On the concept of local invariance in the theory of spin glasses

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On the concept of local invariance in the theory of spin glasses

I.E. Dzyaloshinsk, G.E. Volovik

To cite this version:

I.E. Dzyaloshinsk, G.E. Volovik. On the concept of local invariance in the theory of spin glasses.

Journal de Physique, 1978, 39 (6), pp.693-700. �10.1051/jphys:01978003906069300�. �jpa-00208802�

ON THE CONCEPT OF LOCAL INVARIANCE IN THE THEORY

OF SPIN GLASSES

I. E. DZYALOSHINSKII and G. E. VOLOVIK

Landau Institute for Theoretical

Physics, Vorobyevskoe

^{shosse 2,} Moscow 117334 U.S.S.R.

(Reçu

^{le 3}

février

^1978,

accepté

^{le 6 mars}

1978)

Résumé. ²⁰¹⁴ Nous étudions la

possibilité

d’appliquer ^le concept d’invariance d’échange ^locale

(champ

^de Yang-Mills

[3])

^{à la}

description

^{des verres de} spin. ^Partant de l’invariance locale microsco-

pique (discrète) ^{et du} concept de frustration étudié _par Toulouse [8] ^{et Villain} [7], ^{nous montrons} qu’à

l’échelle macroscopique ^{un verre de} spin ^est naturellement décrit _par des

champs

^de Yang-Mills

continus. En particulier, ^les champs électriques ^et magnétiques s’interprètent ^comme des densités et des courants de disclinations. Nous analysons ainsi les différents types ^{d’ordre à} longue distance,

le spectre d’oscillations et les éventuelles transitions de

phase

dans de tels systèmes.

Abstract. ²⁰¹⁴ We consider the

possibility

^of applying ^the concept ^{of local} exchange ^invariance

(Yang-Mills

^fields

[3])

^{to describe} spin glasses. ^{We start with the}

microscopic

discrete local invariance and frustration concept ^studied by ^Toulouse [8] and Villain [7] and show that on the macroscopic

level a spin glass ^is naturally ^described by continuous Yang-Mills fields. In particular, magnetic ^and electric fields have the meaning ^of density ^{and current} of disclinations.

We also study types ^of long-range order, oscillation spectra ^and possible phase transitions in such systems.

Classification Physics ^Abstracts

75. 0H

1. ^- While

reading

the works on

spin glasses (e.g.

the brilliant _survey

by

^Anderson

[1])

^{it is difficult}

to abstain from the

temptation

^{to use a}

concept

^of local

exchange

invariance

(LEI)

^to describe this

peculiar

^{state. LEI leads to} strong

degeneracy

^{of the}

ground

^{state of a}

spin glass.

^If the local symmetry

were exact, the energy functional would remain invariant not

only

under simultaneous rotation of all the

spins by

^{the same}

angle,

but also for rotation of

only

^one

spin

with all the rest

stationary.

^{In fact}

however,

^{LEI can}

only

^be

approximate (see below).

This means that with the rotation of one

particular spin

^its

neighbourhood adjusts

itself in such a way that in the

leading approximation

^{the energy of the}

system ^{will not}

change.

^{But even} this truncated local invariance and the

essentially

^weaker

degeneracy

which comes with it

permits

^{us to}

reproduce

^a

majority

of

qualitative

features of

spin glasses

^{- absence of}

conventional

long-range order, mictomagnetism,

^etc...

The mathematical tools of _any

theory

with conti-

nuous local

invariance,

^{such as}

LEI,

^are

widely

^known.

They

^are the so-called

Yang-Mills

^fields

[3]

^{and can be}

introduced as follows. We shall describe the

spin glass by

^{the vector field}

S(x)

^{of unit}

length : ^S2

^{= 1}

[2].

The _energy of the system ^{is some}

general

functional of the field

S(x)

^and its derivatives

ôS/ôx;, 02S/0Xi ôxk,

^...

To make this functional local-invariant one has to introduce

Yang-Mills

^fields

which under infinitesimal

coordinate-dependent

^rota-

tion in

isotopic

space

0(x)

^are transformed as

while under the same rotation the field

S(x)

^{is trans-}

formed in the usual _{way :}

The space derivatives of S _may enter a local inva- riant functional

only

^{in a} certain combination with fields

bi

^as so-called covariant derivatives

which under rotations

(2), (3)

^are transformed as the field S itself. The

Yang-Mills

^fields

bi

may appear in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003906069300

694

the energy functional

by

themselves in

S-independent

combinations

which are

obviously,locàl-invariant

^{too :}

If besides this _space local invariance one

requires

^the

time local

invariance,

^we introduce the

Yang-Mills

field

transforming

^as

under

time-depèndent

^rotations

0(t).

The covariant time-derivative is

When this new field a is present ^another

locally

invariant

quantity

^{comes in to}

play :

Formulae

(1)-(9) expressing

^LEI

generalize

^the

regular gauge-invariance

of Maxwell

electrodynamic.

The

Yang-Mills

^fields

bi

^{and a are}

analogues

^{of vector}

and scalar

potentials,

^and

fik

and gi ^are

analogues

^of

magnetic

and electric fields

[4].

However,

^{in solid state}

physics

^{LEI does not exist}

on the

microscopic

^level

(contrary

^to the situation in

particle physics).

^It

might

emerge

only

^{as the} symmetry of some

macroscopic quantities averaged

^{over suffi-}

ciently large

volumes. This

averaging

^{will automa-}

tically provide

^the

physical meaning

^of

Yang-Mills

fields. Moreover

according

^to this definition local invariance for

macroscopic quantities

will be violated at small

enough

^distances

(at sufficiently large

^gra-

dients).

^{That means that in the}

expansion

^of energy in power of the

gradients of,

say, average

(macroscopic) spin m(x) only

^{the first term} may be

locally

^invariant

(- (Vim)’),

^{while the} next terms will

display ordinary

derivatives instead of covariant ones.

2. ^- To understand how to derive the

macroscopic

relations of this kind we shall consider a certain model of a

spin glass : Heisenberg spins

^{on a frustrated}

lattice. This is a model with the standard

Heisenberg

Hamiltonian with

nearest-neighbour

interaction :

S(a) being three-component

^{classical vectors of unit}

length.

^The

exchange integral J(a, b)

^{has - with a}

certain

probability

^-

only

^{two values J =} + ^{1. If} the

probability

distribution is chaotic

enough,

^one

hopes

that this model will

display

^the

properties

^{of a}

spin glass.

The system ^described

by

^the Hamiltonian

(10)

exhibits the local discrete invariance

(LDI) :

The _group

(11)

^{is a discrete}

[5] analogue

^{of the}

Yang-Mills

group

(2)-(3).

^This group is the

only microscopic

^local symmetry ^{in the}

physics

^of

spin systems.

The concept ^{of LDI in the}

theory

^of

spin glasses

^has

been

employed extensively by

^Mattis

[6],

^Villain

[7]

and Toulouse

[8].

^In

particular,

^{Toulouse has}

provided

a

complete description

^{of the}

ground

and excited states of the

glass

^{in terms of a} frustration network.

His results could be formulated as follows.

Let us describe the

spin

orientation

by

^rotation

angle 0(a) depending

^on the lattice

site,

^{so that}

where

So

^{is a constant unit vector.}

According

^to Toulouse, ^{the field}

0(a)

^varies

fairly smoothly

^almost

everywhere

except

[9]

^{in the}

vicinity

of the network of

singular

^{lines and}

singular planes connecting

^{them -}

the frustration network. Therefore we may

employ

^an

intermediate

semi-macroscopic picture :

^{the field of}

the unit vector

S(x),

the derivatives of which are

given by

^formula

Vector fields S and 0 are

singular

^on frustration lines and have a

discontinuity

^on

planes

^stretched upon these lines.

Frustration lines thus defined are disclinations in the fields S and 0 similar to disclinations in

liquid crystals.

^This

similarity, however,

^is

superficial

^because

in

liquid crystals planes

^stretched upon disclinations

are not

singular

^at

all ;

their actual arrangements ^and

shapes

^are

completely

irrelevant

provided

disclination lines are

given.

^{In our case} these surfaces exhibit real

physical singulàrities.

^The

discontinuity

of S and 0 on

such a surface is determined

by

^the

probability

^distri-

bution of the values

of J(a, b).

And in this respect ^our disclination in

spin glass

^{is a twin} brother of the disclination studied

by

^the

theory of elasticity [lo].

Now we can take the last step ^and average ^our

picture

^{over scales}

larger

than distances

between

disclinations. This

procedure

is well known in the

elasticity theory

and consists of

going

^{over to a conti-}

nuous distribution of disclinations. Let us define

[11]

a mean

angle

of rotation

cp(x)

^{and tensor}

b,(x)

^similar

to the distortion tensor of the

elasticity ^theory (see [10]).

The latter is not

generally speaking

^a

gradient

^of any function and has the same transformation

properties

as the

Yang-Mills

^fields

bi

ⁱⁿ

(1), (2).

^{Further we}

introduce the

density

of disclinations

(see [10])

(Eikl

^{is the}

antisymmetric

^unit

tensor),

^{which in a linear}

approximation

coincides with

fik of (5),

^{and the current}

of disclinations

which in the linear

approximation

is identical with _gi

of (9).

For the fourth

Yang-Mills

^{field a of}

(6)

^{one can}

usually

^assume

(see [lo])

The

explicit

form of the

equations

^{of motion}

depends

on further

assumptions

^{about the}

properties

^{of our} system.

3. ^- We start with the case when there is no conven-

tional

long-range order,

^{i.e. we assume that the}

macroscopic averaging

^{either of the}

spin S(x)

^itself

or any of its momentsa xi xk ^...

S(x)

^will

give

^zero.

Then the

Yang-Mills

^fields

bi

and rotation

angle

(p ^are the

only

^variables

describing

^our system. ^{Such a state} could be

naturally

^{called a}

simple

^{or true}

spin glass.

It is an exact

analogue

^{of usual atomic}

glass,

^{i.e. an}

amorphous

media of the

elasticity theory.

^Then

angle

qp

corresponds

^to

displacement, bi

^{to the} distortion tensor

and _Pi to the

density

of dislocations. This

analogy implies

that transition to the

simple spin glass

^{is not a}

thermodynamic phase transition,

but is the result either

of quenching

^{or of}

gradual hardening [12].

Let us first write down the

equations

^{for the case}

when motion of our media is

entirely

^{due to a}

given

motion of

disclinations,

i.e. when

density

^{and current}

of disclinations can be considered as external sources.

This

corresponds

^{to one} of the standard

problems

ⁱⁿ

the

elasticity theory (see [10]).

^{The first two}

^equations

of the

theory

^are

(13)

^and

(14),

from which the law of conservation of disclinations follows :

as well as the

transversality

condition :

To make the system

complete

^one should add the

equation

^which express the conservation of

angular (spin)

^{momentum i in} the absence of an external

magnetic field [13] :

where

llk

^{is current of}

spin-momentum.

^{For the}

spin-

momentum

density

^{it is natural to assume}

Finally providing

^some

phenomenological

^expres-

sion for

Uk (given

for instance

by (20))

^{we come to a}

problem

^{which is}

formally equivalent

^to the standard

problem

^{in the}

theory

^of

plastic

^flow

(see [10]).

The set of

equations (13)-(19)

^{describes a whole}

range of

so-called mictomagnetic phenomena.

^How-

ever we shall not dwell on them

further,

but consider the

simplest possible

^{case when the mean} disclination

density

ⁱⁿ

equilibrium

^{is zero}

[14]

and when disclina- tion motion is

entirely dissipative.

^{Then we can assume}

for

spin-momentum

^{current and}

for the

dissipative

^current of disclinations. From

(13), (16)

^and

(21)

it follows that

Therefore at such motions _pi ^z 0 and the oscilla- tions

of bi

^are

longitudinal :

From this and

(14)

^we get

The

equation

^{of motion for}

spin-momentum

^takes

the form :

When

dissipation

is absent from

(23)

it follows that

and the whole motion is reduced to pure rotation _gp.

The

frequency spectrum

for such oscillations was

studied

by Halperin

and Saslow

[12].

^For

large

^wave-

696

vectors q

(with dissipation

taken into

account)

^there

are three

longitudinal

oscillations of acoustic type

However ^at low

frequencies

^{all three}

Halperin-

Saslow modes become

dissipative :

Without

dissipation

^the

equations

^for

longitudinal

oscillations _may be obtained from the variational

principle

^{with the}

Lagrangian :

It should be varied with the constraint

(25), ^which, using (14),

^we

put

^{in the form}

Thus the fact that the

longitudinal

^{current of dis-}

clinations

equals

^{zero ensures} the absence of dissi-

pation.

In

principle

oscillations could exist in the system ^of disclinations as well. The motion of disclinations in a

spin glass

^{is not}

necessarily

^{of a}

dissipative

^{kind like}

the Peierls

hopping

^of

dislocations.

^For instance there may be oscillations of the value of

angles

^on

singular

surfaces which are of continuous nature. On a macro-

scopic

level these motions occur in the densities and currents of disclinations and are in fact oscillations of the transversal parts ^{of the}

density

pi from

(13)

^and

the current

The

corresponding equations

^{of motion} may be derived from a

Lagrangian

^{of the} type :

Here u is a

quantity

with dimensions of

velocity.

The

constant g

is introduced

by analogy

^{with the}

charge

^of

Yang-Mills

^fields

[3].

^{The last term in}

(30)

comes from

pbf

ⁱⁿ

(27).

^The

Lagrangian (30),

^des-

cribes 6 oscillations

(3 isotopic

components ^x 2 spa-

tially

transversal

oscillations)

^with

equal frequencies :

In contrast to the spectrum ^of

gluons

ⁱⁿ

Yang-Mills theory

^the

frequencies (31)

^{have a} gap ^{- our}

gluons

are massive. The mass guy ^comes from the term

J1bf

which breaks local

exchange (Yang-Mills)

symmetry.

As usual this

symmetry-breaking

generates ³

longi-

tudinal

gapless

Goldstone modes with velocities

proportional

^to

fi ⁽³ Halperin-Saslow

^sounds

(26)).

4. ^- We came to think of a

spin-glass

^{as a dis-}

clination network

starting

^{with a}

microscopic

^model

of a

Heisenberg magnet

^{on a} lattice with random links and then

averaging

^the frustration network cons-

tructed

by

^{Toulouse. It is}

fairly

evident that the same

picture

^{with a} disclination network remains valid for

magnetic alloys

of the MnCu

type

^where

exchange

between

magnetic

^{ions is the} RKKY-interaction :

Therefore ^we shall

approach

^such

systems by

^the

same method.

However in contrast to the

majority

^of

microscopic

theories

starting

with the work of Edwards and Anderson

[15]

^where interaction

(32)

^{was modelled as}

a random

potential,

in what follows we shall substan-

tially exploit

^{the actual law of}

decreasing

^of

J(a, b)

with distance as

R’

5. ^- Let us tum to a

spin glass

^{with a}

well-developed

disclination network but where in addition there exists a kind of a

priori long-range

^{order. We assume}

that the _average value of

microscopic

^field

( S(x) >

which can

again

be considered a unit vector field

m(x)

is different from zero. Now we are in a

position

^to carry out the _programme mentioned in the first section of the _paper and introduce this

ferromagnetic long-range

order in a

locally-invariant

^{manner. Thus we assume}

that the term

quadratic

ⁱⁿ derivatives of m, ’i.e.

(am/ôxi)2

^{will enter the} energy functional

only

^{in the}

form of covariant derivatives

(Vim)2

^from

(4),

^{that is}

only together

with the field

bi. Physically

^{it means that}

in the

leading approximation

^the energy of _magne- tization m is determined

by

its orientation with

respect

to the disclination network

only

^{in the same} space-

point

and that this _energy does not

change

^{when the}

magnetic

^{structure which is defined now}

by

^both

disclination

density

pi and

magnetization

^m

undergoes

a local rotation. The

breaking

^{of LEI in this}

approxi-

mation comes from the same term

pbf

^{in the} energy functional

(27).

The additional

breaking

^{of LEI will}

come

only

^from

higher

derivatives of m.

This kind of LEI manifests itself for instance in

degeneracy

^{of the}

ground

^{state of a}

spin glass

^and

consequent

change

^{of the} energy

dependence

^{of finite}

samples

^on their dimensions. If the

inhomogeneity

^of

spin

orientation on the

sample

surface is

sufficiently

smooth and does not

produce

^new disclinations

inside,

^then without local invariance the _energy of the

sample

^{would be}

But with local invariance

present

^the energy

and the

part

of it which is

asymptotically proportional

to the dimension L can be made

equal

^to zéro

by

^an

appropriate

rotation of the disclination network. The

degeneracy

of this kind takes

place only

^{with a}

sufficiently developed

disclination network. The condi- tions of its appearance have been discussed

by

^Tou-

louse

[8].

^Its existence has been

indirectly

^confirmed

by

Toulouse and Vannimenus in a numerical _expe- riment

[16].

Let us tum now to a small oscillation around

homogeneous magnetization

orientation mo

(m’

⁼

1).

The transversal part ^{of the}

Lagrangian (30)

^which

describes oscillations of disclination

density

^{does not}

change

^{with a}

long-range

^{order m. The}

Lagrangian

^of

longitudinal

oscillations of disclination network and oscillations of m ⁼ mo ⁺ m’ is now of the form

[17].

It reduces to

(27)

^when

magnetization

oscillation is absent and becomes the standard

Lagrangian

^{for the}

spin

^{waves in}

ferromagnets

^{when the}

compensating

field _(p is

equal

^{to zero.}

Simple

calculations show that

(33) provides

⁴

gapless

modes. The first 3 modes

correspond

^to oscillations of the

quantities

They

^are

practically

identical with the 3

Halperin-

Saslow sounds and have the same linear spectrum

(26).

The fourth mode

corresponds

^to oscillations of the

magnetization

^{m’ and its}

frequency

^{in our}

approxi-

mation is

equal

^{to zero :}

The energy of the

spin wave

^may become finite

only

as a consequence of

breaking

^{the local}

invariance,

which should manifest itself in

higher

derivatives. The

simplest assumption

^is

However,

ⁱⁿ

reality

the situation is more

compli-

cated since the

compensation

^{in terms} of lowest order

means that various

physical long-distant

interactions which _may exist in our

system

^become

important.

In

particular,

ⁱⁿ

alloys

^{with an RKKY} interaction

(32)

one should expect ^essential

non-analyticity

^{of the}

expansion

in derivatives and

consequently q3-depen-

dence of the

spin-wave

spectrum.

To convince ourselves let us turn to the

diagram

^of

figure

^{1 for a}

low-temperature

energy

expansion

ⁱⁿ

powers of m which

corresponds

^to the standard mean-

field

theory (cf. [1, 15])

and consider their contribution to the _energy of

spin

fluctuations mq ^at

small q [18].

Diagram (a)

^is

analytic

^at

small q [19]. Diagram (b)

^is

of

special

^{interest to us. It} represents the effective interaction of the form

FIG. 1. ^- Open ^{circles denote m} the average magnetization ^{at a} point ^{x and summation over all} impurities. ^Lines represent ^RKKY

interaction (32).

and at

large

distances behaves like an

ordinary

Van der Waals interaction -

1 IR’.

^Its contribution to the _energy is

evidently proportional

^to

q3. Diagram (c)

is

again analytic

^{at small} q, and the lowest non-

analytical

^{term in}

diagram (d)

^is

q3. Finally

^dia-

gram

(e) gives J,,ff - 11R 12

^and

non-analytical

^contri-

bution to the _{energy -}

q9.

^{It is} easy ^to make sure

by

such an

analysis

^{that the}

leading non-analytical

^term

is indeed

q3.

Thus, assuming

^{that the}

term qz

^is

compensated

thanks to local invariance we may write for the

spin-fluctuation

energy

where k and I are constants, and for the

spin

^wave

energy

something

^like

698

with _y from

(33).

^The

density

^{of states with such a} spectrum ^is

independent

^of energy and the corres-

ponding

^heat

capacity

^is

proportional

^to temperature :

It is of interest to

point

^{out that}

computer

calculations for the model of a

spin glass

^{with RKKY} interaction

performed by

^{Walker and Walstedt}

[20]

^have

provided

a similar result ^- 4

gapless

^{modes and the}

density

^of

states

independent

^of energy.

6. ^- Now we can

explain why

^{we have been so}

cautious at the

beginning

^{of the}

previous

section and called our

magnetization

^{m an a}

priori long-range

order. The

point

is that the fluctuations of _magne- tization determined

by

^energy

(34)

^{are so}

strong

^that

they destroy

^this

long-range

^{order. In this} respect ^the situation in our model of a

spin glass

is like that found in the vector models on 2-D lattices studied once

by

Berezinskii

[21].

The

intensity

of fluctuations with a

given

^wave

vector is

and the

corresponding integral

which determines the fluctuations

of magnetization

^at

point

^R

diverges logarithmically.

^{Therefore we} have here the

case of so-called lower

marginal dimensionality

^and

calculation of the

magnetization

correlation function should follow Berezinskii. His method

gives something

like

The

long-range

order thus

disappears

^but the critical

fluctuations,

.which decrease as a power of distance

(instead

^of

exponent),

^are

preserved throughout

^the

low-temperature region.

The idea that

spin glass

^is a state at lower

marginal dimensionality

^{has been}

expressed by

^Anderson

[1].

He has also discussed

physical

consequences of this situation

[22]

^{and we shall not dwell} upon it. We add

only

^{that in an external}

magnetic

^{field H formula}

(39)

will

give

^an

expression

^for

susceptibility :

Unfortunately

^{the nature of the}

phase

^transition

to our state is

absolutely

unclear. Moreover there

are

fairly

strong ^{reasons to} believe that such a tran- sition does not exist at all. The

point

^{is that as}

Polyakov

has

proved (see

e.g.

[23])

^{in all 2-D vector}

^models,

except ^{XY-model which was studied}

by Berezinskii,

the transition

temperature Te is exactly equal

^{to zero.}

The

tendency

^to

long-range

correlations which is so

characteristic of these systems will manifest itself

only

^{in an}

exponential dependence

^{of the} correlation radius

Re

^on

temperature ;

Actually, Re

^{is the distance at} which the correlation

given by

^formula

(39)

^{becomes of order}

of unity.

Thus at low temperatures ^the

spin-glass,

^{in our}

model behaves like a 1-D

Ising

system for which the formation of

magnetic

clusters of

exponentially large

dimensions is characteristic.

Certainly

^{there is}

always

a

possibility

^of

preserving

^the

phase

transition

by introducing

^finite

anisotropy.

7. ^- Let us now come back to the system ^we started our

investigations

^with,

Heisenberg spins

^on

a

regular

lattice with random

nearest-neighbour

interaction. The situation here is more subtle because the direct

long-range

interaction which in

alloys

gave

us

q3-term

^{in the} energy is absent from the system.

The

only

^{source of}

long-range

interaction now is the

exchange

^of quanta ^of

Halperin-Saslow

^sound

(26).

Appropriate

calculations show that at

a

q3 -term

reappears but with a coefficient

equal

^to temperature ^T

by

^{order of}

magnitude :

The calculations involved

exactly repeat

^{the cal-} culations in the

theory

^{of the}

dependence

^{of thin film}

chemical

potential

^on the width of the film which is due to classical fluctuations of

electromagnetic

^field

and sound waves

[24].

The

spin

^wave spectrum ^{is now}

analytic

and the

corresponding

^heat

capacity

The

magnetization

correlation function is now

described

by

^{the formula}

Considerations similar to those which led us to

(41)

will now

give

^{for the} correlation

length :

Thus at low temperatures ^the

Heisenberg

^model

on a 3-D frustrated lattice is

equivalent

^{to a}

regular

^1-D

vector model.

8. ^- Let us

study

^other

possible

types ^{of a}

priori long-range order,

^{that is}

antiferromagnetically

^ordered

spin glasses.

^{As has} been noted

by

^Andreev

[13] they

arise as result of

averaging

^of

higher spatial

^moments

of the

microscopic

^{field S. Thus the ordered state} may be defined

by

^tensors

In

particular

^{Andreev has} studied the

example

^{of a}

spin glass

^with

which is

formally equivalent

^{to the}

B-phase

of super- fluid He3.

It is not difficult to

adopt

the formalism of local invariance for the

antiferromagnetic

^{state. It is}

sufficient to introduce covariant derivatives of the

field, li

^-

Vkli,

^{etc. and to take into account two terms}

quadratic

ⁱⁿ derivatives of the _{energy -}

(Vkli)-(Vkli)

and

(Vili)2. Everything

^{which was} said above about correlation functions remains valid and the formulae for

spin

^wave spectra may

easily

be written down if

one takes into account that in

antiferromagnets

^the

relation _{a)4 -}

Efluc

^{is to be}

replaced by c4 N Eflur.

That

gives

^for

magnetic alloys

and

For the

Heisenberg

^{model on a} frustrated lattice

The

phase diagram

^{for a} model with frustrated bonds in the variables T and _c, where c is the concen-

tration of bonds of

antiferromagnetic sign,

^{will look}

like

figure

^{2. At small c there is a}

ferromagnetic

state

(F).

^{Further on at a certain value of c the} system becomes

locally symmetric

^{and turns into the state}

SgF

described in sections 5-7.

Similarly

^{at c close to one}

the system ^is

antiferromagnetic

^{and with c}

decreasing

it turns into the state

SgA

considered in this section.

The

marginal

^{case c =}

1/2

^{has much in common with}

the case of a

magnetic alloy

^{with RKKY} interaction.

Which of the two

possible

^states

(SgF

^or

SgJ

^{will take}

place depend

^on subtleties of

impurity

correlation at small distances.

9. ^- Our main result ^- the

expression

^{for the}

Landau

Ginzburg

energy ^at low temperature

(33)

which exhibits LEI - is in a certain sense

equivalent

to the

equation

for the molecular field obtained

by Thouless,

^Anderson and Palmer

[15]. They

^both

provoke

anomalous increase of fluctuations in the

spin glass

^{state and lead to lower}

marginal

^dimen-

sionality

type of behaviour

(cf.

^remarks

by

^Ander-

son

[1]).

FIG. 2. ^- Phase diagrams ^{for a} model with concentration (’ of

antiferromagnetic ^bonds.

References

[1] ANDERSON, ^P. W., Survey of ^Theories of Spin Glass, 1976.

[2] ^{We use vector} symbols ^and operations only ^for components of spin S(x) ⁱⁿ isotopic space. For coordinates and other vectors in regular space ^{we use} only ^{their Cartesian} components xi, ~/~xi, ^etc...

[3] YANG, ^C. N., MILLS, ^R. L., Phys. ^{Rev. 96} (1954) ^191.

[4] ^We present ^{here a} continuqus ^{version of the} theory ^{with LEI.}

It can easily ^be generalized ^{for the case of} spins ^{on a}

discrete lattice (see e.g. survey of KADANOFF, L., Rev.

Mod. Phys. ⁴⁹ (1977) 267).

[5] ^{This is} generated by localization on a lattice of group Z2,

the discrete Abelian _group of two elements (see ^{also the}

survey by Kadanoff mentioned above).

[6] MATTIS, D. C., Phys. ^{Lett. A 56} (1976) ^421.

[7] VILLAIN, J., ^J. Phys. ^{C 10} (1977) ^1717.

[8] TOULOUSE, G., Commun. Phys. ² (1977) ^115.

[9] ^{Inside a} frustration network, between its singular planes, ^one certainly ^{could come} across some local impurity ; ^islands with rapidly changing orientations of S. Their existence, however, ^{will still not} affect the averaged picture.

[10] ^A good survey of the current situation with disclinations in the theory ^of elasticity ^{has been written} by ^{de Wit} (see

R. DE WIT Linear Theory of ^Static Dislocations ed. by Simmons, ^Y. A., de Witt, R., Bullough, ^R. (Nat. ^Bur.

Stand. U. S.) ^{Vol. 1} (1970) p. 651).

[11] ^{We restrict ourselves to} linearized version of the theory.

700

[12] ^This picture ^{of a} spin glass (in ^{the absence of} disclination)

was introduced by Halperin and Saslow (see ^HALPE-

RIN, B. J., SASLOW, ^W. W., Hydrodynamic theory of spin ^{waves in} spin glasses ^and other systems with ^non- collinear spin orientation preprint (1977)).

[13] Generalization for the case of non-zero magnetic ^{field is} fairly simple (see ^{ANDREEV, A. F., Zh. E.T.F. 74} (1978) 785).

[14] ^{This does not mean the} absence of disclinations on a micro- scopical level.

[15] ^{EDWARDS, S.} F., ANDERSON, ^P. W., J. Phys. ^{F 7} (1975) ^965.

SHERRINGTON, D., KIRCKPATRICK, S., Phys. Rev. Lett. 35

(1975) ^1792.

THOULESS, ^D. J., ANDERSON, ^P. W., PALMER, ^R. G., ^Philos.

Mag. 35 (1977) ^593.

[16] VANNIMENUS, J., TOULOUSE, G., ^J. Phys. ^{C 10} (1977) ^537.

Actually they ^{have studied} only ^{the 2D} Ising ^model.

However it seems feasible that a numerical experiment

would provide similar results for the 3D Heisenberg ^model.

[17] ^{We have} given ^a simplified form of the Lagrangian. To genera- lize it in a trivial _way one might insert different coefficients in terms describing oscillations of _~ parallel

(~~, 03BB~, 03BC ~)

and perpendicular (~~, 03BB~, 03BC~) ^to mo.

[18] Similar considerations of the non-analytic character of _q-

dependence of the fluctuation _energy has been made by

D. Khmelnitskii (private communication).

[19] ^The Fourier-component ^{of the RKKY} potential ^has only Kohn-singularity at q ⁼ 2 pF :

J(q)RKKY ~ (q - 2 pF) In | q - 2 pF| ^.

[20] WALKER, ^L. R., WALSTEDT, ^R. E., Phys. Rev. Lett. 38 (1977)

514.

[21] BEREZINSKII, V. L., Sov. Phys., ^{JETP 32} (1971) 493.

[22] ^{It still} remains unclear whether it is possible ^{in our model with}

substantial long-range interaction to introduce any ^reaso- nable concept ^of upper marginal dimensionality. ^{If one tries}

to approach ^this problem formally ^and apply the criterion of marginality directly ^to the correlation function (37),

one then obtains d ⁼ 6 in accordance with the results of HARRIS et al. (HARRIS, ^A. B., LUBENSKY, T. I., CHEN,

J. H., Phys. Rev. Lett. 35 (1976) 1792).

[23] ^{MIGDAL, A.} A., ^{Zh. ETF69} (1975) ^1455.

[24] See e.g. DZYALOSHINSKII, I. E., LIFSHITZ, ^E. M., PITAEVSKII, ^L.

P., Adv. Phys. 10 (1961) 165 ;

Cf. also DZYALOSHINSKII, I. E., DMITRIEV, S. G., KATS, E. I., Sov. Phys. ^{JETP 41} (1976) ^1167.