THE THREE-DIMENSIONAL ROTATOR SPIN GLASS MODEL

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THE THREE-DIMENSIONAL ROTATOR SPIN

GLASS MODEL

A. Holz

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-888

THE THREE-DIMENSIONAL ROTATOR SPIN GLASS MODEL

(!S)

A. Holz

Institut fur Theoretisehe . Physik, Freie Vniversitat Berlin, 1 Berlin 33, Germany

Résumé.- On donne une décomposition approximative des fonctions de corrélation dans la phase verre de spin (SG) du modèle rotateur en 3 dimensions (3D) avec des interactions aléatoires J^i = ± J, qui démontre l'effet des excitations thermiques sur les configurations de spin gelées.

Abstract.- An approximative decomposition of the correlation functions in the spin glass (SG) phase of the 3 dimensional (3D) rotator model with random couplings J.y = ± J is given showing the effect of the thermal excitations on the frozen-in spin configuration.

In the following we will study the 3D rotator model with random couplings J.. = ± J. This problem has first been studied by Villain /l/ and its con-nection with the notion of frustration as explained by Toulouse 12/ and Toulouse et al. /3/ and applied to the 2D- and 3D Ising model by Kirkpatrick / 4 / has been established. The main emphasis of this contribution will be on the process of thermal un-locking /3/ of a supposedly frozen-in spin configu-ration.

We consider the isotropic Hamiltonian

H = - I J., cos (<J>.- <)> ) + J I , (1)

<i,j>

J J

<i,j>

where Sr = (cos cj> , sin <j> ) , and summation is over nearest neighbors only. If p is the fraction of antiferromagnetic (AF) bonds in a ferromagnetic matrix then it is sufficient to study the interval 0 < p £ 1/2 as follows from symmetry. The phase space of the problem can be divided into sectors given by the metastable states of Eq. (1). In an "harmonic" approximation one obtains /5/

H ( { n . . , m. . , N . . } ) ^ J Y c q2\li lb + E '

ii ij ij L • q - q

. 2T T T d r . . d r . ( 1 - C O S q . ( r . - r . ) ) N .L. i j L L Jr c q2

X-3

5 ^ °j (2)

Here N.. = 0(1) if the bond ij has a ferro (anti-ferro) magnetic coupling, m.. = 0(1,-1) if the two spins coupled by the AF bond ij are in ferro (AF)

(*)

Work supported by Deutsche Forschungsgemeinschaft under SFB 161

magnetic state, n.. = 0, ± 1, ± 2 Furthermore

a. and C. are defined over the abbreviation J J

Hi-

d

-

H

H°U

(,

"

2

"

(

^

})(w

^>

)

J

cj

I F ,

( 3 ) + I \ 0? / d r . J

where a given bond configuration {N.,} is expressed by {£.., C..} . Here C, is the frustrated loop

bound-1 bound-1 F l

ing the surface £. located on the dual lattice. The

1 F

surface normal fo J, points along the direction of the AF bonds it cuts through and •which belong to nonfrustrated plaquettes / 2 / . Similarly {£^° ,C.°} represent the ordinary vortex loops of the rotator model and its respective surfaces. The strength and

F orientation of the loops is characterized by 0. = 0, ± 1, and a.°= 0, ± 1,±2 Furthermore N.(Z.° ,{££>) = 1 if l.° is tangential to one E? , where i = i(N.), out of the set {£«}at r., other-wise it vanishes.

The diagonalization leading to Eq. (2) has been done approximately for configurations with c < 1, where (z : coordination number)

c

= ' -hi J.

N

ij <H»ijl> w

E' = - 1 2 T T2J T N . . ( l - | m . . | ) n ? . +

+ 23 I N. .(1-lm. .|)+8ir

2

J 7 N. . x

<ifj> >•' ^ < i , j > ^

(H-Dn..

(nij +

-f-N..)

(3)

In the following c ". 1 will be assumed implying The two-point correlation function of the

E' % 0 and the spin configuration with Cf? = 0 will problem is given by (T and Conf stand for thermal be ignored. This can be expected to be a good appro- and configurational average)

ximation for the ground state configurations and its low energy thermal excitations in the presence

F of an infinitely extended Ci

.

Here

F

where

k0

and

Q.

are arbitrary surfaces bounded by

j J

the ordinary and frustrated loop, and as follows from the form of Eq. (2)

Summation is performed over compatible with a given set

-

F

all {CiIc which are degeneracy is obtained from the invariance of Eq. Lo

{ C :

)

,

i.e., also closed (2) with respect to {oi

,ISTI

+

1-02

(-0;) and

Cf surfaces without frustrated loop have to be in- corresponds to a local reflection of the spin con- cluded and no tangential contact of two different figuration on u

.

If in addition an infinite number

F

cR

-

C, is allowed. Furthermore

I',

is the volume enclosed of degenerate states is present then the SG phase F

by

i2.

+

cF

and the sign of

:F

is irrelevant. Eq.(8) is stable only if by no finite thermal excitation

I j J

has some features in common with the correlation the system can move into other ground states be- function of an Ising type model for a "temperature"

B

= ln((1-p)/p) and it measures essentially the P

disorder imposed onto the system by-frustration. In the SG regime, i.e. Bp<

BpC

,

and a rough estimate

1

gives

-

h pc 4

10 h

,

one expects

The second and third term of Eq. (6) will be discussed in the light of the degeneracy of the ground state of a SG. The Goldstone modes described by the first term of Eq. (2) are a manifestation of the rotational degeneracy of the ground state with respect to a direction u in the plane.

_-

A

further

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F

where X(!,~',{C~})

is the thermal susceptibility of

the frozen-in loop configuration appearing in the

second term of

Eq.

(6). The SG transition occurs at

BSG

where

x

diverges such that the second term of

Eq.

(10) which is proportional to < < , s ~ > { > ~ ~ ~ ~

for

r

=

0 vanishes. Assuming for

6+BSG

References

/ I /

Villain, J.,

J.

Phys. C.

:

Solid State Physics

10 (1977) 4793

;

ibid

11 (1978) 745

-

/2/ Toulouse, G., C o m n . Phys.

2 (1977) 115

/3/