Frustration in highly random magnetic systems

2145

Physica 109 & 1lOB (1982) 2145-2147 North-Holland Publishing Company

FRUSTRATION R.N.

IN HIGHLY

RANDOM

MAGNETIC

SYSTEMS

BHA-IT

Bell Laboratories, Murray Hill, N.J. 07974, USA A numerical method is used to investigate the scaling behavior of positionally random king and quantum Heisenberg spin l/2 systems in two and three dimensions with short range antiferromagnetic interactions. A study of the distribution of couplings, as high-lying states of the system are eliminated, exhibits qualitatively different behavior in the two cases. This bears upon the possibility of frustration and spin-glass freezing in these systems at low temperature.

the

1. Introduction

scene.

range Recent

experiments

semiconductors, romagnetic failed

a positionally

3d Heisenberg

to detect

tibility related

[l) on insulating

n-doped

random

any cusp in the magnetic

(spin-glass magnetic

antifer-

spin l/2 system,

have

suscep-

behavior) or other exchangeordering. This is despite going

down to very low temperatures (-5 mK), much below the median nearest neighbour exchange or the bare

exchange

percolation

naively some kind of expected [2] to develop results

can be explained

basis of numerical how spin

the

l/2 system

conventional cluster

in three

on the

[3], which show

dimensions which

have been

below down

the

median

to where

spins have been frozen the Curie value) [3].

orders

of

near-neighbour

almost

99%

(susceptibility

of the = 1% of

2. Scaling procedure In order

to confirm

the importance

(the large

condensation

tum

effects

inert

character

of singlets)

of quanenergy

in the highly

and

random

netic

from

do not allow so successful quantum

Heisenberg spin and the high dis-

order - i.e., near-neighbour exchange couplings differing by orders of magnitude, pairs (or clusters) of strongly coupled spins have a large condensation energy. Thus, as the temperature is lowered, the strongly coupled parts of the system condense into inert singlets, with a very small polarizability, and effectively “disappear” from 0378-4363/82/0000-0000/$02.75

magnitude coupling,

for temperatures

differs

in fitting the experimental data [l, 21. The crucial point is the extreme nature of the antiferromagnetic l/2 system. Because of this

paramagnetic

of long-

and the system

system, similar scaling calculations have been performed for the corresponding antiferromag-

[4] and also why simple

[l, 2,5]

remains

generation

or correlations

Heisenberg

quantum

spin glasses freezout

successfully

studies

random

calculations

for a spin-glass

quite

suppresses

where

“frustration” may be in the system. These

scaling

highly

threshold,

This

interactions

@ 1982 North-Holland

Ising

The starting positioned

system

in two and three

dimensions.

point is a system of up to 10 000 spins randomly

in a d-dimensional

cube

(not on lattice sites, but a continuum, like the Heisenberg system appropriate for doped semiconductors) with periodic boundary conditions, with a Hamiltonian:

H =

C

J(r;,)SiSj

I-Cj

where with

romagnetic moments.

p;S;

1

I

Si take J(r)

+h C

on the values

= Jo exp(-2rla)

t1/2,

and to start

is uniformly

antifer-

(J > 0) and ,u, = 1 are the magnetic The system is thus characterized by a

2146

R.N.

single

dimensionless

c = nad, where

bonds

up to a fixed,

Typically is analogous

Heisenberg

system,

high-lying sidering

excited

pair;

Ising

remaining

represented magnetic

two

by a single moment (where

the couplings) (Jjc- J,(). This arbitrary couplings within

by con-

means of the

can

then

be

Sk = l/2

with

the

couplings

of the

e of i and j for the Ising

the eigenvalues

is easily procedure

system

this

configurations spin

where

of the

pair of spins (a j)

states

pk = 0. The

new spin k to neighbours Hamiltonian

removal

case),

are linear

shown to in general

in

be Jkl = generates

magnetic moments and sign of the Jkp; however, the Hamiltonian remains

the Ising spin

l/2 subspace

x exp[-fdc h^($)],

fol-

quantum

at each stage. For the initial

of the two parallel the

the

(eq. (1)). The

generalization of the above formulae arbitrary decimation in the procedure

systems

P”

are

procedure

for

magnetic

up to 50

of the system

coupled

antiferromagnetic

removal

keeping

scaling that

states

of spins.

low, threshold

i.e. systematic

the strongest

in the system (the

to

in highly random

parameter

density

this means

for each spin. The

lowed

I Frustration

concentration

n is the actual

All couplings retained.

Bhatt

for an followed

is:

c = nad, and fd is a numerical

depending

on the dimensionality

(f, = I, fi = ~14, f3 = 7r/6). As the system is scaled, the maximum coupling is reduced in magnitude, and the distribution of nearest “nearest

neighbour neighbour”

couplings coupling

given stage of the scaling the width

largest

coupling

of the

scale alluded

=

Pi -

sign(A,)pj.

Jkp = J;, - sign(Jij))J,r, where

J >O

(ferromagnetic)

(CO) refers

to

procedure

distribution

as

spin).

the

logarithmic

(on

to), W, is a measure

The

of the disorder

is given by Wo = [r(l + 2/d) W” P(1 + l/d)]“2/(fcd)1’d. Fig. l(a) shows the variation of W as a function of the maximum coupsee ref. 3) J,,,,,, for the Ising

ling (temperature,

as well as the quantum

Heisenberg

as the system

The distribution

is scaled.

dramatically

(2b)

quantum practically

Heisenberg invariant.

is a direct

consequence

coupling.

is defined

that

(and frustration) in the system - the larger the W, the more disordered the system, and less likely the spins are to be frustrated. The initial width

(24

antiferromagnetic

also changes (the of any spin at any

involving

lings for the Ising system E*k

constant

of the system

with

the latter-the

system

in 2d

of coup-

can be seen to narrow

scaling,

in

contrast

to

the

system, where it remains This difference in behavior

freezing

of the quantum of pairs

nature

or clusters

of into

singlets effectively removes the coupling between them and the remaining spins even though the

3. Results and discussion

bare interaction may be large compared to the interactions among the remaining spins. The Ising clusters, on the other hand (though they may

The initial system for values of c appropriate for doped semiconductors has a very wide distribution of near neighbour exchange couplings, extending over several orders of magnitude in J [l-3]. Consequently, it is convenient to look at the distribution of the logarithm of the nearestneighbour coupling P(ln Jo/J) which, for the initial system, is given by

themselves be predominantly magnetically inert (j..~= 0) for the antiferromagnetic case), have a degenerate ground state (corresponding to flipping of all spins in the cluster). This enables them to serve as a communicating medium for interactions among the more weakly coupled spins. Thus, the Ising system does not have the “exchange dilution” property [2] of the quantum Heisenberg system, and is therefore unable to

R.N.

1

(0)

10

2 D!MENSIONS

Rhatt

I Frustration

‘2’.

l

random

Y

0.0

map&

7147

systems

10-l -

(C=OOB) .

?

in highly

~POQ

fi

c!*o.o*&V aQP

1;

‘I0

&Q 1o-2 A.

05

- ISING

O.-QUANTUM HEISENBERG

10-3 I

0 p

IO.5

I

1o-4

10-J

I

I

lo-2

10-l

1

lo-

lb)

3 DIMENSIONS

(C=OO41

0 ? ;:

D

\ \ \

,E 10-4 -

a

a

\

1

\ \

.

.

--

3d ISING ---

-__

\

‘\

n a -ISING

‘\

10-5 -

0 -QUANTUM HEISENBERG

‘\
HEISENBERG \ ‘,

o10-4

‘\. 40-e 0 10-3

10-z

10-l

I

1

10

20

1

-. -

J

30

ff’

Jmax’Jo

Fig. I. Logarithmic width of the distribution versus maximum coupling (temperature) for Heisenberg and king systems in 2d and 3d.

of couplings the quantum

Fig. 2. Maximum coupling as a function tion of the spins left for the two systems

of the inverse in 3d.

frac-

, escape scaled, to.

the development of frustration like its Heisenberg counterpart

This

evident

difference

Ising

explicitly

scaling

behavior

is

of frustration

in

hospitality where

for development

system

in 3d is demonstrated

in fig.

2 which

plots

the

this

the was

Aspen

Center

for

Physics,

written,

are

gratefully

ac-

knowledged.

more maximum

coupling in the scaled system versus f-’ (f is the fraction of original spins left). For the Ising system, the coupling strength no such behavior is evident

tends to “hang up”; for Heisenberg spins.

Acknowledgements

References 111 K. Andres,

R.N. Bhatt, P. Goalwin, T.M. Rice and R.E. Walstedt, Phys. Rev. B24 (1981) 244. 121 R.B. Kummer, R.E. Walstedt, S. Geschwind, V. Narayanamurti and G. Devlin, Phys. Rev. Lett. 40 (1978) 1098; J. Appl. Phys. 50 (1979) 1700. [31 R.N. Bhatt and P.A. Lee, J. Appl. Phys. 52 (IYXI) 1703: Phys. Rev. Lett. 48 (1982) 344. [41 V. Cannella PI R.N.

Helpful

of

also in 3d (fig. l(b)).

The potential the

in the

as it is appears

discussions

with

P.A.

Lee

and

the

Bhatt

and J.A. Mydosh, Phys. Rev. B6 (1972) 4220. and T.M. Rice, Phil. Mag. B42 (1980) 859.