To cite this version:

B. Shastry,

S. Shenoy.

TIME DEPENDENT GINZBURG - LANDAU EQUATIONS

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-891

TIME DEPENDENT GINZBURG - LANDAU EQUATIONS IN SPIN GLASSES

B.S. Shastry and S.R. Shenoy

School of Physios, University of Hyderabad, Hyderabad 500 001, India

Résumé.- A l'aide des équations de Ginzburg-Landau dépendant du temps, combinées à des idées de modes couplés, nous associons la décroissance de l'aimantation rémanente à la faible décroissance du paramètre d'ordre. Nous trouvons d'intéressants effets pour le cas de la relaxation en champ ma-gnétique fini.

Abstract.- Using time - dependent Ginsburg - Landau equation and ideas related to mode coupling, we relate the decay of the remanent magnetization to the slow decay of the order parameter. In our model we find interesting new effects for the case of relaxation in a non - zero final magnetic field.

Spin glasses exhibit several remarkable dyna-mic effects. For example, the relaxation time for

the remanent magnetization, after the sudden switch-off of a field, is unusually large III, (y minutes). The equilibrium theory of ThoulessjAnderson and Palmer 111 gives a mean - field free energy that has an inflexion point rather than a minimum as a func-tion of the order parameter q. Anderson has specu-lated /3/ that this analytic structure, which implies unusually weak restoring forces, might lead to a

slow relaxation of q(t) for all T < T and not just sg

at T = T . (Here T is the spin glass transition sg sg

temperature). The spin glass order parameter is not, however, directly measurable. In our model we show that the slow relaxation of q(t) is reflected in a slowing down of an experimentally measurable quan-tity, namely the magnetization m(t).

As usual one postulates phenomenological equa-tions for q(t) and m(t) analogous to the time - de-pendent Ginzburg - Landau or Langevin equations M /

\ i(t) = - H

(q

'

m

'

h)

• r

q

<t> (o

T , i < t > - - ! £

( q

-

h )

•*.<*> ( »

Here T , x are relaxation times, and * , J are m' q ' q' m random forces. The latter are irrelevant for the long time relaxation of m(t) and q(t) from non-zero initial values to equilibrium and will hence forth be omitted. Here m is a scaled magnetization rela-ted to the extensive magnetization by M = (y m —) ;

D v

fl being the sample volume and v being a typical ato-mic volume. The magnetic field and free energy are

scaled as H = (k„ T /u„)h and F = (£2/v) k„ T $

B sg B B sg respectively. We w r i t e down a model f r e e e n e r g y f u n c t i o n a l a s a sum of c o n t r i b u t i o n s . $ = $ + $ + $ + $ , (3) q m mq mh

$ is taken as the TAP - like form /5/. q

.TAP 1 -2 2 2 e 3 1 „ ^ .

*

q

= j

£ q

I

£ q

« '

i q

i

£ !

£ = (T - T)/T (4) sg sg

which is an extremum (point of inflextion) at the mean - field value of the order parameter 12/. (Our calculation applies for q ^ q , the equilibrium

va-3$

lue, for which - -5— < 0 acts as a restoring force). dq

The m-q coupling is of a form dictated by the nature of q as the trace of a second rank tensor /6/ in the replica picture : q = <S? S7> a + B. The other terms are of the standard form, and we have

* = - m2q(b + cq) ; $ = f m2 ; 4 = - mh (5) mq m z mh

The $ term may also be obtained by the following mq

consideration. From the replica definition, q would be non-zero in a ferromagnet, where <S.> ^ 0. To de-fine an order parameter unique to the spin glass, we must subtract out such effects, implying that in

the presence of a magnetization q ->• q + m2. With this substitution, the TAP free energy $ yields 0 of the above form with b = - £2, c = 2£, with

mq m

a - a - £ , the overall sign of $ is then nega-mq

tive, as is physically reasonable. The coefficient "a" enters the susceptibility X and must be positi-ve, from stability considerations. It may be taken

to be a = a + q a , incorporating the "frustration effect" characteristic of the spin glass, where X

14 - T.2

decreases as T decreases.

The time dependence of q(t) and m(t) are now

obtained from a substitution of (3)-(5)

into

(1)

and

(2). We assume the system is in equilibrium in an

initial field hi suddenly reduced at

t = 0

to a fi-

nal value h .q(t)

and m(t) then decay from initial

f

_{3 s}

values q(o) and m(o) to values q and m determined

3f 3

. .

by hf (m

,

q are obtained by setting m

=

q

= 0

and

h =

h

) .

We thus obtain equations for the deviations

f

Z

:(t) =

q(t) -q

,

m(t)

=

m(t)

IU*,

from the final va-

lues. Since we are only interested in long

-

time

behaviour in low fields, we retain only terms upto

order hf and

in the equations.

The solution for q(t)

is

t > T

/gq(o)

4

where

T~~~ =

Tq/2 hf

E

6 ;

For strictly zero field hf

=

0, the solution of

Kumar and Barma q

%J

2

is regained. For hf

f

0

the

Ct

general solution for the magnetization has two com-

ponents m

=

m

+

mg,

with m (m

)

dependent on the

A

A B

initial (final field). Both can be analysed natural-

ly in two different time regimes since two times

scales

T

/ E

and

T~~~

govern the problem. (We confine

9

ourselves to

t

> > T ~

throughout). Then in the inter-

mediate time (IT), regime t

<

reff

n

;

n

"

0.1

when

Tq/€t

C1

m

(t) =

X hi t exp(-t/xT

) ;

a

=

2 T

((1

+&)IT,

A

m

9

( 8 4

I

(8b)

In the long time (LT) regime which appears only for

hf

f

0,

t

> TI' T~~~ ; '1'2.

2.5 whenqeexp(-t/r eff

)

Thus the two components of magnetization behave qui-

te differently

(i)

m (t) is proportional to hi, re-

A

latively insensitive to hf in both time regimes,

and has a time decay only indirectly determined by

q(t)

(ii) mg(t) depends sensitively on the final

field

(%

hf in the IT regime,

%

hi in the LT regime)

151, and directly foloows the behaviour of q(t).

In

particular a distinct "crossover" is expected in the

behaviour of m(t) at

t %J T

Details of the beha-

eff'

viour depend, of course on the particular form of

@

chosen. For example, going beyond mean field the-

q

ory might change q *l/t in the IT regime to

q%

l ~ t l - ~

0

5

v

5 I. This would be reflected in the

1

behaviour of m(t).

For

v

=

-,

one then obtains a

2

slower behaviour of m (t) in the IT regime, with

A

competing exponentials. The main point, however, is

that there can be a slowing down of q(t) and hence

m(t),

even far from T

sg'

We identlfy mB of equation (8) with the rema-

nent magnetization observed by Guy (1)

since the

jump in magnetization on switching off a field

(hi

C= 10

Oe) is of the order of h./h

(hi is earths

1

f

field

-

0.4 Oe). We suggest that further experiments

be performed with varying hf in order to test our

ideas

1 7 1 .

References

/ I /

Guy, C., J.Phys.

(1975) L 242

/2/ Thouless, D.J., Anderson, P.W. and Palmer, R.J.,

Phil. Mag.

2

(1977) 593

/3/ Anderson, P.W., AIP Conference Proceedings on

Magnetism and Magnetic materials (1977)

141

Ma, S.K.,

Modern theory of Critical Phenomena,

Benjamin, W.A., Reading, Mass (1976)

/5/ Kumar, D. and Barma, M., Preprint (1978)

/6/ Harris, A.B., Lubensky, T. and Chen, J., Phys.

Rev. Lett.

2

(1976) 415