SUPER-EFFECTIVE FIELD THEORY AND EXOTIC PHASE TRANSITIONS IN SPIN SYSTEMS

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SUPER-EFFECTIVE FIELD THEORY AND EXOTIC

PHASE TRANSITIONS IN SPIN SYSTEMS

M. Suzuki

To cite this version:

M. Suzuki.

SUPER-EFFECTIVE FIELD THEORY AND EXOTIC PHASE TRANSITIONS

Colloque C8, Suppl6ment au no 12, Tome 49, dkcembre 1988

SUPER-EFFECTWE FIELD THEORY AND EXOTIC PHASE TRANSITIONS IN

SPIN SYSTEMS

M. Suzuki

Department of Physics, Faculty of Science, University of Tokyo, Hongo, Bmkyo-hu, Tokyo 113, Japan Abstract. - The ordinary mean-field approximation is extremely extended so that even exotic phase transitions such as spin glasses and chiral orders may be treated analytically. This new scheme is very powerful even in estimating true fractional critical exponents, if we combine this with the coherent-anomaly method (CAM).

Introduction

Recently the present author [l] proposed a new scheme, so-called coherent-anomaly method (CAM) to study the non-classical critical behavior of the relevant system, using generalized meanfield approximations. The keypoint of the CAM theory is t o note that the coefficient Q (T,) in

diverges systematically or coherently as the degree of approximation increases, namely that

where T: denotes the true critical point of the system. When the physical quantity Q (T) is expected to have the following power-law singularity

near the ture critical point

T:,

we may assume that

in the "critical region of approximations"

,

namely for 6 (T,)

-

(Tc

-

T:)

/

Tc*

<<

1. According to the general theory of CAM, we have [I, 21 the following coherent anomaly relation

cp=1+?1. ( 5 )

Thus, the fractional critical exponent cp is estimated by studying the coherent anomaly in generalized mean- field approximations. Many applications [l t o 151 of the CAM have already been reported to find that the CAM is very useful and that the convergence of it is rapid.

The coherent-anomaly exponent

+

can be estimated approximately even from two mean-field-type approx- imations when the exact critical point T: is known,

namely , .

j = 2, and ~ , ( j ) denotes the approximate critical point of the j-th mean-field approximation. When T,* is un- known, we need, at least, three mean-field-type a p proximations to estimate both Tc*and

+

may be esti- mated by solving the following transcendental equa- tion

(TJ~)

-

T,*) (T,(')

-

T:)'

-

(T,(~)

-

T;*)'+' = 0 (7) with ,u = log (XI

/

x.2)

/

log ( ~ 2

/

XS)

.

Then,

+

can be estimated again from (6) with the solution Tc* of

(7). More approximations with smaller values of S (T,) yield [3-101 better estimates of $.

The discovery of the above CAM has encouraged the present author t o try t o find generalized mean- field effective-field approximations in which even exotic phase transitions can be treated. The main purpose of this paper is to present such a generalization, namely so-called the super-effective field theory [16-191.

1. Basic idea of the super-effective-field the- ory

The ordinary effective-field theory is, in most cases, constructed by decoupling the original Hamiltonian, as in Weiss' theory on ferromagnetism and in the BCS theory of superconductivity. Thus, exotic phase tran- sitions such as chiral orders [20-28, 171 and spin glasses I29-42, 81 can not be treated in the ordinary mean-field theory, because effective Hamiltonians to describe such exotic phase transitions are not derivered from decou- pling the relevant original Hamiltonians. One of the essential points in our new theory [16-191 is to intro- duce a generalized effective field which is independent of decoupling. For this purpose, we consider a finite cluster as shown in figure 1. Each domain (say Dj) in figure 1 is designed to be the support of an operator, say Qj, which is a possible order parameter defined in the domain D j

.

The partial Harniltonian correspond- ing t o this generalized cluster is denoted by H,1 and we consider the following generalized effective-cluster

(') Hamiltonian

I ? = & -

hQk

Here X j =

R

( ~ 2 ) )

and Sj = 6

( ~ 2 ) )

for j = 1 and _{k e n} (8)

JOURNAL DE PHYSIQUE

Fig. 1. - A general super-effective cluster [17]. Each super- effective-operator Qj is defined in each domain Dj.

with the super-effective field A k conjugate to a pos-

sible order parameter Qk or ~ k

e

E O ~ Q ~ , where ~ i j

denotes the "modular factor" to take into account the (hidden) symmetry of the system. The symbol 80 de- notes the boundary domain of the cluster 0 , namely 80 = Dl

+

Dz

+

-

.

.

+

D,. The super-effective fields

{ A $ ) are determined by the self-consistency condition

that

(Qo) = ( ~ k ) (9)

for all k. As far as the linear terms in { A k ) are

concerned, the above equations (9) are expressed by Kubo's canonical correlation functions (Qj; Q k ) c l de-

fined by

with

~ t= )eiz?.Icl/EQke-izxcl/fi

(11) The possible global order parameter of the system is then given by

The corresponding critical point

Tc

is given 1171 by the zero of the determinant of the following homogeneous equation

for 1 = 1,2,

...,

z, and for Xk = P A k .

2. Decoupled density-matrix formalism of t h e

super-effective-field t h e o r y

It is more convenient t o formulate the super- effective-field theory in terms of direct correla- tions {(QjQk)) instead of the canonical correlations {(Qj; Q k ) )

.

For this purpose, we introduce the follow-

ing decoupled effective density-matrix

instead of the effective Harniltonian (8), where we have

and

Then, we find easily [17] that the 'Lcritical point equation" (13) is replaced by

for

t

= l , 2 ,

...,

z, where ({A, B)) denotes the sym-

1

metrized direct correlation

-

(AB $. BA)

.

2

3. Response functions i n t h e super-effective- field approximations

In order t o combine the present super-effective-field theory with the CAM theory [I-151, we formulate the response function of the order parameter Q to an ex-

ternal field A as

where

( Q ) ~ = ~r Q e-pfi

/

D e-Pfi ₍₁₈₎ and

f i = f i - ~ Q .

(19)

In the present super-effective-field theory, we obtain easily the following expression

where { x k ) is the solution of the foll~ming linear equa-

tion

in the super-effective Hamiltonian formalism. -411 the above canonical correlations should be replaced by symmetrized direct correlations, in the decoupled- density-matrix formalism.

4. T h e super-effective-field theory of spin glasses

effective-field h, between the relevant two clusters a negative divergence [32, 331 of the nonlinear suscep (0, Q) at the boundaries

an

of these clusters. Then, tibility ~2 defined in

the super-effective Hamiltonian of this cluster is given

by ~ = x ~ H + x ~ H ~ + . . . (29)

= H!:'

+

-

As S ~ S : , (22) for the magnetization m in the presence of a uniform

i ~ a n _{magnetic field} H . It is well known that the nonlinear

where { S j ) and S j denote the spin variables of the

_{

'1

two real-replica systems and ' d 2 ) , respectively. It should be noted that the super-effective field in (22) is applied only at the boundaries 8 0 , in a big contrast to an auxiliary field of the same form already used in the previous papers [37-391. Furthermore the present super-effective-field term in (22) plays a role of a mean- field in each preferable direction at each site. The spin Si at the site j feels a super-effective field ASSj = Hi and conversely the sl;in S; feels a super-effective field H; = ASSj. Thus, the systems ( { S i ) ) and ( I S j ) ) play a mirror image to each other, and consequently the super-effective fields { H i ) and Hi

{

'1

are determined by the so-called bootstrap mechanism. The order parameter of spin glasses is defined by

Q = C Q ~ m d Q,=s,s; (23)

i in the present formulation.

The self-consistency equation for this system is given by

((Qo))J =

( ( Q ~ ) ) J

1 P4) namely

( ( ~ 0 s ; )) = (

(sis;

)) J , (25)

where j E Sf2 and ( . . . ) , denotes the average over the random distribution of the interaction J. The equation to determine the spin-glass transition point Tsg is given by

corresponding to the general expression (13). More explicitly we obtain

-

susceptibility ~2 is related to the spin glass suscep-

tibility Xs, defined by the response of the spin-glass order parameter Q = S ~ S ; with respect to a con- jugate field A appearing in the form -AQ in the total Hamiltonian. More explicitly we have X 2 = -/3xSg,

and Xs, =

flp;xq,

where XQ is given by (20). Thus: we arrive [17] at

when all the boundary sites are equivalent to each other. Here we have

When the boundary sites are not necessarily equiv- alent to each other, we have

and { x k ) is the solution of the following linear equation

4.2 APPLICATION TO THE * J M O D E L IN THREE DIMEN- SIONS. - The Weiss mean-field critical data are given [8, 16, 17, 301 by

x!r)

( T ) z

L p ( ~ ) ;

p ( W ) = 0.08333

($)

(34)

E

for j E 8 0 . This determines Tsg in our super-effective-

with kBT:(,W) = 2.4495 J , and E = (T

-

T,(,W))

/TJ(,W).

field theory, when all the boundary sites are equivalent

to each other and consequently the super-effective field The Bethe approximation shown in figure 2b yields

is unique. Otherwise we have [8, 16, 17, 331

x s g = ( P C L ~ B ) ~

-

('

-

( ( s O s i ) : l ) J for l = 1, 2,

...,

z , corresponding to the general formula 2 2 1

+

tanh2 K

JOURNAL DE PHYSIQUE

a

b

c

Fig. 2.

-

A systematic series of clusters with a super-effective-field Hea, where a, b and c corresl>ond to the Weiss, Bethe and present approximations, respectively.

for z = 6, where

K

= J / kBT. Thus, we obtain the following critical data

xsg

(T) _E ;

g ( B )

= 0.16141

(g)

(36) with

ICBT,(,B)

= 2.0780 J , and E = (T

-

T$))

/

T::).

Next we consider the cluster shown in figure 2c. From the general formula (30), we obtain

with z = 6. Here t = tanh K and

2

2t4 2 (t2

-

ts)

--

2t2

'=-+[

(1

+

t4I2 i t *

]

I+.' (38) Thus we finally arrive at

(S)

/

T(S)

with ~BT::) = 2.0640 J, and E = (T - Tsg

)

_sg

.

These three "canonical approximations" yield a re- markable coherent anomaly shown in figure 3. This gives the estimates of T& and y, defined in

xsg

(T)

-

(T

-

T,",) = -,, a6

T , * g ~ 1 . 2 J / k ~ and y , ~ 2 . 9 . (40) These agree very well with other estimates [40 to 421.

By the way, we obtain one more critical datum from the square cactus approximation by Katsura and Fujiki

[43, 441 as

with ~BT;,"") = 2.03304 J and t. =

(T

-

T;,'))

/

~2').

This critical datum is rather close to the criti- cal data for the Bethe and "square-Bethe" approxima-

Fig. 3.

-

Coherent anomaly of the f J Ising spin glass.

Here the T,

-

Xj1'19

plot is given; a, lj and c correspond to the clusters a, b and c in figure 2, respectively. This yields T,* ? 1.2 J

/

k~ and 7, N _ 2.9.

tions, (36) and (39), but this result (41) is omitted in figure 3, because the square-cactus-tree approximation may be another "canonical series of approximations" different from the present one shown in figure 3. 5. Super-effective-field theory of chiral orders

.4s a typical application of the super-effective-field theory, we study here the chiral order [20-281, partic- ularly in the antiferromagnetic quantum X Y model

on the two-dimensional triangular lattice. Here { u j )

denote Pauli operators.

where E U ~ = *l (modular factor) and Qijk denotes the helicity or chirality of the triangular unit cell (ijlc)

.

Following Fujiki and Betts [26, 271, we consider here a

chirality operator of the form

Q . . 2 j k - (

-

( T z . X Uj

+

U j X Ulc

+

Ck X 0i)'

/

2&. ₍₄₄₎

In order to construct a super-effective-field theory of this chiral order, we have t o consider, at least, a 6-spin cluster shown in figure 4.

Fig. 4. - A super-effective cluster [17].

According to the general formula (16) based on the decoupled-density-matrix formalism, the transi- tion point Tc is determined by the solution of the equa- tion

3

C

(({Qo,

Q ~ } ) ~ ~

-

({Ql,

Q ~ } ) ~ ~ )

= 0 (45) k=l

1

with the direct correlation ({A, B}) =

-

(AB

+

BA)

.

2

From the symmetry of the cluster, the equation (45) is rewritten as

As it is still complicated to calculate the correlation ({Q1,Q2)) in the 6-spin cluster, we make the following simple Kirkwood approximation

Then, equation (45) is simplified as

using another approximation (Q:) II(Q:)

.

This gives [17] the critical point Tc = 4.1 J

/

k ~ . The susceptibil- ity XQ (T) is given by

It will be possible t o apply the CAM theory t o the present problem and to estimate T: more accurately, if we make super-effective-field approximations for larger clusters. This study will be reported elsewhere.

6. Summary and discussion

In the present report, we have explained the basic idea of the super-effective-field theory and its appli- cations t o spin glasses and chiral orders. We have thus demonstrated how t o construct such a general- ized meanfield theory as can be applied even to exotic phase transitions. Furthermore we have shown that it is possible to study the true criticality of such exotic phase transitions using the generalized effective-field theory.

Dynamical problems can also be studied using the super-effective-field theory. Thus, the combination of the CAM and the super-effective-field theory will be very useful as a unified theory of phase transitions.

Acknowledgments

The present author would like to thank N. Kawashima, Y. Kinoshita and N. Hatano for collab- oration on the present problem.

This study is partially financed by the Research Fund of the Ministry of Education, Science and Cul- ture, and also by Central Research Laboratory, Hi- tachi, Ltd.

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