Time correlation functions in the a-cyclic XY model. I

Physica 69 (1973) 259-272 ~ North-Holland Publishing Co.

T I M E C O R R E L A T I O N F U N C T I O N S IN T H E a - C Y C L I C XY MODEL. I P. M A Z U R a n d T H . J. S I S K E N S

lnstituut-Lorentz, Rijksuniversiteit te Leiden, Leiden, Nederland

Received 22 J u n e 1973

Synopsis Explicit expressions for the time correlation functions o f the z c o m p o n e n t s o f two spins a n d the time a u t o c o r r e l a t i o n function o f the z c o m p o n e n t o f the m a g n e t i z a t i o n are derived both for the c-cyclic a n d the c-anticyclic X Y model.

1. Introduction. The time correlation functions of the z components of two

spins and the time autocorrelation function of the z component of the magnetization have been derived previously by Niemeijer ~) for the so-called c-cyclic anisotropic X Y model. These correlation functions only represent correctly the corresponding correlation functions of the cyclic spin X Y model (the so-called a-cyclic X Y model2)) in the limit of an infinite number of spins. The purpose of the present paper and a subsequent one is to calculate these correlation functions for the finite a-cyclic model with the aim of studying later the ergodic properties of this model, which is somewhat more physical than the finite c-cyclic model. Since the hamiltonian of the a-cyclic problem has the property that it acts in a subspace of the space of states, as if it were the c-cyclic hamiltonian, and in the orthogonal complement of this subspace, as if it were the c-anticyclic hamiltonian, we shall first, in this paper, calculate the correlation functions for the c-cyclic and the c-anticyclic hamiltonian as if these were valid in the whole space of states. In a forthcoming paper we shall combine these results in order to obtain the correct expressions for the correlation functions for the a-cyclic problem. The method used in this paper differs from Niemeijer's approach and does not require an explicit diagonalization of the hamiltonians and calculation of their eigenvectors. In section 2 we formulate the problem and give the form of the c-cyclic and the c-anticyclic hamiltonian. In section 3 we introduce hermitian operators ~i and fli (i --- 1. . . . . N) which are linear combinations of the fermion creation and annihilation operators. In terms of these operators the dynamical problem is solved for both models. In section 4 we evaluate the z component of 259

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the magnetization and the internal energy for both models. The expressions for these quantities reduce to traces of functions of cyclic and anticyclic matrices. The cyclic/anticyclic nature of these matrices enables one to express these quantities in a simple way in terms of the parameters of the system. Finally, in section 5, we derive in a similar way explicit expressions for the various time correlation functions for the c-cyclic and the c-anticyclic model. 2. F o r m u l a t i o n . We consider a linear chain of N spins ½ subjected to an external magnetic field B along the z axis. The spins have nearest-neighbour interactions. The hamiltonian of the system (the so-called X Y model) is N

H ---- E [(1 + 7 ) S jxS j +x t

y

.v

z

+ (1 - 7)SiSj+t - B S j ] ,

(1)

j~l

where periodic boundary conditions have been taken. Performing the transformation S]. =

(a;

+ aj)/2,

S~ =

(a;

-- aj)/2i,

S jz = ajt aj -- ½ ( j = ! . . . . .

N),

(2) and subsequently the transformation

aJ =

exp ire

_c ICk ~J (j--2,

....

N),

a,* = c~*,

(3) aj=exp

-i~c~c

k cj

(j = 2 . . . . . N),

k=l

a 1 ~--" C 1

the hamiltonian becomes ( c f refs. 1, 2) N

N

H = ½NB -- B Z c~cj ~- ½ Z [ ( c ~ c j + l j=l j=l - ½ [ ( c ~ c , + ? c ~ c l ) + b.c.]

-]-

* ]IcjCj+l) "~ h.c.]

exp i r c ~ c J c j \

j=l

+ I

.

(4)

/

The c operators are fermion operators. Thus the operators c~cj (j = 1 , . - . , N) are occupation-number operators having eigenvalues 0 or 1. Therefore the operator P, defined as P = exp

it \

c cj j=l

,

(5)

/

has eigenvalues - 1 or + l, and can also be written as N

P = 17 (l - 2c;c,1.

(6)

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FUNCTIONS

IN THE a-CYCLIC

XY MODEL.

|

261

It follows immediately that p2 = 1 (unit operator).

(7)

Since P is hermitian, it is easily seen that the space of states can be decomposed into two orthogonal subspaces, one belonging to the eigenvalue + 1 of P, and one belonging to the eigenvalue - 1 . It can furthermore be checked that H and P commute: [/4, e ] = 0.

(8)

It follows that in the subspace belonging to the eigenvalue + 1 of P, the hamiltonian can effectively be described by N

/4 = / 4 + ,

= ½uB - B E c t~ j=l

-

1 , ~.[(cNc,

N-I

+ ½ Z E(c~j+, +

~c;c~+,) + h.c.]

j=t

+ W~H) + h.c.].

(9)

and in the subspace belonging to the eigenvalue - 1 by N

N

* i /4 = H _ , = ½NB - ~ E ~ c j + ½ Z [ ( ~ e j + , + ~ej~j+,) + h.c.]. 1=1

(10)

j=l

The problem corresponding with the hamiltonian (4) is the so-called a-cyclic problem (cf. ref. 2). If the hamiltonian (4) is replaced by the hamiltonian H_ t [eq. (10)] in the whole space of states, one is dealing with the so-called c-cyclic problem. However, when dealing with the a-cyclic problem one has to realize that H and H_ 1 are effectively identical only in a subspace of the space of states, viz. the one belonging to the eigenvalue - 1 of P. If the hamiltonian (4) is replaced by H+ 1 [eq. (9)] in the whole space of states, one is dealing with, what we shall call, the c-anticyclic problem. The validity of H+ t for the a-cyclic problem is restricted to the subspace belonging to the eigenvalue + 1 of P. The purpose of this paper is to compute spin-spin correlation functions and the autocorrelation function of the magnetization for the models characterized by the hamiltonians H_a and H+I, respectively, as if these hamiltonians were valid in the whole space of states. In a forthcoming paper we shall separate in the expressions thus obtained the contributions from the two subspaces belonging to the two different eigenvalues of P, and subsequently combine the relevant parts (for H+ t the part produced by the subspace belonging to the eigenvalue + 1 of P, and for H_ t the part produced by the subspace belonging to the eigenvalue - 1) in order to obtain the correlation functions for the a-cyclic model.

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3. Solution of the dynamical problem for the c-cyclic and the c-anticyclic hamiltonian. The hamiltonians H + l and H_ I [cf eqs (9) and (10)] can be expressed as follows: N

+ ~ + h.c.]} + ½NB H~ = Z {c,+(Ao)J~ + [~j(o~)j~c~

(a = - I ,

+I),

(II)

j,k=l

where the N x N matrices A. (a = - 1 , + 1 ) are symmetric and the N × N matrices D. (a = - 1 , + 1 ) are antisymmetric. F u r t h e r m o r e A ~ and D_I are cyclic, and A+~ and D+~ are anticyclic (cf. appendices A and B). The elements of the matrices A. and D. are given by

(A,~).

= - B6., 0 + z (1. ,6l

- ~6.,~ =N - cr6.,N_~)

+6.._~

(a = - 1 , + 1 )

(12)

and ( o . ) . = ½~(6.,, - ~ . _ ,

- o6.,,_N + az.,N_,)

(~ = - 1 ,

+1),

(13)

where n (which equals the difference o f the indices) runs from 1 - N to N - 1 and where 6 u denotes K r o n e c k e r delta. N o w we introduce new operators c~i and fli (j = 1. . . . . N), defined by ~j =

(cj + c~),

/~j =

i 2 (ci - c*-)

( j = 1,

N).

(14)

It is easily seen that the ~ and fl operators are hermitian

~j = ~ ,

/~ = / ~

(j = 1. . . . . N),

(15)

and obey the following a n t i c o m m u t a t i o n rules

{~,, ~j} = {fl,, flj} = 3ij,

{~i, fir} = 0

(i,j = 1 . . . . . N).

(16)

In terms o f the ~ and fl operators the hamiltonians (11) read N

Z . = i X ~j(S.)j,~k

(~ = - 1 , +1),

(17)

j,k=l

where the matrices S. are defined by So = a s -

O~ (~ = - 1 , +1).

(18)

It is easily seen that 5_ 1 is cyclic and 5;+ 1 is anticyclic. If .g. is the transpose of 5. (a = - 1, + 1), the hamiltonians can also be written as N

H . = - i ~. /3j(g.)jk~ k (a = - 1 , +1). j,k=l

(19)

TIME CORRELATION FUNCTIONS IN THE a-CYCLIC X Y MODEL. I

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The equations of motion of the e and fl operators are dk =i[H~,C~k]'

~k = i [ H ~ , f l k ]

(tr = --1, + l ; k

= 1. . . . .

(20)

N).

With (17) and using the anticornmutation rules (16), we obtain N

dk = ~(S,~)ktfit

(tr = --1, + l ; k

= 1. . . . .

(21)

N),

l=1

and in a similar way N

/~k = - Z('S-)k, at

(tr = --1, + l ; k

= 1. . . . . N).

(22)

1=1

Introducing the matrices M~ (~r = - 1 , + 1) defined as M~ = S.S. = .g~S. = k 2 - Dz,

(23)

(the M, are symmetric; M_ x is cyclic and M+a anticyclic), the solution of the equations of motion is given by N

~j(t) -- Z [(cos M,~kt)jk ~Zk

.Jr_

(S~rM; ½ sin M~ t)jkflk]

k=l

(tr = - 1 , + l ; j

= 1. . . . . N),

(24)

=

(25)

N

flj(t) = Z [(cos M~t)jkflk -- ( S ~ M ; ~ sin M~t)ykCtk] k=l

=

- 1, + 1 ; a

1 .....

N).

The correlation functions to be computed in the following sections, will be expressed in terms of the elements of matrices which are functions of the matrices S_ i, -S- 1 and M_ 1, and S + 1, S + 1 and M + 1 for the two models, respectively. These elements can, due to the cyclic/anticyclic nature of the matrices S~, S~ and M,, easily be expressed in terms of the eigenvalues of these matrices. These eigenvalues can, in their turn, easily be expressed in terms of the elements of S~, ~;~ and M~, i.e., in terms of 7 and B. Using the properties given in the appendices A and B it follows that the eigenvalues 2., k of Sa, k a r e given by 2.,k = ( c o s ~ b . , k - - B ) + i T s i n q ~ o , k

(a=

--1, + l ; k

= 1. . . . , N ) ,

(26)

where C~_,,k = k 2 n / N

(k = 1 . . . . .

(27)

N)

and ~ + , . k = (k -- ½)2n/N

(k = 1 . . . . .

N).

(28)

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P. MAZUR AND TH, J. SISKENS

It follows immediately that the eigenvalues J~.,k of So are given by ~-., k = (COS ~bo, k -- B) -- i~ sin ~b., k = 2o,k * (a = - - 1 , + l ; k

= 1 .....

N),

(29)

= 1. . . . .

N).

(30)

A2,k of t~la by

and the eigenvalues

2 = 2~,k~,k = (cos ~b., k - B) 2 A~,k

+ ~)2 sin 2 (~)a,k

(a = - 1 , + l ; k

4. The z component of the magnetization and the internal energy. The z c o m p o nent o f the spin at site j reads in terms o f the c operators

S~ = ½(c~ + cj)(cj - cJ),

(31)

and in terms o f the ~ and fl operators

S; = io~jflj.

(32)

The z c o m p o n e n t of the magnetization o p e r a t o r reads N

M: = i ~ ~jflj.

(33)

j=l

The canonical average of M z is thus given by N

(p~MZ> = i~,

(a = - 1 , +1),

(34)

j=l

where < • • • > stands for quantum-mechanical trace and where p . is defined as p . = exp ( - f l H . ) / ( e x p ( - f l H . ) )

(a = - 1 , + 1 ; fl = 1/kT).

(35)

In appendix C it is shown that

(P~ajflk) = ½i(5.M~tanh½flA4~)ja

(a = --1, + l ; j ,

-~- = ½~jk

--1, +l;j,

(0" =

k = 1. . . . .

k = 1. . . . .

N).

N),

(36) (37)

F o r the z c o m p o n e n t o f the magnetization we obtain N

= i ~ = - 3 Tr (5~M.-½tanh ~fl/~l~)l j=l

= - ¼ Tr [(S. + 5.)NI~-~ tanh ½flNl~] = -½~, p= I

(~c°s ~b~-'e - B)tanh½flA.p Aa, p

(tr = - l ,

+l),

(38)

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where the sign of the A,.p is arbitrary and chosen positive. In the thermodynamic limit we obtain for the magnetization per spin 2~

lim 1 =

4 ~ f d~b (cos q~ - B)tanh ½fld(~)

N-.. N

0

A(q~)

(a = - 1 , +1) (39)

where A(~b) = [(cos q~ - B) 2 + ?2 sin z ~b]~-.

(40)

Thus, in the thermodynamic limit the magnetization per spin is the same for the c-cyclic and the c-anticyclic case. For the internal energy E~ we obtain N

N

E. =- = i Z = i Z (~S.)kJ j,k=l

j.k=l

N

= -½ ~ (rJ~)kj(S, M2 ~ tanh ½flM~)jk = --½ Tr (M~ tanh ½tiMe) j,k=l N

= --½ZA.,ptanhkflA~,.

(41)

(a = - 1 , +1).

p=l

In the thermodynamic limit the internal energy per spin is given by 2It

lim (E./N) = - (1/4x) ~ dq~ A(q~) tanh ½flA((a), N~oo

(42)

0

both for the c-cyclic and the c-anticyclic model.

5. The time-dependent correlation functions of the z components of two spins and the autocorrelation function of the z component of the magnetization. For arbitrary j and k (j, k = 1. . . . . N) the correlation function can be written as [cf. (32)] = --

(~ = -- I, + I).

(43)

With (24) and (25) we obtain


= -

Z [cos M t)k,(cos

t)k.
#j #m>

/,m=l -

-

(cos M~t)kt(S. M~"½sin M~ t)k,

+ (5, M; ~ sin M~ t)k,(COS M~ t)km -

(S.M; ½sin M.t)k,(g.M.

sin

fljflt~m>].

(44)

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P. MAZUR AND TH. J. SISKENS

In appendix D the elements of the tetradics X(~), defined by (x~))j~ - , (X (3)~}jklm ~ , (x~5),)jk,m - O ~ P j / ~ k ~ , / L ) ,

~x(~>~ k er .Ijklm ~- (Pa flj O~kOLlO~m>,

tX~4)~ \ ¢r ]jklm ~ , (~ -- - 1 ,

+~;j,

k, l, ~ -- 1 . . . . .

N),

(45)

are computed explicitly. In terms of the elements of the tetradics X~~) the correlation function (44) reads =

N Z

l,m=l -

-

--

[ ( c o s M~t)kl(COS Mat)km(Xa ½ (3) )jjlm

(cos M~t)u(S. ~ ~ Ma-.I- sin M~t)km(X(~2))jjtm (S. M2 ~ sin M~t)k,(COS M~t)km(X(4))..lm JJ

+ (s~

M~- ~ sm " M~t)kl(S ~ M2 I- sin M~t)km(X.(3) ~ )jj~,],

(46)

where use has been made of the anticommutation rules (16). Substituting the explicit expressions for the elements of X(~), X(~3) and X(~4) obtained in appendix D, into eq. (46), we obtain after performing the summations and appropriate rearrangements

(paS~S~,(t)) = (p~Sj)(p~S~): + ¼(cos M~t + i t a n h 1-2flM~sin M~-~t)kj2 -

¼[(i sin M~t + tanh ½flM~ cos M~t)½(Sa + S~)M2~]2j

~- 1 ~ + ¼[(i sin M~t + tanh ½flM~ cos M~t)~(S~ - S~)M~*]2~

(a = - 1 , +1).

(47)

For the autocorrelation function R~(t) of the z component of the magnetization we obtain

Ra(t) - (1/U)((p~M~M:(t)) -- (p~M~) 2) N j,k=l

.

= (1/4N) Tr (cos M~t + i tanh ½flM~ sm M~t) 2 ½ 21~($~ + Sa)2M~-1] (1/4N) Tr [(i sin M~t + tanh ½flM~ cos M~t)

-

- ( 1 / 4 N )

T r E ( i s i n M ~ t

+ tanhkflM~cos Mot) " 21~(so_

L)~M;']

(ff = - 1 , ÷1).

(48)

And after straightforward calculations we get

R~(t) = (1/4N) Tr [¼(S~ +

Sa)2M~ "1

cosh-2 lgriMe ]-k

- (1/4N) Tr [¼(5~ - 5~)2M2~(1 + tanh 2~flM~)~ cos 2M~t] -

(i/2N) Tr [¼(5a - S~)2M~-' tanh~zv,ata~',~sin 2M~t] ( a - - - 1 , +1).

(49)

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In terms of the eigenvalues we obtain instead of (47)

= + 4

Y~ [ ( c o s A~, ~ t p=l

+ i tanh

½flA~,psin A~,pt) exp i (j

-- k)q~,~,p])2/

1 I1 ~ ((isinA,,pt+tanh½flA~,cosA~,pt)

4

p=l

× (C°S__~_*,p- B ) e x p i ( j _ Aa. p 1 [1 4

pt+tanh½flA,

~ ((isinA~ p=l

x y sin qS~,, exp i(j -

k)q~,.p)3 2

'

p cosA, pt) '

k)d~,p

'

(a = - 1, + 1),

(50)

ao., p

and instead of (49) R,(t) =

1 N ((cos49~,p-B)2cosh-Z½flAa,p) 4Np~=, A2,p 1 ~ ( '2sinEq~" " ) +4N--p=1 ~p 'p(1 + tanh 2½flA.,p) cos2A.p, t i + ~

~ ( y2sin2~b~pt ) ~i-A.,p-- '- anh ½flA., p sin 2A.,p t

p= 1

(a

=

- 1, + 1).

(51)

The expression for the autocorrelation function of the z component of one arbitrary spin follows immediately from (50) by putting j = k. The last term on the r.h.s, of (50) then vanishes for reasons of symmetry. In the thermodynamic limit we obtain the well-known expressions derived by Niemeijer t) for the c-cyclic case. It is clear that the results for the c-cyclic and the c-anticyclic case are the same in this limit.

6. Final remark. In a forthcoming paper we shall combine the relevant parts of the correlation functions for the c-cyclic and the c-anticyclic model in order to obtain the correlation functions for the a-cyclic model (cf. section 2). From the discussion in this paper it is already clear, that in the thermodynamic limit the results for the a-cyclic model will be equal to the results obtained in this paper for the c-cyclic and the c-anticyclic model. Acknowledgment. valuable discussions.

The authors are grateful to Professor S. K. Kim for

268

P. MAZUR AND TH. J. SISKENS APPENDIX A

I. Definition. The N × N matrix A is called cyclic, if Aii=

Aj-i

(i,j=

1. . . . .

(A.1)

N)

and A, = A,_N

(n = 1 . . . . .

1).

N--

(A.2)

2. Properties. 1) I f A is a cyclic N x N matrix, and U is a unitary N × N matrix, defined by Uik = (1/x/N) exp ( - 2 n i j k / N )

( j , k = 1. . . . .

(A.3)

N),

we have

(UtAU)jk =

2k6jk

(j, k = l .....

(A.4)

N),

with N--I

2~ = ~ A, exp ( - 2 n i n k / N )

( k = 1. . . . .

(A.5)

N).

n=O

By inversion of (A.5) it follows that N

2)

A, = ( l / N )

Z2kexp(2gink/N)

(n = 1 -- N . . . . .

N--

1).

(A.6)

k=l

3) I f A and B are cyclic N × N matrices, AB is cyclic, and A and B c o m m u t e . 4) I f A, B. . . . are cyclic N × N matrices with eigenvalues 2k, /q . . . . (k = 1..... N) respectively [ e l (A.5)], and if f is an analytic function, we have N

[ f ( A , a . . . . )], = ( l / N ) Z f ( 2 k , I~k . . . . ) exp ( 2 ~ i k n / N ) k=l

(n = 1 -- N . . . . .

N--

1).

(A.7)

APPENDIX B 1. Definition. The N x N matrix A is called anticyclic, if Aij

-~-

.4j_ i

(B.1)

( i , j = 1 , ' " ", N )

and A , = -- A , _ N

(n = 1 . . . . .

N -

1).

(B.2)

2. Properties. 1) If A is an anticyclic N × N matrix, and O a unitary N × N

TIME

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IN THE a-CYCLIC

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1

269

matrix, defined as

Ujk = (1/x/N) exp [ - 2 7 t i j ( k - ½)/N]

(j, k = 1 . . . . .

N),

(B.3)

we have

(UtAU)jk = ).k6jk

(j, k = 1 . . . . .

N),

(B.4)

with N--I

2k = Z A, exp [ - 2 r d n ( k

- ½)/N]

(k = 1 . . . . .

N).

(B.5)

n=O

F r o m inversion of (B.5) it follows that N

2)

A, = ( l / N ) ~ 2k exp [27tin(k - ½)/N]

(n = 1 - N . . . . .

N - 1).

(B.6)

k=t

3) I f A and 8 are anticyclic N x N m a t r i c e s , A8 is anticyclic and A and B c o m m u t e . 4) I f A, 8 . . . . are anticyclic N x N matrices with eigenvalues 2k, Pk . . . . (k = 1. . . . . N), respectively [cf. (B.5)], and i f f is an analytic function, we have N

[ f ( A , 8 . . . . )], = ( I / N ) ~ f ( 2 k , ~k . . . . .

) exp [2rfin(k - ½)/N]

k=l

(n = 1 - N . . . . .

N-

1).

(B.7)

APPENDIX C In this appendix we shall c o m p u t e the elements of the matrices X, Y and Z, defined as

(j,k=l

. . . . . N).

(C.i)

(For simplicity of notation we omit the subscript a.) It is easily seen, using the a n t i c o m m u t i o n rules (16) and the invariance of traces under cyclic permutation, that

xjk = ½([p, .jqBk),

(C.2)

Yjk = ½6ik + ½([P, ~j]~k),

(C.3)

z;k = ½6jk + ½ (J, k = 1 . . . . . N).

(C.4)

Using the well-known equality #

([p, o~j]fln) = i I (p~ij(--i2)flk)d2, 0

(C.5)

270

P. MAZUR AND TH. J. SISKENS

we have, using (21) and (25), Xjk = ½i I {PdJ(--i2)tk) d2 = ½i ~ Sit I ( P t t ( - - i X ) t k ) d 2 0

1=1

0

N = ½i Z Sit 5 [cos (--i2M~)]lmd2(pflmflk) I, ra=l

0

N

--

½'" Z

sj, I[ M- sin

l,m=l

"

(-12M)]lmd)-
0

N # = ½i E SJ l I [cosh ~M~]lm d2 Z,, k l,m=l

0

N - ½ E

l,m=l

smh " aM " ]Zm d2 X,, k

sJ, I [ 0

N --

k1" ~

Sjt(M -~ sinh

/,m=l

t i m 2') l m Z m k

N -- ½ E Sjl[SM1, m = l

1( c O s h t M ½ -- U)]/m X m k '

(C.6)

where U = N x N unit matrix. In matrix notation we obtain X = ½i(SM -~ sinh tM~) • Z - ½(cosh t M ~ - U). X.

(C.7)

In a similar way we find Y = ½U - ½i(SM -½ sinh t M ~ ) • X - ½(cosh t M ¢ - U). Y

(c.s)

Z = ½O - ½i(SM -~ sinh tiM+) • X - ½(cosh tim ~ - U). Z.

(C.9)

and

F r o m (C.7) it follows that X = i[(cosh t M~: + U)- 1SM -~ sinh tim ÷] • Z.

(C.lO)

Inserting (C.10) into (C.9) we obtain

Z=kU.

(C.11)

Substituting (C. 11) into (C. 10) we have X = ½ iSM -~ tanh ½tiMe;

(C.12)

and finally, by inserting (C.12) into (C.8), we find

Y=kU.

(C.13)

TIME CORRELATION FUNCTIONS IN THE a-CYCLIC X Y MODEL. I

271

APPENDIX D In this appendix we shall c o m p u t e the elements of the tetradics X (1., X t2~, X ~3~, X (4) and X (s), defined as X jklm (1) =

, X(2) jktm

= ,

X(3) jklm : ,

X(4) jklm = ,

X(5) jklm = ,

( j , k, t, m = 1 . . . . .

(D.1)

N).

( F o r simplicity of notation we omit the subscript a.) Using the a n t i c o m m u t a t i o n rules (16), the invariance o f traces under cyclic permutation, and furthermore using the results of appendix C, we have X jklm (1) = ½<[P' O~j]O~kO~lO~m>~- 4Ijklml

,

(D.2)

where the tetradic I is defined by Ijklm = 3jkt~lm -- t~jlt~km ~- 6jmgkt,

(D.3)

X(2) jura = ½<[P, fljJO~k % am>,

(D.4)

X(3) l jm (~kl, jklm = ½<[ D, fl j]O~k O~lflm> ~- ~(~

(D.5)

X(4) jklm = ½<[P, O~j]flkfllflm>,

(D.6)

X(5) jklm = ½<[P, flj]flkfllflm> -~ @Ijklm 1 ( j , k, 1, m = 1 . . . . .

N).

(D.7)

Along the lines of appendix C it is easily found that X (a) = ½i(SA4 - ~ sinh flA4~) - X (2)

-

-

½(cosh fib4 ~ -

U ) - X ( ° + ¼I

(D.8)

and X ¢2) = - ½i(.~M -½ sinh time) • X (') - ½(cosh tim ~ - U)- X ~2).

(D.9)

F r o m (D.9) it follows that X (2) = - i(SM -~ tanh ½tiM½) • X °).

(D.10)

Inserting (D.10) into (D.8) we obtain X <') = ¼I,

(D.I 1)

and it follows immediately that X ¢2) = - ¼i(.~M-~ tanh ½flA4~) • I.

(D.12)

In a similar way we have f r o m (D.6) and (D.7) X (s) = ¼I

(D.13)

272

P. M A Z U R A N D T H . J. S I S K E N S

and X (*) = ¼i(SM -~ tanh ½tiMe) • I.

(D.14)

A l o n g the lines o f appendix C we have for X(3): N X(3) jklm :

¼5jm 5k I _ ½i ~, p=t N

(SM- ~ sinh flM~)jp(poq, O~k 0£l tim)

½ Z (cosh tiM½ -- U)j~

--

p=l

N

= ¼6jmOkl + ½i ~ (SM -~ sinh p=l

(2) flM¢).ip X,.pkl

N

½ ~ (cosh tiM{

-

p=l

mljp ~,T, y ( pklm 3) • -- V

(D.l 5)

It is easily seen that N N --(3) = l(~jm 6k, ÷ ½i ~ (~;M --~ ½ Z ( c o s h t i m ½ ÷ U)jp Xpklm p=l p=l

..(2, sinh tiM ,-)jp ampk,

(D.16) or, N p=l

(cosh 2 ~H 1 r~u½~ tvl )jp V'(3) ~pklm N

= ¼6jm6~,, + i ~ (.SM-& sinh ½tiM¢ cosh ½flM~)jp "'V(2)mpk,"

(D.171

p=l

A n d finally we obtain

X(3) nklm

N

: ¼Z j=l

(cosh- 2 ½flM~)n j (~jm t~kl N

+ i 2

j,p=l

--(2)

( c ° s h - 2 ½flM~).J(~M-~ sinh ½tiM~ cosh ½flM)i p A,.vkt N

= ¼ ( c o s h - 2 ½flMaZ)nm6kl +

i ~ (SM -~ tanh

l~A/l~"tJnp Z" y ( mpkl 2) =21Jtvl

p=l

¼(cosh -2 ~-/~M ' *),,m6kt N

+ ¼ ~ (SM - * tanh ½flM*).p(SM -~

tanh ½flM½)mq[qpkl

p,q=l

(n,k,l,m REFERENCES 1) Niemeijer, Th., Physica 36 (1967) 377. 2) Lieb, E., Schultz, T. a n d Mattis, D., A n n . Physics 16 (1961) 407.

= I.....

N).

(D.lS)