737
Progress of Theoretical Physics, Vol. 46, No. 3, September 1971
Difference Equation Solutions to the One-Dimensional
Ising Model with General Spin
H. SILVER and N. E. FRANKEL
School of Physics, University of Melbourne
Parkville, Victoria 3052, Australia
(Received February 23, 1971)
The one-dimensional Ising model with general spin S and either open or periodic
boundary conditions has been formulated as a difference equation of order 2S + 1 with nonzero magnetic field B. This difference equation has the characteristic equation of the transfer matrix, V, as its indicial equation. When B=O, for the open chain, the order of the
difference equation is reduced to [S+1]. It is shown that the correlation functions for the
open chain satisfy similar difference equations. This result explicitly displays the lack of
translational invariance of the open chain. For several cases it is shown that the partition
function can be represented as the difference of two Chebyshev· polynomials of the second
kind.
§ 1.
Introduction
The spin 1/2 Ising chain has been extensively investigated because it can
be solved exactly and has application to some interesting problems in statistical
physics. 1 )' 2) The periodic chain for arbitrary spin S can be directly formulated
as a matrix problem. Suzuki et al. 3) obtained solutions to this problem for a few
values of S together with series expansions of the partition function and the zero
field susceptibility. The limiting case s~oo was exactly solved by Thompson. 4)
In this paper we study both the open and the periodic chains for arbitrary
spin S. In § 2 we use a simple inductive technique, introduced by Leff and
Flicker 5 ) and by Frankel and Rappaport/) to show that the partition function for
the open and periodic chains satisfies a linear difference equation that is closely
related to the transfer matrix V. The open chain can also be directly formulated
as a matrix problem, 6) and Dobson7) used this method to study the many neighbour, open, arbitrary spin chains. In § 3 we show that for the open chain in
zero external field the order of the difference equation is reduced from 2S + 1
to [S + 1]. In § 4 we show that this corresponds closely to the reduction of order for the periodic chain used by Suzuki et al. 3) and generalized by Dobson. 7 )
Marsh 2) expounded in detail the matrix method for obtaining correlation
functions for the periodic Ising chain in an external field. Frankel and Rappaport1) and Stephenson8) have calculated correlation functions for the Ising chain
with next-nearest neighbours. In § 5 we show that the correlation functions for
the open, spin S chain satisfy difference equations analogous to the one satisfied
738
H. Silver and N. E. Frankel
by the partition function. Except for the degenerate case S = 1/2, B = 0 when
the correlation functions are independent of the position in the chain, this result
clearly exhibits the difference between the open chain and the closed chain. In
Appendix B we prove that far enough away from the first spin the open chain
correlation functions tend to the corresponding closed chain functions.
This paper therefore presents a complete study of the partition functions and
correlation functions for all nearest neighbour Ising chains. The difference equation technique has been used to investigate these chains as it provides a direct
and natural way to compare and contrast the various different chains as well as
readily displaying the analytical structure of these models. In particular, in § 6
we show for certain models that the solutions of these difference equations can
be expressed in terms of orthogonal polynomials.
§ 2. Pa:rtition function
Consider an open, one-dimensional Ising chain of N spins si each of spin S.
In an external magnetic field B and with nearest neighbour interactions only, the
total energy of the system is given by
N
N-1
HN {si} = - J ~ sisi+l- fJ.oB ~ si .
i=l
(2 ·1)
i=l
The partition function ON is given by
ON=~ e-flHn{si}
{Si}
(2·2)
where
and
We show that ON satisfies a linear difference equation of order 2S + 1 by an
inductive technique : in Eq. (2 · 2) we sum over the first spin s1 and find
ON= :E eK 2 sXN-1 (s),
(2·3)
8
where
(2·4)
and
"£..f(s)
=
f(S) + /(S -1) + .. · + f(O)
+ ··· + f(-
(S -1))
+ f(- S)
8
for integral spin
=f(2S) +f(2S-2)
+ ... +f(- (2S-2)) +f( -2S)
for half-integral spin .
(2·5)
Difference Equation Solutions to the One-Dimensional Ising Model
739
Now we sum over s1 in Eq. (2 ·4) and find
XN (s) = ~ eKiss'+K2s' XN-1 (s').
(2·6)
s'
Equation (2 · 6) is a system of 2S + 1 simultaneous linear difference equations for
XN (s). They can be written compactly using matrix notation as
(2·7)
where XN = (XN (s)), U = (eK ss'+K 2s'). It is easy to show (cf. simultaneous linear
differential equations) that each component XN (s) of XN satisfies the 2S + 1 order
linear difference equation
1
[det(U-e·I)]XN(s) =0,
(2·8)
s
where e is a linear difference operator: exN (s) = XN+1 (s). If
is half-integral,
Eq. (2 · 3) expresses ON as a linear combination of the 2S + 1 functions XN-l (s)
each of which satisfies the difference equation (2 · 8). ON must therefore also
satisfy this difference equation. If S is integral then QN = XN (0) satisfies Eq.
(2 · 8). In both cases
(2·9)
u-
Equivalently p u( e) QN = 0, where p uO.) = det (
).1) = II~!~ 1 (A- Ai) is the characteristic equation for the matrix U whose eigenvalues are Ai (i = 1, 2, · · ·, 2S + 1).
Equation (2 · 9) has the general solution
28+1
ON=~ AiAiN·
(2 ·10)
i=l
The Ai are functions of K 1 and K 2 fixed by the 2S + 1 partition functions ("initial conditions") Q2, Qs, · · ·, Q28+2·
For the closed (periodic) chain (i.e., SN+I=S1) we have the well known
resule)
28+1
QN=Tr VN = ~/.LiN
(2 ·11)
i=l
where V = (eK 1 ss'+<K2f2 )(8+s')) is the symmetric transfer matrix whose eigenvalues
are f.Li (i=1,2, ···, 2S+1). We can also express Eq. (2·11) as a 2S+1 order
linear difference equation:
(2 ·12)
The 2S + 1 partition functions Q2, Q8, • • ·, Q28 +2 ensure the correct solution.
valently Pv(e) QN=O.
The matrices U and V are similar:
Equi-
(2 ·13)
where V1 =
(Oss'e-<K2f 2 )s').
U and V therefore have the same set of eigenvalues
H. Silver and N. E. Frankel
740
Ai (i = 1, 2, · · ·, 28 + 1) and the partition function for both the open and the closed
chain satisfies Eq. (2 ·12). For the open chain QN = ~~~t 1 Ai).iN, whilst for the
closed chain QN= ~~~t 1 AiN· The Helmholtz free energy per spin, ¢, is given by
- {3¢ = limN->oo [log QN/N]. Since ).1 is non-degenerate (as Vis non-negative) for
both the open chain and the closed chain, - {3¢ =log ). 1, and the thermodynamic
behaviour of the infinite chain is independent of the boundary conditions. Dobson7>
proved this result for the more general arbitrary spin Ising chain with a finite
number of neighbours.
We exhibit difference equations for a few values of S: S = 1/2,
V= ( eK1+K2 .e-K1 ),
\ e-K1
eK1-K2
Pv(A) =). 2-2eK1 cosh K 2).+2 sinh 2Kt,
ON+ 2-2eK1 cosh K20N+1 + 2 sinh 2K1QN= 0.
(2 ·14)
This equation has been previously exhibited by Frankel and Rappapore> and by
Leff and Flicker. 5> For the closed chain we have the well-known result QN=
A1N + A2N where
).1=eK coshK2+ Ve 2K1 cosh 2K2-2 sinh 2Kt
2
2
2=eK1 coshK2- Je K1 cosh K2-2 sinh 2Kl.
1
>).
For the open chain it is convenient to put QN=2N coshN- 1K 1 coshNK2QN and we
find
QN+2- (l+at)QN+l+at(1-a2 2)QN=0,
where ai =tanh Ki.
Equation (2 ·15) has the solution
QN= A1(1N + A2(/,
where
for all T,
A1 =
A 2=
1
2~1
2
_ (1 + 3ala2 2 + a1 2a2 2 - a1 + (1 + a1a2 2) .JLf),
.J L1
1
-
2(2
2
.J L1
2
(1 + 3ala22 + a1 2a2 2 - a1- (1 + a1a2 ) .JLf).
For S= 1
eK1+K2 e(1j2)K 2
V=
e<l/2)K2
(
e-K
1
1
e-Cl/2)K
2
(2 ·15)
Difference Equation Solutions to the One-Dimensional Ising Model
741
We find that the spin 1 open and closed chain partition functions satisfy the third
order linear difference equation
QN+S-
(1 + 2 sinh K 1 cosh K 2 + 2 cosh K1 cosh K2) QN+2
+ 2 (sinh 2Kl +sinh K 1 cosh K2 +cosh K1 cosh K2- cosh K2) QN+l
(2 ·16)
§ 3.
Partition function-zero external field
We consider the same open Ising chain as in § 2 except that now B = 0, so
K 2 = 0. If we set K 2 = 0 in Eqs. (2 · 3), (2 · 4) and (2 · 6) we find
(3 ·1)
where
XN (s) =
N-1
~ eKssl
II eKSiSi+l
{s,}
t=l
(3·2)
and
(3·3)
The number of these equations can be reduced by grouping XN (s) and XN (- s).
We put
i=
lsi.
(3·4)
The cases S half-integral and S integral now have to be treated separately.
(A)
Half-integral spin
Using Eq. (3 · 4), Eqs. (3 ·1) and (3 · 3) can be written
(3·5)
(28+1)/2
YN(i) =2 ~ cosh[(2i-1) (2j-1)K] YN-l(j).
(3·6)
j=l
In matrix notation, Eq. (3 · 6) becomes
YN=2AYN-1'
(3·7)
where YN= (YN(i)), A= (cosh[(2i-1) (2j-1)K]), 1<i, j<(2S+1)/2. Equations (3 · 5) and (3 · 6) are formally analogous to Eqs. (2 · 3) and (2 ·17) and we
immediately conclude that QN satisfies the linear difference equation of order
(2S+ 1)/2.
[det(2A-e·l) ]QN=O.
Equivalently P A (e/2) ON= 0.
Equation (3 · 8) has the general solution
(3·8)
742
H. Silver and N. E. Frankel
(3·9)
where ai (i = 1, 2, · · ·, (2S + 1) /2) are the eigenvalues of A and the Ai are functions of K fixed by the (2S+ 1) /2 partition functions, Q 2, 0 3, • • ·, Q2+( 2S+l)/2 •
We illustrate these results for a few values of S:
For S= 1/2
A= (cosh K),
P A (A)
=
cosh K- A. ,
QN+l-2 cosh KON=O.
(3 ·10)
The partition function Q 2 = 4 cosh K fixes the general solution as ON= 2N coshN- 1K.
For 8=3/2
cosh K cosh 3K)
A= (
cosh 3K cosh 9K '
PA (A.)= ). 2- (cosh K +cosh 9K) A.+ (cosh K cosh 9K- cosh 23K),
ON+ 2-2 (cosh K +cosh 9K) QN+l + 4 (cosh K. cosh 9K- cosh 23K) QN= 0.
(3 ·11)
It is convenient to put ON= 2N coshN-lKQN, and we find
(3 ·12)
where
a(K) = cosh 3K- -3+4 cosh2K'
coshK
{3 (K) = _£osh gK
coshK
=
9-120 cosh 2K + 432 cosh4K- 576 cosh 6K + 256 cosh 8K.
Equation (3 ·12) has the solution
QN = A1[31N + A2[32N,
where
/.) = 1+{3+ .JLf>(.) =1+[3-.JLf
Pl
L1 = (1 -
2
f3Y +
fJ2
(2a)2>0
2
for all T ,
Q 2 =1+2a+[3 and Qs=l+2a+2a 2 +2a{3+[3 2
A1 =
fix A1 and A2:
~[1/2
+a+ 2a 2 + a[3 + [3 2/2- [3 +
2
/31 vJ
A2 =-
'
vLf (1/2 +a+ [3/2)],
2
~[1/2+a+2a
+af3+{3 2 /2-[3- .JJ(1/2+a+f3/2)].
2
/32 vJ
Difference Equation Solutions to the One-Dimensional Ising Model
(B)
743
Integral spin
Using Eq. (3 · 4), Eqs. (3 ·1) and (3 · 3) become
s
QN =
ON-1
+ 2 2:
YN-1 (i)'
t=l
(3 ·13)
s
YN (i) =
QN-1
+2
2: cosh ijKYN-1 (j).
j=l
(3 ·14)
These can be written in matrix notation as
(3 ·15)
where
B=(~
E= (cosh ijK)
and
1<i, j<S.
Since QN is the first component of ZN we conclude that QN satisfies the linear
difference equation of order S + 1:
(3 ·16)
Equation (3 ·16) has the general solution
(3 ·17)
where ri (i = 1, 2, · · ·, S + 1) are the eigenvalues of B.
We illustrate these results for a few values of S:
For S=1
2
)
2 cosh K '
B=(l
\1
PB(A) =A 2 - (1+2 coshK)).+2(coshK-1).
We make the change of variable y = 1-1/ cosh K, put ON= coshN- 1I(QN and find
QN+2- (3-y)QN+1+2y(l-y)QN=0.
Equation (3 ·18) has the solution
·where
3-y+J~
2
>r2-
J = 9y 2 - 14y + 9 ,
3-y-J~
2
,
(3 ·18)
744
H. Silver and N. E. Frankel
A = -~ 17y 2 - 36y + 27 + (9 - 5y) .Jj
1
.JL1 (3- y ) 2 + 2 (3- y) .J L1 + L1 '
A
= ~2
.Jj
(17y 2 -36y+27) + (9-5y)
(3- y y- 2 (3- y) .Jj + L1
.Jj
For S=2
B=
(
1
1
2
2 cosh K
2
)
2 cosh 2K
1
2 cosh 2K 2 cosh 4K
and we find QN Satisfies the third order equation
ON +a- (1 + 2 cosh 2K + 2 cosh 4K) QN+ 2
+ 2 (cosh 4K +cosh K cosh 4K- 2 cosh 22K +cosh K- 2) ON+1
-4 (2 cosh 2K +cosh K cosh 4K- cosh 4K- cosh22K- cosh K) QN = 0 .
(3 ·19)
§ 4. Reduction of order for open chain (B=O)
In Eqs. (2 · 3) and (2 · 6) we can expand QN in the functions YN (i) and ZN (i),
where
YN(i) =XN(s) +XN( -s),
(4 ·1)
ZN(i) =XN(s) -XN( -s).
Equation (3 · 4) is a special case of this orthogonal transformation of the vector
XN. The transformation corresponds to expanding the exponentials eK 1881 and eK 2 .~
in hyperbolic functions and is given by Dobson. 7) We find QN satisfies the
2S + 1 order linear difference equations
P2a(e)QN=O
for half-integral S,
(4·2)
PD(e)QN=o
for integral S ,
(4·3)
where
C=
(
R(l/2)
Tc1;2)
s(l/2) ) ,
Ucl/2)
1
2a'
0
2Tcl)
2/3' )
D= ry 2R 0 > 2Scl) .
(
2Ucl)
1<i, j<S.
S 0 /2), T(l/2) and U(l/2) are Rcl), Scl), Tel> and Ucl) with i ( j) replaced by
2i-1 (2j-1) and 1<i, j<(2S+1)/2.
R(l/2),
a=(coshiK2),
/3=(sinhiK2),
ry=(ri),
ri=1,
1<i<S.
Difference Equation Solutions to the One-Dimensional Ising Model
745
Clearly the matrices C and D are similar to the transfer matrix V and the
B = 0 limit of Eq. (2 ·12) is the B = 0 limit of Eq. ( 4 · 2) for half-integral spin
and the B = 0 limit of Eq. ( 4 · 3) for integral spin. But
c<B=O>
=
(A Fo ),
\0
where A and B are defined in Eqs. (3 · 7) and (3 ·15),
F= (sinh[ (2i -1) (2j -1) K]),
<i .<2S+1
1'
2
' )_
G =(sinh ijK),
l<i, j<S.
Hence
p V(B=O)
(A)
p V(B=O)
(A) =
=
(A) P211 (A)
for half-integral spm ,
(A) P2G (A)
for integral spin ,
p2A
p B
and so the B = 0 limit of Eq. (2 ·12) 1s
PFCe/2)PACe/2) oN=o
for half-integral spm ,
(4·4)
Pa(e/2)PB(e)QN=o
for integral spin.
(4·5)
In § 3 we proved that for the open chain these equations (of order 2S + 1) can
be reduced to equations of order [S+1], viz., PA(e/2)QN=O, PB(e)ON=O. For
the closed chain in zero field the partition function satisfies Eq. (2 ·12) but it
does not satisfy any smaller order difference equation. However, due to the factorizations in Eqs. ( 4 · 4) and ( 4 · 5) we can write for the zero-field closed chain
partition function
(28+1)/2
QN =
~
i=1
(2ai)N +
(28+1)/2
~
(2/f.i)N
for half-integral S,
(4·6)
for integral S,
(4 ·7)
i=l
where the ai are eigenvalues of A, /ii of F, ri of B and lh of G. Thus the
closed chain eigenvalue problem reduces to two eigenvalue problems or order S
and S + 1 respectively for integral spin and two eigenvalue problems of order
(2S + 1) /2 for half-integral spin. For B=/=-0 this is the starting point of Suzuki
et al.'s 3> perturbation expansion for the largest eigenvalue of V.
We illustrate these results for a few values of S:
S=1/2
A= (coshK),
F= (sinh K),
(4·8)
H. Silver and N. E. Frankel
746
8=3/2
A_ ( cosh K
cosh 3K
sinh K
cosh 3K ) ,
cosh 9K
sinh 3K)
F= (
sinh 3K sinh 9K '
ON= (2 cosh K)N[C+St JJ~r +
1
+ (2 sinh K)N [ ( + ();
JJ;)
e+e;
N
JJ,n
1
+ ( + () ~
JJ;) NJ,
(4. 9)
where
J1 = (1- t1Y + (2aY>O ,
J2 = (1- &) 2+
cosh 3K
a=----
/1= cosh 91(
coshK
for all T,
(2rY>O
r=
coshK
sinh 3K
sinh K '
()=sinh 9K
sinh K '
S=1
B=(11
2
2 cosh K
)
'
G= (sinh K),
where
y=1-
1
'
cosh K
for all T.
§ 5. Spin correlation functions
The (k+1)-spin correlation function
finite open chain is defined by
([r,z+r 1 , ... ,z+r~c)
(O<rl< .. ·<r~c) for the
(5 ·1)
where
(5·2)
Provided
Z> 1
we find on summing over s1 in Eq. (5 · 2)
rj{r, l+rw .. , l+rk) = 2: eK sx!J:}, l+r
2
1 -J,
... ,
l+r~c-1) (s),
(5·3)
8
where
(5·4)
Difference Equation Solutions to the One-Dimensional Ising Model
747
Similarly we find for X/i.z+rp· .. ,L+rTc)(s):
X((,l+rr,···,l+rk) (s) = ~ eK 1 ss'+K 2 s'X/f_~l~l+rl--l,···,l+r~c-1) (s').
(5 ·5)
s'
The N-dependence of Eqs. (5 · 3) and (5 · 5) is eliminated by dividing by ON
and taking the thermodynamic limit N-H>O. We define
Equations (5 · 3) and (5 · 5)
becom~
(5 ·6)
(5. 7)
From Eq. (2 ·10)
QN-~= l
ON
}.l
{1 + A2 ( }.2) N-1 + ... + A2S±_!_(l2J+I) N-1} I
Al
}.l '
Al
}.1
(5 ·8)
as 1V-"oo.
Now we define (rl,···,rk) Uz =All.. <szSz+rr' .. Sz+rk), (rl, ... ,rk) VL (s) = ;,lL. (ru·",rk)XL (s) and
find Eqs. (5 · 6) and (5 • 7) reduce to the set of simultaneous linear difference
equations
0
'
(5 ·9)
(5 ·10)
These equations are identical in structure to Eqs. (2 · 3) and (2 · 6) if we make
the correspondence l~N, Uz<~QN, Vz (s) <~XN (s). we immediately conclude that
(rp···,r~c) Uz satisfies the linear difference equation of order 2S + 1
[det (V- ei) J(ru···,Tk) UL = 0'
Equation (5 ·11) has the general solution
(5 ·12)
The Bi (i = 1, 2, · · ·, 2S + 1) are functions of r1, · · ·, r1c and T. The infinite open
chain correlation function <szsz+r
Sz+rk) is therefore given by
1
,
• • ·,
H. Silver and N. E. Frankel
748
(szSz+r
1
,
( J..2 (T) \ z
.
_
)
Sz+r 1)-B1(rh ... , rk; T) +B2(r1, ···, rk, T)
J..1(T)
... ,
+ ··· + B 28+1 ( r1,
J..28+1(T))z
· · ·, rk; T) ( --;:(T) .
Bh · · ·, B 28 +1 are fixed by the 2S + 1 correlation functions (s1s1 +r
(s2S+h ···, S2s+1+rk). When B=O we find (cf. § 3)
<SzSL+r '
1,
Sz+rk
• • ·,
) = B 1+ B 2(' aa2) z+ ' .. + B (28+1)/2 (ac2s+1)/2)
___o__a:___:_~ z
1
1
(s,s,H, .. ·, s,H.> = B, + B,( ~: )' + ". + Bs+t ( r;~t
)'
(5 ·13)
1
,
• • ·,
s1+rk), · · ·,
for half-integral S ,
for integral
s.
(5 ·14)
Except for the trivial case S=l/2, B=O, Eqs. (5·12), (5·13) and (5·14)
explicitly display the lack of translational invariance (i.e., l-dependence) of the
infinite open chain correlation functions. Clearly
(5 ·15)
as l---H>O
and far enough into the chain the correlation functions become independent of
the position in the chain. If J.. 2 (a 2, r2) is the second largest of the eigenvalues
J..i (ai, ri) then
(szSz+r SL+r
1
2
,
···,
Sz+rk)r-vB1Cr1, ···, rk; T) +B2(r1, ···, rk; T) (
J..2(T)
J..1(T)
)z
(5 ·16)
as l->oo
and <szSL-1-rl, .. ·, Sz+rk> tends monotonically to Bl as l-~oo.
For the closed chain (translationally invariant)
(szSz+r
1
,
···,
Sz+rk)=B(rh ···, rk; T).
(5·17)
In Appendix B we prove that B (rh .. ·, rk; T) = B 1 (rh .. ·, rk; T). Clearly the
open chain correlation functions are much more complicated to calculate exactly
than are the corresponding closed chain ones. The latter calculation has been
.expounded in detail by Marsh 2l for the S= 1/2 case. For arbitrary spin S we
show in Appendix A that the closed chain pair correlation function (szSL+r) is
given by:
<SzSL+r >=
y•2
-"11
'\' ().38+1)r
'\'2(A2)r
------- ·
+ ''. + -"128+1
+ -"12
J.1
J..1
When B = 0, .£11 = 0 (as there can be no long range order for T>O).
simplest case S = 1/2, B = 0,
(5 ·18)
For the
Difference Equation Solutions to the One-Dimensional Ising Model
749
and for r= 0, <slsl+r) = 1, so 1:11 = 1 and we recover the well-known result <slsl+r)
= tanhr K. For higher values of S we have to diagonalize V in order to find .J:li.
In general, the specific correlation functions needed to fix the Bi (rl> · · ·, rk; T)
for the open chain can be determined by the inductive technique which results
in a complicated hierarchy of simultaneous algebraic equations. For the particular
case S= 1, however, we can find (forB= 0) the pair correlation function <slsl+r)
by a simple variant of this procedure.
In this case
(5 ·19)
where
r
1
= cosh K 3 - Y + .JL[
2
1
cosh K
y=l-~--
'
r". = cosh J( 3 -
Y-
2
.JL{
'
and
and B2 are determined from <s1s1 +r) and <s2s2 +r ). These are easily calculated
by using the exponential expansion valid for spin 1 :
B1
cKs=coshK(1+as-y
where a= tanh K.
1
+
~- 1 f)
(5 ·20)
(Equivalently eKs = 1 + (sinh K) s+ (cosh K -1) s2 ) .
Now
(5·21)
Summing over s1 in Eq. (5 · 21) w8 find
N
1}(l,l+r)=
2 a cos h
TT
N-l
(5 ·22)
.1'-'Yj(l,r).
Repeated use of Eq. (5 · 22) gives
r;~, 1+ r) = (2a cosh
KY r;~:I) ,
(5 ·23)
where
and we find
2
r;~:I) = 2 cosh KQN-r-1- 2y (1- y) cosh KON-r- 2
•
Combining Eqs. (5 · 23) and (5 · 24) we find
1
1
1j~,l+ r) = 2T+ ar coshT+ K (QN-r-l- y (1- y) cosh KQN-r-2)
(5 ·24)
H. Silver and N. E. Frankel
750
so
and
(5 ·25)
(5. 26)
Summing over s1 m Eq. (5 · 26) we find
r;~, 2 +r) = (3- y) cosh K r;(i:I~- r)
,
so
f'N
'-,(2,2+1")=
(3 -y ) cos h K
ON-1 f'N-l
0~-~.,(l,li-1")
-N
and
3-y
<s3s2+r) = -(~--~- · <slsl+r) .
(5. 27)
We substitute Eqs. (5 · 25) and (5 · 27) into Eq. (5 ·19) and find
Finally we have the general expression for the spin 1 infinite open chain pair
correlation function in zero field,
(5 ·28)
Since B 2 <0, for a given value of r, spins further into the chain are more correlated than those close to the first spin s1. By Appendix B, the corresponding
closed chain result IS
(5 ·29)
and for r=O
Difference Equation Solutions to the One-Dimensional Ising Model
(sl2)=
751
}_+2~: v'A
=2/3
for T= oo,
=1
for T=-.:::0.
§ 6. Analytic structure of some partition functions
In the preceding chapters we have shown how the partition functions for
the open and closed chains for arbitrary spin S and the correlation functions for
the open chain satisfy linear difference equations. If the difference equation is
written
then QN can always be found as a sum of the type
(6 ·2)
where P(),i) = 0 (and Ai = 1 for the closed chain). Because the difference equations are linear their solutions can also be obtained by a transform technique.
This reveals how some particular (viz. order 2) partition functions can be represented as sums of Chebyshev polynomials of the second kind.
We form a generating function for ON and then find ON from a particular
expansion of it. If
(6·3)
then
ON+q + alQN+q-1 +
· · · + aqQN =
0.
(6·4)
The generating function F(z) is defined by
00
F(z) =
:E Qn+2zn
n=O
F(z) =
:E Qn+lZn
for the open chain ,
00
for the closed chain,
n=O
inside the circle of convergence.
Since
we easily solve for F(z) from Eq. (6 · 4):
F(z) =
where for the open chain
Q(z) ,
zqP(1/z)
(6 ·5)
H. Silver and N. E. Frankel
452
and
{1
zqP(1/z)
=II (1-zAi)
i=l
so
ON is the coefficient of zN- 2 in this rational function.
We can recover Eq. (6 · 2)
if we use the geometric series
1
lzl<-.
A1
We can also expand a product of two linear factors in the denominator as a
series of Chebyshev polynomials of the second kind, Un (x) which are defined
by the expansion
(6· 6)
We find simply that
(6·7)
This gives a useful representation for QN when P(A) is a second order polynomial. In this case
F(z) =Q2+ (Qs+alQ2)z
(1 - ZA1) (1 - ZA2)
= [Q2 + (Os + a1Q2) z]
A2 ) ( J A1A2tzn,
:E Un (A~+
2 A1A2
n=O
so
[Q 2U _2 (A1 +J.2) + Qs+alQ2 U _3 (A1 +J.2 )]
Q = (JIT)N-2
12
N 2 J A1A2 '
J A1A2
N 2 J A1A2
N
where we define U-1 (x) =0.
(6 · 8)
We illustrate Eq. (6 · 8) for a few values of S:
8=3/2, B=O (open chain),
2
2
QN= ( J {3- a )N- [ (1 + 2a + {3)
UN-2( 2 ~; {3 a2)
-2J {3-
a 2 UN-s( 2 ~;! a 2)],
(6·9)
where
cosh 3K
a= coshK '
{3- cosh 9K.
coshK
Difference Equation Solutions to the One-Dimensional Ising Model
753
S= 1, B= 0 (open chain),
QN=(V2y(1-y) )N- 2[(9-5y) UN-2 (
-3V2y(1-y) UN-3(
2
2
J
3
-=-Y. )
2y(1-y)
J 3 -y . )]·,
(6 ·10)
2y(1-y)
where
1
y=1---coshK
8=1/2, B=/=0 (open chain),
2
2
2
QN = ( J a1 (1- a2 ) )N- [ (1 + a1a2 ) UN-2 (
2
J a~~~ a 22 )
)
(6 ·11)
where
S= 1/2, B=/=0 (closed chain),
. h 2 K 1)N- 1[ 2eK
0~N = (2 Sill
1
; .
h K 2 U N-1
COS.
- 2-v 2 smh 2K1 UN-2
(eK
J
(ei(V cosh
. K2)
·
1
2 smh 2K1
cosh K 2 \ ]
.
) .
2 smh 2K1
1
(6 ·12)
Similar expansions can also be found for a few correlation functions. In all
cases the arguments of the orthogonal polynomials are outside the usual range
[-1,1].
§ 7. Discussion
In this paper we have seen that the use of difference equations IS a natural
technique for studying Ising chains.
In a straightforward manner this technique can be used to formulate and
study Ising models in higher dimensions. However, now due to the prohibitive
order of these equations, it is expected that a detailed analytical study would be
much more difficult. We are at present studying the finite two dimensional nearest neighbour Ising model with a finite external magnetic field.
Acknowledgement
One of us (H.S.) would like to acknowledge the financial assistance of the
Commonwealth Government through a Commonwealth Postgraduate Award.
H. Silver and N. E. Frankel
7'54
Appendix A
Pair correlation function
The direct matrix formulation is used to derive an expression for the pair
correlation function for the arbitrary spm closed chain. This correlation function is defined by
f'N
'-::.U,~+r)=
N
Q N -1 'l/(~.~+r)'
(AI)
where
=
~
StSl+r
{·~t}
N
II eKt8i8i+t+(K2f2)(8i+Si+t).
(A2)
i=l
The transfer matrix V and the diagonal matrix S are defined by the matrix elements:
Equation (A2) can be written
= ~
(VL-lsvr SVN+l-L-r)ss
s
=
Tr cvl-lsvr SVN+l-~-r)
= Tr cvN-r svr S).
(A3)
Now QN=Tr VN, so
(A4)
The symmetric matrix V is diagonalized by some orthogonal matrix L:
L'L=I,
L'VL=A,
and Eq. (A4) can be written
N
_
(u.~+r)-
Tr(L'VN-rsvrSL)
Tr(L'VNL)
_ Tr (AN-r .SAr17)
Tr AN
where
.S=L'SL,
Difference Equation Solutions to the One-Dimensional Ising Model
Now we take the thermodynamic limit N->oo.
755
Then Cfz.z+r)--'J-<SzSz+r) and
<szSz+r) = A1-r Tr (r;.SAr .S)
where
1/iJ
=0
unless i = j= 1 and r; 11 = 1.
We easily find
(AS)
Appendix B
Equality of open and closed chain correlation functions
We prove the equivalence of the open chain pair-correlation function <szsz+r)
m the limit l--'J-oo to the closed chain function <szSz+r). For the open chain,
N-1
"ffN
-"'
•t(l,l+r)-£.....1
s l s l+r
{8i}
II
N
eKISiSi+i
i=1
II
eKzSi
i=1
(Bl)
If we define the matrix E which has the matrix elements
Equation (Bl) can be written
r;fz. l+r) = Tr cv~- 1 svr svN-~-r E)'
(B2)
QN = Tr cvN- 1E).
(B3)
and we find
Introducing the orthogonal matrix L we find
Tr (A~- 1 .SAr .SAN-~-rF)
I,(Z, l+r)Tr (AN-1F)
rN
where F= L' EL.
_
(B4)
Then
(B5)
Now we take the thermodynamic limit N--'J-oo and find
Finally we take the limit l--'J-oo.
Then
(B6)
H. Silver and N. E. Frankel
756
smce rJFr; = F 11 r;. This completes the proof which is very easily generalized to
multi-spin correlation functions.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
N. E. Frankel and D. Rappaport, Prog. Theor. Phys. 43 (1970), 1170.
]. S. Marsh, Phys. Rev. 145 (1966), 251.
M. Suzuki, B. Tsujiyama and S. Katsura, J. Math. Phys. 8 (1967), 124.
C. J. Thompson, ]. Math. Phys. 9 (1968), 241.
H. S. Leff and M. Flicker, Am. ]. Phys. 36 (1968), 591.
E. W. Montroll, J. Chern. Phys. 9 (1941), 706.
]. Dobson, ]. Math. Phys. 10 (1969), 40.
J. Stephenson, Can. ]. Phys. 48 (1970), 1724.
K. Huang, Statistical Mechanics (John Wiley & Sons, Inc., New York, London, 1963),
p.346.