Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant D_PQ

17 Pages·0.22 MB·English

by  R. J. Baxter

Spontaneous magnetization of the
0
superintegrable chiral Potts model:
1
0
calculation of the determinant D
2
PQ

n
a
J
R.J. Baxter
1
1
Mathematical Sciences Institute, The Australian National

]
h University,Canberra, A.C.T. 0200, Australia, e-mail: none
c
e
m 21 December2009
revised 11 January 2010
-
t
a
t
s Abstract
.
t
a For the Ising model, thecalculation of thespontaneous magnetization leads to the
m
problemofevaluatingadeterminant. Yangdidthisbycalculatingtheeigenvaluesinthe
- large-lattice limit. Montroll, Potts and Ward expressed it as a Toeplitz determinant
d
and used Szeg˝o’s theorem: this is almost certainly the route originally travelled by
n
Onsager. For the corresponding problem in the superintegrable chiral Potts model,
o
neither approach appears to work: here we show that the determinant D can be
c PQ
[ expressed as that of a product of two Cauchy-like matrices. One can then use the
elementaryexactformulafortheCauchydeterminant. Oneofcourseregainstheknown

2 result, originally conjectured in 1989.
v
9
4 KEY WORDS: Statistical mechanics, lattice models, transfer matrices.
5
4
1 Introduction
.
2
1
ExtrapolatingfromtheIsingcaseandfromseriesexpansions,Albertinietalconjectured
9
in 1989 [1] that the order parameter or spontaneous magnetization of the solvable N-
0
state chiral Potts model is
:
v
Xi Mr = (1−k′2)r(N−r)/2N2. (1.1)
r Here r is an integer, between 0 and N, and k′ is a parameter that is “universal”
a
in the sense that it is the same for all rows and columns of the lattice, even for an
inhomogeneous model where the rapidities vary from row to row and from column to
column.[2] Here we consider the ferromagnetically ordered phase of the system, where
0<k′ <1. This k′ is small at low temperatures (high order), and tendsto one at the
critical temperature (vanishing order), so we can regard it as a “temperature”.
Theauthorwasabletoderivetheresult(1.1)in2005,[3,4]usingananalyticmethod
based on functional relations satisfied by generalized order parameters in the large-
lattice limit.
1
Iftheverticalrapiditiesofthehomogeneousmodeltakeaparticularvalue,orifthe
thoseofamodelwithalternatingverticalrapiditiessatisfyacertainrelation,[5,6]then
we obtain the “superintegrable” case of the chiral Potts model. For this case, Gehlen
and Rittenberg[7] showed that thehorizontal and vertical components of thetransfer
matrix satisfy the “Dolan-Grady” condition,[8] which ensures that they generate the
Onsager algebra. This is the algebra generated by the transfer matrices of the Ising
model.[9, eqs. 59-61], [10, eq. 4.12], [11] - [18] Onsager used it to calculate the free
energy of theIsing model.
InfactthesuperintegrablechiralPottsmodellooksverymuchliketheIsingmodel.
(ForN =2itistheIsingmodel.) IfoneconsidersthemodelonacylinderofLcolumns,
withthespinsonthetopandbottomrowsfixedtothevaluezero,thentherow-to-row
transfermatricescanbereducedfrom dimensionNL todimension 2m,wheremisnot
greater than L. Like the Ising model, the partition function is a matrix element of a
direct product of m matrices, each of dimension two.
For the Ising model, one can calculate the correlations quite explicity as determi-
nants by using free-fermion operators,[19, 20] or equivalently by writing the partition
functions directly as pfaffians.[21]
If the partition functions of the N-state superintegrable model resemble those of
theIsingmodel,thenperhapsthecorrelationsarealsosimilar, andcanbeobtainedby
similar methods. In particular, theorder parameter can be definedas
M = hωrai, (1.2)
r
where r=1,2,...,N −1,
ω = e2πı/N, (1.3)
and a is a particular spin inside the lattice. We can use this definition for a finite
lattice: we doso herein. Weonly expect thesimple result (1.1) to betruein thelimit
whenthelatticeislargeandthespinaisdeepinsideit. Itshouldthenbeindependent
ofthevaluesoftherapidityparameters, soshouldbethesameforthegeneralsolvable
model as for the superintegrable case.
We take the spins to have the values 0,...,N −1 and σ = {σ ,...,σ } to be the
1 L
set of spins in a horizontal row of the lattice. Let u be the NL-dimensional vector
b
with entries
(u ) = δ(σ ,b)δ(σ ,b)···δ(σ ,b) (1.4)
b σ 1 2 L
and S thediagonal matrix with entries
r
L
(Sr)σ,σ′ = ωrσ1 δ(σj,σj′). (1.5)
j=1
Y
Then with these boundaryconditions, (1.2) can bewritten in terms of therow-to-row
transfer matrix T as
u†TmS Tnu
M = 0 r 0 , (1.6)
r u†Tm+nu
0 0
wherem(n)isthenumberofrowsbelow(above)theparticularspina. Wehavechosen
atolieinthefirstcolumn: sinceweareusingcylindrical(cyclic)boundaryconditions,
this is no restriction.
ThetranfermatrixT commuteswithahamiltonianH. Forconvenience,wereplace
(1.6) with
u†e−αHS e−βHu
M(1) = 0 r 0 . (1.7)
r u†e−(α+β)Hu
0 0
In the limit of m,n large, the only eigenvectors of T entering the RHS of (1.6) are
thosecorrespondingtotheN asymptoticallydegeneratelargest eigenvalues. Thesame
2
is trueof (1.7) in thelimit of α,β large and positive. Hencethese limits of (1.6),(1.7)
must be thesame.
Let R be theoperator that increases every spin in a row byone: its elements are
L
Rσ,σ′ = δ(σj,σj′ +1). (1.8)
Yj=1
Since RN =1, its eigenvalues are 1,ω,ω2,...,ωN−1. Let V (for P =0,1,...,N −1)
P
be thespace of vectors v such that
Rv = ωPv. (1.9)
ThenthefullNL-dimensionalspaceistheunionofV0,...,VN−1. LetvP bethevector
N−1
v = N−1/2 ω−Pbu . (1.10)
P b
b=0
X
Then v and e−βHv are in V . If
P P P
Q=P +r , modN, (1.11)
then S e−βHv is also in V . Vectors in different spaces V , V are orthogonal. It
r Q P P Q
follows that, for b=0,...N−1,
N−1
u†e−αHS e−βHu = N−1 ω−PbW (α,β) (1.12a)
b r 0 P,P+r
PX=0
N−1
u†e−αHu = N−1 ω−PbZ (α), (1.12b)
b 0 P
P=0
X
where
W (α,β) = v† e−αHS e−βHv , (1.13a)
PQ P r Q
Z (α) = v† e−αHv . (1.13b)
P P P
EachLHSof(1.12)isthepartitionfunctionofalatticewherethetopspinsarefixed
to the value 0, the bottom to the value b. If b 6= 0 this ensures that there is at least
onemis-matchedseam(betweenphaseswheremostspinsarezeroandmostspinshave
value b) running horizontally across the lattice. If ζ is the interfacial tension, (which
we expect to be independentof L) then this will make each partition function smaller
then that for b=0 by a factor e−Lζ.[22, section 7.10]
In the limit of L large the ratio of these expressions for b 6= 0 to their values for
b = 0 will therefore become zero. From (1.12), it follows that in this limit Z (α) is
P
independentofP,whileW (α,β)dependsonP,Qonlyviatheirdifferencer=Q−P.
PQ
We show at the end of section 5 that these assertions are certainly true in the limit
when α,β,L all tend to infinity.
The numerator in (1.7) can therefore be replaced by W (α,β), for any P,Q sat-
PQ
isfying (1.11). The denominator can be replaced by Z (α+β), for any P, but if α is
P
also large, each Z (α) is of the form Keνα, where K,ν must be independent of P, so
P
we can more symmetrically replace the denominator by [Z (2α)Z (2β)]1/2, giving as
P Q
our final expression for thespontaneous magnetization
W (α,β)
M(2) = PQ . (1.14)
r [Z (2α)Z (2β)]1/2
P Q
The three expressions M ,M(1), M(2) are equal in the limit L,m,n or L,α,β all
r r r
becoming infinite.
3
Previous calculations for finite L,α,β
In [23, 24, 25], we have looked at the problem of calculating W (α,β),Z (α), and
PQ P
henceM(1),M(2),algebraically, forthesuperintegrablechiralPottsmodel,withfinite
r r
L,α,β. Werefer to these papers as I, II,III, and prefix theirequations accordingly.
The calculation of Z (α) is straightforward, being a minor adaptation of the par-
P
tition function calculations of [5, 6], and is given in paper II. The real problem is to
calculate W (α,β), or equivalently theratio
PQ
W (α,β)
D (α,β) = PQ . (1.15)
PQ Z (α)Z (β)
P Q
If we also define
Z (α) = Z (2α)/Z (α)2, (1.16)
P P P
then (1.14) becomes
D (α,β)
M(2) = PQ . (1.17)
r [Z (α)Z (β)]1/2
P Q
In I we considered the N = 2 case, which is the Ising model. We used Kaufman’s
spinoroperators (Clifford algebra) [26] tofirstwrite D (α,β) (forP =0,Q=r=1)
PQ
as the square root of an L by L determinant in I.4.59. Then in section 6 of I, eqn.
I.6.29 and I.7.9, we further reduced this result to an m by m determinant (with no
square root), where m≤L/2. With obvious modifications of notation to allow for the
working of the later papers, and taking ρ=0, x=1 in I.3.5, I.3.6 (also in II.5.23 and
II.5.25), this result can bewritten as
D (α,β) = det[I −X (α)E B X (β)E B ], (1.18)
PQ m P PQ PQ Q QP QP
whereI istheidentitymatrixofdimensionm,X (α),E ,X (β),E arediagonal
m P PQ Q QP
matrices,B isanmbym′ matrix,where|m−m′|=0or1,B =−BT andB
PQ QP PQ PQ
is orthogonal in thesense that
BPTQBPQ =Im′ if m≥m′, BPQBPTQ =Im if m≤m′. (1.19)
We used the first (L by L) form to take the limit α,β → +∞: the result of course
agreed with that of Yang[20]and Montroll, Pottsand Ward[21], and with (1.1) above.
In II we considered the superintegrable chiral Potts model and showed that the
NL-dimensional matrices in (1.13a) could be replaced by ones of lower dimension. In
particular,theHinthefirstexponentialcouldbereducedtodimension2m,where(for
P =0,...,N−1),
(N−1)L−P
m=m = integerpartof . (1.20)
P N
(cid:20) (cid:21)
Similarly, the second H could be replaced by one of dimension 2m′, where m′ = m ,
Q
and S by a 2m by 2m′ matrix S . (The p,q,Sr of paper II became P,Q,S in
r PQ red PQ
paper III and herein.)
We went on to conjecture that (1.18), (1.19) also applied to the superintegrable
model, with fairly obvious generalizations of the definitions of the X,E,B matrices.
Weobservedthatthisconjectureagreedwithnumericaltestsperformedto60digitsof
accuracy.
These calculations involved sets of m quantities θ ,...,θ definedby
1 m
cosθ = c = (1+w )/(1−w ), 0≤θ <π, (1.21)
j j j j i
4
where w ,...,w are thezeros of the m-thdegree polynomial ρ (w) given by
1 m P
N−1 1−zN L
ρ (zN) = z−P ωnP , (1.22)
P 1−ω−nz
Xn=0 (cid:18) (cid:19)
taking w=zN. Let c=(1+w)/(1−w) and,for all complex numbersc,
P (c) = N−L(c+1)m ρ (w). (1.23)
P P
Then P (c) is thepolynomial with zeros c ,...,c , i.e.
P 1 m
m
P (c) = (c−c ). (1.24)
P j
Yj=1
Similarly, we can definem′ quantities θ1′,...,θm′ ′ and c′1,...,c′m′ by replacing P by Q
in theabovethreeequations. Theρ (w),P (c) herearethose of paperIII,which are
P P
theP(w), P˜ (c) of II.2.17, II.2.18 and II.6.4.
p
InIIIweshowedthatboththeS andS matricessatisfied variouscommutation
r PQ
relations with the hamiltonians, in particular that S satisfied III.3.39 and III.3.40.
PQ
We conjectured in III.3.45 that the elements of S were simple ratios of products of
PQ
trigonometric functions of the θ . Wesuggested that this result applied for any values
i
oftheθ andθ′,notnecessarilythosegivenby(1.21)and(1.22). Further,ifwedefined
i i
D byIII.3.48, then it was also given as a determinant by III.4.9 and III.4.10, again
PQ
for arbitrary θ , θ′.
i i
Outstanding problems and progress
It therefore appears that there is indeed an algebraic route to calculating D . How-
PQ
ever, thereare still three outstanding problems to beovercome:
1. To prove that the elements of S are given by III.3.45, and hence D by
PQ PQ
III.3.48.
2. To further prove that D is given as a determinant by III.4.9 or equivalently
PQ
III.4.10.
3. Tocalculate thedeterminantIII.4.9in thelimit L,α,β→∞soas toregain the
knownresult(1.1). ThishasnotpreviouslybeendonedirectlyevenfortheN =2Ising
case: in paper I we calculated (1.1) from theexpression for D as thesquare root of
PQ
an L by L determinant, using Szeg˝o’s theorem. This theorem was derived in reponse
to the first (unpublished) derivation of the Ising model spontaneous magnetization by
Onsager and Kaufman.[27,28] It was later used my Montroll, Potts and Ward.[21]
Progress has been made. We have proved that the expression III.3.45 for S
PQ
satisfies the commutation relations III.3.39 - III.3.41. From numerical calulations for
smallN,L(N,L≤6),itappearsthattheserelations (whicharelinearintheelements
of S , with many more equations than unknowns) determine S uniquely. If so,
PQ PQ
then III.3.45 and III.3.48 haveto be correct.
InpaperIIIwedefinedy ,y′tobetheelementsofthediagonalmandm′-dimensional
i i
matricesY =X (α)E ,Y′ =X (β)E . BoththeexpressionIII.3.48forD asa
P PQ Q QP PQ
2m+m′-dimensional sum, and the expression III.4.9 as an m-dimensional determinant,
are rational functions of the c ,c′,y ,y′. One can take all these variables to be arbi-
i i i i
trary and can verify that the denominators are identical. One can then prove that
thenumeratorsarealso thesame byarecursivemethodusingthesymmetriesandthe
fact that if cm = c′m′, then each expression simplifies to one with m,m′ replaced by
m−1,m′−1.
InsomewaysthehardestofthethreeproblemsistotakethelimitL,α,β →∞. The
determinant for D is a hugely smaller calculation than the original 2L-dimensional
PQ
5
sumsin (1.13a),(1.13b),butit isstill ultimately infinite. Wehavesucceeded in calcu-
lating the determinant in the limit α,β →∞ as a simple product, for finite L,m,m′:
the key trick is to note that when α,β →∞ the matrix sum in (1.18) can be written
astheproductoftwoCauchy-likematrices. Theresult(1.1)followsbythentakingthe
L→∞ limit of the product. It is this calculation we report here. Wehope to publish
thework on thefirst two problems later.
2 The matrices
WeshallneedthedefinitionsoftheX andE diagonalmatrices. FromII.3.16andII.7.4
−k′sinθ sinh(Nαλ )δ
[X (α)] = i i i,j , (2.1)
P i,j λ cosh(Nαλ )+(1−k′cosθ )sinh(Nαλ )
i i i i
where
λ = (1−2k′cosθ +k′2)1/2. (2.2)
i i
The matrix X (β) is defined similarly, with P,α,θ ,λ replaced by Q,β,θ′,λ′.
Q i i i j
FromII.6.18andII.6.19,thematrixE isanmbymdiagonalmatrixwithentries
PQ
[E ] = e(P,Q,i)δ , (2.3)
PQij i,j
where, for 0≤P,Q<N and P 6=Q,
e(P,Q,i)= sinθ if P <Q and m′ =m−1
i
= tan(θ /2) if P <Q and m′ =m
i
= 1/sinθ if P >Q and m′ =m+1 (2.4)
i
= cot(θ /2) if P >Q and m′ =m.
i
These equations cover all cases. The matrix E(Q,P) is defined similarly, with P,Q
interchanged, m,m′ also interchanged, and θ replaced byθ′.
i i
B is an m by m′ Cauchy-likematrix with elements
PQ
f f′
(B ) = i j . (2.5)
PQ ij c −c′
i j
Given c1,...,cm, c′1,...,c′m′ with |m−m′| ≤ 1, there is a unique way of choosing
f1,...,fm, f1′,...,fm′ ′ so that BPQ satisfies the orthogonality condition (1.19). The
working is given in section 6 of II. We remark in section 4 of III that it is true for
arbitrary c ,c′. Let
i i
′
m m
a = (c −c′), a′ = (c′−c ), (2.6)
i i j i i j
Yj=1 Yj=1
′
m m
b = (c −c ), b′ = (c′−c′),
i i j i i j
j=Y1,j6=i j=Y1,j6=i
then theresults II.6.8, II.6.13, II.6.16 can bewritten as
f2 = ǫa /b , f′2 = −ǫa′/b′, (2.7)
i i i i i i
where ǫ=±1is independentof i.
(Fortheparticularvaluesofc ,c′ givenby(1.21)and(1.22),weobservenumerically
i i
that f2 and f′2 are positive real if we choose ǫ=1 if P <Q, and ǫ=−1if P >Q.)
i i
A quantitythat we shall need is
(c −c ) (c′ −c′)
∆m,m′(c,c′) = Q1≤i<j≤m mii=1 jmj=Q′1(1c≤ii−<jc≤′jm)′ j i . (2.8)
Q Q
6
3 The function Z (α)
P
ThepartitionfunctionZ (α)is,fromII.3.16andII.5.38,orfromIII.3.27andIII.3.29,
P
m λ cosh(Nαλ )+(1−k′cosθ )sinh(Nαλ )
Z (α) = e−µPα i i i i , (3.1)
P λ
i
Yi=1
where
µ = 2k′P +(1+k′)(mN−NL+L). (3.2)
P
When α is large, Z (α) has the form Cegα,where C,g are independentof α and
P
m
g = g = −µ + λ . (3.3)
P P i
Xi=1
Hencefrom (1.16), Z (α)→C−1 as α→∞.
P
Set
XP =XP(∞), xi=(XP)i,i , ZP =ZP(∞)=C−1 (3.4)
and let c,θ,λ be variables related to oneanother by
c = cosθ = (1+k′2−λ2)/2k′, (3.5)
thenthec ,θ ,λ of(1.21),(2.2)arerelatedinthesamemanner. Insteadofviewingx ,
i i i i
Z ,etc. astwo-valuedfunctionsofthec ,wecanregardthemassingle-valuedrational
P i
functions of theλ . Then
i
k′2−(1−λ )2
x2 = i , (3.6)
i (1+λ )2−k′2
i
m m
4λ
Z = i = (1+x2). (3.7)
P (1+λ )2−k′2 i
i
Yi=1 Yi=1
Analogous relations apply,with P,λ ,x replaced by Q,λ′,x′, respectively.
i i i i
Anotherfunction that we shall useful is
m (λ+λ )/2
R(λ) = i=1 i , (3.8)
m′ (λ+λ′)/2
Qj=1 j
together with the elementary identityQ
′
∆m,m′(λ2,λ′2) 2 = 2(m−m′)2 m R(λi) m 2λ′ R(λ′) . (3.9)
(cid:20) ∆m,m′(λ,λ′) (cid:21) i=1 2λi ,j=1 j j
Y Y
4 Calculation of D
PQ
For any m by m′ matrix A, and m′ bym matrix B, it is truethat
det(Im+AB)=det(Im′ +BA), (4.1)
so from (1.18),
DPQ(α,β) = det[Im′ −XQ(β)EQPBQPXP(α)EPQBPQ]
= D (β,α). (4.2)
QP
This symmetryalso follows directly from thedefinitions (1.5) -(1.15),thefact that H
is hermitian and Sr†=S−r.
7
Withoutlossofgenerality,wecanthereforerestrictourattentiontothecaseP >Q,
when m≤m′.
Thenfrom(1.19)wecanwriteI in(1.18)asB BT . RememberingthatB =
m PQ PQ QP
−BT , we can then write (1.18) as
PQ
D (α,β) = det[UBT ], (4.3)
PQ PQ
where
U = B +X (α)E B X (β)E . (4.4)
PQ P PQ PQ Q QP
Define
y =[X (α)] e(P,Q,i), y′ =[X (β)] e(Q,P,j), (4.5)
i P i,i j Q j,j
then from (2.5) theelements of U are
f f′(1+y y′)
U = i j i j . (4.6)
ij c −c′
i j
In general we do not know how to calculate the determinant of such a matrix.
However,ifwetakethelimitsα,β→+∞andexpressc ,c′,y ,y′ asrationalfunctions
i j i j
of λ ,λ′, we findthat a factor λ +λ′ cancels out of theRHS of (4.6). If m′=m,the
i j i j
result is Cauchy-likematrix, and one can calculate the determinant of U.
Hereinafter we take the limitα,β→+∞,so
y =x e(P,Q,i), y′ =x′ e(Q,P,j), (4.7)
i i j j
where xi,x′j aregiven by(3.4). Wewrite DPQ(∞,∞) simply as DPQ. TheintegerL is
still finite.
The case P > Q, m = m′
The simplest case is when m = m′ and all matrices are square. so from (1.19) and
(4.3),
D = detU/detB . (4.8)
PQ PQ
Cauchy-like matrices
If A is the m by m matrix with entries
1
A = , (4.9)
ij c −c′
i j
then it is a Cauchy matrix and its determinant is ∆ (c,c′), using the definition
m,m
(2.8).[29,eq. 2.7]Anymatrixwithelementsoftheform(2.5)issaidtobeCauchy-like,
and has determinant
m
detB = ∆ (c,c′) f f′ (4.10)
PQ m,m i i
i=1
Y
for all f ,f′. We have in fact chosen the f ,f′ so that B is orthogonal, so has
i i i i PQ
determinant±1. However,theform(4.10)isconvenienthereasthef ,f′ productswill
i i
cancel out of (4.8).
8
The determinant of U
From (4.8), we still have to calculate the determinant of U. Its elements are given by
(4.6), so U is not in general Cauchy-like. However, in the limit α,β→∞ we findthat
acommonfactorcancelsfromthenumeratoranddenominatorof(4.6),andU becomes
Cauchy-like. Wecan then evaluate its determinant by parallelling (4.9) - (4.10).
The case P >Q,m=m′ is the fourth one listed in (2.4), so
(E ) =cot(θ /2)δ , (E ) =tan(θ′/2)δ . (4.11)
PQ ij i ij QP ij i ij
From (2.1), takingthe limit α→+∞,
−2k′sinθ
xi = [XP(∞)]ii = (1+λ )2−ki′2 . (4.12)
i
Noting that sinθ cot(θ /2)=1+c =[(1+k′)2−λ 2]/2k′, it follows from (4.7) that
i i i i
1+k′−λ
y = − i . (4.13)
i 1−k′+λ
i
(A factor 1+k′+λ has cancelled.)
i
The calculation of y′ is similar, except that now we usesinθ′tan(θ′/2)=1−c′ =
j j j j
[λ 2−(1−k′)2]/2k′ to obtain
j
1−k′−λ′
y′ = j . (4.14)
j 1+k′+λ′
j
Thus
2(λ +λ′)
1+y y′ = i j . (4.15)
i j (1−k′+λ )(1+k′+λ′)
i j
Also,
c −c′ = (λ′2−λ 2)/2k′. (4.16)
i j j i
We see that the factor λ +λ′ cancels out of (4.6), leaving
i j
−4k′f f′
U = i j (4.17)
ij (1−k′+λ )(1+k′+λ′)(λ −λ′)
i j i j
soU isaCauchy-likematrix,similartoB ,butwiththedenominatorc −c′ replaced
PQ i j
by λ −λ′. Analogously to (4.10), its determinant is
i j
m −4k′f f′
detU = ∆ (λ,λ′) i i . (4.18)
m,m (1−k′+λ )(1+k′+λ′)
i i
i=1
Y
Also, from (4.16), we can write (2.5) as
−2k′f f′
(B ) = i j (4.19)
PQ ij λ2−λ′2
i j
so detB is also equal to
PQ
m
detB = ∆ (λ2,λ′2) (−2k′f f′). (4.20)
PQ m,m i i
i=1
Y
The f ,f′ cancel out of theratio (4.8),leaving
i i
∆ (λ,λ′) m 2
D = m,m . (4.21)
PQ ∆m,m(λ2,λ′2) (1−k′+λi)(1+k′+λ′i)
Yi=1
From (1.17), (3.7), (3.8) and (3.9) it follows that
′
R(1+k′) m R(λ′)
M(2) 2 = j=1 j . (4.22)
r R(1−k′) m R(λ )
Qi=1 i
(cid:0) (cid:1)
Q
9
All cases
When P > Q and m′ = m+1, we can still write D as in (4.3). However, U and
PQ
B areno longer squarematrices, so wecan longer simply takeproductsor ratios of
PQ
determinants, as in (4.8). Even so, we can still calculate D (and hence M(2)) by
PQ r
adding a row to B and U to make them square Cauchy-like matrices. The matrix
PQ
UBT in (4.3) is then m′ by m′, but is of upper-block-triangular form. The top left
PQ
block is the original m by m matrix UBT , while the lower-right block is the one by
PQ
one unit matrix. Hence thedeterminant (4.3) is unchanged and can be evaluated as a
product of thetwo m′ bym′ determinants.
We do this in the Appendix. The result (A18) is the same as (4.22), except that
thefactor R(1+k′) is inverted.
From (1.17) and (4.2), M(2) is unchanged by interchanging P with Q, m with m′
r
and the λ with the λ′. From (3.8), this inverts the function R. We can use this
i j
symmetry to calculate M(2) in theother two cases.
r
For thefour cases (2.4), definethefactor G by
G = R(1−k′)R(1+k′) if P <Q and m′=m−1
= R(1−k′)/R(1+k′) if P <Q and m′=m
= 1/[R(1−k′)R(1+k′)] if P >Q and m′=m+1 (4.23)
= R(1+k′)/R(1−k′) if P >Q and m′=m,
then we find that
′
m m
M(2) 2 = G R(λ′) R(λ ) (4.24)
r j i
,
(cid:0) (cid:1) Yj=1 Yi=1
for all four cases.
5 The limit L → ∞
Theresult(4.24)isexactforfiniteL. Thelast stepinthecalculation istoletL→∞.
For this we shall need for the first time herein the particular definition (1.22) - (1.23)
of thepolynomial (1.24).
Let c,λ be two variables related as are c ,λ in (3.5). Noting that λ2 −λ2 =
i i i
2k′(c−c ),it follows from (3.8) and (1.24) that
i
R(λ)R(−λ) = (k′/2)m−m′P (c)/P (c). (5.1)
P Q
Using (1.22), (1.23),this can bewritten as
R(λ)R(−λ) = [k′(c+1)/2]m−m′zQ−PW(λ), (5.2)
where
W(λ) = W (λ)/W (λ), (5.3)
P Q
N−1
1−z L
W (λ) = 1+ ωn(P+L) . (5.4)
P ωn−z
Xn=1 (cid:16) (cid:17)
Wehave cancelled then=0 term in (1.22) from theratio (5.3).
These z, w, care related to λ by therelations (1.22), (1.23),(3.5). In particular,
c−1 (1−k′)2−λ2
zN =w= = . (5.5)
c+1 (1+k′)2−λ2
10