Equilibrium scaling laws for layered spin glasses

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Equilibrium scaling laws for layered spin glasses

M. Thill, H. Hilhorst

To cite this version:

M. Thill, H. Hilhorst. Equilibrium scaling laws for layered spin glasses. Journal de Physique I, EDP

Sciences, 1993, 3 (10), pp.2041-2062. �10.1051/jp1:1993231�. �jpa-00246851�

Classification

Physics

Abstracts

75.10N 75.50L 75.70F

Equilibrium scaling laws for layered spin glasses

M-J- Thill and H.J. Hilhorst

Laboratoire de Physique Th60rique et Hautes Energies

(*),

Bitiment 211, Universit6 de Paris-

Sud, 91405 Orsay Cedex, France

(Received

15 December 1992, revised and accepted 18 March

1993)

Rdsumd. On considbre _un systbme de couches parallbles de largeur _w d'un _verre de spin m6tallique

(comme

le Cumn), s6par6es _par des couches _non magn6tiques (Cu) de largeur £.

Dans le cadre d'un groupe de renormalisation h _un paramdtre _on d6duit des lois d'6chelle _pour la temp6rature critique _en fonction de _w et de £, et pour la longueur de corr61ation _en fonction de w, de £, et de la temp6rature T. Une 6chelle de longueur Lcr, au-dell de

laquelle

le couplage entre

couches magn6tiques devient important, joue _un r61e essentiel. Cette 6tude statique pr6cdde _une

6tude de ph6nomdnes dynamiques dans des _verres de spin multicouches.

Abstract. We consider _a system of parallel layers of _a metallic spin glass

(such

as

Cumn)

of width _w separated by nonmagnetic layers

(Cu)

of width £. Within the context of _{a one-}

parameter renormalization _group description _we derive scaling laws for the critical temperature

as a function of _w and £, ^{and for the} correlation length _{as a} function of _w, £, and the temperature T. An essential r61e is played by _{a crossover} length scale L~r beyond which the interlayer coupling

becomes important. This static study is _a preliminary to the study of dynamic phenomena in

layered spin glasses.

1 Introduction.

Virtually

_no exact results exist _on

spin glasses

in the

physical

dimensions d

= 2 and d

= 3.

The most

complete

and successful

approximate description

is the

scaling theory developed by

Mcmillan

ill, Bray

and Moore [2,

3],

^{Fisher and} Huse [4, 5], ^and

Ney-Nifle

and Hilhorst [6,

ii.

The

scaling theory

rests _{on a} number of

hypotheses,

of which the main _one is the existence of

an effective

coupling

constant

K(L), dependent

_on the

length

scale

L,

which satisfies

scaling

laws _{near zero} temperature and _near

criticality.

On the basis of these

scaling

laws others _may be derived and _a full picture ^of the

spin glass phase

has been

developed.

The

scaling theory

(*) Laboratoire associd

au Centre National de la Recherche Scientifique.

can be based _{on a} renormalization _group

picture. Augmented

with certain extra

hypotheses,

it _can also be extended [8] ^to

spin glass dynamics.

Fisher and Huse [9] have extended the

scaling description

to _a two-dimensional

spin glass layer

of

arbitrary

width _w. Such two-dimensional

spin-glass

systems ^have _no critical tem-

perature, ^but nevertheless show

experimentally spin-glass-like

^behaviour at low temperature

[10-13].

^{Therefore their}

study

is both

interesting

^{and of}

experimental

relevance.

In this _{paper we are} interested in _{a more}

complicated spin glass

system ^{that has also been} studied

experimentally, namely

_a three-dimensional

spin glass consisting

of metallic

spin glass layers (e.g. Cumn)

of width _w that alternate with

nonmagnetic layers (e.g. Cu)

of width £.

Experimentally

_w ^{and £ take values from about 20 lL} up

[10-13].

For _{w - oo we recover} the

purely

three-dimensional

situation,

and for I _{- oo} at _w fixed the _case of _a

single spin glass layer.

At temperatures

sufficiently high

_so that the correlation

length (

is less than the width _w of _a

single layer,

^the

layered

system ^{should be}

expected

to show three-dimensional

(3D)

behaviour.

As the temperature ^T goes down

enough

_so that

(

exceeds _w, there will be _a temperature

regime

in which the system behaves _{as a} collection of

isolated, independent layers,

but

only provided

that the

interlayer

distance £ is

large enough

with respect to the

layer

thickness _w;

otherwise this 2D temperature

regime

is absent. For all £ and _{w, as} T is lowered

further,

the interaction between the

layers eventually

^{renders the} system

effectively three-dimensional; (

will thereafter increase without bounds and

diverge

at the

"system

critical

temperature" Ta,

whose value lies below the critical temperature T~ of the

homogeneous

bulk

spin glass.

In this work _we

analyse

the various _crossovers that take

place

_{as a} function of temperature and of the two characteristic

lengths

_w and

I,

and _express

them,

where

possible,

in terms of

scaling

laws. Clear

experimental

evidence for _a double 3D-2D-3D _{crossover comes} from the

aging experiments by

Mattsson et al.

ill]

and

by Granberg

et al. [10]. ^{In this} paper we

do not

directly

address these experiments, in which the time

dependence plays

_an essential r61e, but limit ourselves to

equilibrium

properties. The _reason is that these

by

themselves

are

sufficiently complicated

and

interesting,

and require that _we

distinguish

several different

regions

in the space of the three

independent

parameters _w,

I,

and T. For the _{same reason,}

we

ignore

_{any crossovers} other than the dimensional _ones,

although

the

possibilities

for other crossovers, such _{as a crossover} from _an

Heisenberg spin glass

at small

lengths

to _an

Ising-like spin glass

at

larger lengths

_{or a} slow _crossover between

length

scales where the

spin glass

interaction is

effectively long-

and

short-ranged

_{[4, 9,}

14],

are numerous [15].

We obtain _our results within the established

scaling theory, using

the

language

of _{a one-} parameter renormalization _group.

Scaling

laws in the variables _w,

I,

and T will exist in those limits in which the renormalization _group

trajectory

_passes

infinitely

close _{to one or more} of the

following

fixed points:

(I)

^the 2D _zero temperature ^fixed

point;

(it)

the 3D _zero temperature fixed

point;

and

(iii)

the 3D critical fixed

point.

In those _cases where _an intermediate 2D

regime

exists, a

particularly

important rble is

played

by

_a crossover

length

scale L~r

beyond which,

_upon

renormalizing,

the final

effectively

3D

behaviour sets in. In section 2 _we recall

briefly

the renormalization

theory

for _a bulk system and _a

single layer.

In section 3 _we consider _a

layered

system; we determine the _crossover

length

L~r

beyond

which the

interlayer

interaction is _no

longer

_a

negligible perturbation.

We also discuss the

system's

anisotropy near

criticality.

^In section 4 _we determine

scaling

laws for the

system's

correlation

length (

and critical temperature Ta. We consider in particular the limits of _a

nearly

three-dimensional system

(I

_{not too}

large,

whence Ta not too far below

T~),

and _a

nearly

two-dimensional system

(I

^rather

large,

and T~ way below

T~).

2. Renormalization of _a bulk system ^and a

single layer.

2. I ONE-PARAMETER RENORMALIZATION GROUP. We

begin by considering

_a

homoge-

neous spin

glass

in dimension d

= 2 or d

= 3. We _assume that the effective interaction between

two

adjacent spin regions

of linear size L _can be

represented by

_a random

coupling

constant of

rms value

J(L)

_or

(including

_a factor

p

₊

I/kBT) by K(L).

Here and henceforth _{we express} all

lengths

_as

multiples

of _a

typical microscopic

distance. We shall also write

K(L)

_e Km when L ₌ 2".

(2.1)

Presumably

Km _can be calculated from the

microscopic coupling

K _e

Ko

via _a dimension

dependent

renormalization transformation

Kn+i

"

R(Kn). (2.2)

The transformation R has the fixed

points

K

= 0 and K

= oo. For Km > I, one has in

spatial

dimension d

= 2 _or d

= 3,

Kn+i

t

2YdKn (2.3)

with y~ ^< 0 and _{y3 >} 0. In d

= 3 the transformation has, in

addition,

_a critical fixed point K = K~, _near which _one has to linear order

Kn+i K~

_{m 2YC}

(Km K~). (2.4)

In the arguments ^{that follow} we shall _use

equation (2.3)

in the entire

low-temperature

domain

and

equation (2.4)

in the critical

region

K~ £ K £

2K~

for d

= 3.

(2.6)

Throughout,

the bare

coupling

K ₌

pJ

will be the variable

representing

^the temperature.

As _a direct

application

_{we may}

find,

at low temperature, the two-dimensional correlation

length (2(K) by iterating

the

microscopic coupling

_a number _nj of times such that

Km m 2"tY2K

r~ 1

(2.7)

or

f2(K)

₊ ^2nt

r-

Ki/'Y2' (2.8)

Similarly,

the correlation

length (3(K)

in the three-dimensional critical

region

is obtained from

(Km K~( c~ 2"tYC

(K

K~

r~ K~

(2.9)

or

~ -i/y~

(3(K)

₊ 2"t

r~

)- _Kc -1) ^(2.10)

For K £ I _we have

(2(K)

r~ I, ^{and for K} £

)K~

we have

(3(K)

r~ I.

Approximate

numerical values for the three exponents _{y2, y3,} and y~ in the ^case of _an

Ising spin glass

have been obtained

by

several methods. We cite those from reference [16]

y2 " -0.23

y3 = 0.27

(2.ll)

y~ = 0.36.

Metallic

spin glasses,

like

Cumn,

_are

anisotropic Heisenberg

spin

glasses,

with

experimental

critical exponents ^that appear to agree

moderately

well with the

Ising

exponents _{[2] and}

which, according

to several authors [9,

14],

^should be

equal

to the

Ising

values.

In what

follows,

the

symbols (2

and

(3

will continue to denote the correlation

lengths

of

homogeneous

two- and three-dimensional systems.

2.2 A _{SINGLE LAYER} OF WIDTH w. A further illustration of the _use of the RG equa-

tions

(2.3)

and

(2A)

is

provided by

the determination of the correlation

length ((w,K)

of _a

single

spin

glass layer

of width _w. The bare

coupling

K is iterated

according

to the three- dimensional transformation

(3DRG)

_as

long

_as n < nw, where _{w e} 2"«, ^and

according

to the

two-dimensional transformation

(2DRG)

for _n > nw. The correlation

length (

+ 2"t is reached when Km becomes of order I It will _appear that four different temperature

regimes

have to be

distinguished (see Fig. I).

For _a

given

width _{w a}

special

r61e is

played by

the temperature

regime

^{around K~ for which} w £

(3(K). Together

^with

(2.10)

the relation

defining

this

regime

becomes

~

l £ W~YC

(2.12)

K~

We consider _now the effective

coupling K(L)

in these temperature

regimes.

^As

before,

L _e 2" will denote the

length

scale of interest.

For

large

bare

couplings,

2K~ £

K,

the effective

coupling

is first to be iterated

according

to the 3DRG

(2.3).

For _n ^< _{nw we} find

K(L)

_m 2"Y3K

=

LY3K,

L £ _w.

(2.13a)

When

iterating

more than _nw times, the 2DRG

(2.3) applies.

For _n ^> _{nw we} get

K(L)

_t

2~"~"W~Y2K(w)

= LY2wY3~Y2K, w £ L.

(2.13b)

The correlation

length ((w, K)

is the solution of

K(L)

rw I and hence [9]

((w, K)

rw

w~+%Kii, _(2.13c)

In the zero-temperature ^limit

(w

fixed and K _-

oo)

this is in agreement with the

purely

two-dimensional result

(2.8).

For bare

couplings

such that I + W~YC £

£

1l 2, the effective

coupling

can first be iterated n~

times

according

to the 3DRG

(2.4).

For _n ^< nj ^we find

K(L)

K~ _t

2"YC(K K~)

= LYC

(K K~),

L 1l

(3(K). (2.14a)

L

w --'-

-,---,

'~

Ki _K~

Fig. 1. Renormalization flow of the effective coupling

K(L)

at length scale L of _a simple layer. We

distinguish four different temperature regimes by the change in behaviour while renormalizing. For very large initial couplings

(K

» Kc, 1), _one renormalizes with the 3D low-temperature exponent ^until L = w, then with the 2D low-temperature exponent. In _an intermediate regime

(1+

W~YC £

fi

_{£ 2,}

2),

one starts renormalizing according to 3D critical laws, continues with the 3D

low-tempe~ature

exponent ^when the effective coupling leaves the 3D critical region, and terminates with the 2D low- temperature exponent when exceding length scales L

= w. This temperature regime _{goes over} into

a temperature regime around the critical coupling Kc of the _pure system

((fi -1(

1l _W~YC) at the real coupling K such that the effective coupling attains L _{= w} exactly when

leaving

the 3D critical

region

(3).

For

couplings

in that regime

(4

and

5),

one does not leave the 3D critical region before attaining L ₌ w, where _one starts to renormalize according to 2D low-temperature laws. For higher

temperatures

(fi

£ 1- W~YC), the effective coupling becomes of order 1 before L

= w is reached, _so

that the

correlat~on

_{length is roughly the}

one of _{a pure} system; the renormalization flow is negligible

for _our purposes.

By

the definition of

(3(K),

_one has

K(()

rw

K~.

Next,

one iterates _n n~ ^times

according

to the 3DRG

(2.3).

For n~ ^< n <

nw we find

K( L)

_~

^2(n-M)Y3

K~

-

~~lK~1~~

Kc

f3(K)

£ L £ _W.

(2.14b)

Finally,

_we

proceed

_as for

large

bare

couplings

to get ^for n >

nw

K(L)

_t

2("~"W)Y2K(w)

=

LY2wY3~Y2(3(K)~Y3Kc,

_w 1l

L, (2.14c)

which

gives

_[9]

((w, K)

_t

w~+%(3(K) ^fi

~10~~~

^~

l)~

(~ i~d)

j~

c

Around the critical

coupling,

when

£ _II

1l_{W~YC, one} first iterates up to _nw t1nles

according

to the 3DRG

(2.4).

This

yields

the

e#ective _coupling

K(L) K~

_t

2"YC(K K~)

=LVC(K-K~),

L£w.

(2.Isa)

Hence

K(L)

is of order

K~

for all L £ _w. The next few iterations nlake

K(L)

decrease

rapidly

and hence

K(L)

1l

K~,

_w £

L, (2.lsb)

and

((w, K)

rw w.

(2.lsc)

Finally,

for snlall bare

couplings, £

£ I W~YC, the

layer

width _w is

larger

than the 3D bulk correlation

length (3(K),

and

henci

one can understand without RG arguments ^that

1(w, K)

=

i~(K). (2.16)

This

conlpletes

the discussion of the

single layer.

The results

(2.13c)

and

(2.14d)

_are due _to Fisher and Huse [9] ^we have rederived thenl here for coherence of

presentation.

In what follows the function

((w, K)

will continue to denote the correlation

length

of _a two-dimensional

layer

of width _w. It is _an

increasing

function of K whose

behaviour, according

to the temperature range, is

given by equations (2.13c), (2.14d), (2.lsc)

and

(2.16). Eventually,

_we observe that the formulas for

large

bare

couplings

and for bare

couplings

around the critical

coupling

_may

be obtained _as l1nlits of the _ones for _{I + W~YC} £

£

£ 2

by putting (3

~ l for 2K~ £ K and

(3

rw w for

(£ _II

£ W~VC

3.

Layered spin glass:

_crossover

length

and

anisotropy.

3. I INTRODUCTION. We consider _{now a} three-d1nlensional system ^of metallic

spin glass layers (say Cumn)

of width _w,

separated by nonmagnetic layers (pure Cu)

of width I. Such

a systenl has _a critical temperature that _we shall call

Ts(w,I),

located between _zero and the critical temperature T~ of the _pure 3D system. For I _{- oo} with _w

fixed,

_{we recover} isolated 2D

layers;

for _{w - oo, we} get the pure 3D system. Hence

Ts(w, oo)

= 0 and

Ts(oo, I)

₌ T~. In the l1nlits 1- _oo and

/or

_{w - oo}

scaling

laws should be

expected.

In this section _we shall

again

make _use of RG arguments to address the _new

problem

of how to take into account the

interlayer

interactions and to discuss the effects of

anisotropy.

Roughly speaking,

when _one blocks

spins

at a scale L ₊ 2", the

interlayer

interactions will be

negligible

_as

long

_as L is

sufficiently

small.

However,

when L gets

larger

than _a certain

length scale,

the total renormalized interaction of the

spins

in _a

particular

block with the

spins

in all other

layers

becomes

comparable

to the

nearest-neighbour

interaction between that block and its

neighbouring

^{blocks in the} same

layer.

We shall call this _crossover

length

L~r e 2"Cr It will be important ^when

considering

the correlation

lengths

and the

systenl's

critical temperature in the _next section

(see Fig. 2).

Finally,

it will be shown in this section that _near

criticality,

whenever _our RG treatnlent leads to _a nontrivial critical temperature

Ks,

the anisotropy in _a

layered

spin

glass

_can be

described in terms of _two correlation

lengths:

_one

parallel

and _one

perpendicular

to the

layers,

denoted

by ((w,I, K)

and

(I(w,I, K) respectively.

L

wst

~-i ~-i ~-i

s c

Fig. 2. Renormalization flow of the effective coupling

K(L)

at length scale L of _a layered spin glass. For L < Lcr, its behaviour is the _{same as} the _one of _a simple spin glass layer with the respective

temperature regimes

(see

Fig. 1). In particular, the flow is two-dimensional in the region between the dashed line L

= w and the solid _curve representing Lcr. Kc is the critical coupling of the _pure system, KS the _one of the layered system. We _see the mapping of the system's critical region around KS _onto

the _one of the _pure system.

3.2 INTERLAYER INTERACTIONS AND THE CROSSOVER LENGTH

L~r.

Let

Ho

be the

Hanliltonian of _a system ^of

noninteracting spin glass layers.

In _our case, there _{are, nlore-} over, RKKY interactions between

spins

in different

layers.

^Since the RKKY

potential

vanishes

quickly enough

with

increasing distance,

the

interlayer

interactions _can be treated _{as a} pertur- bation of

Ho.

Let V be this total

perturbation.

Then _{one may} write the

partition

function _as

ii?]

Z ₌

e~°(~~+~(~)

=

£ £ T(s', s)e~°(~)+~(~~ (3.la)

~ ~,

~~ ~~~~

Z #

£(£ _T(S',

_S)e~°~~~)

£~j))~~~~~~~~~~'

~~ ~~~

s, _s ~ ^'

where

T(s', s)

is the transfornlation

operation

^for the renormalization of the

original spins

_s

to block

spins

s'. The fraction in

(3.lb)

_may be considered _as the renormalized

perturbation

, ,

£ jn(/ ~)~Ho(s)~V(s)

~V ^(s s '

(3 ~~)

£~ T(s', s)eH°(S)

^'

in which e~(~~ is

weighted

with _a

"probability

function"

~'~~~ ~~ij~~~i~~~s)

~~'~~~

We _can write _to first order in V

e~'

=<

e~

_>ot

e<~>°+... (3.3)

JOURNAL DE PHYS>QUE -T 3, N' 10, OCTOBER 199~ 75

where <>o indicates the _average with respect to

Pa, (s).

When

calculating

the renormalized

perturbation V',

_{we use} that to zeroth order the

layers

are

independent,

_so that

< V >ot

~j _J,j

_{< s, >o< sj >o}

(3.4)

1,J

where the _sum is _on all

pairs

of

spins

I and

j

in different

layers

and

J,j

is the RKKY

coupling

J;j

₌

Jo ~°~(~" ^(3.5)

r~~

with r;j the distance between the

spins

I and

j

in nlicroscopic units and

Jo

the _energy scale.

For the

nonperturbed

system, one blocks

spins

_as in section 2.2. to get an effective

coupling K(L)

between blocks of linear size L. The total

interlayer

interaction has _at each

stage

^of

renormalization L

= 2" the form

V(L)

=

~j JIj(L)SI(L)Sj(L) (3.6)

1,J

where

SI(L)

= +I _are the Lth level block

spins,

and I and J _are in different

layers.

The system of

layers

cannot be treated _{as one} of

independent layers

^when the _{mean square}

coupling

of _a

block spin

SI(L)

with the block

spins

J of the _same size in all other

layers,

~~~~~)

"

(~ ^Ij(L))~

=

£

j~

~

~~(L)

J

'

(3.7)

gets ^{of the} same order _as the _square of the

intralayer nearest-neighbour coupling K~(L). Thus,

Lcr is to be defined _as the solution of

AK~(L~r)

₌

K~(Lcr). (3.8)

In the

Appendix,

_we find

explicit expressions

^for both members of this

equation,

and obtain its solution

Lcr(w,I,K).

This solution is

interesting especially

when it is

larger

than _w. For Lcr _{> w, a} double _{crossover occurs as a} function of

length

scale

L,

first from 3D to 2D behaviour

at L

= w, and next back from 2D to the 3D behaviour _at L

= L~r. A solution L~r iS _{w appears}

to exist

only

in the

spin glass phase

and in the temperature

regime (3

^it _w around Kc. In the

spin glass phase

when 1 1l

(3

1l _w its

expression

is

(see

the

Appendix):

f f~ ^l~Y3 ~m

~#

L~r _{rw w}

maz(w,1) _(~

2

,

(3.9a)

W W

subject

to the condition

~ i-y3

([)

~_

~3 ^~ ~ _~>

(~.~~)

3

for

(3

i~ _w the correct

expressions

are obtained

by setting (3

'~ w in

(3.9). Moreover, by putting (3

_rw I, one obtains from

equation (3.9a)

the

low-temperature

limit of L~r. In

(3.9),

_~~

is the correlation function exponent at

criticality,

whose

Ising

value is estimated [18] ^to be

q~ = -0.3.

(3.10)

From

(3.9)

and

(2.ll)

it follows that the exponent ^of

(3

in

(3.9a)

takes the value 1- y3

(1 ~~)/2

₌ 0.08; this exponent ^will reappear in several of the

scaling

laws below.

Finally,

the notation

maz(z, y)

in

(3.9a)

is shorthand for _a function that describes _a snlooth _crossover between zll _y and _y £ _z.

At fixed I and _w,

equation (3.9b) gives

the temperature

regime

outside of which _one has

L~r

^£ _w, I-e- the double 3D-2D-3D _crossover does not _occur. In

particular,

it follows that when I and _w

satisfy

^the

inequality1

1l

w~i~,

_{the double}

crossover does not occur at any

temperature.

3.3 ANISOTROPY. The

layered

system is

anisotropic.

We _{can express} the

anisotropy by

nleans of _a

scaling

law in the l1nlit that the RG

trajectory

_{passes very} close to the 2D _zero

temperature ^fixed point. This will be the _case for L~r » w. When _upon renormalization _a

length

scale L~r » _w is

reached,

the systen1can ^{be considered} as

consisting

^of

rectangular

blocks with _a

nearest-neighbour coupling K(L~r)

both inside and

perpendicular

to the

layers.

The linear d1nlension of these blocks is L~r in the two directions

parallel

to these

layers,

^and we

shall denote

by L$

the _one

perpendicular

to the

layers.

From

(3.9a)

_we deduce that

L~r

» w

for I _{- oo} at fixed _{w or} for I _{- oo, w - oo} while

(

- 0.

Taking

account of the fact that

w may be

larger

than I in the last _{case, we} have

L$

rw

maz(w,1). (3.lla)

Hence the

anisotropy

at the _moment of _crossover is

given by

LS

~_

'~~~(~°,0

~~ ~~~~

Lcr Lcr

which _may be made

fully explicit

with the aid of

(3.9a).

If _one renormalizes

further,

_one attains

a

parallel length

scale

L,

to

which,

within the above

picture

of

rectangular blocks,

is associated

a

perpendicular length

scale Ll _such that for all L _> L~r

~~

=

~~ (3.12)

If the

system's

correlation

length

in the

parallel

direction

((w, I, K)

is

larger

than the _crossover

length

Lcr, there will therefore then be _a second correlation

length (I(w, I, K), perpendicular

to the

layers,

and their ratio will be

given by (3.12).

4.

Layered spin glass:

critical temperature ^and correlation

length.

In this section _we calculate the

"systenl

critical

temperature" K~(w,I)

and the correlation

length ((w,I,K)

for _a

layered

spin

glass

in various

scaling

limits with _w

and/or

1

beconling large.

^{As stated} in section 3.2, we have

Lcr

1l _w for ill

w~i~,

so that in that _case the

RG,

within _our

approach,

is three-dimensional for all L and the critical temperature

Ks(w, I)

does not differ from the critical temperature K~ of _{a pure} 3D spin

glass. (We

^have

neglected

devi- ations from

purely

3DRG that stem from the

anisotropic

blocks that

bridge

the

nonmagnetic interlayers

in the

length

scale interval L~r £ L 1l

w.)

However, _we shall be able to deternline

nontrivial expressions for the critical temperature

Ks(w, I)

and the correlation

length ((w, I, K)

when _w ~₂ £1.

Let _us thus _now consider the behaviour of the system correlation

length ((w,I,K)

for _an

increasing coupling

constant K. We understand without RG argunlents ^that as

long

as the

correlation

length

of _a

single layer ((w, K)

is not yet of order

Lcr

_we have

j(w, I, K)

_m

j(w, K)

for

j(w, K)

S Lcr.

(4.1)

When

inserting ((w,K)

from

(2.13c)

and

(2.14d)

and Lcr from

(3.9a)

in the

validity

condition and

solving

for

K,

_{we see} that

(4.I)

is valid for

couplings

K smaller than _a fraction of the

critical temperature

Ks(w,1)

deternlined below. For

larger couplings

K

(see Fig. 2),

_we have

t(W,I, K)

'~

~crt3(K(~cr))

~°~ ~cr s

t(W, K) (4.~)

From

(4.2)

_{we can} deduce the critical tenlperature

Ks(w, I).

It _occurs when

((w, I, K) diverges,

I-e- when

K(L~r)

= K~.

Indicating explicitly

that Lcr

depends

_on the

(bare) coupling

constant,

we have for

Ks

the

equation

K(Lcr(Ks))

₌

Kc. (4.3)

We _can make this

equation explicit by

nleans of

(A.21)

and

(3.9a)

to get, since

Lcr

iS _w,

~~'~~~~

C(w, I)

₌

maz(w, )lwV3~~~~~j

^~~~~_w~3~~~

(4.4b)

and in which (3 is

taken

in Ks. In _order to olve

(4.4a) for Ks(w, I), _{we first} ^the

limit

~~~'~~

J~

~

~~Y3+Y2

y~~~(~ f)f~Y3~Y2~l

^{~ ~~} ~

~)

j~ ^'

c

which is consistent

only

if the RHS is _» I. That is, for _w £

I,

_we nlust have

w~~w

_£

1,

(4.6a)

whereas for 11l _{w we} get ^the contradiction

l 1l _{w 1+}

~

_Y2

(4.6b)

When

addressing (4.4a)

for I ₊ W~YC £

£

_{£ 2, we get} with

fi

_m I

~l'~~

I

rw

[C(w,I)]i~ _(4.7)

As _p is small

positive,

we have with 1 1l

(3(Ks)

1l _w the

validity

condition

i s

c(w, i)

s wv.

(4.8)

f

=

wi+@

_/

^'t

='°

/ / /

Ks

~~ ^,

Kc /

14.I oa)

, '

,

/~~-vc

s ~' s 2

, 1<~

/ 14.I ob)

/ / /

/ t ,_

= w 2

/ /

K,

(I-I(

~ W y^~

)4.loo)

w

Fig. 3. Validity _ranges for the expressions of the system critical temperature

Ks(w,t).

The _respec- tive validity domains of

(4.10a), (4.10b),

and

(4.10c)

_are separated by the solid _curves t

=

w~+d~

and t ~m

= w ²

For wll

I,

this leads _to

~~

~

l £ _{w Y2}

,

(4.9a) imposing

I £ _w

gives

w~P

_{£ 1.}

_(4.9b)

We _see that the above solutions of

(4.3)

_are the _ones for

w~l~

_{1l 1.}

_Recalling

_{that Lcr 1l}

w

for all temperatures ^when ill

w~i+,

we get in summary

(see Fig. 3)

Ks

(,1)

_~

^w~% ~'

~°~~~

_~

~~ ~~ ~~~~

~l'~~

I

rw

[C(w, I)]~i~, w~i~

_{£ ill}

w~+@, _(4.10b)

c

~~

l'~~

m I, 11l

w~i~ _(4.10c)

c

These expressions show the

expected

result that

Ks(w,I)

increases when I increases _{or w} decreases.

Furthermore,

_we see from

(4.2) that,

for

w~i~ _1l1,

we can write

_j

((w, I, K)

rw L~r

~(~~~

^{l ~~}

,

£

~~~~~~

£

2, (4.ll)

for the correlation

length

_near the critical

point

Ks. For _very

large Ks,

I-e- for

w~+I

_£

_I,

we have from

(4.10a), (3.9a)

and

(2.13b) )

=

£

_so that

equation (4.5)

becomes

((w, I, K)

rw w

~ ^~

Y3)

~~~~~

( ~,

_£

(

_{£ 2,}

(

» 1.

(4.12)

W

s

2

s c

In the intermediate _case, I-e- for

w~i~

_{1l 1 1l}

w~+d~,

we

expand )

_{I in equation}

(4.ll)

around K ₌

Ks

to first order and get ^~

((w, I, K)

rw w~

maz(w,1) ~ ~~~~~~(

l

~,

_£

(

_{£ 2,}

~

m 1,

(4.13a)

W s s c

where

~ I + flc

y

r 1 +

~ ~~ _~~~

~~ ~~~~

~ =

~~~ (l

⁺

/

jy~

~

2 ^~' ~ _~~

The

validity

condition of

(4.13a)

_comes from the domain of

validity

of the

expansion

^of

(4.ll).

When _one considers the interval of tenlperatures T

=

£K-I

around Ts where the correlation

length

scales _as in

(4.

II

),

I-e- with the three-dimensional exponent

,

we see from the

validity

conditions of

(4.12)

and

(4.13a)

that for

w~i+

_{1l the considered interval is}

_{given by}

1l

)

_1l

2.

(4.14)

s

This is

evidently

true for ill

w~+~

_{where within}

our

approach

Ts ^differs

negligibly

front T~.

So the temperature interval of three-dimensional critical behaviour is of order Ts ^{and decreases} in width with

increasing

I. It vanishes in the limit I _{- oo as}

expected.

Finally,

_{one can}

easily

_recover the

expected

limits of

Ks(w, I)

and

((w,I, K)

_{as a} function of _w, I in the

following

_cases:

(I)

_w

finite,

I

= oo Here Ks

= oo, I-e- Ts

= 0, and

((w, I, K)

=

((w, K)

from

(4.I),

which _are the characteristics of _a

single layer.

(ii)

w = oo, I finite Here Ks

= K~ and

((w,I,K)

=

(3(K)

front

(4.10c),

which _are the characteristics of _{a pure} 3D system.

5 Conclusion.

Metallic

spin glass layers separated by interlayers

of _pure metal have been studied experimen-

tally by

several authors. In this paper we have obtained theoretical

scaling

laws for the critical temperature of such _a system as a function of the

layer

width _w and the

interlayer

width

I,

and for its correlation

length

_{as a} ^{function of} _w,

I,

and the temperature T. Crossovers from three- to two-dimensional behaviour and back _to three-dimensional behaviour _are discussed. Such

crossovers

clearly

_appear in

experimental

studies of the

time-dependent magnetization [lo, iii.

The present static

study

is _a

preliminary

to the

study

of

dynamic phenomena

in

layered spin

glasses.

Appendix.

Determination of the _crossover

length L~r.

We determine in this

appendix

the _crossover

length

L~r introduced in section 3. Two steps are

involved:

first,

_we calculate the _{meat square}

interlayer coupling AK~(L); second,

_we equate this

quantity

to

K~(L)

_as

prescribed by equation (3.8).

It is clear from the main text that

only

_crossover

lengths

L~r

bigger

than the

layer

thickness _w lead _to nontrivial

results,

because

otherwise _{one never} stops

renormalizing

as in 3D. We will _see in what follows that this

happens only

for

couplings £

_{iS I W~VC.}

_Therefore,

for _reasons of

convenience,

_we discuss smaller

couplings

at the end ~of this

appendix only.

A-I- THE MEAN SQUARE INTERLAYER COUPLING

hK~(L).

As stated in the main text,

we calculate in this section the _{mean square}

coupling AK~(L)

of _a block

spin SI(L)

in _a fixed

layer

due to the _presence of the other

spin glass layers.

The total

interlayer

interaction has at each

stage

^of renornlalization L

= 2" the form

V(L)

=

£ JIJ(L)SI(L)Sj(L) (A.I)

I,J

where

SI(L)

= + I _are the Lth level block

spins,

^and I and J _are in different

layers.

As the renormalization of the order parameter ^reads

< s, >o=

B~/~(K)SI (A.2)

~~~ _~ ~ ~ ~~~

jJ2( j~)

=

~ ^~~ ~~~~~ _~ ~'

~

~ _~~

(A.3a)

1, _f > 1,

in d

= 3,

B~(K)

= I

(A.3b)

in d

= 2, we get ^for the renormalized

coupling

constant after _one renormalization step,

sunlming

on I e I and

j

e

J,

f

=

B~(K) £ _{J( (A.4a)}

1,J or

J)j

=

B~(K)2~~J(, (A.4b)

if Ii

j(

does not vary nluch. After _n renormalization steps we obtain

J)j(L)

=

B~(K~"~~))B~(K~"~~~).. B~(K) ~j £ _J( (I) (A.5)

,eI jeJ

where

K("~

_{is the}

intralayer coupling

between

nearest-neighbour

blocks. The scheme _as shown in

figure

I is interrupted when the

interlayer coupling

gets

nonnegligible. Furthernlore,

in the 2D

regime,

I-e- for L ~t _w, the blocks L3 _become blocks wL~. To calculate the _{mean square}

coupling AK~(L)

of _a block

spin SI(L)

with the block

spins

J of the _same size in all other

layers,

I-e- the

quantity

that is to be

conlpared

to the _square of the renormalized

nonperturbed

nearest-neighbour coupling K~(L),

_we have to determine

/~K~(L)

"

£ JIJ(L)~) ^(A.6)

Jilayer of ₁

wst;Lsw wst;tsL

t

~

, i

, i

, i

, ⁱ

i '

i

w -

~ »

L

Fig.

4

Fig.

₅

Fig. 4. Calculation of

AK~(L)

for _case I: _w £ t; L £ _w. For _w £ t, the interaction of _a given spin

s; in the indicated domain at length scale L with spins outside its layer is mainly due _to spins in the shaded _cone.

Fig. 5. Calculation of

AK~(L)

for _case I: _w £ t; _w £ L. For

w £ t and _w £ L, the _{mean square}

coupling

of the considered block spin is due to

nearest-neighbour

interaction between the cubes that constitute the block spins in the two layers.

and since _cross terms cancel

AK~(L)

=

B~ (K("~~))B~ (K("~~) )...B~ (K) £ £ @ _(A.7)

;eI jeJ

We calculate

F(L, w,1)

=

£ £ @ _(A.8)

;eI jeJ

first. As

long

_as L is

sufficiently snlall,

_{we can} limit ourselves to the effect _on

SI

due _to the block spins Sj in _{a cone} with _an

angle

of order I

(see Fig.

4 and

Fig. 6),

_as the

magnitude

of the RKKY interaction between

spins

_s,, I _e

I,

and

spins

_sj,

j

_e

J,

in _a

particular layer

is of the _same

order;

the

spins

_sj outside the _{cone will}

change

the

prefactor

of the result

only.

Inside the _cone, it is sufficient _to consider _one

nearest-neighbour layer,

shown shaded in the

figures.

The

nearest-neighbour layer

_on the other side will at nlost contribute _a factor 2, and all

layers

further _away will

again s1nlply change

the

prefactor

of the

result,

due to the

rapid

convergence with distance of the RKKY interaction.

Beyond

a certain

length scale,

_one starts to block spins that _are outside the shaded _cone. In that case, the _{mean square}

coupling AK~(L)

is

mainly

due to

nearest-neighbour coupling

_as shown in

figure

5 and

figure

7. We have to

distinguish

two cases,

depending

_on whether the

interlayer

^distance I is

larger

_or smaller than the

layer

width _w.

Case I _w £ I. For

length

scales L 1l _w 1l

(see Fig. 4),

the block spin SI ^of size L3

interacts with ^'~

~

blocks of the _same size at a distance

I,

_so

L

~3~f2 ~3~

F(L, W,1)

~

~

^"

i,

^{L £ W £ ~.}

^(A.9a)

For _w

£

L

1l1,

_we

get similarly

~( L, w,1)

~

~~j/~~ ~~~~'

^{~ ~ ~ ~}

^~~

^~~~

When ill

L,

_{one can} consider the block

SI

_as

consisting

^of blocks of size wl~ _{that interact} each with the

corresponding nearest-neighbour

block in the other

layer (see Fig. 5).

We obtain thus

~ ²

~~

_~~2

~2~2

F(L, W,1)

'~

(7) _~

"

i,

^{i ~l L.}

^(A.9~)

In _{summary, we can} write for _case

F(L,

_w,

I)

rw

)min (, )). _(A.lo)

tsw;Lst

I i

Fig. 6. Calculation of

AK~(L)

for _case II: t £ _w; L £ t. For tll _w, the interaction energy between two neighbouring layers ^is mainly due to the interaction between spins within _a distance t from the

surface. The spin _s, interacts predominantly with the spins in the shaded _area.

Case II I £ _w. When I becomes smaller than _w, the interaction _energy between _two

neighbouring layers

is

mainly

due _to the interaction between

spins

close to the surface. As

long

_as the

length

L of _a block

spin SI(L)

is smaller than

I,

it interacts

predominently

with block

spins

in _{a cone} of width I in the

nearest-neighbour layer

that _are within _a distance I from the surface

(see Fig. 6). However,

when L

gets larger

than

I,

the

interlayer

distance I becomes

negligible

and

SI(L)

has

mainly nearest-neighbour

interaction with the

corresponding

block spin in the other

layer.

Because of the

rapid

decrease with distance of the RKKY

interaction,

it is still the

spins

at the surface that determine

AK~(L), though.

For L £ I

(see Fig. 6),

the f3

block

spin

SI of size L3 interacti _with

~ blocks of the _same size at _a distance

I,

_so

L

F(L, w,I)

_rw ~3_~, L £ I.

(A.lla)

tstu,tsL

L

Fig. 7. Calculation of

AK~(L)

for _case II: t £ _w; t £ L. For t £ _w, the interaction energy between two neighbouring layers is mainly due to the interaction between spins within

a distance t from the surface. The nearest-neighbour interaction between the cubes that constitute the block spins is due to the interaction between spins in the shaded _areas.

For1 1l

L,

it is

again only

the

nearest-neighbour coupling

that

prevails (see Fig. 7).

Al-

though

the block is

larger

^than

I,

it is still the

spins

within _a distance I from the surface that

predominate,

^{for the} renormalization of the order parameter is local and

J;j

decreases

rapidly

with distance.

So, similarly

to _case

I,

_{we can} consider the

large

blocks _as

consisting

of cubes of size 13 and that it is those cubes at the surface that determine

mainly

the interaction between the

large

blocks.

So,

~

F(L, W,1)

_~

,

l 1l L.

(A. lib)