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

Physica 71 (1974) W-578

0 North-Holland Publishing Co.

TIME-CORRELATION IN THE

FUNCTIONS

a-CYCLIC XY MODEL.

TH. J. SISKENS

II

and P. MAZUR

Imtituut-Lorentz, Rijksuniuersitait te Leiden, Leiden, Nederlmd

Received 6 August 1973

Syncipsis Explicit expressions are derived for the free energy per spin, the z component of the magnetization per spin, the internal energy per spin, the time correlation functions of the z components of two spins and the time autocorrelation function of the z component of the magnetization for the finite a-cyclic XYmodel. It is shown tbt in the thermodynamic limit these expressions reduce to the expressions derived previously by various authors starting from the finite c-cyclic XYmodel.

1. Infroduckz. In a previous pap&) (to be referred to as I*) we derived explicit expressions for the time-correlation functions of the z components of two spins and the time-autocorrelation function of the z component of the magnetization for the c-cyclic and the c-anticyclic XY model (cf. refs. 1,2,3). The purpose of the present paper is to calculate these correlation functions for the cyclic spin XY model, the so-called a-cyclic XY mode13). In section 2 projection operators P, 1 and P_ 1 = 7 - P+ 1 are introduced. These operators have the property that P+&’ = P+lS+, and P_,&’ = P_+El, where S is the a-cyclic, A@+1the c-anticyclic and ~6, 1 the c-cyclic XY hamiltonian. Furthermore P+l and El commute with all three hamiltonians and with the z component Sj” of each spin j (= 1, . . . , IV). In section 3 a common orthonormal basis of eigenstates of SO (0 = - 1 or + 1) and P = P+l - El is constructed, Making use of these representations in a formal way, it is shown in sections 3-7 that canonical averages for the a-cyclic XY model can be written essentially in terms of the expressions derived in 1 for the c-cyclic and the c-anticyclic XY model. In section 4 expressions are given fur the partition function and the free energy per spin. * Equations and sections of I, referred to in this paper, are preceded by the prefix I. 560

TIME-CORRELATION

FUNCTIONS

IN a-CYCLIC

XY MODEL.

II

561

In section 5 the z component of the magnetization per spin and the internal energy per spin are calculated. In section 6 the time correlation functions of the z components of two spins are evaluated. Finally, in section 7 the time autocorrelation function of the z component of the magnetization is calculated. In the thermodynamic limit the free energy per spin, the z component of the magnetization per spin, the internal energy per spin, the time cor‘relation functions of the z components of two spins and the autocorrelation function of the z component of the magnetization are shown to reduce, respectively, to the same values for the a-cyclic, the c-cyclic and the c-anticyclic XY model. 2. A formal expression for the time currelation functiun of the z cumponents of two spins in the a-cyclic XY model. The hamiltonian (1.4) of the a-cyclic XY model can be written as

c%@ = iC1

+ +h(P + l),

(1)

where Z_ 1 is the hamiltonian (1.10) of the c-cyclic XY model, P is the operator defined by (1.5) (and also given by (I.6)), and where h is defined as h =-

((c;c, + rCfY& + hx.).

(2)

It follows immediately that the hamiltonian (1.9) of the c-anticyclic XY model can be expressed as t% +1

S-1

=

+ h.

(3)

Let us introduce hermitian operators P, l and P, 1, defined as P + 1 = * (1 + P) P __1=

(1 = unit operator),

(4)

917 - P).

(5)

With the property (1.7) of P, it is clear that P+ 1 and P, 1 are projection operators :

P; = P# Furthermore P+&1

(0 = -1,

+I).

(6)

it is easily checked that = P_1P+1 = 0.

(7)

Since P commutes with =C1 and h, we have also [Pb, S_,]

= [PO, h] = 0

(0 = -1,

+l).

(8)

TH. J, SISKENS AND P. MAZUR

562

From (1) and (4) it follows that fl can also be written as 2

= St1

(9)

+ hP+,.

Using the projection operator character of P+l and P_ 1 and the properties (7) and (S), we find the following properties P,Z

= Puc#,

+l),

(a = -1,

(10)

and more generally, if f is an analytic function,

Since

we can write the time correlation function of the z components arbitrary sites j and k as follows

(es,“s,’(0) =

@P

= ;

(-16%) WP

_

c

b- -1,fl

s;sk” (0 (P+,

(-W) WP

(P+1

of two spins at

+ p- 1))

+ P-1))

(-BX)

m(f)

WY

(13)

where the partition function 2 is given by

c

z=

@P

(-Pm

--1,$-l

m*

(14)

Using (8) and (11) it follows immediately that 2 can be rewritten as

zu=-1,

c

@XP +1

ww

p,>*

(15)

The partition functions for the c-cyclic and the c-anticyclic model, Z_ 1 and Z +11 respectively, are defined as

zb =

WP

(-pa>

(a = -1,

+l).

(16)

Substituting (4) and (5) into (15) we obtain, with (16), Zu=-1,

c

fl

32, (1 +

tT <@UP))9

(17)

where ea is the density operator defined by (1.35). We shall now consider the other traces in the last member of eq. (13).

TIME-CORRELATION

FUNCTIONS

IN a-CYCLIC

The operator P reads in terms of the oc- and p-operators p=

XY MODEL.

II

[c$ (1.6) and (1.14)]

-2 “fi&jPjm j=l 0 i

From the anticommutation {P,o1j} = {PJj)

563

(18) rules (I. 16) it follows that

= 0

(j=

l,...,N).

(19

Since s; = iajpj

(20)

(j = 1, . . ..N)

[cJ (1.32)], it is immediately seen that [P, $1 = 0

(j = I, .*, iv),

(21)

l

and consequently [PO,

sj”] = 0

(a = -1,

+l;j

= 1, ... JO l

(22)

Using (8), (11) and (22) we obtain

(a =

-1,

+a

SZ(t;o) =: exp (iX,t)

(23)

si exp (-is&

(a = - 1, + 1).

so,we

(24)

may write the time correlation function of the z components of two spins at sites j and k as

or, substituting (4) and (5) into (25), as

where the time correlation functions (e&s: (l; cr)) (cr = - 1, + 1) are the lation functions for the c-cyclic and the c-anticyclic model, evaluated in I.

564

THJ,

SISKENS AND P. MAZUR

For the canonical average of e.g. the operators Sj”, AP, Z, APAP (d) in the a-cyclic XY model, formal expressions similar to (26) can be obtained in an analogous way. 3. A cummon orthonormal basis of eigenstates ofP and Ho (a = - f or + I). The following procedure holds both for G = - 1 and + 1. We introduce operators qa, k and& (k = 1, ..*, N), defined as

(27)

(k = 1, ..*, Iv),

where UC and Vb are real, orthogonal N x N matrices. The orthogonality of Ub and VUensures the fermion character of the VU-operators (cf. appendix). By appropriately choosing Ub and Vb (cf. appendix), the hamiltonian 9, can be transformed into H

C(na,k

a

-

9

(28)

il a,ky

k=l

where the Ata k ‘s are the positive* roots of the quantities A,“,k (k = 1, . . . , IV) defined by (I.iO), and where the operators n,, k (k = 1, . . . , IV) are fermion occupation-number uperators, defined as n up k

=

d,kqa, k

(k = 1, . . . . Iv).

(29

The 2N states 1{n,, k}) (n,, k = 0 or 1; k = 1, . . . , IV), defined in the usual way, form an orthonormal basis of eigenstates of Sam From (19) and (27) it follows immediately that

{?Iu,ky p} = {&k,P) = 0 Let us, as an illustration, follows that ~a,kPIO;b>

=

-pqa,klO;a)

(k = 1, . . . . N).

(30)

consider the vacuum state IO; 0) uf Sb.

=o

all k(=

1, . . . . IV).

From (30) it

(31)

Thus, P IO; a) = ei’# IO; a)

(&a rea1) 9

* By taking the positive roots of theA:,& Zm to be its ground state as well.

(32) = 1, ..,, N) one ensures

vacuum state IO; U>

TIME-CORRELATION

FUNCTIONS

IN a-CYCLIC XY MODEL, II

565

where e’“” = -1 or +1, sinceP2 = 7. In the same way it follows that all ‘eigenr states i{n,, ,3) of Z, are also eigenstates of P with eigenvalues - 1 or + 1. Let the state I(n,, ,>) be a common eigenstate of P and X, with p ( = 0, 1, . . . or N) modes excited. The eigenvalue of P on this state is given by

= (--l)%*, , (1% ,>Ip @%,k}>

(33)

P O;a =
(34)

with

and where use has been made of (30). From (33) it follows that the eigenvalue of P numbers, be written as

on the state 1{n,, ,>) can, in terms of the occupation

((n,, k}l p l(n~# k}> = PO;, exP

-ix f %,

(39

km

k=l

It is clear that the states

(0; 01 P IO;0) = Det SJl&‘” = fi (A,, &,

(36)

&

k=l

We have to distinguish the following five situations* (if y # 0): Nodd,

B<

-1,

Po.-1 = I, ,

Po;+l

N odd,

IBI < 1,

PO;_1 = 1,

PO;+1 = -1;

N odd,

B>

PO._1 = -1: ,

PO.+1 = -1; 1

N even,

ISI < 1,

PO;_1 = -1,

PO;+1 = 1;

N even,

IB1 > 1,

PO;_, = 1,

PO;+1 = 1.

I,

= I;

(37)

* In discussing the various possibilities (37) for the value of the quantity PO; Q [eq, (34)] we have excluded the case y # 0, ISI = 1 (and in fact also the case y = 0, ]Bj < l), since PO; d may then be undefined due to the occurrence of zero-frequency modes. The case y # 0, IBi = 1 should be studied by taking the limits B -+ 3 (or B + - 1) either from below or from above of the ensemble-averaged expressions.

e

or t

tities

n

-

mi

f

TIME-CORRELATION

FUNCTIONS IN a-CYCLIC XY MODEL. II

567

The free energy of the a-cyclic XY model is given by F=

1 --lnZ=

1 --In

B

c

(U” -1, fl

P

426 (1 + c W?)

l

1

The free energy per spin can be written as F -= N

--

1

NP

+ z-1 + z+1 (e+JY - z-1
ln (t(Z+,

1 In + - ln(Z+, Nl W 1

=--

1 - -1n NP

Z+1 + Z-1

(

>

- $l*(l+L&$n$

1 ]n

--

Z+1 - Z-1
I+

I -- --lnZ+, NP

+ Z-1)

1+

Z+1
-

z-1

(e-2) l

w

z+,

(

+L

>

Inspecting the function In (2 cash &?A (&}, it follows, that Iln Z-1 - In Z+J

= O(1)

(for all N),

(44)

and consequently

Z-l/Z,

W)

1 =

(for all N).

(49

Since furthermore the quantities ZJ(Z+ 1 + Z, J (a = - 1, + 1) are bounded, we obtain in the thermodynamic limit, using (4f), 2x F lrm -= ~+oo N l

.

llm ~+oo

1 -lnZ+, Np

= --

1 287c s 0

d$ In (2 cash &?A (4)) c

(46)

Thus the free energy per spin reduces, in the thermodynamic limit, to the same value for the c-cyclic, the c-anticyclic and the a-cyclic XY model. 5. The z component of the magnetization and the internal energy. The canonical average of the z component of one spin at arbitrary site j, reads for the a-cyclic XY model [analogously to eq. (26)]:

TH.J. SISKENS AND P. MAWR

568 The quantities (Q&)

(a = - 1, + 1) are given by (cJ I section 4)

(0 = -1,

+I)

(48)

for arbitrary+ 1, ..*, IV). m, 1 [@A]and m, 1 [@I]are functionals of the function /?A (@), where A(+) is defined by (1.40). The quantities (QJ~P) (0 = - 1, + 1) are given by

where use has been made of the diagonal form of P in the I(n,, ,})-representations, and of (35). Using the equality of the first and third member of (39), we obtain instead of (49)

<@,sfP> = &rp) [

= (&P)

[

ck)exP (- k=l5 &%,k hr,

k

-

8

-

h,

ck)exp (- i

W,

= (~aP)mb[@l + ix]

k=l

(P&k + i4 (%,k -

k=l

(0 = -1,

5 k}]-I

i7c n,,

+l)

i))]-I

(50)

TIME-CORRELATION

FUNCTIONS

IN a-CYCLIC XY MODEL.

for arbitraryj (= 1, .**, N). Here the quantities &P) functionals m, [@A + ix] read explicitly (cf. (48)) m,[~A+in]=

--

1

CN 2N P=I

{c’s &.I) - B,

coth

II

569

are given by (39) and the

+/jA

A brP

(o

89P

=

-1,

+l). (51)

The z component of the magnetization

per spin is thus given by

where 2 is given by (40), Zb by (38), m, [PAIbY(W, (e=P)by(39),andm,[~~+ by (51); (C = - 1, + 1). From (1.39) we know that limm-,[~A]=limm,,[~A]. N+oo N+CQ

4

(53)

So we may write me1 [16rll =

m_,[@A]+h

tP4

(54)

where limdm [/?A] = 0.

(55)

N-,m

Eq. (52) reads explicitly

+

= (Z+1m,1C,6rfl + z-1m-1 r/w (Q_~P) m-, [PA -t

- Zl

+ z+1 (e+1P)m+1 [PA+ id

id}

x (z+1 + z-1 + z+1 - z-1 (e-P>)-’ =

zilz

[/%I] + z

m_l

i

+1

x

1+ i

-1

AmlJAl +

z-1

z+l z+1 +

z+lz;lz_l - z+lz;lz_l
TH. J. SISKENS AND P. MAZUR

570

= - 1, + 1) and m, [/?A + ix] (a = - 1, + 1) are boundlimit, using (41) and (SS),

Since Z&Z +1 + Z-1)(0

ed, we obtain in the thermodynamic . Iim 1. (@kfz) = lim N+m N

m-,

[@I]

=

1’

N+co

27r

=--

1 47c

s

d+ (‘OS ’ - @ tanh *PA (4)

W)

l

fw)

Thus the z component of the magnetization per spin reduces in the thermodynamic limit to the same value for the c-cyclic, the c-anticyclic and the a-cyclic XY model. Defining the functionals h, [&!l] (C = - I, + 1) as

ha WI = + (e,c@,> =

--& &a,p taw16rla,,

(0 = -1, + l)? (58)

we obtain for the internal energy per spin in the a-cyclic XY model the following expression (using the analogues of eqs. (26) and (52) for this case) 1 c tzb {hawi $
+ c kap) haCPA+

idjy

(59)

where the functionals hd [@I + in] (C = - 1, + 1) read explicitly h&l

+ ix]=

- -!- ~&,coth&%& 2N p=l

-1,

(o=

,

For the internal energy per spin we obtain in the thermodynamic the arguments given previously in this section) J\J

;t; =;\t

+ <@*X*1>

= -$JdW(4)

+I).

(60)

limit (using again

tanhw4

ML (61)

0

which result is again the same for the c-cyclic, the c-anticyclic and the a-cyclic XY model. 6. TIze time correlation,functions elf the z components of two spins. Defining the functionals& [/IA] (0 = -1, +l; j, k = 1, . . ..N) as

=

C 1

5

2N P=I

Ccos 46.a,PP

A

-

B,

tanh

SPA

fl,P

1 2

TIME-CORRELATION

FUNCTIONS IN a-CYCLIC XY MODEL. II

x

sin&t)

--

i

x

Ccos 4a.P- B,

exp i 0 - k>LJ sin A,, pt + tanh 2 exp

i

(j

k) 4ur

-

A U# P -- 1 [$

x

571

II

;I ((i sin&t P=

YSWU,. A u*P

P

+ tanh +&4b,P cosA,,,t) 2

(u = -1,

exp i (&I= - w 4b. P

-t-o,

(62)

II

&f eq. UW), we obtain for the time correlation function of the z components of two spins at arbitrary sites j and k the following expression (using (26) and the analogue of (52)):

The functionais g$3 [@l + i7cJ (a = - 1, + I) read explicitly 2

ws 40, B - B) A urP +- :

c&h

*FL,

P

p$ {(cm Au,pt + i coth +/&p =

i

2

x sin A,, J) exp i (j - k) &, J 1

-- i

x

(i sin A,,.t+

(“’ +U* P

A

-- i [$

x

-

B, exp

i

coth @16,P cos ACI J) 2

(j

-

k>

(b ut P

II

U,P

5

p=l

((i sinIl,,,t

Ysin4u,. expi(jA 61P

+ coth +@&

cos A, , J)

2

k)&,

(a = -I, >I

+I).

(64)

572

Tl3.J. SISKENS

AND P. MAZUR

In the thermodynamic limit we obtain for the time correlation function (@is,” (t)) in the a-cyclic XY model the following expression (using the arguments given in

d# {(cos A(+) t + i tad +@A (4) 2 x

sin A($) t) exp i (j - k) 4} 1

d$

(i sin A(4) t + tanh @A (4) cos A(+) f)

11 2

x

("'4

-

B, eqi(j

4?@

-

w

4

(i sin A(+) t + tanh +/3A(4) cos A(+) t)

which result is again the same for the c-cyclic, the c-anticyclic and the a-cyclic XY model. The time autocorrelation function for the z component of one arbitrary spin can easily be found from (63) by putting j = k. The fourth term in the last members of (62), (64) and (65) will vanish then for reasons of symmetry. 7. The time autocurrelation

function

of the z cumpunent

of the

magnetizutke.

Let us first define the functionals i& [@A](a = - 1, + 1) as (cf. (48))

TIME-CORRELATION

FUNCTIONS

and the functionals FH;t [@l] (0

IN a-CYCLIC

a)) -
@OS $0.P

-

W’

cash- 2 */?&, p

2 A fl*R

” ‘ii

c{ N

i

2N (cj:

‘a*’ (1 + tanh2 @AU,p) cos 2&~ fl,P

y* sin2 &, P

P=I

573

II

= - 1, + 1) as

3,;t WI = ; {(Q,M=M’(t;

3.

XY MODEL.

A

2

tanh +/?A#,P sin 211,, .t

(a = - 1, + 1).

fl*P (67)

eq. (1.51)). The functionals r,; t [@I] (a = - 1, + 1) are defined as r,;,

[/?A] = (@,M”W

(t; c)>

(0 = - 1, + 0,

(68)

which can be expressed explicitly by combining (66) and (67). For the a-cyclic XY model we obtain, using the analogues of (26) and (52) for this case,

~;a;$[PA + in] = ---&

pNl

= E1

- B)2 sinhB2 @&a

(cos ‘i:

btP y2

‘;I

>

d’uwp

(1

+

coth2 &3Aa,J cos 24, pt

09P N ++

2N R=I

TV’sin2A

1.- A

2 fl*P

>

I

P coth @Aa, p sin 2A,,,t (a =

-1, +l),

(70)

and where the explicit forms of the functionals Ma [PA + ix] (a = - 1, + 1) can be found from (51). The z component for the magnetization in the a-cyclic XY

THJ. SISKENS AND Pa MAZUR

574

model is given by (c$ (52))

From (69) and (71) it follows that the time autocorrelation function R(t) of the z component of the magnetization in the a-cyclic XY model is given by 1

wo -= i

-
KW~‘(O>

1 c =Fa=-_l,-_1 + 0 (Qap)

92,

Ta.t [

’

[PAI + ’

yb; t [PA +

N

t”a

ix] + 1 N (Ma

[PAD2 [prl + ix])’ >I

1

(72)

1

(73)

2

326

{Ma

[PAI + o (@ap)

Mcr

IPA

+

ixl}

l

The expression (72) fur R(r) can be rewritten as

--

The

k [$ 0=-l,C

+Za (M, [pAI -5 6 (@ap)Ma [PA + ixl} ’ +1

first term on the r.h.s. of (73) gives in the thermodynamic lim -

1

C

Il’+ccz a=-I,+1

l

limit

-$Za {7;6;r [/VI] + Q (gap) F,; t [,Rrl + in]} = lim r+ 1; 2 [,6rll,

iv*09

(74)

where the arguments given in section 5 have been used again. Since the quantities (~,a> (< CN(0 < c c 1)) tend to zero very fast in the thermodynamic limit, it is easily seen that for the second and third term on the r.h.s. of (73) we have

TIME-CORRELATION

1 Z,l

-{

FUNCTIONS IN a-CYCLIC XY MODEL. II

c

+ z-1

u=-1,

+1

WI

UK

II

2 +4N),

575

(75)

where E(N) --) 0 as N --) 00. Defining the functional AM [@I] as

AWN

= M+1 WI - M-1 ~/w,

(76)

it follows from inspection of the function (cos 4 - B) JW?M tanh W

(4)

(cf. (66))

that IdM

IrBrill= @xl)

(for all IV).

(77)

Omitting the symbols [@I] and also omitting the term E(N), the r.h.s. of (75) can be expressed as 1 N

1 z,,

1

+ z-1

a=-1,+1

+ Z-lit&

Z+lM:l

1 -N

z+1

Z+lM+l

2+14-z-1

-

=i

>I >I ZoMb ’

2

+ Z+l (AM)* + Z_,M:l

Z+1 + Z-l

Z+,if_l+ z+l AM + z+, + z-1 =-

+ ZIK1

z+1 + z-1

Z+lMfl + 2Z+,M_lAM

1

+z_p=-1,+1

1

MI,

N

-

=+1

+

Z+, + Z-1

2Z+1

z,, + z-1

2

z_lM_l II

Z +I

iKIAM+

Z+1 + Z-1

zf

I

i&AM-

(LIM)~ - Mf,

(AW2

tz+1 + Z-d2 1

=Nz+1 +-Z-J _(Z,, +Z_J2 1(AM)** Z +1

1

[

Zl

(78)

TH. J. SISKENS AND P. MAZUR

576 Since by virtue of (77)

~+ao 1 Iim N

Z +I

Z,,

Z2

+ Z_,

(Z,. 1 ++H_,,2

we obtain in the thermodynamic

=0 1(dM)2

(79

limit for R(r), using eqs. (73)-(W), 27r

lim R(t) = lim7’,,., [@I] = -!d+ (‘os ’ - @’ cosh-2+/?A (4) N+oo ’ 8x s Iv3a A’(#) 0

2x

+-

1 8x s 0

d+ ‘2::)’

(1 + tanh2 #A (4)) cos 2A (+) f

Therefore in the thermodynamic limit we obtain- also for the time autocorrelation function of the z component of the magnetization the same result for the c-cyclic, the c-anticyclic and the a-cyclic XY model.

APPENDIX

The argument given in this appendix holds both for 0 = -1 simplicity of notation we shall omit the subscript G. The operators qk and 7: (k = I, . . . , N) are defined as

(k = 1, **$Iv), l

7;

=

c

N

i=l

and +I.

For

(A.0

1 -

(V,,a,

-

i y,,&)

J2

where the matrices U and Y are real orthogonal IV x N matrices. We shall show in this appendix, that the orthogonality of U and V ensures the fermion character of the q-operators, and furthermore that U and V can be chosen in such a way that JV assumes in terms of the q-operators the diagonal form (28). First we shall

TIME-CORRELATION FUNCTIONS IN a-CYCLIC XY MODEL. II

577

show that the q-operators are fermion-operators.

=

5 ( ujlUkm (@l&m+ %a%)-

l,m=l

+ ivjl

ukm (@lam +

Using the anti-commutation

&rn@l)

{qj,

qk}

=

i

wjl vkm (LYlpm+ bd}

l

rules (I. 16) we obtain

h, rlk} = (Wjk - (Wljk Using the orthogonality

+

&l vkm (bl~rn + r6mr61)

(j, k = 1,

l -9

N).

(fw

of U and V, it follows that

(A4

0

Trivially

In a similar way it is easily seen that

and thus (~J,Q> = Sjk

(A.7)

(j, k = 1, a9 W l

It is obvious from (A.3) and (A.6) that the orthogonality of U and V is also a necessary condition to ensure the fermion character of the q-operators. In terms of the a- and p-operators the hamiltonian reads H = i 5

OC~S&&.

(A- 8)

j,k=l

Inversion of (A.1) yields

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

(A.9)

578

TK J. SISKENS AND P. MAZUR

In terms of the q-operators X reads (A. 10) We choose Uto be a real orthogonal matrix diagonalizing the symmetric matrix S&

(Km),,

= A&,

(j,k=

l,...,N),

(A.11)

where the &(k = 1, . . . . N) are given by eq. (1.30). Furthermore real orthogonal matrix V by

(usv)jk = A,6,,

cl; k =

I,‘...,

(A. 12)

N),

where rl, = + (&)+ (k N). The transformation (A.10) with above-mentioned way takes hamiltonian

we define the

and V subsequently defined in the form (A. 13)

REFERENCES Mazur, P. Siskens, Th. J., Physica 69 36 (1967) 2) Niemeijer, 3) Lieb, E., Schultz, T. and Mattis, D., Ann. Physics 16 (1961) 407.