Higher-order correlation functions of the planar Ising model II

Physica 93A (1978) 354-384 © North-Holland Publishing Co.

H I G H E R - O R D E R C O R R E L A T I O N F U N C T I O N S OF T H E PLANAR ISING M O D E L II R.Z. BARIEV

Physico-Technical Institute, Academy of Sciences of the USSR, Kazan 420029, USSR

Received 24 January 1978

We compute exactly the many-point correlation functions formed by arbitrary number of spins, disorder variables, fermion operators, energy-density operators and components of the stress tensor for the planar Ising model in the absence of a magnetic field for T < Tc and T > To. It is shown that these correlation functions near the critical point have a scaling form. The scaling functions have been obtained as an expansion suitable for studying large distances between points. The asymptotic behaviour of the scaling correlation functions for distances R ,> ¢: (where ~: is the correlation radius) is determined.

I. Introduction

In a previous paper 1) (which we shall further indicate as I* we have proposed a method of calculating the a s y m p t o t i c behaviour of the higher-order correlation functions for the planar Ising model 2'3) in the absence of a magnetic field. According to this method the many-point correlation function defined according to (1.2.1) can be presented as an expansion for large distances. We have c o m p u t e d the leading terms for the three-point correlation functions f o r m e d by spin operators and energy-density operators. In the present paper the treatment of I will be generalized in two ways. Firstly we shall examine an arbitrary term of the expansion of the correlation function. Secondly we shall consider the correlation function f o r m e d an arbitrary number of basis operators of the Kadanoff 4) algebra. More precisely, we shall suppose that Oi in eq. (1.2.1) is one of the following o p e r a t o r s t : spin operator 1)

o-i = o'm,,,,~,

(1.1)

* Equations from this paper will be preceded by a prefix numeral I. t The operators (!.3)-(1.7) are linear combinations of those used by Kadanoff4). 354

CORRELATIONS OF THE PLANAR ISING MODEL II

355

disorder variable N

2)

~ =/~,..., = I ~ exp(-2K~o'm,-i,~o',~.~),

(1.2)

~=n i

fermion operators 3)

a[=

a +,~i,~ = /~ m m ° ' , . : r i ,

(1.3)

4)

ia~- = ia?~,,.~ = ~,.,.~o'm:~,

(1.4)

energy-density operator

5)

~'i= o'm~,.:,.::i- (~rm.o',.,.+O,

(1.5)

c o m p o n e n t s of the stress t e n s o r 6)

t ~1 = t '~:i N = sh 2K2[o'm~-l.n:rm;.n~- (O'm-i.nO'm.)] -

7)

t~ 2 =

¢r.,:,-lcr,.,. i + (o',...-lo'.,.),

(1.6)

t ,12. : ; __ - sh 2Kz[cr,.i_l,.,o',.:i- (O'm-i.nO',nn)] --O'mi+nilJrmi,ni+l

"t- ( O ' m , n O " . . . .

I)"

(1.7)

In the p r e s e n t p a p e r we shall not consider correlation f u n c t i o n s containing gradients of the o p e r a t o r s (1,1)-(1.7). T h e s e correlation functions m a y be o b t a i n e d by differentiation of correlation functions which do not contain the gradients. It is o u r aim in the p r e s e n t p a p e r to c o m p u t e s-point correlation f u n c t i o n f o r m e d by s~, S 2 . . . . . S7 ( S l + $ 2 + " " " + S7 : S ) o p e r a t o r s or,/~, a +, a - , ~, t n and t n respectively G ':2

.... 7(Rl, R~ . . . . .

R,) =

o'i

#z ~ + i

a,~+l =

$4

S5

$6

$7

x__r!liaT~+li=~7¢,a+;]-]1tsi+i li ~ tsUi 1., ) , _

=

=

(1.8)

w h e r e R~ is the position v e c t o r of the point i with c o o r d i n a t e s (m~, n~) and the p r i m e d n u m b e r s are J s) = ~=i Sk.

(1.9)

T h e correlation function, containing an odd n u m b e r of o p e r a t o r s v a n i s h e s for T > T~ ( T < Tc). T h e r e f o r e , we shall s u p p o s e that 2g = s2+ s3+ s4

isevenfor

2g=s~+s3+s4

is e v e n f o r T > T~.

T < T¢,

cr(tz)

(1.10)

356

R.Z. BARIEV

T h e correlation f u n c t i o n (1.8) was e x a m i n e d p r e v i o u s l y in s o m e special cases. In particular the results for the correlation f u n c t i o n f o r m e d by s spin o p e r a t o r s were r e p o r t e d in refs. 5-7 and for that f o r m e d by the s l spin o p e r a t o r s and the s2 e n e r g y - d e n s i t y o p e r a t o r s in ref. 8. S o m e interesting results for the four-spin correlation f u n c t i o n at distances R <~ ~: were obtained in refs. 9 and 10. F o r earlier r e f e r e n c e s see paper I. L e t us outline the c o n t e n t s of this paper. The main results for the correlation f u n c t i o n (1.8) are given in section 2. Section 3 is d e v o t e d to the m a t h e m a t i c a l f o r m u l a t i o n of the problem. The details of c o m p u t a t i o n are p r e s e n t e d in sections 4 and 5. The rotational invariance of the scaling correlation f u n c t i o n is d i s c u s s e d in section 6. The a s y m p t o t i c b e h a v i o u r of the scaling correlation f u n c t i o n s for distances R >> ~: is d e t e r m i n e d in section 7.

2. S u m m a r y

of results

For the correlation f u n c t i o n (1.8) we have obtained the following results: (a) T < Tc G sis2 . . . . 7(RI, R2 . . . . .

Sl+S~

R,) = ~

<: - Pf(.O(~,,,2 ....... 7)) exp[ T(~,.~2~],

(2.1)

where ~

= ]1 - (sh 2Ki sh

TI,~,,2) =

2K2)-2[ 1/8,

(2.2) (2.3)

r(s,s2)(k), k=2

~ ( , , , s , ...... 7) is the s k e w - s y m m e t r i c 2k' x 2k' matrix with elements

~'~(st,s2)i/= ~ J](s,.s2)ij(k),

i, j = 1 , 2 . . . . .

(2.4)

2k',

k-1

with k' = ½(s2 + s3 + s4) + s5 + s6 + ST:

(2.5) 7r

T(~t,~2)(k) = -

- ~

~ { l l , 12 . . . . .

dO,..,

1k+2=--12,

~k+l =-- ~l"

EI;,+,(#~,) r= I

x H t,#.~t,2(Cbr, 'br÷l)P .... l,

lk+l =-- II,

d~bk

Ik} n-

here

~r

71-

(2.6)

C O R R E L A T I O N S O F T H E P L A N A R ISING M O D E L II

•r

g-l~s~,~2)ii(k)= -

{I 2, l .....

lk}

f

357

~r

ddh..,

-or

f

dd~k

fi

E;g,+,(~b,)

r= 1

-or

k-I

i × I-I Ht,l,+d,+2(~r, ~r+l)Pr, r+l(t[]i< 5~]<,

(2.7)

r=l

where

~= 1)

sh K1 -

sign(m t,Q ch K,g+(ei*')/g_(ei*l),

II=Sl+I, 2)

for i = l ( 1 = 1 , 2 . . . . . s2),

ch Klg+(e i * ' ) - sign(m t,t2) sh Kdg_(ei¢"), I~ = s~ + 1, for i

3)

4)

3)

-~- l (l = 1 , 2 . . . . .

s3),

s3+s4+21-1

I~=s~+l,

fori=sz+

s3+ s4+2ss+2s6+21 (I = 1,2 . . . . . s7),

(1=1,2 .....

ss),

sh Kmg+(e i * ' ) - sign(mt.Q ch Kdg_(ei~'.),

(2.8c)

Im=s~+l,

fori=s2+

s3 + 1 (l = 1, 2 . . . . .

I~=s~+l,

fori=s2+

s3+s4+21

/l=S~+/,

fori=s2+s3+s4+2Ss+21-1

s4),

(l = 1 , 2 . . . . .

ss),

( / = 1 , 2 . . . . . s6),

[ch Klg÷(ei*0 - sign(m qQ sh K1/g_(ei4")][1 - exp(iSt)],

(2.8d)

for i = s2+s3+s4+2ss+21 ( / = 1,2 . . . . .

s6),

[sh K~g+(ei*0- sign(mt~Q ch Kdg_(ei*')][l - exp(-i$1)], for i = s 2 + s 3 + s 4 + 2 s s + 2 s 6 + 2 1 - 1

(2.8e) ( / = 1 , 2 . . . . . s7).

ch KI + sign(mlklk+~) sh Klg_(ei*O/g÷(ei*O,

lk+l=sl+l, 2)

s 2

fori=sz+

ll=s~,+l,

1)

=

(2.8b)

lm=s~+l,

l~= s;+l, 5)

(2.8a)

forj=l

(1= 1,2 . . . . .

(2.9a) S2) ,

sh Klg_(ei*0 + sign(m ikt,+~)ch K1/g+(ei*O,

(2.9b)

lk+a=s~+l,

forj=s2+l(/=

lk+l=s~+l,

forj=s2+s3+s4+21-1

lk+~=s~+l,

f o r j = s : + s 3 + s 4 + 2 s s + 2 s 6 + 2 1 ( l = 1,2 . . . . . ST),

1,2 . . . . . s3), (1= 1,2 . . . . .

ss),

ch Klg_(ei*0 + sign(m Iktk+~)sh Kdg+(ei*O,

lk+~=s~+l, lk+,=S'4+l, lg+l=s'5+l,

forj=s2+s3+l(l=l,2

(2.9c) . . . . . s4),

f o r j = s 2 + s 3 + s 4 + 2 1 ( l = l , 2 . . . . . ss), f o r ] = s 2 + s a + s 4 + 2 s s + 2 1 - - 1 ( 1 = 1 , 2 . . . . . s6),

358

4)

R,Z. BARIEV

[sh K,g_(ei~0 + sign(m ,~,~+,)ch Kdg+(eie'O][l - exp(-i4~k)l, l~+~=s's+l,

5)

fori=sz+s3+sa+2s~+2l(l=l,2

.....

(2.9d) s6),

[ch K~g_(ei~0 + sign(m ~+t) sh K#g+(ei*O][ 1 - exp(i~bk)], lk+~=s~+l,

fori=s2+s3+s~+2ss+2s6+21-1

(2.9e)

( l = 1,2 . . . . .

s7).

In eqs. (2.6)-(2.9) we used the following abbreviations: Eb,+~( & ) = Eu,+~( & ) [ e x p ( i & ) ] a ~ , [ e x p ( - i & )]a,,+,,

(2.10)

E U . ~ ( & ) = e x p [ - y ( d 0 1 m u.~[ - in u,+~&],

(2.11)

ch y(&) = ch 2K~ ch 2 K * - sh 2K~ sh 2K~' cos &;

mjk = m k - - m j # O ,

_/,k= 1,2 . . . . .

nik = n k - - n i ;

sh 2 K ~ = (sh 2K~)-% (2.12) (2.13)

s,

/i t={10 f o r / = s , + l , s , + 2 . . . . . s'4, s ~ + l , s ; + 2 . . . . . s/, f o r / = 1,2 . . . . . s~,s'4+ 1. s ~ + 2 . . . . . s'~.s'6+ 1, s ~ + 2 . . . . .

(2.14) s'

Hu,~J,..(&b &2) = 1 + sign(m i;,,) sign(m t,+l~,,)g_(ei'~Og+(e i~2) x [g+(ei~)g_(ei%] -1. sign(m) =

{+1 1

(2.15)

for m > 0 for m < 0 '

(2.16)

g_+(ei4') = [(1 - al,2 e±i*)/(1 - o~2,~e±i'~)] 112,

(2.17)

al.2 = z~t(1 - z2)/(1 + z2);

(2.18)

z~.2 = th K1.2,

P~.~=[1-exp(-i&~+iq%-rt)]

-~,

r/~+0.

(2.19)

E~,.~,..... ~k~d e n o t e s the s u m m a t i o n o v e r all l,., each index assuming the values 1,2 .....

s;. ( l i # 1~+0.

(b) T > T,. G s,s: . . . . 7(Ri, R~ . . . . .

Rs) = ~

- p f ( t / % ~ . ~:...... 70 exp[

T~,.~2~],

~/R. = [1 - ( s h 2Ki sh 2K2)21I/s,

(2.20)

(2.21) (2.22)

=

q%,.,2 ....... 7~ is the s k e w - s y m m e t r i c 2 k " × 2k" matrix with elements

q/(,~l,s:~ii = "~--1lI)'(sl's")ii(k );

i, j = 1, 2 . . . . .

k" = ½(s~ + s3 + s4) + s~ + s6 + s7,

2k",

(2.23) (2.24)

C O R R E L A T I O N S O F T H E P L A N A R ISING M O D E L II

T~l,~9(k ) = - ( -

(q, 1

i

•,~}

i

dck~..,

--~r

359

k

dqbk ~ Eld,,,(C~r)

--¢¢

x H~,+,,,+~(~b,, d~+,)P ,.,+,,

(2.25)

here lg+~---- 1~, lg+2-~- 12,

~k+l-- ~b~,

(2.26)

(1:

¢r

¢r

/

f

--~r

--or

k-I

where

~= l)

sh K~ + sign(m~lt2) ch K~ g*(ei*9/g*-(ei¢'~), l~=l,

2)

3)

4)

fori=l(l=l,2

. . . . . s0,

[sh K~g*(e i*~) + sign(m z~t2)ch K~/g*__(ei4'l)] e -i•l,

l~=s~+l, l~=s'4+l,

for i = s ~ + l ( / = l , 2 . . . . . s3),

l~= s~+l,

for i = s l + s 3 + s 4 + 2 s s + 2 s 6 + 2 1

fori=sl+s3+s4+21-1

(2.28b)

( l = 1 , 2 . . . . . ss), (l = 1,2 . . . . . ST).

ch K~'g*(e i'¢'~)+ sign(m ~l,) sh K~Ig*_(ei~"),

l~=s~+l,

for i = s l + s 3 + l

ll=s'4+l,

fori=s~+s3+s4+21(/=l,2

l~=s'5+l,

fori=sl+s3+s4+2ss+21-1

( / = 1 , 2 . . . . . s4), (2.28c)

. . . . . ss), ( l = 1 , 2 . . . . . s6),

- [sh K~'g*(e i~') + sign(rn t~2) ch K~/g*(eiC'q][1 - e x p ( - i ~ 0 ] ,

l~=s'5+l, 5)

(2.28a)

fori=s:+s3+s4+2s~+21

[ch K~'g*(ei~9 + sign(m~t2) sh K~/g*(ei'~q][1 - exp(-i~b0],

l~=s;+l,

(2.28d)

/ = 1 , 2 . . . . . s6),

fori=s~+s3+s4+2ss+2s6+21-1(l=l,

(2.28e)

2 . . . . . sT).

~= 1)

ch K~" - sign(m nk~k+~)sh K~g*(ei'~qlg*(eie'q,

lk+l=l,

forj=l(l=l,2

. . . . . sO,

(2.29a)

360

2)

3)

4)

R.Z. BARIEV

[ch K~'g-*(e i**)- sign(m tkl~+,) sh lk+l = S~-'t- I,

for j = s~+ l (l = 1,2 . . . . .

s3),

lk+l = S'4 + l,

forj=st+s3+s4+21-1

(l=l,2

lk+l = S~ + l,

for j = st + s3+ s4+ 2s5+ 2 S 6 + 21 (l = I, 2 . . . . .

sh K ~ g * - ( e i ~ ' q - s i g n ( m ~ + ~ ) c h

.....

lk+l=S;+l,

forj=s~+s3+l(l=l,2

for j = s~+ s3+ s4+21 (l = 1,2 . . . . .

lk+l "= S~5 + 1,

for j

=

s 1+

(2.29b)

ss),

s7),

icbk),

K2*/g+(e*

l~+~ = s~ + 1,

.....

s4), (2.29c)

s_0,

$3+ $4"~-2S5 + 21 -- 1 (1 = 1, 2 . . . . .

S6),

- [ch K~'g*-(e i~) -- sign(m tktk+~)sh k~/g*(ei~'k)][1 - exp(i~bk)], lk+l = S'5 + I,

5)

K~/g*(ei¢'~)] e i ~ ,

forj=sl+s3+s4+2ss+21(l=

1,2 . . . . .

(2.29d) s6),

[sh K~'g*_(e i ~ ) - sign(m tktk+~)ch K~/g*(ei~'q][1 - exp(i~bk)], lk+! = S~ + 1,

for j = s~ + s3+ s4+ 2s~+ 2 S 6 + 21 - 1 (l = 1,2 . . . . .

(2.29e) s7),

with E~,+t(&) = El,#+l(~b)[exp(i~b)]'~'[exp(-i~b)]'~,+~, {~

A~=

for 1 = 1,2 . . . . . s~, s~+ 1, s~+ 2 . . . . . s~ forl=s'4+l,s'4+2 . . . . . s'5, s ~ + l , s ~ + 2 . . . . .

(2.30) (2.31)

s'

H~r+,l,+2(~bl, ~b2) = 1 + sign(m l~+~) sign(rn l,.+llr+z)g*_'(ei4'l)g*+(e i'~2) x [g*(ei~l)g*_(ei~2)] -l,

(2.32)

g*(e i~) = [(1 - a~J e-+i~)/(1 - ot~.] e±i~)] 1/2.

(2.33)

T h e a b o v e results hold f o r all t e m p e r a t u r e s . On the basis of these results we have s h o w n that in the limit Rij = (m2/sh 2 K 2 + h 2 / s h 2 K 0 1 / 2 ~ oo,

(2.34)

sc-l = [2(ch 2K1 ch 2K2 - sh 2Kl - sh 2 K 2 ) ] m ~ 0,

(2.35)

RiiHj are arbitrary fixed n u m b e r s ,

(2.36)

but

the correlation f u n c t i o n can be written in the scaling f o r m , i.e. G '''z .... 7(R1, R2 . . . . . R.) = ~)2~IR.(sCVsh 2K~c) -s' s I

s 2

- -

× fs, s2 .... 7({R#/~}),

(2.37)

where S' = ½(S3 + S4) + S5 + 2(S6 +

$7),

f~,sz .... 7({Rd~}) are the scaling f u n c t i o n s [in w h i c h superscript + ( - )

(2.38) denotes

CORRELATIONS OF THE PLANAR ISING MODEL II

361

T > T¢ (T < T¢)] and the subscript c denotes the value of the function at the critical temperature T¢. Furthermore, we have f o u n d explicitly fsls2 ....

7({Rij/~}) = Pf(ca (s~,,:...... 7)) exp[t s~)],

(2.39)

where co

t(s9 ~=2t(~)(k).

(2.40)

=

oJ(,~.s2...... 7) is the skew-symmetric 2 k ' x 2k' matrix with elements too~)ii = ~__~oJ(~gij(k); i, j = 1, 2 . . . . . 2k',

(2.41)

with

t(,~×k) = -- ~1 ( ~ ) k

f

~

{11,12.....

dUl

. . .

f du, ~' el:,+,(u,)

Ik}

x th ½(u, - u,+l + iOl:,+~l,÷2); l~+l-- ll, 1~+2-- 12, Uk+, = Ul, ca ca (i)k-I ~ f (2~r) ~12,13..... Ik~-~

f

(2.42)

r~_ k

-oo

k-l x ~ th ½(u, - u,+~ + iO~,+~,+2)lt ~lI~<,

(2.43)

where

1)

1,

(2.44a)

2)

sh Kicv-i- sign(m 1,12)ch Kl¢v~,

(2.44b)

3)

ch Kl¢v~- sign(ml,12 sh Kl~v~

(2.44c)

4)

i[ch Kicv~f - sign(m t,t2) sh Klcv~] sh(ul - i01:2),

(2.44d)

5)

- i [ s h Klcv~i - sign(mt~12) ch Klcv~] sh(u~ - iOl,12),

(2.44e)

llJ<= 1)

1,

(2.45a)

2)

sh Ktcv~ + sign(m,~lk+.) ch Kl~v~,

(2.45b)

3)

ch K~¢v~ + sign(mt,lk+~) sh Kacv~,

(2.45c)

362

R.Z. BARIEV i[ch Kl~v~ + sign(mlkjk+~) sh Kl~v~] sh(uk -- i01~+,).

4)

5) - i [ s h K~.t', + sign(m ~ , ) ch KlcV~-] sh(/,k -- i0ld~. ) .

(2.45d) (2.45e)

T h e c o n n e c t i o n s b e t w e e n l~ and i in (2.44) and lk+~ and j in (2.45) are the s a m e as in (2.8) and (2.9) respectively. T h e a b b r e v i a t i o n s are as follows:

e~,~,+,(u,) = ( - I ) a ' / . + , sign(m t,~,+,) exp(-R~,~,+~ ch Ur/~),

(2.46)

V~ = [1 --+sign(m t~t~+,)sh(u~ - i0~it~+)]~/2

(2.47)

0~/,~ = tg-~[n~+l(Sh 2K2)~/~/m ~/~+,(sh 2K~)~/~],

(2.48)

Ot/,+~,+2 is the orientation angle b e t w e e n the v e c t o r s R~,+,~, and Rtr+~,r+,.

[-Tr < 0~,#+~,.~

=

(Rt,+~t,Rt,.t~,+,) < "n'],

A t;,+, is the n u m b e r of spin o p e r a t o r s and d i s o r d e r variables the c o o r d i n a t e s of which (mi, n~) satisfy the conditions

njIm,,~l < nitr.,/[mitr+,l,

sign(mla) = sign(mit,+,).

(2.49)

A b o v e the critical t e m p e r a t u r e we have f o u n d f*+,~2.... 7({Ri/~}) = Pf(Om.~2 ....... 7)) exp[t(s~)],

(2.50)

w h e r e t~s+~ is still given by (2.40) and (2.42) and qJ,,~2 ..... sT~ is a skews y m m e t r i c 2k" x 2k" matrix with e l e m e n t s q/(~)~j = ' ~ q/,~)~i(k);

i, j = 1, 2 . . . . .

2k",

(2.51)

k=l

with

6(,'~,)ij(k) = ~ ~L, (2¢r) (12,t3..... lk~

dut . . .

"

k

duk

~=I

el,#+,(Ur)

I

x 1-I th ½(u, - ur+, + iO~,~r+,~,+2 )1I ~IU>

(2.52)

r=l

1)

1,

(2.53a)

2)

ch Kl¢V~ + sign(mt,~2) sh Kl~v'{,

(2.53b)

3)

sh K t ~ v ~ + s i g n ( m ~ , Q c h

(2.53c)

4)

- i [ s h Kl~v~ + sign(m i~t2)ch Kicv~] sh(ul - i01~12),

(2.53d)

5)

i[ch KI~v~( + sign(m~,~2) sh Kl~v~] sh(ul - i01~12),

(2.53e)

l~

1)

I.

K~vT,

=

(2.54a)

CORRELATIONS OF THE PLANAR ISING MODEL II

363

2)

ch Kl¢v~- sign(mt~l~+,) sh Klcvi,

(2.54b)

3)

sh Kl~v~- sign(ml~tk~) ch KlcVk,

(2.54c)

4)

- i [ s h Kl~v~ - sign(m i~+~) ch Kl~v~]

5)

i[ch Kl~v~- sign(miklk+) sh Kl~v~] sh(Uk -- i01~÷).

sh(//k

--

i0~klk+~),

(2.54d) (2.54e)

The connections between l~ and i in (2.53) and lk+l and ] in (2.54) are the same as in (2.28) and (2.29), respectively.

3. Mathematical formulation of the problem It is known "'~) that the correlation function of the two-dimensional Ising model can be expressed in terms of many-point correlations of fermion operators. To perform this transformation we use the following relations ~r,.,._lo',.. = ia ,,namn, ÷ -

trm-LnO',,, = ch 2 K ~ - i sh 2K~aLna + m,tl+[ ~

(3.1)

/~,../~r~+l,.= ch 2K~ - i sh 2K~a,..a,...+ -

(3.2)

and /z.,.~,~,.+l = ia,..a.,,.+.+

Using Wick's theorem we can express the many-point correlation of fermion operators in terms of Pfaffians of the s k e w - s y m m e t r i c matrices Q. The elements of these matrices are two-point correlations of fermion operators Qij = -Qji = (LiLi}. Thus we have ( L 1 L 2 . . . LEM) = Pf(Q),

Pf(Q) = [det(Q)] lt2,

(3.3)

where the L~ are the linear combinations of the operators a ~,,. The Pfaflian of a s k e w - s y m m e t r i c 2 M x 2 M matrix Q is defined by ~2) Pf(O) = ~ ' ( - I)P(Lk, Lk2)(L~3Lk,)... (Lk2u_,Lk2~),

(3.4)

where the sum runs over the permutations P = k . k2. . . . . k2~ which satisfy kl < k2, k3 < k4. . . . . k2M-1 < kzM and k~ < k3 < • • • < kzM-~. The expressions for the two-point correlations of operators a~,. in the most general form were obtained by Kadanoff ~3) (see also the references cited there) ~r

((a:'"la:2"2)

(a:'"'a=~"2}~=(27r)-' f d~E,2(~)

364

R.Z. BARIEV /sign(m 12) / +i sh 2K~ sh 2K~' sin ~b/sh Y(4~) x~i(sh 2K1 ch 2K~' ,~ \ - c h 2K~ sh 2K~' e )/sh y(~b)

-i(sh 2K~ ch 2K~' , \ - c h 2K~ sh 2K~' e-~)/sh y(~b) sign(m12) .~ - i sh 2K~ sh 2K~' sin ~b/sh T(~by

(3.5) In the future we shall need the expressions for correlations of the following linear combinations of the operators b~.. = ch K~a~,.w-i sh Kta~..

c~,. = ch

K~a~.

-7-i sh

(3.6)

K~a~,,.~.l.

(3.7)

The expressions for the correlations of the operators a~.., b~.., c~.. which will be used further on in this paper can be easily obtained from (3.5). For example we have lr

-'n"

× { sign(mlz) \i[q~(~b)]-'

- i~0(~b) ), sign(mlz)

(3.8)

7r

[ (c +,",c+'":) (c+'"'c~'2"2)~= \(c ;.,., c +..2.:) (C-~,.,Cm:.:)/

(27r)-' f d~bEt2(~b) ~r

× { sign(m,z) \-iw*(~b) e i~ where ~o(~b) = g+(ei*)g_(ei*),

i[w*(~b) ei~] -''] sign(m ~2) /'

(3.9)

(3.10)

w*(~b) = g*(ei*)g*(ei*).

(3.11)

Using (3.1)-(3.5) in (1.8) we can express the correlation function in terms of Pfaffians~Z). Since there is no simple way to compute these Pfaffians, the representation for the correlation function in such a form is not convenient. Therefore, we reformulate the computation of the correlation function (1.8). Following refs. 1, 14, 15 the correlation function (1.8) is determined by considering the s + s~-point correlation function. To do this we replace the spin and disorder variables by the following expressions (,~ .....

)= ~

1 1

lira (,~.~.,~ ........

),

(3.12)

(3.13)

CORRELATIONS

OF THE PLANAR

ISING MODEL

II

365

below the critical temperature Tc and .....

1

> =

(3.14)

m,,+,...>,

tim

(~ . . . . . ) = 1 lim (~,t~m.,+,-- .) k/x/,--,®

(3.15)

above rc. Here (+) is the spontaneous magnetization +'t+) and Qz) is the transform of (~) under the Kramers-Wannier transformation 4) (~)

=

(3.16)

T > To'

0 T T ¢ "

(3.17)

Taking into account that (crm.)2=(~m.) 2= 1 and using (3.1) and (3.2) in (1.5)-(1.7) and (3.12)-(3.15) we obtain the fermion representation for the correlation function (1.8) (a) T < Tc G +'s2.... ' ( R , R2 . . . . . Rs) s I

nr

= (or) -'~ l i m . . , lim ( l ' I ~ (ib +~..... b m..,,,+~) nl-'~

s~ " + ~ % r = l

s-~

×

nr

1"-[ ( s h K , b ~ , . + i c h K t b ~ , o . , ) ~ ( i b ~ , . , + , b ~ . , ÷ ~ ) r=sl+l

s~

x

x

sl

s~

. . . . +la-. . . . + l - (a+,,,,,a-,,,,,)) r=~+ a +m,.nrr=Hs3+l ia-,,,.n,, r=~+t i( a +

r -=Hs~ . +1

i[a . . . . ( . . . . -

a+

$

..... ,)

a+

-

x

(aT,,,(a+,,,,, - a,,,+O)] +.

_

+

i[-(a ~,,~, - a . . . . +l)a . . . . +l + {(a~,.-i - a 2~,)a~.)]). r--

1

(b) T > T¢ G *'.2.... '(RI, R2 . . . . . R+) Sl

=

(~>-+~ lim • . . lira®(11 (ch g ~ c "t ''®

nr

",'-+~ XT='I

s+

+. . . .

+l

+i

sh K ~ c . . . . )

nr-I

× ~ (icT,,,..,,,+,c+,,,.,+,+') I"[ ~ (icT,,,.,',+'c+-',,-.",+"+') r=$1+l

(3.18)

366

R.Z. BARIEV si X

s~

r -__SHE+ 1

a m+ ,,n

r

si

r -=sl~[3+1

ia-,n,,nr r=~s4+1 l'( a m r+,

n r + l a m r -,

nr+l

-- ( a , n+n a m n- ) )

- a

.+ ...

s~

x I-I

i[a~,.,,~(a+,.,,. -

• =s~+ 1

a .+ ...

+1)-

(am.(a~..

i))]

s

×

r -=$1~6+ 1

i [ - ( a ? , . , - a .--. . . +Oa .+ . . . +l+((am,n i - - a m_, ) a , ~+, ) ] . --

(3.19)

A p p l y i n g W i c k ' s t h e o r e m to the right-hand sides of (3•18) and (3.19) we obtain the Pfaflian r e p r e s e n t a t i o n for the correlation f u n c t i o n u n d e r consideration• N o t e that the eqs. (3.18) and (3•19) are valid up to a sign. To obtain the c o r r e c t sign it is n e c e s s a r y to multiply the right-hand sides of (3.18) and (3.19) by the f a c t o r s (__l)N(ms,+l,ms,+2 . . . . . .

~:~)

(3.20)

and ( - 1 ) u(m,, m2...... ,,, ms+. . . . . ~+2...... ,2

(3.21)

respectively. H e r e N ( m b m 2 . . . . . mR) is the n u m b e r of " d i s o r d e r " in s e q u e n c e ml, m2 . . . . . ink. H o w e v e r , we omit these f a c t o r s not to o v e r l o a d the calculation [see h o w e v e r (6.8)].

4. Calculation C o n s i d e r first the correlation f u n c t i o n for T < Tc i.e. the Pfaffian w h i c h is o b t a i n e d f r o m (3.18)• Following I this Pfaflian is e x p r e s s e d as an e x p a n s i o n [see the d i s c u s s i o n p r e c e d i n g (1•3.3)] GSlS2 . . . . 7(RIR2 . . . . .

Rs) = (Or) s~ k--~' Fk,

(4.1)

w h e r e Fk is the sum of terms, e a c h of w h i c h contains k interpath pairings [k' is d e t e r m i n e d by (2.5)] k l k 2 . .. ks4•

Fk = ~ Fk

"

(4•2)

FRk ' ' 2 , ' ~ = ( < r ) - Z : ~ l i m . . . x

s~ I__

lim ~

~--0

k~

s~ •

+ ....

-. . . .

n (s.

K~b+

r=Sl+l k,

x ( i c h r /~Zl v, .-. . . +a6),I-,,1"]" (lb • + . . . . +,.rb -. . . . +or)

. . . . +.~),"

CORRELATIONS OF THE PLANAR ISING MODEL II

367

s

a+ a+ x ( . . . . -- . . . . +1)] ~:~+1 [--i(a 7 ~ . , - aT,~.,+l)a +~,,+1]) sI

s~

X ~ A("; ) . . . . x

I-I tA(*')

,t A ("p~ ~. . . .

)~-'~.

(4.3)

w h e r e the pairings in the angular b r a c k e t s are m a d e o n l y b e t w e e n o p e r a t o r s belonging to different paths. In (4.2) the s u m m a t i o n is taken o v e r the values k4, k2. . . . . k,~ = 0, 1, 2 . . . . w h i c h satisfy ~__~k, = k - k'.

(4.4)

In (4.3) the first sum runs o v e r all values ~ , = 0 , 1 ; the s e c o n d sum is p e r f o r m e d o v e r a~ and //~ u s e d in the p r e s e n t s u m m a n d . T h e s e indices a s s u m e those values 0, 1, 2 . . . . . rt, - 1 w h i c h satisfy

a [ < a[+l, #[
(4.5)

In (4.3) r~(") t ~ 2.... ~ ~a2...0k d e n o t e s the following e x p r e s s i o n k

A ~ 2 .... ~ al~2--.~k (n)

~, ai+O~

( -1)i°~

=

II-k

x (,_~_(ib+ ...~bT...+,,)).

(4.6)

H e r e the indices a l , a~ . . . . . a ' - k (/3~,[3~ . . . . . /3'-k) are o b t a i n e d f r o m the set 0, 1, 2 . . . . . n - 1 omitting the al, a : . . . . . ak (/34,/32 . . . . . /3k). C o n s i d e r (4.6) for k = 0 ))

.-4 -

lb..,.+jb,...+j

.

(4.7)

A p p l y i n g W i c k ' s t h e o r e m to (4.7) and taking into a c c o u n t that

(b~.b~,.,)=(b?..bL.,)=O,

for n ~ n '

(4.8)

we obtain /~111--4 ~nn+j) / _ . =. (ib ~.+sb = det A ("~, " .

(4.9)

w h e r e A (") is the n × n Toeplitz matrix with e l e m e n t s ~r

a.., = (ib+.b~,..,) = a._., = (2rr) -~ !- d4~ exp[i(n - n')4~]q~(40.

(4A0)

368

R.Z. BARIEV

The limiting value of this determinant is connected with the magnetization u'12) lim det A (")= lim

(tr,,,,,O'm.,,+.)

= (~r)2.

(4.11)

The expression (4.6) differs from (4.7) by the absence of the operators b~,.+o,, + + bm..+~2. . . . . bm..+~k and b~,.+0,, b~,.._02. . . . . b~,..+0k. Therefore, (4.6) can be obtained from the determinant (4.9) by omitting the k rows ( a ~ , a 2 . . . . . a k ) and k columns (/3./32 . . . . . ilk). Then (4.6) is the minor of the determinant (4.9) k

--~,-2 a(") . . . . klo,o2

.

t

o~ = ( - 1 ) ' ~ ' ~ ' + ° ' a ( " ) ( ' ~ ; ~ ' ' ' ' ~ n - ~ , --

.

(4.12)

/3.-k]

\/3;/3~.

Applying Jacobi's theorem '6) to the right-hand side of (4.12) and using (4.11) and the relation between the cofactors and the elements of the inverse matrix we have "-~]~2 a(") .... k Io,02... Ok = (ffmn£Tm,n+")E

l

I xPr'(n)

i",(n)

r,(n)

~--"1 I O,al'l 02a2 . . . 1 0 k a k ,

where the sum runs over all permutations P = a ] , e t 2 . . . . . elements of the inverse of the matrix A ("). Then we have

(4.1 3) /-'(m

Otk. - - t ~

are the

I1--1

E

a ~ _ ~~(n) 0 ~ = 8~,

( a , y = 0, 1, 2 . . . . .

n-

1).

(4.14)

0=0

Using theorem 2.1 of ref. 17 it can be shown that lirLmF ~ = F~,

(4.15)

where F ~ are the elements of the inverse of the semi-infinite Toeplitz matrix with elements ( 4 . 1 0 ) a~_oFov = 6~.

(4.16)

0=0

To solve the equations (4.16) we use the W i e n e r - H o p f Is) method. The result is rnin(a,0)

rao

=

E s=0

.r~-~r0-~, (1) .(2)

(4.17)

~r

,y~,2)

(2rr) -1 ] d~g-+(ei#) exp[~ ia~b].

(4.18)

--Tr

Using (4.13) and (4.15) in (4.3) we obtain +0~+1)

CORRELATIONS OF THE PLANAR ISING MODEL II

369

s~ X r =s~Ii+ i

(sh Klb+ .,+a~)"(i ch

k,

Klb- +[J~)l - ' r

~

~

× tI~=(ib +,,,~r+a~bm,,n,+~)rffi~s+l a +,,n,rffi~+ ia~,n, s~ ~ x =~H+(ia+..+,aT....+,) =~[5+ [ i a -. . . .

(a + .... -

s

a+ ....

+1)

kr

x =~+,[-i(a~'."r-aT".".+')a+.".+']~ fitI'~= ,=,

x

H (Fo,q)"(F~o) H',

(4.19)

r=$1+l

where just as in (4.3) the pairings in the angular brackets are made only between the operators belonging to different paths and the first sum runs over all values E , = 0 , 1. The second sum runs freely over all values a~,[3J= 0, 1,2 . . . . which are used in the present summand. It is both useful and convenient for the analysis of (4.19) to develop a diagram technique. To do this we introduce the following rules: 1) the operators b~. and a~m. are represented by the different points [see fig. l(a), l(b), l ( c ) a n d l(d)]; 2) the interpath pairing is represented by a straight line connecting the points belonging to different paths [see fig. l(e)];

a) ( ~

c) ( +1

g) @

Ic)

m~

mSN I

rrl~n~-d,

Fig. 1. Definition of the graphical symbols.

÷

O. mn

370

R.Z. BARIEV

3) F ~ is represented by a wavy line connecting the points belonging to the same path [see fig. l(f)]; 4) F0~ and F~0 are represented by the wavy loops [see fig. l(g)]. We call the two different points, connected by a wavy line, an internal vertex and any other point an external vertex. Then an arbitrary term of the expansion is represented by a class of diagrams with 2k external and k - k' internal vertices. The diagram is called 19'2°) disconnected if it is possible to separate the points of the diagram in two or more parts such that there is no line joining a point of one part with a point of the other. Otherwise the diagram is connected. Then the expansion (4.1) is the sum over all diagrams (connected and disconnected). The contribution of disconnected diagram is the product of contributions of its connected parts. It follows from the theorem of connectivity ~9'2°) that this expansion can be expressed in the form (2.1)-(2.5), where

Fk

/](~,s,,,,(k) = ~

O(.~,.s2)ii(con)(klk2k ks~)"

(4.20)

j.~(sl,s2)iJ(klk2.k. . ks½) s~ k,.

s½ G +,,,n,+,~;b~,,.nr÷O~,)OiO~) ~= ~ Fo~,~;. < k T('"~'(c°n)(k,k2... k,,)'

× ,=--~l~Ii÷1~= (ib T(<~"2)(k) = ~

s~ s~ k~ X [ I (ib+.~r+~fb . . . . +Of)) ~ t~- r.f~f,

(4.21) (4.22)

(4.23,

,=Sl+l

The subscript (con) denotes the part of the corresponding expression which is represented by the connected diagrams. The summations in (4.20) and (4.21) are taken over the values k~, k2. . . . . k~ = 0, 1,2 . . . . which satisfy k~ = k - 1

(4.24)

kr = k

(4.25)

r=l

and

s~ r=l

CORRELATIONS OF THE PLANAR ISING MODEL II

371

r e s p e c t i v e l y . In e x p r e s s i o n (4.21) O~ d e n o t e s the f o l l o w i n g o p e r a t o r s : 1)

~

(sh Ktb,,~,,+~) + " (t" ch K l b -. . . . +~8)1-, ( F 0 ~ ) • ( F ~ 0 ) i-, ,

~=0,t

w h e r e r = st + l (l = 1, 2 . . . . . 2)

+ a rn~n r

3)

ia ~ , , ,

4)

+ a m.nr+ t ~

5)

ia m~,,+,,

6)

i a ~ , , r,

7)

a +

r=s~+l

sz) f o r i = I r e s p e c t i v e l y ,

(1=1,2 .....

r=s~+l(l=l,2 r=s~+l

.....

for i = s 2 + l ,

$4)

fori=s2+s3+l,

( l = 1,2 . . . . .

r=s'4+l(l=l,2 r=s~+l

. . . . - a +. . . . +t,

s3)

.....

(1=1,2 ..... r = s s +p l

ss)

for i = s 2 + s 3 + s 4 + 2 1 - 1 ,

ss)

fori=s2+s3+s4+21,

s6)

fori=s2+s3+s4+2ss+21-1,

(1=1,2 .....

s6)

f o r i = s2+ s3+ s 4 + 2 s s + 2 / , 8)

-i(a~,..r-a~,~.r+0,

r=s'6+l

( l = 1.2 . . . . .

s7)

for i=s2+s3+s4+2ss+2s6+21-1, 9)

+

a . . . . +t,

r = s~ + l (l = 1,2 . . . . .

$7)

f o r i = s2 + s3 + s4 + 2s5 + 2s6 + 21.

(4.26)

T h e g r a p h i c a l r e p r e s e n t a t i o n s of eqs. (2.3) a n d (2.4) are g i v e n in fig. 2. A f t e r s u b s t i t u t i n g (3.5) a n d (3.8) into (4.21) a n d (4.23) the s u m s in t h e s e e x p r e s s i o n s can be c a l c u l a t e d . U s i n g (4.17), (4.18) a n d the m e t h o d c o m m o n l y u s e d in d i a g r a m techniquem), w e h a v e e x p [ - i a ~ b t + ifl~bElF~ = g (ei*')g+(e i.2) ~ e x p [ - i ( ( b t - ~b2)s] a,~ =0

(4.27)

s =0

= g (ei*')g+(ei*2)Pt2.

T(s~) sz) =

=

4-

(

~

+

-4- - ' ,

Fig. 2. Graphical representation of the equations (2.3) and (2.4).

(4.28)

372

R.Z. BARIEV

Using (3.8) and (4.27) we can establish a correspondence between the elements of diagrams and the elements of the integrand. This connection is presented in fig. 3. The complete integrand is the product of its elements. To obtain the contribution of the diagram it is necessary to perform the integration over all 4); between -~r and ~r with the factor (2~r) -1. Note the important property, which will be used frequently in the following. If two diagrams differ only by the type of one internal vertex, then the ratio of the corresponding integrands is independent on the general part of the diagrams. This ratio is H tll213(~bl,~b2)- 1.

(4.29)

The graphical representation of this property is given in fig. 4, where the dots denote the general part of the diagrams. Consider the expression (4.20) for /2(sl.s2)0(k). Calculate at first the general part of this function which is independent of the operators O~ and Oj. The graphical representation of a typical term of expression (4.20) is given in fig. 5, where 11 and lk+~ are the indices of the operators O~ and O i respectively. The

~_(e~,O %(e ~¢~)p~z 4

4

2

2.

Fig. 3. Contributions of the diagram elements.

Fig. 4. Graphical representation of the property (4.29).

C O R R E L A T I O N S O F T H E P L A N A R ISING M O D E L II

373

Fig. 5. Graphical representation of the quantity (4.30).

analytic expression corresponding to this diagram is k-I

k-I

( - 1)k-l[g-(ei~t)g+(ei~)]-t ~ E~,÷,(d),) ~ P,.r+I.

(4.30)

All other 2k - l - 1 diagrams are obtained by changing the types of the internal vertices. Then using the property (4.29) we have for the summary contribution k-I

k-1

(-1)k-l[g-(ei¢')g+(ei¢t)] -1 ~= E[3,+1 ~

H ~a,+,tr+2(d), ~ r + l ) P r,r+l-

(4.31)

The expression (4.30) corresponds the II~, kr! labeled diagrams which are obtained from fig. 5 by permutations of the indices l~ belonging to the same path. This factor is cancelled by that of expression (4.21). Note that the summation in (4.21) is replaced by that over the l~ which assume the values 1, 2 . . . . . s~ (l~g 1~+1). Finally we have for D(,,.~2) the expression (2.7) where t < g_(e idh)Ehtz((bl)©i,

g+(e

i~k

~

(4.32) j

)El~lk+,(d)k)©<

(4.33)

are analytic expressions corresponding to the pairing O~b~,~2~t2and b+%,k÷Os respectively. To calculate © f and ©s< it is necessary to replace Oi and Oi by the operators (4.26) and to determine from (3.5) and (3.8) the analytic expressions. The result of this calculation is given by (2.8) and (2.9). In the same way the expression for T(<~,.,o(k) can be obtained. The graphical representation of a typical term of expression (4.22) is given in fig. 6. The analytic expression corresponding to this diagram is k

(--1) k-I I-I E[,#+~(ck,)P,.,+t, lk+l 11, Ckk+~----&l.

(4.34)

":

r=l

1

Z

3

x-1

K

Fig. 6. Graphical representation of the quantity (4.34).

374

R.Z. BARIEV

For the summary contribution we have

k (--1)k-I r~__E ~ r l r + l ( ~ b r ) H t , l ~ + , t r + 2 ( ( b r ,

&,,+I)P

r,r+l.

(4.35)

The expression (4.34) corresponds to the ( l / 2 k ) I I ~ kr! labeled diagrams which are obtained from fig. 6 by inequivalent permutation of the indices li belonging to the same path. Finally we have for T( Tc [see (2.20)-(2.33)].

5. The scaling functions Consider the correlation functions in the most interesting region, which is determined by (2.34)-(2.36). The leading contribution to the integrals (2.6) and (2.7) near the critical point comes from the behaviour of the integrand for small values of &r (order of ~-t). T h e r e f o r e we expand the integrands about &r = 0 and approximate "Yr =

(sh 2KE)-l/2N/~-2 + sh 2Kl~b~.

(5.1)

Then we make the change of variables

Y-r = ~(sh 2K1)l/E~r,

(5.2)

and extend the limits of integration to _+oo. Using (5.1) and (5.2) we see that in the scaling limit (2.34)-(2.36)

EL~,+,(ck,. ) -o e~a,+~(x,.),

(5.3)

el,.t,.+l(Xr) = e x p ( - Yrlm ~,t,+~]- in ~,+,xr),

(5.4)

y,= +Vl+x~,

(5.5)

' = m lrlr+l m lrlr+l H

/~X/~2K2,

Irlr+llr+2(~r, ~r+l) ~

nL#+, = n,,~,.+,/~/~2K1,

(5.6)

sign(m l,l,+~)Yr / 1 h t,~,+~l,+2(x,,Xr+l),

(5.7)

h i~,+~t~+2(Xr,Xr+O = sign(m I,#+l)Yr + sign(m 1~+~l~.2)Y~+l,

(5.8)

P r.~+l~ - i ¢ ( s h 2Ki)l/Zp ~,~+1,

(5.9)

P r , r+l = ( X r - - X r + l - -

i'r/) -1,

(5.10)

g+(ei4',-) .-). ~l/2ktc(Vr)-I '

(5.11)

g_(e ior) -> ~-'/2 kcl v +~,

(5.12)

CORRELATIONS OF THE PLANAR ISING MODEL II

375

where kc = (2 sh K I d c h 3 K,¢) TM, v~ = ~/1 - i x ,

(5.13)

Applying (5.3)-(5.12) to (2.6) and (2.7) we have the scaling form for the correlation function (2.37)-(2.39), where the elements of the matrix to .~. s2..... s7) are

to(,~o = ~ to~,gii(k),

(5.14)

k=l ~o

,

(i),-,

X

to(sgii(k) = - (27r-----)(t:.I ..... Ik~

~o

Yl

Yk

-~

-oo

k

× N s ign(rn l~,+,)et~,+,(xr) r=l k-I

x I-I h ~a,,t,+~(x,, xr+l)p,.~+~ tt~
(5.15)

r=l

where 1I'< and LI~ is determined by (2.44)-(2.48) using the inverse of the change of variables (5.18). Likewise from (2.6) we have

t(,~) = ~ t~,,)(k),

(5.16)

k=2 o0

t~s~.)(k) = -

(q,z

.,lk}

00

'Yl -~

.. f

Yk

-00

k

x ~ sign(m ~+~)e~+t(x,)h t~÷~t,÷2(x, x~+Op ~,~+~,

(5,17)

where

lk+t-- 11, lk.2 =- lz,

Xk+l =--Xl.

The form (5.14)-(5.17) is not the most convenient representation for the scaling correlation function. The formulae (5.14)-(5.17) don't exhibit many properties in an evident form. Moreover it is generally accepted that the scaling correlation function depends on the R~i and the angles in the evident form, but not as in (5.14)-(5.17). To write the scaling correlation function in a more convenient form we deform each xq-integration contour into the upper (lower) half-plane for n t~J~÷t< 0 (nt$~+, > 0) as it is shown in fig. 7. The curve C,Qtq+~ is given by the parametric equation =

< Ot~q+~< ~'rr.

(5.18)

376

R.Z. BARIEV

Fig. 7. Contour C used in (5.18).

The singular points of the integrands in (5.15) and points x, = +i and the poles of first order

(5.17) are the branch

x, = x,+1 + i77,

(5.19)

x, = x,_, - 17.

(5.20)

The integrals

over CR and CR vanish

as R + ~0. Then

using Cauchy’s

theorem

we have

/dx-,=1

Res(x, = x,-~- iv) dx, + 27~i[Acocq_,c,q,q+,

cI&+ I

-B

wll+l~‘I& _,

Res(x,

= xq+l +

iv),

(5.21)

where

Jdx, = i

dx,;

-m

co

_f dx, = i

du, ch(u, - ior,,,,,),

(5.22)

-5

cl&*I

C,_, and Cq+, are the integration

contours

for x,_~ and x,+~ respecttvely

=

1 -1 1 0

c, < cjs c,. C,2Cj>CI, otherwise

(5.23)

B cic,c, =

- 1 1 1 0

c, =S cj < CI. C,>CjaCk. otherwise

(5.24)

&,cici

The condition Ci > C’i is understood to be Im(Ci) > Im(Cj), Re(Ci) = Re(Cj) at some fixed point. It is easily shown that this inequality holds at any other point for the curves under consideration if C, > Cj is valid at some fixed point. This follows from the fact that the necessary and sufficient condition for Ci>Ci is sign(mj)

tg 0i < sign(mj)

tg 0,.

(5.25)

CORRELATIONS OF THE PLANAR ISING MODEL II

377

Therefore if sign(m ~i~,)= sign(m ~) the equations C~ < C~,

(5.26)

Ct~ < C~A

are equivalent. Substituting (5.21) in (5.15) we obtain the following recursion formula for the transformation of the integration contours

E (12,13 . . . . .

[

1112...lq-llqlq+l.., lklk+l

lk} °'i~,CoCo. . . CoCoCt~+,~+~ .. C~,~+, J

{ 1112... lq-llJq+l.., lklk+t "~ =

Z,

{/2,13 . . . . . lk}

t°°l" " 0

0 • • •

C C 0

lcllq+1 • • •

C

I

Iklk +l /

[ l l l z . . , lq-dflq+t.., lk-llk+l + ~, t~..... ~ ~ ~ °°iJkCoCo... CoCl~l,+~... Cl~_,t~+,~"

× ~ dl'l-dil'lAc"c°clll. +

E

{12,13..... tk-~

~¢"OiJ(

1,12...lq-llflq+,...lk-llk+l

)

\ C o C o . . . CoCql~+~Cl~l~+~... Clk_~l~+~/

× d~,,tlt,,+~Bc c

(5.27)

CI,A+I,

where

l'12

¢'°ii~C1C2...

Ck]

,i,klf xl

=

~

y,

"

fdx__~=k sign(mla,+l) Yk

Ci k-I X

t e~a,+,(x,) ~ h ~:,+,z,+2(x, Xr+l)p r,+lU~
dillA = 1 + sign(mt~Q sign(mtjtk);

li¢ li¢ lk.

(5.28) (5.29)

Applying transformation (5.27) to each xq we express in terms of to( 1,12...ldk+, ") \ Cl,12Ct2t3 . . . C lkt,+, /"

to(l,lz...ldk+2"~ \ C o C o . . . Co/

Substituting this expression in (5.14) and (5.16) we obtain (2.41), where ~°~s9ijfk) =

(i) k-I ~

'~

{12.13. . . . . Ik}

r/t,,2...,,+ ,

Qo k X f -oo

dUl..,

f duk ~

(--1)Alrlr+lelrlr+l(Ur)

-oo

k-I

x ~ th ½(u, - u,+l + iOt~,+,~,+2)lI~IV<,

(5.30)

378

R.Z. BARIEV

and (2.40) w h e r e

t~)(k) = '-

{1~. 12. . . . . l~}

*l~,l:...l~ f dul..,

dUk

× th ½(Ur -- U,+~ + i'Ot,.t. ,t,+2)',

lk+l~-ll.

1~.+2~-- 12,

(--1)a'/~+,el~+~(u~)

(5.31)

Uk+l~ Ut.

T h u s the p r o b l e m is to calculate the numerical coefficients T/~,~,. ~., and -t/~:,,_...~k-This calculation is p e r f o r m e d using the diagram technique. It is clear that k

~/1~t2. l~.~ = ~ r/ti~+,,

(5.32)

k

rt~,12., t~ = j~= ~lt~lj+~, lk+l =-- Ii,

(5.33)

~/~jli+~; (k)

(5.34)

77 ljlj+ I

=

"~=0

T h e graphical r e p r e s e n t a t i o n of the series (5.34) in the first order is given in fig. 8. For the two first terms we have (0)

~/~jl~+~= 1,

(5.35)

(I)

~/ti~j+t = ~ dt~tltj+~(A CoCoCtlt,~ + Bcoc~,j.,c~,,i )"

(5.36)

It can be a s s u m e d that (k)

k

k

~lh+ , = (--2) Ca, j,j+,

(5.37)

where Ca,k j,j+~ are the binomial coefficients. We have s h o w n that (5.37) is valid for k = 0, k = 1 and k = 2. To p r o v e the validity of (5.37) for arbitrary k we use induction. A s s u m e that (5.37) hold for k. T h e n it can be s h o w n that the validity of (5.37) follows for k + 1. T h u s the validity of (5.37) is established. Using (5.32)-(5.34) and (5.37) in (5.30) and (5.31) we obtain (2.42) and (2.43) with (2.44) and (2.45). The results (2.50)-(2.54) can be o b t a i n e d f r o m (2.20)-(2.33) in a similar manner.

r/o.k. Fig. 8. Graphical representation of the series (5..34).

CORRELATIONS OF THE PLANAR ISING MODEL II

(+) (-)

379

(+) ( - ;

Fig. 9. Graphical representation of the property (6.1).

6. Rotational invariance

Consider the scaling correlation function formed by spins, disorder variables and energy-density operators [i.e. the function (2.37) for s3 = s4 = s6 = s7 = 0]. We now show that this function is rotationally invariant. In our case a function is rotationally invariant if it depends on the distances R 0 and the a n g l e s 19ijk but not on the m o and nli separately. From (2.42) it is clear that tt,9(k) is rotationally invariant up to a sign. Since from (2.43)-(2.45) the to(,gii(k) depend on the angles 0tt~ and 01kt~÷~, they are not individually rotationally invariant. H o w e v e r , these expressions are used in (2.39) only in combinations which are independent of the angles 0. To show this, note that for an arbitrary diagram of the expansion (4.1) it is possible to find another diagram which differs only in the type of two external vertices (see fig. 9). In the expression for the correlation function under consideration these diagrams are used only in a sum. The analytic expression corresponding to this sum is

- i A t~'l~l[2~

I. (I) sh ~t-k

U ~2) .i.. iO

I[%~l~2~)

(6.1)

where

l k(+tl)~ _

1(2)

~l ,

A 1,1213= 1 + [sign(mtlt2) + sign(m~213)]sign(Ol,~2t3)- sign(ml,12) sign(ml213).

(6.2)

Thus an arbitrary term of the expansion of the correlation function is rotationally invariant up to a sign. We now show that the sign of this term is rotationally invariant too. From (2.42), (2.43) and (6.1) it is clear that this sign is determined by the product of the factors of the following form '~__( - 1)

sign(m It(,,lr+l ~,> )(A,,~,,~ ,<,> )P,,÷,, "r "r+l'r+2

(6.3)

where the index i denotes the different factors, .,l(°= 1, 2, .. ,. s(lt/># l,+t),(° {1 P~"=

for l , = s~+ 1, s ~ + 2 . . . . . s~, otherwise.

(6.4)

380

R.Z. BARIEV

The products (6.3) can be of two types: lk+, =-- I~i), tk+~'(i) _= l[i); (a) cyclic, if ,0).

(b) a-cyclic, if 1~)+1# l~i). We will now show that the cyclic product (6.3) is rotationally invariant, using induction. Considering all the possible configurations of three points we obtain 3 1-'I ( - 1)A/ rl r+l sign(m l,#+l)(A Irlr+llr+2)Plr+l r--|

14=--lb 15=--12,

=--(--1)atl~213sign(Oq1213),

(6.5)

where A q~t3 is the number of spins and disorder variables inside the triangle composed by the points 11, 12, 13. The rotational invariance of the cyclic product (6.3) for k = 2 is obvious. For k = 3 it follows from (6.5). Assume that the rotational invariance holds for k. Then it is easily shown that (6.3) is rotationally invariant for k + 1. To do this we use (6.5) and the rotational invariance of the following expression A ql213A qht,A 141213= 411 + sign(Oqt2t3) sign(Oql214)+ sign(Oqt2Q sign(Ot4t213) -

sign(Oql:14) sign(Ow:13)].

(6.6)

Eq. (6.6) can be proved considering all possible configurations of four points. Thus the cyclic products (6.3) are rotationally invariant. For the case (b) all 1~I), tkl+l, m) l~2), 1(2) ,k2+l,..., 1~s212), 1(s212) k,,a+ 1 are different and •

.

.

assume the values Sl + 1, s~ + 2 . . . . . s~. If we multiply the correlation function by s~12

r=~lI-~+I ( - 1)a2'-'.2' sign(m z,-1.2,),

(6.7)

then the a-cyclic products in the expansion of the correlation function become cyclic. T h e r e f o r e the correlation function can be written as a product of two factors. The first factor is (_ l)Ntm,,+,.,,,,+, ......

'P r=sll~+l(-- 1) a~'-'a" sign(m :,-L:,),

(6.8)

and the second is rotationally invariant. The function (6.8) assumes the values -+1 and is not rotationally invariant in general. It is related to the fact that the paths of spins and disorder variables can intersect when the rotation is performed. According to Kadanoff and Ceva 21) it leads to a change in sign. H o w e v e r the function (6.8) is rotationally invariant if sl or s2 is zero. For s2 = 0 it is obvious. For sl = 0 the rotational invariance of (6.8) can be proved by induction.

CORRELATIONS OF THE PLANAR ISING MODEL II

381

In the s a m e w a y the correlation function for T > T~ can be considered. T h u s we p r o v e d that the scaling correlation f u n c t i o n f o r m e d b y spins, d i s o r d e r variables and e n e r g y - d e n s i t y o p e r a t o r s is rotationally invariant up to a sign. T h e sign of the f u n c t i o n is rotationally invariant too if the spins or the d i s o r d e r variables are absent in the correlation function.

7. Asymptotic behaviour As it is s h o w n in I [see the discussion p r e c e d i n g eq. (I.3.6)] the a s y m p t o t i c b e h a v i o u r of the scaling correlation function for the large R~j/¢ is d e t e r m i n e d by the first t e r m s of the e x p a n s i o n (3.1). In this case the integrals (2.42) and (2.43) are easily calculated by the m e t h o d of s t e e p e s t decent. T h e n we h a v e for the irreducible part of the scaling correlation function the following asymptotic behaviour f-(i,.) ,,,~ ....

,({R,;/¢})= g

~ (-1) p rI

~ X~.jt~'"~'~"'~;k' ~=1 tt[,~, t['~ ..... ~,'~1

{it

(7.1)

= I~"~- si, j = ,k~+l-Sl} = 1, 2 . . . . . 2g ,(,)

[ll, 12 .....

Ill 2 • ~'< "

Is

for g # 0

for g = 0

(7.2)

1~]

w h e r e Et~,.~2..... tkJ d e n o t e s the s u m m a t i o n o v e r all 11,12 . . . . . lk e a c h index a s s u m i n g the values 1,2 . . . . . s~, s ; + 1, s ~ + 2 . . . . . s ( l i # li+O. T h e values s ; + l, s ; + 2 . . . . . s are used only once in each s u m m a t i o n in (7.1) and (7.2). T h e first s u m in (7.1) runs o v e r those p e r m u t a t i o n s P = i~, jr; i2, J 2 ; . . . ;i~,j~ which satisfy i, < Jr and it-1 < Jr, the second s u m runs o v e r all values kr = 1,2, • .. which satisfy Er=l kr = s - 1 . T h e sums satisfy also the following condition. If the indices are divided into two g r o u p s {l~')}, {l~ 2)}. . . . . {l~ ,)} and {l~i)}, {1~9},.. . . . tl. (r'"-,)"h there exists at least one index which is used in both groups. H e r e {1~)} = 1~'), 1(2r}, . . . . 'kr+~, i(r) . rl,. r2. . . . . . r~_, are the remaining indices after omitting f r o m the set 1, 2 . . . . . g the indices r~, r2 . . . . . r,. Using p r o p e r t y (6.1) and its analogue for the c o m p o n e n t s of the stress t e n s o r we obtain f r o m (2.42) and (2.43)-(2.45) (--l) k l~[

.

.

Xlf2"k+~=(2~'~'~l~l(-1)a'"'÷'slgntmt~'÷d

, exp(-R~,+,/~) 'R

,e~1,2

k-I

x l'~_ - Al,~,tt,+, q,<

5i~1<,

(7.3)

382

R.Z. BARIEV

where •

+ .

q~=l"

--slgn(mt,t~)~ qt2, --¢2~id2

q'< = 1"

-sign(m~÷,)~+~;

(7.4)

and

for i , j = ! , 2 . . . . . s~; i , j = s , ~ + l . s3 + 2 . . . . . 2g r e s p e c t i v e l y . H e r e

+

~

(7.5)

lklk + I

s2 + 2 . . . . .

s2 + s3 and

i , j = s2+ s~+ l, s2+

Pill i = [sign(m~,t) ch K~c + sh Kt~] sin ~0~,l~ -+ [ch Kt~ - sign(m~Q sh Kt~] cos ½0~r

(7.6)

~,+~ = [sign(m~Q ch Klc - sh Kt~] sin ½01~ -+ [ch K ~ + sign(m~,Q sh Kt~] cos ½0~,le

(7.7)

and

1 (-1) k i3i (_l)a,/~.,

I1l~ . . . I k

2k ( ' 2 ~ - ~ ~=l

×

sign(ml,#+0

exp(-Rt,sr+,/~) ~ . rR /~112 At,#+~l,+,, lk+l =-- lt, I

(7.8)

lk+2=" 12,

It!r+ I t a !

where tg~Ol,l,+~l,+,, -

for lr+l = 1, 2 . . . . . I

I

for Ir+L= S'4+ I . s 4 + 2 . . . . .

Al,#+d,.,.sln~Ol,#.dr+,. <

~.a,4, ~

=,

A t,~,+~1r+2

S" l

S,

~,.~ sin Ol +,t+,- ~ , . , P l l , + ~ l , ~ s i n •

.

for l,+~ = s~+ 1. s~+ 2 . . . . .

(7.9)

s~

sign(m i~,÷,) sign(m i,+,~•+,)[~ ~,+,21~+,~,. sin 0 t a , . , - ~ b~.,?l~,,t,+, × sin 0t,~t,+2] f o r l r + ~ = s ~ + l , s ' 6 + 2

.....

S.

W h e n T > Tc, the irreducible part of the scaling correlation function has the following a s y m p t o t i c b e h a v i o u r +(iv)

R

-

f~,~2 . . . . ~({ o / ~ } ) ZZ(--I)P

H r=l

{i,, j,.} = I, 2 .....

Z

2s',

i, = 1~"~, jr =lkr+l(tr, (r)

ir=l~'~--s2, j, ='~rl ..... lkr+l - - s 2 ( i , , j r = l , 2 X [q, 12. . . . . tsl

Ii1"~... I, 7>-

for g # O

12Jr, r

[1!,1f.i(~. • . . . . . i~,~11

for s = 0,

•

lr = 1 , 2 . . . . . .

S3 + S4);

sO;

(7.10) (7.11)

CORRELATIONS OF THE PLANAR ISING MODEL II

383

where the summations in (7.10) and (7.11) satisfy the conditions of summations (7.1) and (7.2), respectively. From (2.52)-(2.54) we have (_ l)k-I k v,]~~'~"... '~7+,= ~ ~ (_ 1)a,/,÷, sign(m ,~,÷,) k-1

×

(Rld,+l[~)

i/2

]1

(7.12)

l-llrlr+ltr+2~lt t$>~

where q~ = 1;

-~;iI2;

sign(mll~)~:~t2

(7.13)

ql> = 1", 9~+lkt~+l," sign(mt~tk÷~)9~t~.,

(7.14)

and for Ir+t = 1 , 2 . . . . . s~, i , j = s l + l , s ~ + 2 s3 + 2 . . . . . 2s, respectively.

. . . . . s~+s3 and i , ] = s t + s 3 + l ,

sl+

tg2! O l,/,+d,+2, for lr+t= 1,2, . .,S~, • ! 0 t,#+d,+2, fOr/r+l -At,#+dr+2sln2

A>

•

.

=

+

+

S~+I,s~+2,

..

.,S5,'

•

slgn(m ~,#+~)slgn(m t,.,~1,+2)[9~t,~,+~ l,+~l,+~sm Ot,#.~

[fir+lie+2 = .

--927~,+l~?,÷~t,+2sinel,÷~l,+2], for l~+l= s'~+ l , s ~ + 2 . . . . . s6

(7.15)

9A~,+,~ ~+~,+~sin O~,+,l,+2-93~,+,~ ?,+~t~+2sin 0~,+~ for 1,+~= s~ + 1, s~ + 2 . . . . . s. Thus the asymptotic behaviour of the scaling correlation function does not depend on the choice of the operators within the angular dependence for the class of operators Oi. When T < Tc this class contains the spins, the energydensity operators and components of the stress tensor and when T > T¢ it contains disorder variables, the energy-density operators and components of the stress tensor. Acknowledgements

The author would like to thank Dr. M.P. Zhelifonov, Dr. G.B. Teitelbaum and Professor A.M. Polyakov for useful discussions. References

1) R.Z. Bariev, Physica 83A (1976) 388. 2) L. Onsager, Phys. Rev. 65 (1944) 117. 3) C.N. Yang, Phys. Rev. &~(1952) 808. 4) L.P. Kadanoff, Phys. Rev. Letters 23 (1969) 1430.

384 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

R.Z. BARIEV M. Sato, T. Miwa and M. Jimbo, Proc. Japan Acad. 53A (1977) 6; RIMS preprint 234 (1977). B.M. McCoy, C.A. Tracy and T.T. Wu, Phys. Rev. Letters 38 (1977) 793. D.B. Abraham, Phys. Letters 61A (1977) 271. R.Z. Bariev, Phys. Letters 64A (1977) 169. A. Luther and I. Peschel, Phys. Rev. 12B (1975) 3908. H. Au-Yang, preprint ITP-SB-77-41 (1977). T.D. Schultz, D.C. Mattis and E.H. Lieb, Rev. Mod. Phys. 36 (1964) 856. E.W. Montroll, R.B. Potts and I.C. Ward, J. Math. Phys. 4 (1963) 308. L.P. Kadanoff, Nuovo Cimento 44B (1966) 276. R.Z. Bariev and M.P. Zhelifonov, Teor. Math. Phys. 25 (1975) 108. R.Z. Bariev, Phys. Letters 55A (1976) 456. A.C. Aitken, Determinants and Matrices (Oliver and Boyd, Edinburgh, 1959). I.T. Gokhberg and I.A. Feldman, Equations in Convolutions and the Projection Methods of its Solution (Nauka, Moscow, 1971). M.G. Krein, Am. Math. Soc. Transl. 22 (1962) 163. A.A. Abrikosov, L.P. Gorkov and l.Ye. Dzyaloshinski, Quantum Field Theoretical Methods in Statistical Physics, 2nd edn. (Pergamon, Oxford, 1965). G.E. Uhlenbeck and G.W. Ford, in Lectures in Statistical Mechanics Vol. I, J de Boer and G.E. Uhlenbeck eds. (North-Holland, Amsterdam, 1962). L.P. Kadanoff and H. Ceva, Phys. Rev. B3 (1971) 3918.