On the transverse susceptibility χxx of the anisotropic one-dimensional XY model in the presence of a magnetic field in the z direction

Phvsica 76 (1974) 445-485

Q North-Holland Publishing Co.

ON THE TRANSVERSE ANISOTROPIC PRESENCE

SUSCEPTIBILITY

ONE-DIMENSIONAL

OF A MAGNETIC

FIELD

H. W. CAPEL, E. J. VAN DONGEN

xXx OF THE

XY MODEL IN THE

IN THE

z DIRECTION

and TH. J. SISKENS

Instituut-Lorentz, Rijksuniversiteit te Leiden, Leidey, Nederland

Received 7 June 1974

Synopsis

A high-temperature expansion up to order b6 is given for the transverse susceptibility xXx of the one-dimensional anisotropic spin-cyclic XY model in the presence of an external magnetic field B in the z direction. It is found that xXXdepends strongly on the anisotropy parameter y and weakly on B. It is shown that the expression for xXx is different from its analogue in the c-cyclic XY model, even in the thermodynamic limit.

1. Introduction. The one-dimensional anisotropic XY model was introduced by Lieb, Schultz and Mattisl) in 1961. Katsura*) evaluated an exact expression for the free energy, also in the presence of a nonzero magnetic field in the z direction. Niemeijer3) derived the time correlation functions of the z components of two spins. From refs. 2 and 3 one can obtain an expression for the longitudinal susceptibility i.e. the zz component of the susceptibility tensor. In ref. 3 the spin correlation functions have been calculated for the so-called c-cyclic XY model, but in the thermodynamic limit these correlation functions can be shown to be equal to those for the so-called a-cyclic XY model, which is cyclic in the spins. A detailed investigation of the relation between the a-cyclic and the c-cyclic (or c-anticyclic) model has been given in refs. 4, 5 and 6. In ref. 5 it was shown in particular that the time-dependent zz correlation functions in the a-cyclic model can be expressed in terms of canonical averages referring to the c-cyclic or c-anticyclic hamiltonian. The calculation of the transverse susceptibility (i.e., the xx or yy components of the susceptibility tensor) is much more difficult, since, as will be shown later on in this paper, the canonical averages involve both the c-cyclic and the c-anticyclic hamiltonian. This difficulty does not show up in the calculation of the timeindependent xx or yy correlation functions, cf. ref. 1 for T = 0 and refs. 7 and 8 for nonzero temperatures. Time-dependent correlation functions have also been 445

446

H. W. CAPEL, E. J. VAN DONGEN

AND TH. J. SISKENS

considered by McCoy et al. for T = 0 in two asymptotic cases related to the time and the distance between the spinsg). In the present paper we give a high-temperature series expansion for the transverse susceptibility xxx in the thermodynamic limit using expansions for the xx spin correlation functions for imaginary times (in the absence of a magnetic field in the x direction). In section 2 we review some properties of the a-cyclic XY model as given in refs. 4 and 5, which are relevant for the present calculation. In section 3 we express the xx spin correlation functions in terms of canonical averages involving the density operators of the c-cyclic and the c-anticyclic models. However, in the case of xx correlation functions, we must evaluate averages of complicated operators, which contain in particular a factor involving the c-cyclic as well as the c-anticyclic hamiltonian. This factor’prevents a rigorous calculation of the timedependent xx spin correlation functions. In section 4 this factor is investigated in detail and in particular those aspects which are relevant for the calculation of the high-temperature expansion for xxx up to a certain order. In section 5 we calculate the transverse susceptibility up to order p6, (,8 = l/kT). In the calculation use is made of the thermodynamic Wick theorem in a version due to Bloch and De DominicislO). In section 6 we give a few graphs and also a discussion on some aspects which can be of interest, such as the dependence on the inverse temperature p, the magnetic field B and the anisotropy parameter y. In section 7, we give a high-temperature expansion for the susceptibility xxx of the c-cyclic XY model. The result turns out to be different from the transverse susceptibility in the a-cyclic model, even in the thermodynamic limit. This demonstrates in an explicit way the difficulties which can arise, if one uses the c-cyclic model for the calculation of physical quantities, even in the thermodynamic limit. 2. Review of the a-cyclic XY model. We consider a linear chain of N spins 4 in an external homogeneous magnetic field B along the z axis. The spins have anisotropic nearest-neighbour interactions. The hamiltonian of the system (the socalled XY model) is given by Jr = f

j=l

[(l + y) s;s;+,

+ (1 - y) sysJy+1 -

=$I,

(2.1)

where

s;,, = s; Performing

and

SG,,

the transformation

Sj” = 3 (aj + aj), SJ =

aIaj - +

= S:.

to paulion

(2.2) operators

Sj’ = (a: - aJ&!i, (j=

l,...,N)

(2.3)

SUSCEPTIBILITY

and subsequently

xxx OF ANISOTROPIC

the Jordan-Wigner

u~=exp(is~$c~c,)cJ,

aj = exp the hamiltonian

assumes

- :

447

,...,

N),

al=ct,, (2.4)

cj,

(j

=

2, *.., N),

a, = Cl,

the form CjCj +

SC? = $NB - B i j=l

XY MODEL

transformation

(j=2

j-l -ix 2 c&zk k=l >

ONE-DIMENSIONAL

+ i

j=l

[(cJc~+ 1 +

YcJcJ+,)

+

h-C.1

[(C&C, + ycfvci)

where the c operators satisfy fermion anticommutation relations. The model under consideration is cyclic in the spin operators and consequently in the a operators. It is therefore called the a-cyclic XY model. It is, however, not cyclic in the c operators. By neglecting the last term in the right-hand side of (2.5) one obtains the so-called c-cyclic XY model, which is much easier to handle. For certain quantities it can be shown that in the thermodynamic limit the c-cyclic and the a-cyclic XY models give the same results, c$ refs. 4-6. (This can of course not be true for finite systems.) In the present paper it will be shown that this simplification is not always correct, even in the thermodynamic limit. For this purpose we shall present a detailed investigation of the transverse susceptibility, corresponding to an infinitesimal magnetic field in the x direction. The a-cyclic hamiltonian (2.1) can be written 2

= 2-l

+ fh(P

+ 1),

where the c-cyclic hamiltonian .%-I

=&NB-

B;c:c,

(2.6) 2-r

+

is given by + yc:cj’+l) + h.c.1

(2.7j

_i=l

and where h and P are defined as h = - [(&cl

+ y&fj

+ h.c.]

(2.8)

and P = exp (inj$lcIcj). I

(2.9)

H. W. CAPEL, E. J. VAN DONGEN We now define operators

P,=-&(l

AND TH. J. SISKENS

P, for G = f 1 by (2.10)

+aP),

where 1 is the unit operator. P,” = P,

The operators

P,, are projection

operators,

i.e.

(0 = kl),

which is obvious

(2.11)

from the trivial relation

P2 = 1. In addition

(2.12)

they have the following

POP_, = 0 [PC,

(CT= -1,

l),

((r =

-1,

z-11 = 0

[PO, 121= 0

(d = -1,

it follows immediately P,Y? = P,.Yrc

%I=

(2.15) since P commutes

that (cr = -1,

(2.17)

+1),

of the so-called

c-anticyclic

model, defined by (2.18)

&?_I + h. ifSis

= Pb_W,)

an analytic

function,

(a = -1,

then

1).

In order to evaluate physical quantities involving a-cyclic XY model, one can introduce the projection relation CPc 0

with X._ 1 and h.

(2.16)

is the hamiltonian

More generally, POfW)

(2.14)

I),

+ hP1,

= 2-1

where S1

(2.13)

1).

Eqs. (2.14) and (2.15) are obvious, From the relation z

properties

= 1.

(2.19) canonical operators

averages for the P, by the trivial

(2.20)

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

449

Using the projection and (anti)commutation properties of the operators P,, averages for the a-cyclic XY model can always be expressed in terms of averages with respect to the c-cyclic and the c-anticyclic hamiltonian. Before we do this, we shall briefly review the c-cyclic and c-anticyclic XY model. These models can be conveniently described in terms of hermitean operators 01~ and /Ij (j = 1, . . . . N), which are linear combinations of the c operators, viz.,

-!- (Cj +

ocj =

CJ),

J2

Bj=~(Cj-C:)

and which have the anticommutation {&i:i,OLj}

=

dij3

{(xiypj}

= 0

(i,j

The hamiltonians

PO =

i

{Bi,r6j)

#,,

F j,k=l

(SJjk

where the matrices (sn)jk

=

=

(2.21)

(j=l,...,N)

relations 6ij,

= 1, ...) N).

(2.22)

(G = - 1, + 1) can now be expressed

as

(2.23)

&j@k,

S, (S_,

is cyclic and S1 is anticyclic)

are given by (2.24)

(sc)k-j

and (SO)” = -Bd,,,

+ + (1 + Y) S”,, - -To (1 + Y) ~“,I-,

+ t(1

- Y)&*-I

- +o(l

Herenrunsfroml -NtoNThe dynamics of both models erators. We have the relations

o(~(Z, 0) = ;

(2.25)

- Y)d,,N-I.

1. can be solved easily in terms of the LYand /I op-

[(cash M$t)jk &k - i (S,M,+

sinh M,~T)~,ok],

k=l

(0

=

Bj (t, 0)

-1,

1;j

= 1, . ..) N);

=,il[(cashM%)jkbk + i (SCM,*sinh

((7 = -1,

l;j

= 1, . ..) N);

(2.26)

M$T)~, o(k], (2.27)

H. W. CAPEL,

450

E. J. VAN DONGEN

where 0 (r, a) for an arbitrary

operator

matrix

M, = S,&, = $,S,

(CT= -1,

TH. J. SISKENS

0 is defined by

0 (z, c) = exp (z%“,) 0 exp (-TX”,) and where 3, is the transpose

AND

(CT= -1,

l),

(2.28)

of S,, and M, is defined by 1).

(2.29)

The matrices M, are symmetric (M_, is cyclic and M,, is anticyclic). The relations (2.26) and (2.27) can be obtained from eqs. (24) and (25) in ref. 4 by substituting t = -ir. The reason for introducing the 01 and ,!I operators lies in the simple expresfor sion (2.23) for the hamiltonian X0,. This expression is not only convenient the solution (2.26)-(2.27) of the equations of motion, but, as we shall see later on, it enables us to apply Wick’s theorem in a convenient way without performing the transformation to the representation in which z+?‘~is diagonal. We finally express the components of the spin operators in terms of the LXand /3 operators. Using (2.3) (2.4) and (2.21) we have

s;=fL,

s; =

sjx=

(~$&lk)$-

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

_L,

(j=2 ,..., iVj?

J2

ST = iajaj

(2.32)

relations

(2.22) it follows that the operator

P, defined

zjfil U/i) ajBjl>

anticommutes tp,

(2.31)

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

From the anticommutation by eq. (2.9), and reading

P

(2.30)

with the 01 and /3 operators, /lj) = 0

&j}

Consequently,

(2.33)

P anticommutes

{P, S,x> = 0, [P, SJ] = 0

(P,SJl

Le.,

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

with SJ” and SJ and commutes =0

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

(j=

I,...,

Iv),

(2.34j with Sj’: viz., (3.35) (2.36)

SUSCEPTIBILITY

For the projection

xxx OF ANISOTROPIC

operators

ONE-DIMENSIONAL

P, we therefore

have the following

XY MODEL

451

properties

P,S;’ = S,“P_,

(C = -1,

1;j

= 1, . ..) N),

(2.37)

P&

(a = -1,

1;j

= 1, . ..) N),

(2.38)

= sjyP_b

P,SJ = S,‘P,

(Q = -1,

1;j

(2.39)

= 1, . ..) N).

From eqs. (2.14), (2.15), (2.18), (2.19), (2.28) and (2.37), (2.38) and (2.39) we obtain the relations P,S,” (T)

P, exp (~2”) ST exp (-t%)

E

= exp (t&J

S,“P_, exp ( -zs)

= Sjx(t, 0) exp (z%,) (0 = -1,

1;j

= exp (t&J

PCS,” exp (-&)

= exp (z%“,) ST exp

exp (-t&J

( - t%_,) P_,

P_,,

= 1, . ..) N);

(2.40)

and similarly

P,J,Y (7) = Sj’ (z, 0) exp (UP”,) exp (-X%-J (a = -1,

P-, ,

1; j = 1, . . . . N);

P$J (t) = s; (7, 0) P,

(2.41)

(C = -1,

1; j = 1, . . . . N).

(2.42)

Note that the anticommutation relations (2.35) lead to an additional factor exp (~2,) exp (-t&+ _,) which contains the c-cyclic as well as the c-anticyclic hamiltonian. This factor prevents in general an exact calculation of time-dependent correlation functions involving the x and y components of the spins, in contrast with the zz correlation functions which have been calculated by Niemeijer3). 3. Transverse susceptibility of the a-cyclic XY model. In the presence of a magnetic field B, in the x direction, the hamiltonian of the system is given by 2 - B,M,, where Z is the hamiltonian of the XY model given by eq. (2.1) and M, = I:= 1 SF is the x component of the magnetization operator. The transverse susceptibility per particle for B, = 0 in the thermodynamic limit is defined by the relation xXx 3 x = lim B,+O

lim (/?N)-l N-cc

5

In Tr [exp -,!I (2 x

- B,M,)].

(3.1)

4.52

H. W. CAPEL, E. J. VAN DONGEN

AND TH. J. SISKENS

From eq. (3.1) one can derive the relation*

In eq. (3.2) use has been made of the notation = Tr 0,

(3.3) 0 exp (- ts)

O(t) = exp (~2) for an arbitrary

operator

e = exp (-Bs)

,

(3.4)

0 and

-‘,

(3.5)

is the density operator corresponding to the hamiltonian (2.1). In view of the translational invariance of the hamiltonian (2.1) and the boundary condition (2.2), the susceptibility can be written

(3.6)

The right-hand side of eq. (3.6) contains important contributions k = N, N1, N2 ,.... These contributions can be related k = 2, 3,4... by using the KMS property

W CT> B) =
operators

(es: (7) SC-,> = and the susceptibility x

=

<@

I dt

from the spins to those from

(3.7)

3

A and B. From (3.7), (2.1) and (2.2), we have


S;(z) S;)

The right-hand side of eq. a-cyclic hamiltonian. We eq.(2.20) the averages can c-cyclic and the c-anticyclic

4 s;+2>

(p = 0, 1,2, . ..)

(3.8)

is given by

+ 2D$1
(3.9)

(3.9) contains canonical averages with respect to the do not know these averages a priori, but by using be expressed in terms of averages with respect to the hamiltonian.

* Here it is assumed that fhe limits in eq. (3.1) can be interchanged.

SUSCEPTIBILITY

xxx OF ANISOTROPIC

By using eq. (2.19) the spin correlation expressed

ONE-DIMENSIONAL

functions

appearing

453

XY MODEL

in eq. (3.9) can be

as

WC (9 ST+1) =

(c zo~PbP~)1- l 0

cCJz, G?YpJ:(4 s;+1)

x

(P = 0, 1,2, . ..I.

(3.10)

where

Z, =


(d = -1,

(3.11a)

1)

and

en = Zi

l

(C = -1,

exp (-/WJ

(3.11b)

l),

are the partition function and the density operator of the c-cyclic and c-anticyclic models, respectively. After applying eq. (2.40) with P, to S:(t) and with P_, to S,X+, successively and using the explicit relation (2.10), eq. (3.10) can be written as


C Z, WC CT,4 exp CT%,>exp (-72-J d

;

S,X+,(1 + UP)>

zo(@a(1 + m> (3.12)

(p = 0, 1, 2, . ..). By using eq. (2.30), the right-hand the operators o( and /l. The result is

(es (4

side of (3.12) can be expressed

in terms of

s;+1) = i c 77 GA?(1 + m>1-I 0

x 4 C Z, Gwl CT,4 exp&@“,Iexp(-r&J [ 0

x

i

jfilWi)ajPj) &+I (1 + UP>> 1

(p = 1, 2, .*.). (3.13)

n ..

(For p = 0, the factor . should be replaced by the unit operator.) One of the apparent complications in the right-hand side of eq. (3.13) is the occurrence of the factor exp (z%,,) exp (-z&?-J which contains the c-cyclic and the c-anticyclic hamiltonian. Such difficulties do not appear in the case of zz correlation functions (@f (T) S,l+ 1>, CJ the difference between eqs. (2.41) and (2.42). In the special case of time-independent correlation functions with t = 0, the right-hand side of eq. (3.13) can be evaluated by using a modified version of

454

H. W. CAPEL, E. J. VAN DONGEN

AND TH. J. SISKENS

Wick’s theorem due to Bloch and De Dominicislo), which does not involve ordered products of operators. This version can be formulated as follows. Consider a one-particle hamiltonian

time-

(3.14)

where the qr are fermion operators. Let Al, A,, . . . . AZ, be arbitrary linear combinations nihilation operators 7; and l;lk, and let e = exp (-/3*) density operator, then

WI&

a**AZ”)

of the creation

and an-

(exp (-BzF))-’

be the

=j,,jr,~,j _1(-1)‘(p).a*
(3.15)

where j, runs through the values 2,3, . . . , 2n; iz is the smallest integer different from 1 and j, ; j, can have all values different from 1, j, and i2 ; i3 is the smallest integer different from 1, j,, iz , j, ; and so on. Y(P) is the sign of the permutation 1 j,

i2 j, a-.

12

***

(

i,

2n -

j, 1 2n > *

The right-hand side of eq. (3.15) is by definition the pfaffian consisting triangular array of averages (QAiAj) with 1 I i < j < 2n. Hence

WI&

..a A,,)

= Pf (A,,

A,,

of the

. . . , A,,)

2,,--1A2n). (@A

(3.16) The relation between the Wick theorem and pfaffians was first noted by Caianielloll). An elegant proof of the modified version eq. (3.15) was given by Gaudin12) and this formulation has also been used in previous treatments of the XY model, cf. Lieb, Schultz and Mattis’) and McCoy et ~1.~~~). If z = 0, the right-hand side of eq. (3.13) can be evaluated in a straightforward way by noting that for each value of 0, i.e., the c-cyclic case and c-anticyclic case respectively, the operators OLand /I can be expressed as linear combinations of the

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

455

fermion creation and annihilation operators ri,, qka that diagonalize the hamiltonian 2, , cjI, the appendix of ref. 5. The result is then a pfaffian consisting of the averages (e,AiAj), where the Ai and Aj correspond to operators LXand 8. The averages (Q,AJ j) have been calculated previously by Mazur and Siskens4). The result is <@P#j>

=

@ij

Pij

(f~ = -1,1 (a

=



=

(Q,aipj)

= 3i (S,M,‘tanh

-1,

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

(3.17)

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

#?M$ij

(G = -I,

(3.18) 1; i,j = 1, . . . . N),

(3.19)

where the matrices S, and M, have been defined by eqs. (2.24), (2.25) and (2.29). For z = 0 and finite values of /?, i.e., at nonzero temperature, it is obvious that the terms in eq. (3.13) involving the operator P do not give a contribution in the thermodynamic limit. In fact we then have an average of a product in which the number of different operators Bk is of order N. In view of Wick’s theorem and eq. (3.19) this gives rise to a factor which has the order of magnitude (tanh&!IM!)“. (We shall not go into the details of the time-independent correlations which have been calculated by Lieb, Schultz and Mattis’) for T = 0 and by McCoy et al. for nonzero temperatures7*8).) We now consider eq. (3.13) in the case that z # 0. If we may neglect the factor exp (ts”,) exp (-t#_,), there is no problem. Using the explicit solution (2.26) for &I (t, o) and Wick’s theorem, the correlation function (3.13) can be expressed in terms of a finite pfaffian, since in the same way as above the operator P does not contribute in the thermodynamic limit. However, it will be shown in the present paper that it is by no means justified to neglect this factor. Finally, it may be noted that McCoy et aLg) have also studied the behaviour of time correlation functions (QS: (r) S,X+, ) with t = it for T = 0 in the two limiting cases p + co, t 4 p, and t + co, p 6 t, respectively. The effect of the factor exp (TV?“,) exp (-t#_,) could be avoided by considering four-spin correlation functions of the type (gS,” (t) S,“(T) S,XS:). Such correlation functions contain an even number of S” operators with the same time dependence, Then from eqs. (2.14), (2.15), (2.18), (2.19) and (2.37) it can be seen that P,SI” (z) s;(r)

= si” (z, 0) s; (t, 0) P,

(3.20)

and by using Wick’s theorem the four-spin correlation functions can be expressed in terms of pfaffians. By choosing j and I in such a way that the distance between the spins j and i and between the spins 1 and k is sufficiently large, one can argue that in the thermodynamic limit the decoupling

(es: CT)s;c4 s,“s;>

N


s,“>
m

(3.21)

456

H. W. CAPEL, E. J. VAN DONGEN

AND TH. J. SISKENS

will lead to correct expressions for the spin correlation functions. The results obtained by McCoy et al. are based on asymptotic properties of Toeplitz determinants in the limit that the number of rows and columns tends to infinity. In the following section we shall investigate the factor exp (TX,,) exp (-tX_,).

4. Time-dependent factor. In this section we consider in detail the time-dependdent factor exp (TX”,) exp ( --cJ?_,) occurring in eq. (3.13) for the spin correlation function. From (2.18) we have the relation

a?_, =

S,, - oh

(G = -1,

(4.1)

1)

and in order to handle the time-dependent factor expansion to the operator &‘_,. The result is

exp(tP,,)exp(-tZ’_,,)

(u = -1, Since it can details We

= 1 + f@jdtr]dz, q=1 0

we can apply a Dyson-series

0

(4.2)

1).

the operator h is only a local hamiltonian, with a finite operator norm Ilhll, be shown that the right-hand side of eq. (4.2) is always convergent. Some of the proof will be given in appendix A. write

h=

c

(4.3)

v,%BN+l-,,

&=l,N

where VI

=

-i(l

+ y),

vN =

-i(l

-y).

(4.4)

After inserting eq. (4.3) in eq. (4.2), the qth order of perturbation can be written as sum of the 2q terms which are characterized by the values Ed, Ed, . . . , cq corresponding to the q factors h. Consider now the contribution of one of these 24 terms to the numerator in the right-hand side of eq. (3.13). It can now easily be seen that the terms with the operator P do not give a contribution in the thermodynamic limit. In fact in view of eqs. (2.26) and (2.27) we can also apply Wick’s theorem to the time-dependent operators LX(t’, cr), /3 (t”, o) and an arbitrary term of the resulting pfaffian contains at least N - p - q different operators /?, which in view of eq. (3.19), would lead to a contribution with the order of magnitude (tanh $pM’)N-P-q.

SUSCEPTIBILITY

After neglecting

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

the terms with the operator

457

XY MODEL

P, the correlation

function

(3.13)

can be written

+

zq-I

2 8 jdt,

g=l

1

TdT, ... ,s dt,

0

0

v,,v,, . . . vEq

El*E~,...,E~=~,N

. . ..~.Eq(24)0))~N+1-Ea(t~,a),nl,Bl, ....%.pa,%J+l) )I (4.5) The pfaffians occurring in eq. (4.5) are defined by eq. (3.16), if the density operator Q is replaced by en, i.e., the density operator corresponding to the c-cyclic and c-anticyclic models, for o = - 1, + 1 respectively. In the right-hand side of eq. (4.5) we can also omit the factor x,, Z,/(Z, + Z- 1). This follows from the fact that each pfaffian Pf, occurring in eq. (4.5) multiplied with the corresponding factor (T”is independent of 0, i.e., CT”Pf, (. ’ .)

independent

of C.

(4.6)

In order to prove eq. (4.6) we use Wick’s theorem and we consider an arbitrary term of the pfaffian. This term involves a product of CJ+ p + 1 averages (QJ (z, cr.)A’ (T’, o)>, where A=ni

or

A’ = aj

pi;

For the subscripts

or

pj.

(4.7)

i and j we have two possibilities:

a>

i is an index of the order

1,

i.e.

i = U(l),

if

i 4 N;

b)

i is an index of the order

N,

i.e.

i = O(N),

if

N - i < N.

(4.8)

Then by using eqs. (2.25)

(2.26)

(2.27) and (3.19), it can be seen that

<@,ACT,4 A’ CT’> 4>

independent

of (T, if i and j are both O(1) or O(N), (4.9a)

<@,A (~2 a) A’ (T’, Q))

proportional

to G’,

if i = 8(l), j = Lo(N), or if i = O(N), j = O(1).

(4.9b)

H. W. CAPEL,

458

E. J. VAN DONGEN

AND TH. J. SISKENS

If we now consider an arbitrary term of a pfaffian corresponding to the qth order of perturbation, the number of averages (4.7) that are proportional to G, c$ (4.9b), is equal to q - 2Q, where Q is the number of averages (e,A(t,o)A’(r’,a)) where i and j are both U(N). From this eq. (4.6) is obvious since c?Jo~-~~ = 1. From eqs. (3.9) and (4.5) with the factor cg ZJ(Z, + Z_,) omitted, we have a formal expression for the transverse susceptibility per particle. The actual evaluation of the right-hand side of (4.5) however, is extremely tedious. First of all, the number of pfaffians as well as the number of terms contained in each pfaffian increases rapidly with the order of perturbation. Secondly, the actual calculation of the factors
x =

c &Pm,

(4.10)

m=l

where m I Y. The spin correlation series in t and ,!l, i.e.,

functions

and in view of the condition satisfying

(@ST (t) S,X+1) can

m 2 v, we can restrict

l+/&u4v-1.

be expanded

ourselves

as power

to values of ;1 andp

(4.12)

This has three advantages. First of all, we can restrict ourselves to a limited number of orders q of perturbation, in fact q 4 v - 1. Secondly, we can restrict ourselves to a finite number of neighbours, since, if q is limited and p is too large, we would have too many different operators Bk, each leading to an average (eaLxiok), which in view of (3.19) is at least of order p = (kT)-‘. Finally, in the calculation of the averages (@,A (r, o) A’ (r’, a)), the matrices (cash Mzr), (SCM,’ sinh Mzr) and (S,M,* tanh +/3M%), occurring in eqs. (2.26), (2.27) and (3.19) can be expanded as power series in t and p, respectively, and all matrix elements up to a certain order v can be calculated analytically. Although the calculation of the correlation functions subjected to the condition (4.12) is straightforward using eq. (4.5) it will turn out to be convenient to use a slightly different method which avoids the multiple integrations over We shall define a “Liouville” operator s0 by its action on an arbit1,72, ...‘T,. trary operator A, uiz. LZcA = [So,

A] + Aah

(c? = -1,

l),

(4.13)

459 i.e. Y”, consists of the commutator with SC?,,and a multiplication the operator oh. From the definition of the operator 0 (r, C) = exp (ts”,) we have the equation $

O(t,c)

(c = -1,

exp (-zS_-,)

on the right by

(4.14)

l),

of motion

= W”,, O(t,Cr)]

and in view of the definition

(4.13), the formal

solution

function

1)

(4.15)

of (4.15) is given by (4. 16)

((T = -1,l).

0 (t, G) = exp (2,~) For the spin correlation

((r = -1,

+ O(r,o)oh

(3.13) we have the expression

Here use has been made of the fact that the terms involving the operator P and the factor x0 Z,/(Z, + Z_,) in eq. (4.5) can be omitted in the thermodynamic limit, c$ the discussion below eq. (4.4) and eq. (4.6). This implies in particular that the right-hand side of eq. (4.17) should be independent of CT.This conclusion can be verified explicitly by using eqs. (4.9a) and (4.9b) and the fact that an arbitrary term of 2:: which contains m operators CCand ,4 with an index O(N) is proportional to (T”. This statement can be proved easily by induction. The various powers of P’, can be calculated directly using the commutation relations (2.22) Lx,,

a.il = -i f (Sa)jIcBk,

(4.18)

k=l

[x09

Bjl =

ikfl(SJk_i

ak

(G

=

1;j

-1,

= I, . . . . N).

(4.19)

As a result, we have P’, = ah = -2io %‘,z = [S,, = (SE,

(S_,CX,@~ + S,&,,!?,),

(4.20)

ah] + a2h2 + Sf) - 2aB (S_,

+ 2aS-1Sl

(+G-~

- S,) (01~0~~- bJN)

+ BILl

- a2aN - B2/3d (4.21)

- 8S--1&44~31, 22

= [X”,, &]

+ zzah

(a = -1,

1).

(4.22)

460

H. W. CAPEL,

Here S_1

E. J. VAN DONGEN

S1 are

notations

(S,), =

(So)_, = s-1 = &(I + y),

AND TH. J. SISKENS

for

matrix

s, = 3 (1

elements

- y).

(4.23)

The explicit formula for L.Z’p,” is rather complicated and is given in appendix B. We now consider the general structure of 92. By using the anticommutation relations (2.22) each operator 0~~or pL which occurs twice can be eliminated to give a factor ++, the sign depending on the number of interchanges that are needed to bring the two operators next to each other. Then 9: can be written

xJ=c

Ir,I i,
c

ci’~;2_.l,;j,“. j,aip12

2

,..
j,-cjic

...

cxiJjl ... pj,.

(4.24)

... -cj,

If n increases, the number of operators k + I involved in a term on the right-hand side can increase. This is due to the multiplication on the right by the operator ah; in each step two additional operators mlBN or o(&r will be added and if these operators do not occur in the original term, the number of operators will increase by 2. On the other hand, the number of operators LXand ,!l cannot increase, if we take the commutator [A?“,, . ..I. Hence, for increasing values of n, the powers 9: can contain many operators LXand /I, which can be considered as many-body interactions within the context of the c-cyclic and the c-anticyclic XY models. This feature explains the complicated structure of the time-dependent correlation functions (QS: (z) S,X+1). The operators 9: have the following two general properties: a)

b)

if n is even, then k = 1 is even,

(4.25a)

if n is odd, then k = I is odd;

(4.25b)

consider

a nonvanishing

term in (4.24) with indices i, , i2, . . . , ik;

jl , . . ..Jl..

From these indices 3 (k + I) indices are of order U(1) the other 4

(k + l)indices are of order O(N).

(4.26)

The proof of eqs. (4.25) is trivial by induction. The proof of eq. (4.26) is somewhat more involved and will be given in appendix C. We finally consider the contribution of an arbitrary term on the right-hand side of (4.24) to the spin correlation function (4.17). The contribution is proportional to

Using eq. (2.26) we can expand 01~(z, cr) as a power series in z. Consider now the contribution due to an operator yi occurring in the rth-order term of this expan-

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

461

sion. (Obviously yi = ai, if r is even and yi = pi, if r is odd.) Then the contribution to the spin correlation function is proportional to

In order to evaluate the average in eq. (4.28), we eliminate all operators K and p which occur twice. Then we are left with an average of a product of operators LX and /3, each occurring once. The average can be different from 0, if the number s of different operators b is equal to the number of different operators 01. Then the result can be expressed in terms of a 2s-dimensional pfaffian, each term of which contains s contractions (Q,aipj). The pfaffian will have the order of magnitude p” = (/CT)-“, provided that it contains at least one term so that in all the contractions (e,aipj), j - i = 0, ) 1, -t(N - 1). If this is not the case the pfaffian will have at least the order of magnitude p+2. Thus one can evaluate in a systematic way all coefficients c?L in eq. (4.11) satisfying il + ~1 2 v - 1, for a certain value of v. (It may be noted that the 2s-dimensional pfaffian obtained after removing the operators which occur twice, can also be written in terms of an s x s determinant, cf. Lieb, Schultz and Mattis. [In our calculation this will not be an important simplification, since in general in a sufficiently large pfaffian only a few terms will have an order in /3 that is low enough to satisfy eq. (4.12).]) In the following section we shall calculate the transverse susceptibility up to order 8” (v = 6). We then can restrict ourselves to the terms ~“/3” in the spin correlation functions satisfying 3, + p I 5. 5. Correlation functions. In this section we investigate the spin correlation functions (4.11) and we shall restrict ourselves to the terms P/3” with 3, + ,U 5 5. Instead of writing down complicated expressions for various powers of _Yz, we shall select the terms that are relevant from our point of view, i.e. a term in the right-hand side of eq. (4.24) is called a “relevant term”, if it can give a nonvanishing contribution ~~~‘3’~” with L + ,u I 5 to some spin correlation function <& (t) S,“+ 1). Obviously, in eq. (4.17) the terms 9: with n > 5 are irrelevant. We now consider 9:. In the average (4.27) we must take the O-order term in the expansion of a1 (z, 6) in powers of z, i.e. ml (z, a) = cxl. Then after eliminating the two operators (Ye occurring in eq. (4.27), all operators (II and /3 should occur twice, otherwise we would have at least one additional factor of order /I = (kT)-‘. This is not possible, however. [From (4.25) it follows that each term of 9: contains an odd number of operators CCand an odd number of operators ,9; from (4.26) it follows that at least one of the indices il ... ik, j, . ..j. occurring in (4.27) must be of order U(N) and if CX,(z, C) = 01~ each operator with an index O(N) occurs only once.] Hence, 9: is “irrelevant”. We now turn to 32. 92 itself is of order x4, so that we only can have an additional factor t, arising from the first-order term in t in the expansion of o(~ (t, cr),

462

H.W. CAPEL,

or an additional 1).

E.J. VAN DONGEN

AND

TH.J.

SISKENS

factor /I, arising from one average (Q+x~/~~), where j - i = 0, rfi I,

First of all 3% has a constant term which‘does not contain operators oc and @. This term can give a contribution z-c4 to the autocorrelation function (4.27) with p = 0 and a contribution z:t”/3 to the nearest-neighbour correlation with p = 1. Obviously, the constant term can give no contributions %ts, since the first-order term in the expansion of o(r (t, o) involves operators B1, p2, /IN and we are left with at least one average (e&3j) which would lead to an additional factor p. We now consider the terms which contain two operators oliaj. Obviously, from (4.20, j must be an index of order 6(N). The first-order contribution of 0~~(t, (r) involves operators 8, so that we are left with an operator aj, where j = O(N), which occurs only once in (4.27) and which would lead to at least one additional factor /3 = (kT)-‘. So we can restrict ourselves to terms with &I (r, 0.) = oil, which lead to one average (~,ol~,Q,) which is proportional to /3. The only relevant term is 01~01,,,,giving a contribution zr4,!I to the nearest-neighbour correlation with p = 1. All terms @t/Y,are irrelevant. If we consider the first-order contribution in t of 01~(z, G), we are left with the operator LY, in (4.27) which would lead to an additional factor p. If we take the O-order contribution, i.e. CX~(t, a) = LYE,the two operators &I occurring in (4.27) can be eliminated. The operator /Ij has an index j = O(N) and a nonvanishing contribution can only occur in an average (e ,,0,J~,+ 1), where p 2 1 and this average is at least of order O(j3”). Finally we can consider operators (x~cx,/?J?~which contain two indices of order O(N). We can restrict ourselves to the terms with 0~~(r, 0) = ~1, leading to an average (Q+x~~~J, where the indices j and I are both of order U(N). This gives an additional factor /3. In addition all remaining operators in eq. (4.27) should occur twice. The only relevant terms are those for which i = 2, k = I, i.e. the terms oC2~j~1~~, where j and I are of order O(N); these terms can give a nonvanishing contribution wr”/3 to the nearest-neighbour correlation with p = 1. In addition the difference I - j should have one of the values 1, 0, - 1. If we now look at the complete expression for 9:, given in appendix B, we see that the only terms of this type that can arise after taking the commutator [P”,, . ..I or the multiplication at the right by oh are the terms LX~OI~~,P,,, and LX~C_X&‘~~~_~. Finally 9: contains a few terms involving 4 operators CCor 4 operators p; these terms can never give a contribution. As a conclusion all the terms of 3’2 are irrelevant; the relevant terms in the expansion (4.24) of 9: are given by constant

term,

a201N

2

DL20LN/%pN

>

0(20(N/%!N

- I.

(5.1)

A similar analysis can be given for the terms of 3:. Since we have given the complete expression in appendix B, we shall omit the details. The relevant terms of 93, are

SUSCEPTIBILITY

x,.. OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

463

Finally it can be shown that all the terms of 9: and dp*, as given in eqs. (4.20) and (4.21) are relevant terms. Using the relevant part of the operator exp (9,~) the calculation of the spin correlation function is straightforward. Since the right-hand side of eq. (4.17) has been shown to be independent of (T, we shall from now on restrict ourselves to the value cr = - 1. Using eqs. (5.1) and (5.2), the relevant part of exp 9-,t is given by

The coefficients cC2)and cC3)have been given in eq. (4.21) and appendix B. After substituting c = - 1, we have c1(2) -_ -c:2) = 2B(S_,

CL” = (SZ, + s:>, C2)= c(2) = _p 4

c3

= _ c6(2) =

S

c:3) = S”, + 5s_& cy) = 5S!J,

c(3J) = -

_2s_

+ 2BZ(S_,

1

CJ

- S,), c:2)

I,

=

-8s_1sl;

(5.4)

- S,),

+ S,” - 2B2 (S_, - S,),

BS_l (S_,

-

S,),

cs(3) = 6S!,S,,

cC3) = S-J:, S

ck”’=

BSl (S_ c:3)

=

1

-

S,),

6S_&.

(5.5)

The coefficients cC4)can be calculated by using the result for LZip,” with (r = - 1 in appendix B. ~(04)is due to the multiplication of the terms with a,pN, CY,,$~ by oh. /3N after multiplication by oh and from the term LX~CX~OL~ Cl(4) has a contribution also contributions from (Y~/?~,agpN, 01~/6~,aNp3 and @IN- 1 after taking the commutator [YP,, . . .]. cy) can arise from the terms CX~C+X~~~ and &IJ2,!IN by taking the commutator [Z,, . . .] and also from a2pN after multiplication by oh. Finally ci4) can be obtained from LY~/?~_ 1 after multiplication by oh. The results are cb”’ = [S!,

+ lOS”_,S: + S: + 2B2(S1,

c:4) = [20S3J1 c(24)= -8B(S2J1

+ 4S_$13 + 2B2 (-3S2, - s_,s:>,

- 2S_,S1 + S;)], + 2S_,S,

c:4) = ss”,sf.

+ SF)], (5.6)

H. W. CAPEL,

464

E. J. VAN DONGEN

AND

The spin correlation functions can be calculated eq. (4.17). We then use the expansion, cJ (2.26). iyl

(t, - 1) = LX1-

iT(-%

TH. J. SISKENS

by inserting

(.5.3)-(5.6)

+ &a2 + S-JM

+

h’ w2 (B3 + Q-1) + Ml (a2 + %v) +

-

Ait3KSW3’84 + (SW2P3+ (SW, P2 + (SW, B1 WV-,

+

(SW-,

+

AT” [(M”)O 0~~ +

BN

+

B,-,

+

(SW-3

(M’), 01~ + irrelevant

~ooc,l

&-21 terms]

+ higher orders in t.

(5.7)

Here M,, (SM), and (M2),, 141 I 3 are shorthand which are independent of 0, i.e.

Mj = U4Jk~+w

into

notations

for matrix

elements

(SW, = (%Mo)kk+y, (5.8)

(M’), = (M:)kk+q. After inserting eqs. (5.3)-(5.7) into (4.17) we obtain a large number of terms, each containing an average of a product of operators 01 and /3. Using Wick’s theorem, the averages can be expressed in terms of pfaffians with elements (Q,aipj). To each element we apply the expansion (@&&)

= +i (S,M,” = f

p

tanh +/?M$),_,

(so),-k -

s

&M,),-,

+ g

(S,M:),-,,

(5.9)

foro = -1. For each value of p, (i.e. for each spin correlation function), we can combine the contributions that are proportional to -c”~“. We restrict ourselves to the values 1 and p satisfying il + p 5 5. This restriction reduces the number of neighbours we have to consider. In fact, in this approximation, we have


1)

=

0,

for

p 2 6.

(5.10)

[In this equation it has been assumed that p 5 N - 5, CJ eq. (3.8).] Furthermore, the restriction il + ,u I 5 reduces to a large extent the size of the pfaffians we have to consider. The largest pfaffians have 4 or 5 rows, but they contain only one term which satisfies the restriction. Then we are left with pfaffians corresponding to 3 or less different operators OLand different operators @; such pfaffians have

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

465

at most 6 nonvanishing terms. The calculation of the spin correlation function is now straightforward, but very tedious. Some useful matrix elements are given in appendix

D. The result can be expressed

(eSx(r)

$+I)

as (5.11)

= A, + B,,

where A, is the contribution from the term 1 on the right-hand side of (5.3) and B, is the contribution from the remaining terms. The A, are the correlation functions we would have obtained, if we had neglected the factor eTXu e-rz-P-o in eq. (3.13). The explicit expressions for the coefficients A, are given in appendix E. The coefficients BP are given in appendix F. With these results the evaluation of the spin correlation functions with p i 5 is simple, and the result is

(@s;(z) S,“) =

(@s:(t)

(&-

ST,,

(5.12)

s;> = 5 St,,


(Q;(t)

-5

= -5

S” 1 + &

S,“) = $

(z) S;)

S”,

= -0

+

(5.13)

- 5

(3s: 1 + 5S3,S:

(2S4,

S_ 1 + 2

[St,

(P2t - BT’) (S-J:

+ 4S2,S;

+ 3B2S9,

+ 2S_ $7: + B2 (2S-1

-

(5.14)

+ 4B2S3J,

(5.15)

+ &)I

B'SJ

16 - -$

[S?,

+ B2(6ST1

+ 6S3,S:

+ 3S_1S14 + B4(3S_1

+ 9S:J,

+ 12S_J;

- z

[3S3&

+ 3S_,S:

- 2B4S,

+ s

[S!,S:

+ 3S_&

+ B4S_,

+ B2 (-2S2J1

+ 2S_,S:)]

+ 2SJ

+ 3$)] - B2 (3S!&

+ S:)]

466

H. W. CAPEL, E. J. VAN DONGEN

AND TH. J. SISKENS

- B4 (S-,

[2S3,$ - B2 (St,S,

@S;(t)

S;)

(5.16)

+ 3$)],

-8 r2) (B’ + 2s;)

= 4 - (‘r

+ 5

+ 2S_&

+ 2S1)

[4S: &

+ 2s:

+ B4 + B2 (2S1 1 + 4S_ 1S1 + 6S:)]

+!?$g+($-&L) + SS: + B4 + B2 (2S!,

x [4S2&

With the correlation transverse susceptibility

xxx

=+,!-j-@2S_1

+ $

+ -&

-

+ 6S:)]. (5.17)

functions (5.10) and (5.12)-(5.17) we can calculate the (3.9) per particle in the thermodynamic limit. The result is + $6Si,

(-2S3,

[5S4_, -

[-6S1,

-2$-B’)

+ 3S_&

+ 2B2S_,)

16S2_,S:

+ B4 + B2 (- 13St1 + &

+ 4S_1S1

+ 3s:

+ 4S_1S1

+ 22S1,S:

17B4S_1 + B2 (24&

+ 6S:)]

- 33S_,S:

- 38S!,S,

- 7OS_&)].

(5.18)

Remarks. The yy component of the susceptibility tensor x,,,, can be obtained from the expression for xXx by the substitution y -+ -y, which is equivalent to the interchange of S_1 and S1. tensor are The off-diagonal components xxv, xxz and xyZ of the susceptibility since the quantumequal to zero. For xXZ and xYZ this is easily appreciated, mechanical traces involved contain an odd number of OLand /3 operators. For xxv one can easily check that all terms in the high-temperature expansions for the correlation functions are purely imaginary, and therefore cannot contribute to xXY.

SUSCEPTIBILITY

xXx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

467

6. Results and conclusions. In this section we shall give a few graphs of the transverse susceptibility xXx US. one of the parameters /?, y of B (keeping the other two parameters fixed), and discuss some salient features*. In fig. 1 the susceptibility is plotted US. the inverse temperature values of y, keeping B fixed at B = 0.

B for various

Fig. 1. The transverse susceptibility x us. the inverse temperature p in the absence of an external field B for various values of the anisotropy parameter y. The dashed curves represent the exact expressions (6.1) and (6.2).

y=

-l..O

0.2 -0.5 0.0 0.5 0.1 -

1.0

-

X t 0. O&

I 0.5 Fig. 2

I 1.0

-To y

I

0

1.0

Fig. 3

Fig. 2. The transverse susceptibility x vs. the external field B at 6: = 1 for various values of the anisotropy parameter y. Fig. 3. The transverse susceptibility x vs. the anisotropy parameter y in the absence of an external field B for various values of the inverse temperature /3. * Although the calculations given in this paper hold for arbitrary values of y, we shall, when interpreting the results, restrict ourselves to the interval - 1 < y 5 1, i.e. the antiferromagnetic XY chain.

468

H.W. CAPEL, E.J. VAN DONGEN

AND TH.J. SISKENS

The drawn curves have been obtained by using the high-temperature expansion up to p6, the dashed curves represent the exact expressions for xXx at y = - 1 (the upper dashed curve) and at y = 1 (the lower dashed curve) in the thermodynamic limit. These exact expressions are xXx(7 = 1, B = 0) = $be-@

(6.1)

xXx(7 = -l,B=

(6.2)

and 0) = $tanh&!? + $flcosh-2$/3.

Eq. (6.1) is the (longitudinal) susceptibility at B = 0 for the Ising model. Eq. (6.2) can be obtained from xzz at B = 0 and IyI = 1 for the XY model, by using e.g. eq. (39) of ref. 4. For B < 1 the agreement of the high-temperature expansion with the exact expressions is excellent. For 1 < p I 1.3 there is a slight discrepancy which increases to an amount of 2% at ,Q= 1.3. We note that for y = 1 there is a maximum at p = 1.0 for the exact expression as well as for the expansion. For y = 0.8 the susceptibility reaches its maximum at about /3 = I. 1. For y = 0.5 the maximum is found at about /? = 1.3. For lower values of y the maximum will occur at higher values of b.

Fig. 4. The transverse susceptibility x vs. the inverse temperature j3 for various values of the external field B; the anisotropy

parameter y is equal to 1.

For y = - 1 the value of /I for which x reaches its maximum can be estimated from the exact expression (6.2). This value is /? = 2.40. The curves of fig. 1 refer to the case B = 0.A change of the magnetic field in the range 0 I B 4 1.2 would modify the picture only slightly. This is shown in fig. 2, where we plot x vs. B for various values of y, /l being fixed at fi = 1. The variation in the range 0 I B I 1.2is at most 6%, and for lower values of /? the variation is even smaller. The dependence of x on the anisotropy parameter y, however, is very strong, e.g. at B = 1 and B = 0 the susceptibility for y = - 1 is 2.3 times as large as the susceptibility for y = 1. This strong dependence is shown in fig. 3, where x is plotted VS.y for various values of p, keeping B = 0.Note that the steepness of

SUSCEPTIBILITY

xXx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

469

the curves is the more pronounced as /? increases. It is of interest to compare these dependences (weak on B, strong on y) with the behaviour of the longitudinal susceptibility xZZ. To this end we have estimated xZZfor various values of y, p and B, by using a high-temperature expansion of eq. (39) of ref. 4 up to B’. Firstly, we note that xZZdoes not depend on the sign of y. But apart from this, it appears that for p = 1 and 0 I B I 1 the longitudinal susceptibility xZZ decreases only by at most 4 % when Jyl varies from 0 to 1; this change is even less for lower values of p. This contrasts clearly with the dependence of xXx on y (see fig. 3). It is obvious that for establishing the anisotropy parameter of an XY system, the transverse susceptibility is a much more useful quantity to be measured than the longitudinal susceptibility. As for the dependence on B, it appears that at p = 1 the longitudinal susceptibility xZZdecreases by about 20 %, when B varies from 0 to 1. This contrasts with the very weak dependence of xXx on B (see fig. 2). Finally, in figs. 4 and 5, we plot xXxVS.j3 for various values of B, keeping y = 1 and y = 0.5 respectively. One should realize that for the values B = 1.4 and B = 1.8 the expansion we have used, starts to be less reliable at B values of about 0.9.

Fig. 5. The transverse susceptibility x vs. the inverse temperature /? for various values of the external field B; the anisotropy parameter y is equal to 0.5.

It seems, however, correct to conclude that the value of #I for which x reaches its maximum decreases to lower values of /? with increasing B, and furthermore that the maximum value of x decreases too, when B increases. 7. The susceptibility in the c-cyclic XY model. In this section we shall investigate the susceptibility for the c-cyclic XY model, i.e., (7.1)

470

H.W. CAPEL,

E.J. VAN DONGEN

AND TH.J.

SISKENS

where Q,, cf. (3.1 I), for u = - 1, + 1 is the density operator corresponding to the c-cyclic and c-anticyclic hamiltonian zO, given by eq. (2.23). We shall show that the susceptibility (7.1) is different from the XX susceptibility (3.2) in the a-cyclic XY model, even in the thermodynamic limit. From the KMS property (3.7) it follows that N

lJ

1 j=l

x = j dz lim (l/N) 0

+2;

N+m

j=l

P=I,

c


(5 4 ST+,> .

(7.2)

2. . . . . N-j

In contradistinction to the a-cyclic XY model, that is periodic in the spin components S”, CJ eq. (2.2), we cannot restrict ourselves to the first spin with j = 1. In fact the spin correlation functions appearing in the right-hand side of (7.3) depend on j. In the case that j = 1, the result is simple. Noting that the terms involving P can be omitted in the thermodynamic limit, c$ the discussion below eq. (4.4), and also noting that the factor x0 Z,/(.Z, -+ Z_,) in eq. (4.5) can be omitted due to eq. (4.6), we see that the spin correlation functions (e,S; (r, o) S,X+l), can be obtained from eq. (3.12) by omitting the factor exp (zp”,) exp (-TV@.__.), or equivalently, by replacing the operator exp L?,t by 1, cJ (4.14) and (4.15). Hence the correlation functions (e,S; (r, o) S,“+ i) are given by the coefficients A,, defined by eq. (5.1 I) and explicit expressions for these coefficients have been listed in appendix E. It is not surprising that these correlation functions are different from the a-cyclic correlations (es; (r) Si+ 1). In fact the c-cyclic and c-anticyclic model can be obtained from the a-cyclic model by introducing a disturbance in the coupling between the spins 1 and N and here we investigate the correlation function just at the position of the disturbance. We now consider a general spin correlation function (~$7 (z, u) S,“,,), for sufficiently large values of j, i.e. far enough away from the disturbance by virtue of which the c-cyclic and c-anticyclic model have been introduced. By using the Jordan-Wigner transformation (2.4), or more specifically the explicit expressions (2.30) for the spin components, and also the anticommutation relations (2.22), the spin correlation function can be written

= 3 (@&j(T, 0) (PI “’ Pj_1) (t, 0) PI “. Pj_lPj where P, = (2/i) akPk. [Note that the factor for j = 1, c$ eq. (7.9).]

“’ Pj+p-l”j+p),

(7.3)

(P, ... Pj_l) (z, o) does not occur

SUSCEPTIBILITY

In order to evaluate

Pl

1..

Pj_llfg

=

xxx OF ANISOTROPIC

the right-hand

ONE-DIMENSIONAL

XY MODEL

471

side of (7.3) we use the identity

(S”, - oh + hj)P,

..’

Pj_l,

(7.4)

where

Here h has been defined by eq. (2.8) CJ also eq. (4.20) and S_, and S1 have been given by eq. (4.23). The derivation of (7.4) is given in appendix G. From (7.4) it follows that

Pl ..* Pj-1 exp (-t%“,)

= exp [-t

+ hj)] PI *a*Pj_l,

(Z_,

(7.6)

where also use has been made of the relation (4.1). We now use (7.6), the trivial relations Pj’ = 1 and the fact that the operators P, commute among each other. Then the spin correlation function (e,,S,” (t, a) ST+,) forj> l,j+p I Nisgivenby

(&Sj” (Z, U) ST+,) = + (& exp (t%“,) &jP, ... Pj-1 exp (-t%,) x P, ***Pj_,Pj =

.‘.

pj+p-l"lj+g)

+(eoexpWfJajexp

= +

[-t(X-a

(7, 0) exp (TX”,) exp (-tX_,)

(@n&j

+ hj)l

0~ (t, -0)

(7.7) where Oj

Cry-0)

EE

exp (z&‘-J

exp [--t

(Z-,

+ hj)].

(7.8)

Eq. (7.8) is not valid for j = 1; we then have

(7.9) For Oj (t, -0)

5

we have the equation

O,(t> -0) = [Zf_,,

Oj

of motion

(t, MU)] - Oj (t, VU) hj

(7.10)

H. W. CAPEL, E. J. VAN DONGEN AND TH. J. SISKENS

472

and the formal solution can be written as O,(r, -0)

(7.11)

= exp W-,A,

where the “Liouville operator” operator A, i.e. 9-,,jA

9 _-brj is defined by its action on an arbitrary

(7.12)

= [J%-,, A] - Ahj.

In particular, we have 9_,,,

= -h,,

&,,,

= -[X_,,

(7.13a) hj] + hj”.

(7.13b)

From (7.7), (7.1 l), (4.14) and (4.16) we have the expression <@OS? (r, o) ST+,> = t
ewWAewW-,,j4

*Jj

(

a~+~).

(2/i) w% )

(7.14) In order to evaluate the right-hand side let us first consider the correlation function
ST+,>

=

c 2, ( 0‘

x


(1

+

m)-’

t

c

z7 (r


4

(PI

. ..Pi_.)(z,o)exp(t~,)exp(-z~-,)P1

a.. Pj_IPj ..* Pj+p-lo(j+p (1 + OP)).

(7.15)

Using eq. (7.6) with -C instead of CTand using a similar line of reasoning as in the derivation of eqs. (7.7) and (7.14), it can be shown that
ST+,>

c ( d

2,


~~Z,(@,eXp(~~,J~jP~

d

(1

+

m> 1

...Pj_lexp(-ai_,)

x P, ..’ Pj-1Pj “’ Pj+p-l&j+p (1 + UP)> = 3 C Z,
+ hj)l

SUSCEPTIBILITY x

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

473

Pj 0.’ Pj+p-1aj+n (1 + cP)>

= 4 C z. <@&j (t9 a) Oj (t, O) pj *” pj+p--lolj+p (1 + Op)> CT

In the same way as we did in section 4 it can be argued that the terms involving the operator P can be omitted in the thermodynamic limit for non-zero temperside of atures and furthermore the average (Q~ ‘..) occurring in the right-hand (7.16) can be seen to be independent of C, cf. the discussions below eqs. (4.5), (4.6) and (4.17). The a-cyclic correlation function is then given by

(7.17)

Eq. (7.17) obviously is equal to the correlation function (es: (z) S,“,,) given by eq. (4.17). First of all the right-hand sides of (7.17) and (4.17) are independent of 0, so that we can restrict ourselves to the c-cyclic case with o = - 1. If we then apply the translation 9 over j - 1 lattice sites which transforms in particular the pair of neighbours j - 1, j into the pair of neighbours iV, 1, the operator exp (dp_ I,lt), defined by (7.11) and (7.8) for c = 1, is transformed into the operator F exp

(P__l,jt)

F-l

=

exp (tZ_l)

exp [-z

= exp(tY?_,)exp(-t#,)

(Z-1

+ h)] = exp(Y_lr),

(7.18)

so that the translational invariance is shown in an explicit way by (7.17). If we now compare the second line of the right-hand side of eq. (7.7) with the second line on the right-hand side of eq. (7.16), we see that the operator exp [-z (%_, + hj)] occurring in eq. (7.7) for the c-cyclic correlation function has been replaced by the factor exp [--t (s, + hj)] occurring in eq. (7.16) for the a-cyclic correlation function. This replacement is a direct consequence of the commutation relation (2.37) for ST, which involves both projection operators P, and P-,. As a result the c-cyclic correlation function (Q,S~ (t, o) ST+,> given by eq. (7.14) for j # 1 contains an additional operator exp LZ’~Z,which does not occur in the corresponding expression eq. (7.17) for the a-cyclic correlation function. We now consider eq. (7.14). Assuming that the correlation function (7.14) can be expanded

474

H. W.

a power (j p 2

in t B, we the decoupling (r, 0)

E. J.

= 5

DONGEN

AND

J. SISKENS

for sufficiently

values of

and N

j,

exp (2~))

G?,P_i (r,

exp

C9-c,jT)

(7.19) Eq. (7.19) is proved in more detail in appendix H. In addition, for sufficiently large values of j, if we restrict ourselves to contributions = ?‘/3” satisfying R + ,u I Y - 1, cJ (4.12), the operator exp (+Z_C, jt) on the right-hand side of (7.19) can be replaced by exp (+LZ,, jr) and due to of (7.17), we have the simple relation
(r, g) ST+,> = (e, exp (2~))

(7.20)



for sufficiently large values of j and N - j. Hence the correlation function in the c-cyclic model can be obtained from those in the a-cyclic model after multiplication by the factor (Q~ exp L?~z). The evaluation of this factor is straightforward. If we restrict ourselves to the case that v = 4, “the relevant part” of exp LZ- 1t, i.e., the terms which give contributions z -r”@ with )3 + ,u 5 3, is given by exp (JZ-1t)

= 1 + 2iz (S-1a,/3N

+ S1a,/31)

+ &It2 (SZ,

+ Sf),

(7.21)

so that (Q-~ exp(Z_,t))

= 1 + $(t’

- @x)(S?~

+ S:).

(7.22)

Using eqs. (7.20), (7.22) and the a-cyclic correlation functions the c-cyclic susceptibility (7.3) up to order /I” is given by Xc-cyclic = Xa-cyclic - (p3/48) (S?,

+ S:) + (b4/48) S_1 (S!,

Here Xa_cycricis the a-cyclic susceptibility xc_cyclic

=

tp

-

tB”s-t

+ (p4/48) (-S!,

+

(b3/48)

+ 4S&

(5.16) and (5.17),

+ ST).

(7.23)

given by eq. (5.18). The result is (5SZ1 - 3s:

- B2)

+ 2B2S_1).

(7.24)

Hence, the c-cyclic model cannot be used to evaluate the xx susceptibility of the a-cyclic XY model, even in the thermodynamic limit. For arbitrarily large values of j, the effect of the local disturbance in the coupling between the spins 1 and N

SUSCEPTIBILITY

on the correlation

xxx OF ANISOTROPIC

function

ONE-DIMENSIONAL

(e_,Sj” (t, o) ST+,) is not negligible.

fact that the Jordan-Wigner “time dependent”.

XY MODEL

475

This is due to the

PI (z, o) ... Pj-1 (z, 0) in eq. (7.3) are

operators

APPENDIX

A

In this appendix we prove that the perturbation expansion (4.2) is convergent. Consider the contribution Ccg) of the qth order of perturbation to the correlation function (es: (z) S,X+1). Noting that the terms with the operator P can be neglected in the thermodynamic limit and using the fact that the factor cg Z,/(Z, + Z-,) in eq. (4.5) can be omitted, cf. (4.6), C(@ can be written Ccq) = 5 (exp (-/3&?c)>-‘bdr,b’dr, X (e-(B-OzO~l

e-(r-r,)X~h

...]-drqoq e-“‘-‘2’““,r

,,. e-“4_I-TI)8~/r

,-r&0

where

In order to estimate

I<0102 *.* 0,)l for arbitrary

operators

Cq), we use the Holder I

inequality

for operators

fJ((o~o,)*ey’e~,

(A-3)

k=l

0, and nonnegative

numbers

19~)19,) . . . , fl,, satisfying

k$e;’= 1.

(A.4)

A simple proof of this inequality numbers ii,, &, . . . , As with

was given in ref. 14. For arbitrary

real positive

a, + a, + ... + as= /I and a hermitean (e-po)-1

operator l(e-“lO

(A.5)

0, we have in particular

0, eeAzO O2 ... e-‘T” OS)1 i

the inequality [lOI I/ (IO,Il ..a IlO,)\,

where IjOkI1 is the norm of the operator Ok, i.e. the largest eigenvalue Eq. (A.6) follows immediately from (A.3) by choosing n = 2s,

0;’

= 6;’

= . . . = fj,l

(A.6)

of (OLO,).

= 0, (A-7)

476

H. W. CAPEL, E. J. VAN DONGEN

The simple version (A.6) of the Holder by Ginibre and GruberlJ). From

(A.l)

AND TH. J. SISKENS

inequality

and (A.6) it is now obvious

has been applied

previously

that

Ic(q)I5 3 W/4!) ll%II 11~114 11~11 and all the operator

norms

(A.@

on the right-hand

APPENDIX

In this appendix we give the expression defined by eq. (4.13). It is given by

side of (A.8) are finite.

B

for the operator

_.Y’z, where

ZO is

+ 60s~1s: (BIBzBN~~~ - Bswiv--14 + +#IN [S: 1 + 5S_ 1s: + 2B2 (S-1 - S,)] + LX& [S,3 + 5S21S, - BS-1 (S-1

- S,)

-

BSI

S-A

-

2a2&+1S”1S1

(S,

-

+ 2B2 (S, - S_ J

(~28~

(~32

+

+

~IBN-I)

TV-IA)

+

4NS%

+

- 2LxN_-1/3~S--1S:}.

APPENDIX

%B&S-1

(B.1)

C

In this appendix we prove eq. (4.26) by induction over IZ. Suppose it is true for _Yz, then each term in the right-hand side of eq. (4.24) contains 3 (k + I) operators with an index O(1) and 3 (k + I) operators with an index O(N). In order to prove (4.26) for _Yi+l we first note that the commutation with L%?~does not change the number of different operators OLand /I, i.e. if we take from (4.24) a term T with k + I different operators, then [X0, T] has also k + 1different operators. Suppose [X”,, T] has a term with k + I - 2 different operators. This can occur at least in principle, if in the term T = OLilaiZ **.Ori,pj,... fij,, we have operators (Xi, and @j, such that [&C, ai,] = -i(Sn)iPjq

Bj, + other terms,

(sb)*Pjq # OY

then we have two operators pj, after the commutation can be eliminated using the anticommutation relations.

Cl)

and these two operators However, in that case we

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

477

can write (C.2)

where A is an operator which does not contain OL~,and p,q. Then

where [z,,

ai,bjql = -i F (%hpkBkB_iq + i C (SJkj, k

=

-ikz

(Sa)ipkpkbJg

+

ikFi

(Sb)kjq

4,&k

(C.4)

4,&k

P

4

and the right-hand side of (C.4) is a bilinear form in the operators cs and p. Hence [&‘, T] contains k + 1 different operators (x and fi. In view of the identity

[SF,, T] contains at least _t (k + 1) - 1 operators with index O(N). Furthermore only terms T involving LY~,c+, PI, pN can lead to terms [X,, T] which have a number of operators with index U(N), which is different from + (k + 1). So we can restrict our attention to terms T, in which

i,=l,

or

ik=N,

Or

jr=l,

or

j, = N.

(C.6)

Consider now the product Toh. Only the terms which satisfy one of the four conditions . 11

=l,j,#N;ik=N,j,#l;i,#l,j,=N;ik#N,j,=l,

(C.7)

can lead to terms in which the number of operators with index O(N) is different from the number of operators with index O(1). Using eqs. (C.6) and (C.7), we find

H. W. CAPEL,

478

E. J. VAN DONGEN

AND

TH. J. SISKENS

Here the other terms contain the same number of operators with index U(1) as operators with index 8(N). By using the commutation relations (2.22) and also the relations (S,),, = --OS-~, (S&i = --us,, CJ (2.24) and (2.29, it can be shown that the term between curly brackets in the right-hand side of (C.8) vanishes, so that

2,

II+1

= other terms,

i.e., each term of Y:+’ with index O(N).

(C.9)

contains

as many

operators

APPENDIX

with index O(1) as operators

D

In this appendix we give some matrix elements (M,), = (Ma)k,k+q, (SaMo)qr (Mz), and (S,M& for small values of 141.These matrix elements are independent of a and have been used in the calculation of the spin correlation functions in section

5.

M, = S”, M,

(D.1)

+ ST + B2,

= M_,

= -B(S_,

+ S,),

(SM),

= -B(S:.

+ 2S_,S,),

(SM),

= 2S21S1

+ S: + BZ (S-1

(SM),

= -B3

= -2

(M2), = S!, (SM2)_,

(D-5)

+ S,),

(D.6)

+ 2S_,S,),

[B3 (S-1

+ S,) + B(S!l

+ 4S!,S:

= S:,

+ S:),

+ B2 (2S_,

= -B(S!,

(M’),

(D-4)

+ 2S_,S:

(SM)_,

(D-2)

+ 2S,),

+ S-rS,

= S”,

S-IS,,

(D-3)

- 2B(S2r

(SM)_r

:=

M2 = M-2

+ 6&S:

+ 3B2(2S31

(D-7) + 2S21S1

+ S: + B4 + 4B2 (St, + 3S_,S: + 3S!1S1

+ S_,S,

+ B4(3S_1 + 4S_J:

+ 2S_J:

+ S:)], + S:),

(D.8) (D.9)

+ 2S,)

+ S:,.

(D.lO)

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

APPENDIX

XY MODEL

479

E

In this appendix we give explicit expressions for the contribution A, to the spin correlation functions (es; (t) S,X+1) arising from the term 1 in eq. (5.3). We have A, = 0,

P L 6,

(E-1)

A, = -f$S!,,

052)

A4 = $

(E.3)

St,,

Al=-5

s”,

+ &(3S”,

+ ss”&

+ 4BYQ

- B”t’) S’ 1) + (B”r 64 (E.4)

A2 = 5

+

S’,

(2S’!, + 4&S;

- $

(B’t -

iw

16 - -&

(Sl, + S-J:

+ 3B%!&) -

(’ 3t - P’r”) s-J 1) 32 (E-5)

- B2S1)

[S:,

+ 6&S:

+ 3S_,S’: + B4 (3S_, + 2SJ

+ B* (6S:,

+ 9&Y,

+ 12S_ $7: + 3S,3)]

+@$SS,+3S’,S:+2S_,S:+B4S_,+B2(S’,--3S’,S,-ST)]

+(!?_$x-J

[S5,

+ 4s3,s;

- B4 (S_ I + 2SJ + 2B2 (S:,

&XL_

(” ; ‘“I

(Sz,

+ s_1s’:

- ST)],

+ ST + B*)

+

(E.6) g

_ g

4 x [St,

+

$ >

+ 4S!,S:

+ S; + B4 + 4B2 (S!,

+ S_,S,

+ St)].

05.7)

H. W. CAPEL,

480

E. J. VAN DONGEN

AND

APPENDIX

TH. J. SISKENS

F

In this appendix we give explicit expressions for the contributions BP to the spin correlation functions (& (t) Sf,,) arising from exp (Z1t) - 1 in eq. (5.3). We have B, = 0,

if

B3 = s

(-2c:2’S31

(also ifp I N - 3),

pr4

B2 =

+ c:2’S2,S,)

c:2’S_,S,)

B1 = $

(-4$‘S_

- (Z

+ ($

(F.1)

- J=),

(F.2)

- F),

(F.3)

1 + 2ci4’S1 + ci4’B - c:~‘&)

+($g)[zcB’Si,-cy’(S_,s,-B2)]

+p -$

b2

(2ca’BSr - 2~:~‘s: + c;~‘B~) - s

cys,

+

32

-B”z” (2~;” [St, 384

+ c:~‘B(S?~ + 2S_,SJ

+ 2S_$7:

c~~‘S_~

+ B2 (2S_, + S,)]

- c!+~‘S?J,

- ci2’ [2S! 1S1 + S: + B2 (S-,

+ 2S,)]}

+ (B’r” - Br”) [4@ (S3, + S-J: 128

- PS,)

+ 2cy’B (S: 1 - S:) - 2c’,2’B(S?, + S- 1S1) - 2~‘32’S2~S, - 2~:~’(S:,S, S”, + g

+ $?

B2 (S$

x (S:,

+ 2&S;

_ Sz,S,)

+ S: - B2S_J (2S5, + 4S3,S;

- ($

- c:“B2 (S-,

+ S,)]

+ 3B2S9

+ B’z” _ “) 32 48

+ 2B2Sz1 + B2Sf1S1),

(F-4)

SUSCEPTIBILITY

&

= $

&,“’ -

xxx OF ANISOTROPIC

t4 c:3’x1 --g

+;Cb”+

+ 5

ONE-DIMENSIONAL

(c:3’&K1 -

XY MODEL

481

C:3’S,)

-B2)

(g-g)C!:‘(S-,S,

+ (b"-7") [--c:~'B(S_~ + S,)+ c',t'S_,S, + c~'(S?~+ S: +B2)] 32 i_

$

(S2, -

+ T(S:

s:>- $ + S”,S:

-$(S:,

s”, - g

(St,

- St + 2B2S?, - 2B'S:)

- B2S_1S,)

f 2S2,S: + 2B2S21 -IB2S_1SI)

+$Stl

+ S:,S; - S: +2B2Sf, +2B2S_1SJ.

(F.5)

The contributions B3, B2 ,B, , B, can be calculated using the explicit expressions (5.4), (5.5) and (5.6). The result is B3 = _ (p”’ - p3t2) S” 64

=

(F.6)

Sk

B2 =

B1

1,

_(B”

-

b’)

(F. 7)

S3

16

+

-I

Bz - -$s5,+2s5&

p4’ 192

(2S5, + 4S3,S:

+ 3B2S3J

-S_1S;1+B2(S~1-S?1S1-2S_-1S:-S;)]

[SZ, + 2s3,s:

+ B2 (2S3, + S:,S,

+ 2S_,S:

+ s_,s;

-t- S:)],

03.8)

482

H. W. CAPEL.

$ (/?t - t’)

E. J. VAN DONGEN

(S?, -

s:>+ -g

AND

[-Sl,

TH. J. SISKENS

+ S’: - 2BZ(S$

- ST)]

+~s:+(~-~)[s’,-7s:+282(Si,-S:)l.

(F.9)

Note that the coefficients ~2’ and ~(33)do not occur explicitly in eqs. (F.l)-(F.5). The corresponding operators pz,dN in 95 1 and also OL~/~~in LZ’!., give rise to different contributions to the spin correlation functions, which cancel to give 0.

APPENDIX

In this appendix, Pf-1

G

we prove eq. (7.5). First we note that because

of (2.22)

= 1,

(G-1)

so that Pj-1ZbPj_1

= C%g - 40(j_,/3j_,

Using the commutation

relations

- 2i (Sl&j_zBj__l

Pj_,Pj_lXoPj_,Pj_,

((3.2)

(4.18), (4.19) and again eq. (2.22) it follows that

= A?“, - 2i (S,Orj_,bj

From eq. (G.3) with j -

[Z”,, oLj-1pj-l]*

1 instead = H,

ofj,

+ S-loCj/5j-1) + S-,olj-,/5j-2)*

(G.3)

and (2.22), we have

- 2i (S,aj-Zgj-l

- 2i (SlDLj-Jf9j-2 + 8i~j_zpj-~

+ S-I&j-Igj-2) + S-,LXj-,bj-,)

[SloCj_,i3j + S-IoCjpj-1

+ Slolj__2j3j__l + S_laj-,bj-2] = i7ifm - 2i (Sldj-j/5j - 2i (S1o(j-3~j-2

o(j-Zpj-2

+ S_rajpj-1) f

S-_1(xj-_2~j-3).

(G.4)

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

483

Proceeding in the same way, we find Pz ‘.. pj-,Pj_lXgPj-_1Pj_2

‘*’ Pz

= So - 2i (SloCj_llgj + S_,o1jpj-,)

- 2i (Slo1,/3~ + S-r01&?r).

Applying eq. (G.2) for j = 1 and the commutation we have

(G-5)

relations (4.28) and (4.291,

(0

where h = -2i (S_-l~,/% + S1n,j3r).

(G.7)

From (G.5), (G.6) and (2.22), it follows that PI **. Pj-,2f?,Pj-1

a.* PI = Af,, - ah - 2i (Slaj_l@J + S-i~~~j-1).

(G.8)

Eq. (7.4) is obvious from (G.8) after multiplying both sides by PI ... Pj_1 using eq. (G.l). APPENDIX

H

In this appendix we prove the decoupling (7.19) for sufficiently large values of j. In the proof we shall restrict ourselves to the case that j < +N, j + p < N. (Similar considerations can be given for the case that j > SN, j + p 5 N. Then the decoupling is correct if N - j is sufficiently large.) In order to evaluate the right-hand side of (7.14) we use the expansions

(xj(Z,

0)

=

f

t"-

k,=O

k,!

‘dkl,

(H.1)

03.2) and (H.3)

H. W. CAPEL,

484

E. J. VAN DONGEN

AND TH. J. SISKENS

Consider now the contributions of (7.14) for fixed values of k, , k2, k3. It can be written as a sum of averages of products of operators 01 and /3. These averages can be expressed in terms of pfaffians. Let us choose j so large that j L k, f k2 + k,. Should the decoupling (7.19) be incorrect, we would have nonvanishing contributions from averages
<&B,~“,) 3

or

(H-4)

where the operator with index 1 originates from the expansion (H.2) and the operator with index m originates from (H.l) or (H.3). Without loss of generality we can restrict our considerations to the case that 1 is an index of order O(l), CJ eq. (4.8). The averages (H.4) have the order of magnitude pm--I, or even a higher power of /?. Consider now the contribution for fixed values of kI, k,, k3. The highest index 1 which can occur in 22 is equal to k,. The lowest index m occurring in Ak, or Al;, is equal to j - max (k,, k3), which is at least j - (k, + k3), so that m - 1 2 j - (k, + k, + k3). The contribution magnitude

to the

spin

(H.5)

correlation

function

would

have

the

order

of

(H.6) On the other hand, the spin correlation in z and ,u, i.e.

and in a high-temperature satisfying

function

can be written

= A; &“a”

expansion

as a power series

(H.7) we can restrict

ourselves

3,+/AuIv-1, for some value of v, CJ eq. (4.12). Hence, the contribution (H.6) is irrelevant,

to values I and ,U

(H.8)

if

k,+k,+k,+m-l>v-1

(H.9)

and (H.9) is satisfied if j>v-1, so that for each value of v we can ensure that the decoupling choosing j in such a way that (H.10) is satisfied.

(H.10) (7.19) is correct by

SUSCEPTIBILITY

xxx OF ANISOTROPIC

ONE-DIMENSIONAL

XY MODEL

485

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

Lieb, E., Schultz, T. and Mattis, D., Ann. Physics 16 (1961) 407. Katsura, S., Phys. Rev. 127 (1962) 1508. Niemeijer, Th., Physica 36 (1967) 377. Mazur, P. and Siskens, Th. J., Physica 69 (1973) 259. Siskens, Th. J. and Mazur, P., Physica 71 (1974) 560. Siskens, Th. J., Physica 72 (1974) 123. McCoy, B.M., Phys. Rev. 173 (1968) 531. Barouch, E. and McCoy, B.M., Phys. Rev. A3 (1971) 786. McCoy, B.M., Barouch, E. and Abraham, D.B., Phys. Rev. A4 (1971) 2331. Bloch, C. and De Dominicis, C., Nucl. Phys. 7 (1958) 459. Caianiello, E.R., Nuovo Cimento 10 (1953) 1634. Gaudin, M., Nucl. Phys. 15 (1960) 89. Dunford, N. and Schwartz, J., Linear operators, Wiley Interscience (New York, XI. 9.14 and XI. 9.20. 14) Cape& H. W. and Tindemans, P.A. J., Rep. Math. Phys., to be published. 15) Ginibre, J. and Gruber, C., Comm. Math. Phys. 11 (1969) 198.

1963)