# Multiparameter Statistical Models fromN2×N2 Braid Matrices: Explicit Eigenvalues of Transfer Matrices T(r), Spin Chains, Factorizable Scatterings for All N

Authors: B. Abdesselam; A. Chakrabarti
Journal: Advances in Mathematical Physics
 (2012)

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## Abstract

For a class of multiparameter statistical models based onN2×N2 braid matrices,
the eigenvalues of the transfer matrix T(r) are obtained explicitly for all (r,N). Our
formalism yields them as solutions of sets of linear equations with simple constant
coefficients. The role of zero-sum multiplets constituted in terms of roots of unity
is pointed out, and their origin is traced to circular permutations of the indices in
the tensor products of basis states induced by our class of T(r) matrices. The role
of free parameters, increasing as N2 with N, is emphasized throughout. Spin chain
Hamiltonians are constructed and studied for all N. Inverse Cayley transforms of
the Yang-Baxter matrices corresponding to our braid matrices are obtained for all N.
They provide potentials for factorizable S-matrices. Main results are summarized,
and perspectives are indicated in the concluding remarks.


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## Body

## 1. Introduction

Statistical models “exact” in the sense of Baxter [1] satisfy “star-triangle” relations leading to transfer matrices commuting for different values of the spectral parameter. Crucial in the study of such models is the spectrum of eigenvalues of these matrices. But even for the extensively studied 6-vertex and 8-vertex models based on 4×4 braid matrices (the braid property guaranteeing star-triangle relations), after the first few simple steps, one has to resort to numerical computations. (We cannot adequately discuss here the vast associated literature but refer to texts citing major sources [2, 3].) But the basic reason for such situation is that, starting with 2×2 blocks Tab of the transfer matrix and constructing 2r×2r blocks via coproduct rules for order r  (r=1,2,3,…), one faces increasingly complicated nonlinear systems of equations to be solved in constructing eigenstates and eigenvalues. Even when a systematic approach is available, such as the Bethe Ansatz for the 6-vertex case, it only means that the relevant nonlinear equations can be written down systematically. The task of solving them remains. For the 8-vertex case one explores analytical properties to extract informations (See [4], e.g.). Also the number of free parameters remains strictly limited for such models, including the multistate generalization of the 6-vertex one [2].Our class of models exhibits the following properties.(1)
A systematic construction for all dimensions. One starts fromN2×N2 braid matrices leading to Nr×Nr blocks of T(r), the transfer matrix of order r for N=2,3,4,5,…;r=1,2,3,… with no upper limit.(2)
The number of free parameters increases withN as N2. This is a unique feature of our models.(3)
For all(N,r) one solves sets of linear equations with systematically obtained simple constant coefficients to construct eigenvalues of T(r)=∑a=1N‍Taa(r). This is a consequence of our starting point: braid matrices constructed on a basis of “nested sequence” of projectors [5].The total dimension of the base space on which such equations have to be solved increases asNr. This will be seen to break up into subspaces closed under the action of T(r), thus reducing the work considerably. Evidently one cannot continue to display the results explicitly, as Nr increases. But it is possible to implement a general and particularly efficient approach. This will be illustrated for all N and r=1,2,3,4,5. The generalization to r>5 will be clearly visible. This is the central purpose of our paper. Certain other features will be explored with comments in conclusion.


## 2. Braid and Transfer Matrices

The roots of our class of braid matrices and the transfer matrices they generate are to be found in the nested sequence of projectors [5]. In [5] it was noted that the 4×4 projectors providing the basis of the 6-vertex and 8-vertex models (i.e., the braid matrices leading to those models) can be generalized to higher dimensions—all higher ones (N odd or even). Braid matrices for odd dimensions were exhaustively constructed on such a basis [6], and related statistical models were studied [7]. Then even dimensional ones were presented [8], and the corresponding braid matrices and statistical models were studied [9]. In previous works the eigenvalues and eigenfunctions of the transfer matrices were presented mostly for the lowest values of N, namely, the 4×4 and 9×9 braid matrices. We present below a general approach for all N.Let us just mention that systematic study of the exotic bialgebras that arise from the9×9 unitary braid matrices was constructed in [10]. The dual bialgebra of one of these exotic bialgebras was also presented.We start now by recapitulating the construction of our nested sequence of projectors, leading to the remarkable properties of our solutions.


### 2.1. Even Dimensions

LetN=2n  (n=1,2,3,…). Define ((ab) denoting a matrix with unity on row a and column b)
(2.1)Pij(ϵ)=12{(ii)⊗(jj)+(i-i-)⊗(j-j-)+ϵ[(ii-)⊗(jj-)+(i-i)⊗(j-j)]},
where i,j∈{1,…,n}, ϵ=±, i-=2n+1-i, j-=2n+1-j. Interchanging j⇌j- on the right one obtains Pij-(ϵ). One thus obtains a complete basis of projectors satisfying (with Pij(ϵ)=Pi-j-(ϵ), Pij-(ϵ)=Pi-j(ϵ) by definition)
(2.2)Pab(ϵ)Pcd(ϵ′)=δacδbdδϵϵ′Pab(ϵ),a,b,c,d∈{1,…,N=2n},∑ϵ=±∑i,j=1n(Pij(ϵ)+Pij-(ϵ))=I(2n)2×(2n)2.
Let mij(ϵ) be an arbitrary set of parameters satisfying the crucial constraint
(2.3)mij(ϵ)=mij-(ϵ).
Define the N2×N2 matrix
(2.4)R^(θ)=∑ϵ∑i,jemij(ϵ)θ(Pij(ϵ)+Pij-(ϵ)).
The θ-dependence entering exclusively through the exponentials as coefficients as above. Such construction guarantees [7–9] the braid property
(2.5)R^12(θ)R^23(θ+θ′)R^12(θ′)=R^23(θ′)R^12(θ+θ′)R^23(θ),
where, in standard notations, R^12=R^⊗I and R^23=I⊗R^, I denoting the N×N identity matrix.Define the permutation matrixP=∑a,b‍(ab)⊗(ba). Then
(2.6)R(θ)=PR^(θ)
satisfies the Yang-Baxter equation. The monodromy matrix of order 1 (r=1) is given by
(2.7)T(1)(θ)=R(θ).
The N×N blocks are (with a,b∈{1,…,2n})
(2.8)Tab(1)(θ)=12(emba(+)θ+emba(-)θ)(ba)+12(emba(+)θ-emba(-)θ)(b-a-)≡fba(+)(θ)(ba)+fba(-)(θ)(b-a-),
with
(2.9)mba(ϵ)=mba-(ϵ)=mb-a(ϵ)=mb-a-(ϵ).
Higher orders (r>1) are obtained via coproducts
(2.10)Tab(r)(θ)=∑c1,…,cr-1Tac1(1)(θ)⊗Tc1c2(1)(θ)⊗⋯⊗Tcr-2cr-1(1)(θ)⊗Tcr-1b(1)(θ).
For some essential purposes, it is worthwhile to express (2.8) as (with i,j∈{1,…,n})
(2.11)Tij(1)(θ)=12∑ϵemji(ϵ)θ((ji)+ϵ(j-i-)),Ti-j-(1)(θ)=12∑ϵemji(ϵ)θ((j-i-)+ϵ(ji)),Tij-(1)(θ)=12∑ϵemji(ϵ)θ((j-i)+ϵ(ji-)),Ti-j(1)(θ)=12∑ϵemji(ϵ)θ((ji-)+ϵ(j-i)).
The transfer matrix is the trace on (a,b):
(2.12)T(r)(θ)=∑aTaa(r)(θ).
The foregoing construction assures the crucial commutativity
(2.13)[T(r)(θ),T(r)(θ′)]=0.
This implies that the eigenstates of T(r)(θ) are θ-independent (sum of basis vectors with θ-independent, constant relative coefficients).


### 2.2. Odd Dimensions

From the extensive previous studies [6, 7], we select the essential features. For N=2n-1 (n=2,3,…),
(2.14)n-=N-n+1=n
with the same definition as in (2.1). This is the crucial new feature. For (i,j)≠n, one has the same Pij(ϵ), Pij-(ϵ) as before. But now
(2.15)Pin(ϵ)=12{(ii)+(i-i-)+ϵ[(ii-)+(i-i)]}⊗(nn),Pni(ϵ)=12(nn)⊗{(ii)+(i-i-)+ϵ[(ii-)+(i-i)]},Pnn(ϵ)=(nn)⊗(nn).
Normalizing the coefficient of Pnn(ϵ) to unity, R^(θ) of (2.4) now becomes
(2.16)R^(θ)=Pnn(ϵ)+∑i,ϵ(emni(ϵ)θPni(ϵ)+emin(ϵ)θPin(ϵ))+∑i,jϵemij(ϵ)θ(Pij(ϵ)+Pij-(ϵ))
(i,j∈{1,…,n-1}, i-,j-∈{2n-1,…,n+1}, ϵ=±). There are evident parallel modifications concerning Tab(r)(θ). Reference [7] contains detailed discussions and results for N=3, r=1,2,3,4. We will not repeat them here but will reconsider them later in the context of the general methods to be implemented below.


### 2.3. Free Parameters

A unique feature of our constructions is the number of free parametersmij(ϵ), increasing as N2 with N. For our choice of normalizations (apart from a possible altered choice of an overall factor, irrelevant for (2.5)), the exact numbers are
(2.17)12N2=2n2forN=2n,12(N+3)(N-1)=2(n2-1)forN=2n-1.
This is to be contrasted with the multistate generalization of the 6-vertex model where the parametrization remain fixed at the 6-vertex level ([2] and sources cited here).One of our principal aims is to display the roles of our free parameters concerning the basic features of our models.


## 2.1. Even Dimensions

LetN=2n  (n=1,2,3,…). Define ((ab) denoting a matrix with unity on row a and column b)
(2.1)Pij(ϵ)=12{(ii)⊗(jj)+(i-i-)⊗(j-j-)+ϵ[(ii-)⊗(jj-)+(i-i)⊗(j-j)]},
where i,j∈{1,…,n}, ϵ=±, i-=2n+1-i, j-=2n+1-j. Interchanging j⇌j- on the right one obtains Pij-(ϵ). One thus obtains a complete basis of projectors satisfying (with Pij(ϵ)=Pi-j-(ϵ), Pij-(ϵ)=Pi-j(ϵ) by definition)
(2.2)Pab(ϵ)Pcd(ϵ′)=δacδbdδϵϵ′Pab(ϵ),a,b,c,d∈{1,…,N=2n},∑ϵ=±∑i,j=1n(Pij(ϵ)+Pij-(ϵ))=I(2n)2×(2n)2.
Let mij(ϵ) be an arbitrary set of parameters satisfying the crucial constraint
(2.3)mij(ϵ)=mij-(ϵ).
Define the N2×N2 matrix
(2.4)R^(θ)=∑ϵ∑i,jemij(ϵ)θ(Pij(ϵ)+Pij-(ϵ)).
The θ-dependence entering exclusively through the exponentials as coefficients as above. Such construction guarantees [7–9] the braid property
(2.5)R^12(θ)R^23(θ+θ′)R^12(θ′)=R^23(θ′)R^12(θ+θ′)R^23(θ),
where, in standard notations, R^12=R^⊗I and R^23=I⊗R^, I denoting the N×N identity matrix.Define the permutation matrixP=∑a,b‍(ab)⊗(ba). Then
(2.6)R(θ)=PR^(θ)
satisfies the Yang-Baxter equation. The monodromy matrix of order 1 (r=1) is given by
(2.7)T(1)(θ)=R(θ).
The N×N blocks are (with a,b∈{1,…,2n})
(2.8)Tab(1)(θ)=12(emba(+)θ+emba(-)θ)(ba)+12(emba(+)θ-emba(-)θ)(b-a-)≡fba(+)(θ)(ba)+fba(-)(θ)(b-a-),
with
(2.9)mba(ϵ)=mba-(ϵ)=mb-a(ϵ)=mb-a-(ϵ).
Higher orders (r>1) are obtained via coproducts
(2.10)Tab(r)(θ)=∑c1,…,cr-1Tac1(1)(θ)⊗Tc1c2(1)(θ)⊗⋯⊗Tcr-2cr-1(1)(θ)⊗Tcr-1b(1)(θ).
For some essential purposes, it is worthwhile to express (2.8) as (with i,j∈{1,…,n})
(2.11)Tij(1)(θ)=12∑ϵemji(ϵ)θ((ji)+ϵ(j-i-)),Ti-j-(1)(θ)=12∑ϵemji(ϵ)θ((j-i-)+ϵ(ji)),Tij-(1)(θ)=12∑ϵemji(ϵ)θ((j-i)+ϵ(ji-)),Ti-j(1)(θ)=12∑ϵemji(ϵ)θ((ji-)+ϵ(j-i)).
The transfer matrix is the trace on (a,b):
(2.12)T(r)(θ)=∑aTaa(r)(θ).
The foregoing construction assures the crucial commutativity
(2.13)[T(r)(θ),T(r)(θ′)]=0.
This implies that the eigenstates of T(r)(θ) are θ-independent (sum of basis vectors with θ-independent, constant relative coefficients).


## 2.2. Odd Dimensions

From the extensive previous studies [6, 7], we select the essential features. For N=2n-1 (n=2,3,…),
(2.14)n-=N-n+1=n
with the same definition as in (2.1). This is the crucial new feature. For (i,j)≠n, one has the same Pij(ϵ), Pij-(ϵ) as before. But now
(2.15)Pin(ϵ)=12{(ii)+(i-i-)+ϵ[(ii-)+(i-i)]}⊗(nn),Pni(ϵ)=12(nn)⊗{(ii)+(i-i-)+ϵ[(ii-)+(i-i)]},Pnn(ϵ)=(nn)⊗(nn).
Normalizing the coefficient of Pnn(ϵ) to unity, R^(θ) of (2.4) now becomes
(2.16)R^(θ)=Pnn(ϵ)+∑i,ϵ(emni(ϵ)θPni(ϵ)+emin(ϵ)θPin(ϵ))+∑i,jϵemij(ϵ)θ(Pij(ϵ)+Pij-(ϵ))
(i,j∈{1,…,n-1}, i-,j-∈{2n-1,…,n+1}, ϵ=±). There are evident parallel modifications concerning Tab(r)(θ). Reference [7] contains detailed discussions and results for N=3, r=1,2,3,4. We will not repeat them here but will reconsider them later in the context of the general methods to be implemented below.


## 2.3. Free Parameters

A unique feature of our constructions is the number of free parametersmij(ϵ), increasing as N2 with N. For our choice of normalizations (apart from a possible altered choice of an overall factor, irrelevant for (2.5)), the exact numbers are
(2.17)12N2=2n2forN=2n,12(N+3)(N-1)=2(n2-1)forN=2n-1.
This is to be contrasted with the multistate generalization of the 6-vertex model where the parametrization remain fixed at the 6-vertex level ([2] and sources cited here).One of our principal aims is to display the roles of our free parameters concerning the basic features of our models.


## 3. Explicit Eigenvalues of Transfer Matrices for All DimensionsN and Orders r

We present below a unified approach for allN and illustrate it in some detail for r=1,2,3,4,5. For r>5 the extension will be evident.


### 3.1.  Even Dimensions

In Section 5 of [9] it was pointed out that the antidiagonal matrix
(3.1)K=∑a=12n(aa-)=∑i=1n((ii-)+(i-i))
relates the blocks of T(1) as follows:
(3.2)KTab(1)=Tab-(1),Tab(1)K=Ta-b(1),KTab(1)K=Ta-b-(1).
For N=2 this led to an iterative construction of eigenstates and eigenvalues of the transfer matrix T(r) for increasing r. This works efficiently only for N=2 (the 4×4  R^(θ)-matrix).For our present approach the crucial ingredient shows up immediately in the structure (arising from those of our projectors and the coproducts rules)(3.3)T(r)=12r∑a1,a2,…,ar∑ϵ21,ϵ32,…,ϵ1re(ma2a1(ϵ21)+ma3a2(ϵ32)+⋯+ma1ar(ϵ1r))θ×[(a2a1)+ϵ21(a-2a-1)]⊗[(a3a2)+ϵ32(a-3a-2)]⊗⋯⊗[(a1ar)+ϵ1r(a-1a-r)],
where each ϵ is independently (±). From (3.3), one obtains immediately
(3.4)(T(r))-1=12r∑a1,a2,…,ar∑ϵ21,ϵ32,…,ϵ1re-(ma2a1(ϵ21)+ma3a2(ϵ32)+⋯+ma1ar(ϵ1r))θ×[(a1a2)+ϵ21(a-1a-2)]⊗[(a2a3)+ϵ32(a-2a-3)]⊗⋯⊗[(ara1)+ϵ1r(a-ra-1)].
Consider the basis state
(3.5)|b1〉⊗|b2〉⊗⋯⊗|br〉≡|b1b2⋯br〉.
(1) If any one (even a single one) of the indices bi≠ai+1 or ai+1¯ (cyclic), then the coefficient of e(ma2a1(ϵ21)+⋯+ma1ar(ϵ1r))θ, namely, [(a2a1)+ϵ21(a-2a-1)],⊗⋯⊗[(a1ar)+ϵ1r(a-1a-r)], will annihilate it. On the other hand (2.9) plays a crucial role. Gathering together all the terms with e(mb2b1(ϵ21)+mb3b2(ϵ32)+⋯+mbrbr-1(ϵr,r-1)+mb1br(ϵ1r))θ as coefficient, one obtains the action of T(r) on (3.5). All essential features for the general case can be read off the first few simples examples:
(3.6)22T(2)|b1b2〉=∑ϵ21,ϵ12e(mb2b1(ϵ21)+mb1b2(ϵ12))θ(1+ϵ21ϵ12)[|b2b1〉+ϵ21|b-2b-1〉],(3.7)23T(3)|b1b2b3〉=∑ϵ21,ϵ32,ϵ13e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb1b3(ϵ13))θ(1+ϵ21ϵ32ϵ13)×[|b2b3b1〉+ϵ21|b-2b3b-1〉+ϵ32|b-2b-3b1〉+ϵ13|b2b-3b-1〉],(3.8)24T(4)|b1b2b3b4〉=∑ϵ21,ϵ32,ϵ43,ϵ14e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb4b3(ϵ43)+mb1b4(ϵ14))θ(1+ϵ21ϵ32ϵ43ϵ14)×[|b2b3b4b1〉+ϵ21|b-2b3b4b-1〉+ϵ32|b-2b-3b4b1〉+ϵ43|b2b-3b-4b1〉+ϵ14|b2b3b-4b-1〉+ϵ21ϵ32|b2b-3b4b-1〉+ϵ21ϵ43|b-2b-3b-4b-1〉+ϵ21ϵ14|b-2b3b-4b1〉].
Note that the bars correspond to the indies of the ϵ’s. One has thus ϵij|b-ib-jbkbl〉, ϵijϵki|b̿ib-jb-kbl〉=ϵijϵki|bib-jb-kbl〉 since b̿i=bi. Consequently only even number of additional bars appear on the right. Starting with even or odd number of bars on the left leads to “even” and “odd” closed subspaces. 								(3.9)25T(5)|b1b2b3b4b5〉=∑ϵ21,ϵ32,ϵ43,ϵ14e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb4b3(ϵ43)+mb5ba4(ϵ54)+mb1b5(ϵ15))θ(1+ϵ21ϵ32ϵ43ϵ54ϵ15)×[|b2b3b4b5b1〉+ϵ21|b-2b3b4b5b-1〉+ϵ32|b-2b-3b4b5b1〉+ϵ43|b2b-3b-4b5b1〉+ϵ54|b2b3b-4b-5b1〉+ϵ15|b2b3b4b-5b-1〉+ϵ21ϵ32|b2b-3b4b5b-1〉+ϵ21ϵ43|b-2b-3b-4b5b-1〉+ϵ21ϵ54|b-2b3b-4b-5b-1〉+ϵ21ϵ15|b-2b3b4b-5b1〉+ϵ32ϵ43|b-2b3b-4b5b1〉+ϵ32ϵ54|b-2b-3b-4b-5b1〉+ϵ32ϵ15|b-2b-3b4b-5b-1〉+ϵ43ϵ54|b2b-3b4b-5b1〉+ϵ43ϵ15|b2b-3b-4b-5b-1〉+ϵ54ϵ15|b2b3b-4b5b-1〉].(2) It is important to note that the preceding features are independent ofN—for a given r they are valid for all N. For r>N all the indices (b1,…,br) cannot be distinct. But even for r<N, some or all of them can coincide. This general validity permits a unified treatment for all N.(3) Note also the crucial cyclic permutation of the indices under the action ofT(r): (b1,b2,b3,…,br-1,br)→(b2,b3,b4,…,br,b1). This is a consequence of the trace condition in (2.12) incorporated in (3.3). It will be seen later how this leads to an essential role of the rth roots of unity ei((k·2π)/r) (k=0,1,…,r-1) in our construction of eigenstates and also of eigenvalues as factors accompanying exponentials of the type appearing in (3.6)–(3.9).We continue the study of the particular casesr=1,2,3,4,5 each one for any N. General features will be better understood afterwards.(i)
r
=
1
(3.10)
T
(
1
)
=
1
2
∑
a
,
ϵ
e
m
a
a
(
ϵ
)
θ
[
(
a
a
)
+
ϵ
(
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-
a
-
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=
1
2
∑
i
,
ϵ
(
1
+
ϵ
)
e
m
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i
(
ϵ
)
θ
[
(
i
i
)
+
ϵ
(
i
-
i
-
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,
where a∈{1,…,2n}, i∈{1,…,n}. This is already in diagonal form. Eigenstates and values are obtained trivially.(ii)
r
=
2
We start with (3.6). Separating “even” and “odd” spaces, define (for i≠j when (i,j,i-,j-) are all distinct)
(3.11)T(2)(c1|ij〉+c2|i-j-〉+c3|ji〉+c4|j-i-〉)=υe(c1|ij〉+c2|i-j-〉+c3|ji〉+c4|j-i-〉),T(2)(d1|ij-〉+d2|i-j〉+d3|ji-〉+d4|j-i〉)=υo(d1|ij-〉+d2|i-j〉+d3|ji-〉+d4|j-i〉).
The solutions are, for (c1,c2,c3,c4)=(1,1,1,1),(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1) respectively,
(3.12)υe=e(mij(+)+mji(+))θ,-e(mij(+)+mji(+))θ,e(mij(-)+mji(-))θ,-e(mij(-)+mji(-))θ,
and for (d1,d2,d3,d4)=(1,1,1,1),(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1), respectively,
(3.13)υo=e(mij(+)+mji(+))θ,-e(mij(+)+mji(+))θ,-e(mij(-)+mji(-))θ,e(mij(-)+mji(-))θ.
For i=j, with ϵ=±(3.14)T(2)(|ii〉+ϵ|i-i-〉)=e2mii(ϵ)θ(|ii〉+ϵ|i-i-〉),T(2)(|ii-〉+ϵ|i-i〉)=ϵe2mii(ϵ)θ(|ii-〉+ϵ|i-i〉).
Summing over all i, consistently with the general rule, one obtains
(3.15)Tr⁡(T(2))=2∑i=1ne2mii(+)θ
contributions coming only for i=j. For r=2, the rth roots of unity are ±1 and their role above is not visible very distinctly due to the presence of ±1 also from ϵ. The role of the roots of unity will be more evident from r=3 onwards. The simple exercise above gives all eigenvalues and eigenstates for T(2) for all N  (=2n). For N=2, r=2, writing out the 16×16  T(2) in full and effectively diagonalizing it we have confirmed the results above.(iii)
r
=
3
Now the starting point is (3.7). Define, for the even subset,
(3.16)Vλ=(|i1i2i3〉+ϵ12|i-1i-2i3〉+ϵ23|i1i-2i-3〉+ϵ31|i-1i2i-3〉)+λ(|i3i1i2〉+ϵ12|i3i-1i-2〉+ϵ23|i-3i1i-2〉+ϵ31|i-3i-1i2〉)+λ2(|i2i3i1〉+ϵ12|i-2i3i-1〉+ϵ23|i-2i-3i1〉+ϵ31|i2i-3i-1〉),
where
(3.17)λ=(1,ei(2π/3),ei(4π/3)),
and hence λ3=1, and also 1+ei(2π/3)+ei(4π/3)=0. Corresponding to (i1,i2,i3)⇌(i-1,i-2,i-3), define for the odd-subset,
(3.18)Vλ¯=(|i-1i-2i-3〉+ϵ12|i1i2i-3〉+ϵ23|i-1i2i3〉+ϵ31|i1i-2i3〉)+λ(|i-3i-1i-2〉+ϵ12|i-3i1i2〉+ϵ23|i3i-1i2〉+ϵ31|i3i1i-2〉)+λ2(|i-2i-3i-1〉+ϵ12|i2i-3i1〉+ϵ23|i2i3i-1〉+ϵ31|i-2i3i1〉).
The 8 values of the set (ϵ12,ϵ23,ϵ31) are restricted (see (3.7)) by (1+ϵ12ϵ23ϵ31)=2 (≠0). The remaining 4 possibilities along with 3 for λ (see (3.17)) yield 4×3=12 solutions for Vλ and Vλ¯ each for distinct indices i1≠i2≠i3≠i1. The eigenvalues are both Vλ and Vλ¯:
(3.19)e(mi1i2(ϵ12)+mi2i3(ϵ23)+mi3i1(ϵ31))θ(1,ei(2π/3),ei(4π/3)).
For two of the three indices equal  (i1=i2≠i3,…) the preceding results can be carried over without crucial change. But for i1=i2=i3≡i, say, there is a special situation due to the fact
(3.20)1+λ+λ2=(3,0,0),
for λ=(1,ei(2π/3),ei(4π/3)), respectively. Now
(3.21)Vλ=(1+λ+λ2)|iii〉+(ϵ12+ϵ31λ+ϵ23λ2)|i-i-i〉+(ϵ23+ϵ12λ+ϵ31λ2)|ii-i-〉+(ϵ31+ϵ23λ+ϵ12λ2)|i-ii-〉.
The eigenfunctions and eigenvalues are (for mutually orthogonal basis states)
(3.22)T(3)(|iii〉+|i-i-i〉+|ii-i-〉+|i-ii-〉)=e3mii(+)θ(|iii〉+|i-i-i〉+|ii-i-〉+|i-ii-〉),T(3)(3|iii〉-|i-i-i〉-|ii-i-〉-|i-ii-〉)=e(mii(+)+2mii(-))θ(3|iii〉-|i-i-i〉-|ii-i-〉-|i-ii-〉),T(3)(|i-i-i〉+ei(2π/3)|ii-i-〉+ei(4π/3)|i-ii-〉)=ei(2π/3)e(mii(+)+2mii(-))θ(|i-i-i〉+ei(2π/3)|ii-i-〉+ei(4π/3)|i-ii-〉),T(3)(|i-i-i〉+ei(4π/3)|ii-i-〉+ei(2π/3)|i-ii-〉)=ei(4π/3)e(mii(+)+2mii(-))θ(|i-i-i〉+ei(4π/3)|ii-i-〉+ei(2π/3)|i-ii-〉).
For Vλ¯ one follows exactly similar steps and obtains the same set of eigenvalues (with i⇌i- in the eigenstates). Combining all the solutions above, we obtain complete results for r=3 and all even N.(iv)
r
=
4
The key result (3.8) indicates the following construction (a direct generalization of (3.16)). Define, for the even subset,
(3.23)V(b1b2b3b4)=|b1b2b3b4〉+ϵ12|b-1b-2b3b4〉+ϵ23|b1b-2b-3b4〉+ϵ34|b1b2b-3b-4〉+ϵ41|b-1b2b3b-4〉+ϵ12ϵ23|b-1b2b-3b4〉+ϵ12ϵ34|b-1b-2b-3b-4〉+ϵ23ϵ34|b1b-2b3b-4〉,
and similarly, implementing circular permutations as in (3.16), V(b4b1b2b3),V(b3b4b1b2),V(b2b3b4b1), define
(3.24)Vλ=V(b1b2b3b4)+λV(b4b1b2b3)+λ2V(b3b4b1b2)+λ3V(b2b3b4b1),
where
(3.25)λ=(1,ei(2π/4),ei2⋅(2π/4),ei3⋅(2π/4))=(1,i,-1,-i).
One obtains for distinct (i1,i2,i3,i4)(3.26)T(4)Vλ=λe(mi1i2(ϵ12)+mi2i3(ϵ23)+mi3i4(ϵ34)+mi4i1(ϵ41))θVλ,
where for nonzero results (1+ϵ12ϵ23ϵ34ϵ41)=2. For the 8 possibilities remaining for the ϵ’s and the 4 values of λ, one obtains the full set of 32 eigenstates and the corresponding eigenvalues. For the complementary “odd” subspace formed by
(3.27){|b1b2b3b-4〉,|b1b2b-3b4〉,|b1b-2b3b4〉,|b-1b2b3b4〉,|b-1b-2b-3b4〉,|b-1b-2b3b-4〉,|b-1b2b-3b-4〉,|b1b-2b-3b-4〉},
and the sets related through circular permutations of the indices, one obtains, entirely in evident analogy of (3.24),
(3.28)Vλ¯=V(b1b2b3b4)¯+λV(b4b1b2b3)¯+λ2V(b3b4b1b2)¯+λ3V(b2b3b4b1)¯.
One obtains, for distinct indices, the same set of eigenvalues as in (3.26). For some equal indices, but not all, the preceding results can be carried over essentially. For all equal indices one obtains the following spectrum of eigenvalues (grouping together the even and the odd subspaces):
(3.29)e4mii(+)θ(1,1),[3-times]:e2(mii(+)+mii(-))θ(1,ei(2π/4),ei2⋅(2π/4),ei3⋅(2π/4)),e4mii(-)θ(1,-1),
(In fact these are the same as in (A.13) of [7], since the distinguishing index for odd dimensions signalled in (2.14) and (2.15) is not involved.) We will not present here the 16 orthogonal eigenstates corresponding to (3.29). But a new feature, as compared to (3.22), should be pointed out. For r=3, considering together even and odd spaces, one has for identical indices the eigenvalues
(3.30)e3mii(+)θ(1,1),[2-times]:e(mii(+)+2mii(-))θ(1,ei(2π/3),ei2⋅(2π/3)).
In fact, for each r one obtains, for ϵ=1, ermii(+)θ(1,1) giving
(3.31)Tr⁡(T(r))=2∑i=1nermii(ϵ)θ.
All other multiplets have each one zero sum. Concerning the zero sum multiplets, comparing (3.30) and (3.29), we note that(1)
forr=3, one has only triplets;(2)
forr=4, one has quadruplets and also a doublet (1,-1). This last enters due to factorizability of r (4=2×2).Such a feature is worth signalling since it generalizes.
Whenr is a prime number, the zero-sum multiplets appear only as r-plets (with rth roots of unity). How they consistently cover the whole base space (along with the doublets giving the trace as in (3.31)) has been amply discussed in our previous papers [7, 9] under the heading “An encounter with a theorem of Fermat.” When r has many factors the multiplicity of submultiplets corresponding to each one is difficult to formulate giving, say, an explicit general prescription.(v)
r
≥
5
Another problem concerning systematic, complete enumeration one encounters already forr=5. Starting with (3.9) the generalizations of (3.23)–(3.26) are quite evident. We will not present them here. But as compared to (3.29) one has now for the 32-dimensional subspace (for i1=i2=⋯=i5)
(3.32)e5mii(+)θ(1,1),[n2-times]:e(3mii(+)+2mii(-))θ(1,ei(2π/5),ei2⋅(2π/5),ei3⋅(2π/5),ei4⋅(2π/5)),[n4-times]:e(mii(+)+4mii(-))θ(1,ei(2π/5),ei2⋅(2π/5),ei3⋅(2π/5),ei4⋅(2π/5)),
where n2+n4=6. At this stage it is not difficult to present complete construction. But as r increases, for prime numbers, one has
(3.33)ermii(+)θ(1,1),[n2-times]:e((r-2)mii(+)+2mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),[n4-times]:e((r-4)mii(+)+4mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),⋮⋮[nr-1-times]:e(mii(+)+(r-1)mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),
the sum of the multiplicities of such r-plets satisfying
(3.34)n2+n4+⋯+nr-1=1r(2r-2).
(That the right-hand side is an integer for r a prime number is guaranteed by a theorem of Fermat, as has been discussed before.) A general explicit prescription for the sequence (n2,n4,…,nr-1) is beyond the scope of this paper.The situation is as above for prime numbersr. For factorizable r the presence, in addition, of submultiplets has already been pointed out. For r=4, only such submultiplets were (1,-1). For r=6, one can have (1,-1), (1,ei(2π/3),ei2·(2π/3)) corresponding to the factors 6=2×3, respectively. For r=30=2×3×5 one can have also submultiplets corresponding to 5. And so on.But apart from the above-mentioned limitations concerning multiplicities of zero-sum (“roots of unity”) multiplets and submultiplets we can claim to have elucidated the spectrum of eigenvalues for all(N,r). For the generic case with distinct indices for any (N,r), one has eigenvalues
(3.35)λke(mi1i2(ϵ12)+mi2i3(ϵ23)+⋯+mir-1ir(ϵr-1,r)+mi1ir(ϵ1r))θ,
where (1+ϵ12ϵ23⋯ϵr-1,rϵ1,r)=2 and λk=eik·(2π/r)(k=0,1,2,…,r-1). Here each mikil(ϵkl) is a free parameter. We solve only sets of linear equations with quite simple constant coefficients to obtain the eigenstates and the eigenvalues.


### 3.2. Odd Dimensions

We refer back to (2.14)–(2.16). When the special index n (n=n¯) for N=2n-1 is not present in the basis state |b1b2⋯br〉 and hence in the subspace it generates via cyclic permutations as in the forgoing examples, the foregoing constructions can be taken over wholesale. When n is present, the modifications are not difficult to take into account. The trace is now Tr(T(r))=2∑i=1n-1‍ermii(ϵ)θ+1, since the central state |n〉⊗|n〉⊗⋯|n〉⊗≡|nn⋯n〉 contributes with our normalization 1 to the trace. Various aspects have been studied in considerable detail in our previous papers [6, 7] on odd N. Here we just refer to them.


## 3.1.  Even Dimensions

In Section 5 of [9] it was pointed out that the antidiagonal matrix
(3.1)K=∑a=12n(aa-)=∑i=1n((ii-)+(i-i))
relates the blocks of T(1) as follows:
(3.2)KTab(1)=Tab-(1),Tab(1)K=Ta-b(1),KTab(1)K=Ta-b-(1).
For N=2 this led to an iterative construction of eigenstates and eigenvalues of the transfer matrix T(r) for increasing r. This works efficiently only for N=2 (the 4×4  R^(θ)-matrix).For our present approach the crucial ingredient shows up immediately in the structure (arising from those of our projectors and the coproducts rules)(3.3)T(r)=12r∑a1,a2,…,ar∑ϵ21,ϵ32,…,ϵ1re(ma2a1(ϵ21)+ma3a2(ϵ32)+⋯+ma1ar(ϵ1r))θ×[(a2a1)+ϵ21(a-2a-1)]⊗[(a3a2)+ϵ32(a-3a-2)]⊗⋯⊗[(a1ar)+ϵ1r(a-1a-r)],
where each ϵ is independently (±). From (3.3), one obtains immediately
(3.4)(T(r))-1=12r∑a1,a2,…,ar∑ϵ21,ϵ32,…,ϵ1re-(ma2a1(ϵ21)+ma3a2(ϵ32)+⋯+ma1ar(ϵ1r))θ×[(a1a2)+ϵ21(a-1a-2)]⊗[(a2a3)+ϵ32(a-2a-3)]⊗⋯⊗[(ara1)+ϵ1r(a-ra-1)].
Consider the basis state
(3.5)|b1〉⊗|b2〉⊗⋯⊗|br〉≡|b1b2⋯br〉.
(1) If any one (even a single one) of the indices bi≠ai+1 or ai+1¯ (cyclic), then the coefficient of e(ma2a1(ϵ21)+⋯+ma1ar(ϵ1r))θ, namely, [(a2a1)+ϵ21(a-2a-1)],⊗⋯⊗[(a1ar)+ϵ1r(a-1a-r)], will annihilate it. On the other hand (2.9) plays a crucial role. Gathering together all the terms with e(mb2b1(ϵ21)+mb3b2(ϵ32)+⋯+mbrbr-1(ϵr,r-1)+mb1br(ϵ1r))θ as coefficient, one obtains the action of T(r) on (3.5). All essential features for the general case can be read off the first few simples examples:
(3.6)22T(2)|b1b2〉=∑ϵ21,ϵ12e(mb2b1(ϵ21)+mb1b2(ϵ12))θ(1+ϵ21ϵ12)[|b2b1〉+ϵ21|b-2b-1〉],(3.7)23T(3)|b1b2b3〉=∑ϵ21,ϵ32,ϵ13e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb1b3(ϵ13))θ(1+ϵ21ϵ32ϵ13)×[|b2b3b1〉+ϵ21|b-2b3b-1〉+ϵ32|b-2b-3b1〉+ϵ13|b2b-3b-1〉],(3.8)24T(4)|b1b2b3b4〉=∑ϵ21,ϵ32,ϵ43,ϵ14e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb4b3(ϵ43)+mb1b4(ϵ14))θ(1+ϵ21ϵ32ϵ43ϵ14)×[|b2b3b4b1〉+ϵ21|b-2b3b4b-1〉+ϵ32|b-2b-3b4b1〉+ϵ43|b2b-3b-4b1〉+ϵ14|b2b3b-4b-1〉+ϵ21ϵ32|b2b-3b4b-1〉+ϵ21ϵ43|b-2b-3b-4b-1〉+ϵ21ϵ14|b-2b3b-4b1〉].
Note that the bars correspond to the indies of the ϵ’s. One has thus ϵij|b-ib-jbkbl〉, ϵijϵki|b̿ib-jb-kbl〉=ϵijϵki|bib-jb-kbl〉 since b̿i=bi. Consequently only even number of additional bars appear on the right. Starting with even or odd number of bars on the left leads to “even” and “odd” closed subspaces. 								(3.9)25T(5)|b1b2b3b4b5〉=∑ϵ21,ϵ32,ϵ43,ϵ14e(mb2b1(ϵ21)+mb3b2(ϵ32)+mb4b3(ϵ43)+mb5ba4(ϵ54)+mb1b5(ϵ15))θ(1+ϵ21ϵ32ϵ43ϵ54ϵ15)×[|b2b3b4b5b1〉+ϵ21|b-2b3b4b5b-1〉+ϵ32|b-2b-3b4b5b1〉+ϵ43|b2b-3b-4b5b1〉+ϵ54|b2b3b-4b-5b1〉+ϵ15|b2b3b4b-5b-1〉+ϵ21ϵ32|b2b-3b4b5b-1〉+ϵ21ϵ43|b-2b-3b-4b5b-1〉+ϵ21ϵ54|b-2b3b-4b-5b-1〉+ϵ21ϵ15|b-2b3b4b-5b1〉+ϵ32ϵ43|b-2b3b-4b5b1〉+ϵ32ϵ54|b-2b-3b-4b-5b1〉+ϵ32ϵ15|b-2b-3b4b-5b-1〉+ϵ43ϵ54|b2b-3b4b-5b1〉+ϵ43ϵ15|b2b-3b-4b-5b-1〉+ϵ54ϵ15|b2b3b-4b5b-1〉].(2) It is important to note that the preceding features are independent ofN—for a given r they are valid for all N. For r>N all the indices (b1,…,br) cannot be distinct. But even for r<N, some or all of them can coincide. This general validity permits a unified treatment for all N.(3) Note also the crucial cyclic permutation of the indices under the action ofT(r): (b1,b2,b3,…,br-1,br)→(b2,b3,b4,…,br,b1). This is a consequence of the trace condition in (2.12) incorporated in (3.3). It will be seen later how this leads to an essential role of the rth roots of unity ei((k·2π)/r) (k=0,1,…,r-1) in our construction of eigenstates and also of eigenvalues as factors accompanying exponentials of the type appearing in (3.6)–(3.9).We continue the study of the particular casesr=1,2,3,4,5 each one for any N. General features will be better understood afterwards.(i)
r
=
1
(3.10)
T
(
1
)
=
1
2
∑
a
,
ϵ
e
m
a
a
(
ϵ
)
θ
[
(
a
a
)
+
ϵ
(
a
-
a
-
)
]
=
1
2
∑
i
,
ϵ
(
1
+
ϵ
)
e
m
i
i
(
ϵ
)
θ
[
(
i
i
)
+
ϵ
(
i
-
i
-
)
]
,
where a∈{1,…,2n}, i∈{1,…,n}. This is already in diagonal form. Eigenstates and values are obtained trivially.(ii)
r
=
2
We start with (3.6). Separating “even” and “odd” spaces, define (for i≠j when (i,j,i-,j-) are all distinct)
(3.11)T(2)(c1|ij〉+c2|i-j-〉+c3|ji〉+c4|j-i-〉)=υe(c1|ij〉+c2|i-j-〉+c3|ji〉+c4|j-i-〉),T(2)(d1|ij-〉+d2|i-j〉+d3|ji-〉+d4|j-i〉)=υo(d1|ij-〉+d2|i-j〉+d3|ji-〉+d4|j-i〉).
The solutions are, for (c1,c2,c3,c4)=(1,1,1,1),(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1) respectively,
(3.12)υe=e(mij(+)+mji(+))θ,-e(mij(+)+mji(+))θ,e(mij(-)+mji(-))θ,-e(mij(-)+mji(-))θ,
and for (d1,d2,d3,d4)=(1,1,1,1),(1,1,-1,-1),(1,-1,1,-1),(1,-1,-1,1), respectively,
(3.13)υo=e(mij(+)+mji(+))θ,-e(mij(+)+mji(+))θ,-e(mij(-)+mji(-))θ,e(mij(-)+mji(-))θ.
For i=j, with ϵ=±(3.14)T(2)(|ii〉+ϵ|i-i-〉)=e2mii(ϵ)θ(|ii〉+ϵ|i-i-〉),T(2)(|ii-〉+ϵ|i-i〉)=ϵe2mii(ϵ)θ(|ii-〉+ϵ|i-i〉).
Summing over all i, consistently with the general rule, one obtains
(3.15)Tr⁡(T(2))=2∑i=1ne2mii(+)θ
contributions coming only for i=j. For r=2, the rth roots of unity are ±1 and their role above is not visible very distinctly due to the presence of ±1 also from ϵ. The role of the roots of unity will be more evident from r=3 onwards. The simple exercise above gives all eigenvalues and eigenstates for T(2) for all N  (=2n). For N=2, r=2, writing out the 16×16  T(2) in full and effectively diagonalizing it we have confirmed the results above.(iii)
r
=
3
Now the starting point is (3.7). Define, for the even subset,
(3.16)Vλ=(|i1i2i3〉+ϵ12|i-1i-2i3〉+ϵ23|i1i-2i-3〉+ϵ31|i-1i2i-3〉)+λ(|i3i1i2〉+ϵ12|i3i-1i-2〉+ϵ23|i-3i1i-2〉+ϵ31|i-3i-1i2〉)+λ2(|i2i3i1〉+ϵ12|i-2i3i-1〉+ϵ23|i-2i-3i1〉+ϵ31|i2i-3i-1〉),
where
(3.17)λ=(1,ei(2π/3),ei(4π/3)),
and hence λ3=1, and also 1+ei(2π/3)+ei(4π/3)=0. Corresponding to (i1,i2,i3)⇌(i-1,i-2,i-3), define for the odd-subset,
(3.18)Vλ¯=(|i-1i-2i-3〉+ϵ12|i1i2i-3〉+ϵ23|i-1i2i3〉+ϵ31|i1i-2i3〉)+λ(|i-3i-1i-2〉+ϵ12|i-3i1i2〉+ϵ23|i3i-1i2〉+ϵ31|i3i1i-2〉)+λ2(|i-2i-3i-1〉+ϵ12|i2i-3i1〉+ϵ23|i2i3i-1〉+ϵ31|i-2i3i1〉).
The 8 values of the set (ϵ12,ϵ23,ϵ31) are restricted (see (3.7)) by (1+ϵ12ϵ23ϵ31)=2 (≠0). The remaining 4 possibilities along with 3 for λ (see (3.17)) yield 4×3=12 solutions for Vλ and Vλ¯ each for distinct indices i1≠i2≠i3≠i1. The eigenvalues are both Vλ and Vλ¯:
(3.19)e(mi1i2(ϵ12)+mi2i3(ϵ23)+mi3i1(ϵ31))θ(1,ei(2π/3),ei(4π/3)).
For two of the three indices equal  (i1=i2≠i3,…) the preceding results can be carried over without crucial change. But for i1=i2=i3≡i, say, there is a special situation due to the fact
(3.20)1+λ+λ2=(3,0,0),
for λ=(1,ei(2π/3),ei(4π/3)), respectively. Now
(3.21)Vλ=(1+λ+λ2)|iii〉+(ϵ12+ϵ31λ+ϵ23λ2)|i-i-i〉+(ϵ23+ϵ12λ+ϵ31λ2)|ii-i-〉+(ϵ31+ϵ23λ+ϵ12λ2)|i-ii-〉.
The eigenfunctions and eigenvalues are (for mutually orthogonal basis states)
(3.22)T(3)(|iii〉+|i-i-i〉+|ii-i-〉+|i-ii-〉)=e3mii(+)θ(|iii〉+|i-i-i〉+|ii-i-〉+|i-ii-〉),T(3)(3|iii〉-|i-i-i〉-|ii-i-〉-|i-ii-〉)=e(mii(+)+2mii(-))θ(3|iii〉-|i-i-i〉-|ii-i-〉-|i-ii-〉),T(3)(|i-i-i〉+ei(2π/3)|ii-i-〉+ei(4π/3)|i-ii-〉)=ei(2π/3)e(mii(+)+2mii(-))θ(|i-i-i〉+ei(2π/3)|ii-i-〉+ei(4π/3)|i-ii-〉),T(3)(|i-i-i〉+ei(4π/3)|ii-i-〉+ei(2π/3)|i-ii-〉)=ei(4π/3)e(mii(+)+2mii(-))θ(|i-i-i〉+ei(4π/3)|ii-i-〉+ei(2π/3)|i-ii-〉).
For Vλ¯ one follows exactly similar steps and obtains the same set of eigenvalues (with i⇌i- in the eigenstates). Combining all the solutions above, we obtain complete results for r=3 and all even N.(iv)
r
=
4
The key result (3.8) indicates the following construction (a direct generalization of (3.16)). Define, for the even subset,
(3.23)V(b1b2b3b4)=|b1b2b3b4〉+ϵ12|b-1b-2b3b4〉+ϵ23|b1b-2b-3b4〉+ϵ34|b1b2b-3b-4〉+ϵ41|b-1b2b3b-4〉+ϵ12ϵ23|b-1b2b-3b4〉+ϵ12ϵ34|b-1b-2b-3b-4〉+ϵ23ϵ34|b1b-2b3b-4〉,
and similarly, implementing circular permutations as in (3.16), V(b4b1b2b3),V(b3b4b1b2),V(b2b3b4b1), define
(3.24)Vλ=V(b1b2b3b4)+λV(b4b1b2b3)+λ2V(b3b4b1b2)+λ3V(b2b3b4b1),
where
(3.25)λ=(1,ei(2π/4),ei2⋅(2π/4),ei3⋅(2π/4))=(1,i,-1,-i).
One obtains for distinct (i1,i2,i3,i4)(3.26)T(4)Vλ=λe(mi1i2(ϵ12)+mi2i3(ϵ23)+mi3i4(ϵ34)+mi4i1(ϵ41))θVλ,
where for nonzero results (1+ϵ12ϵ23ϵ34ϵ41)=2. For the 8 possibilities remaining for the ϵ’s and the 4 values of λ, one obtains the full set of 32 eigenstates and the corresponding eigenvalues. For the complementary “odd” subspace formed by
(3.27){|b1b2b3b-4〉,|b1b2b-3b4〉,|b1b-2b3b4〉,|b-1b2b3b4〉,|b-1b-2b-3b4〉,|b-1b-2b3b-4〉,|b-1b2b-3b-4〉,|b1b-2b-3b-4〉},
and the sets related through circular permutations of the indices, one obtains, entirely in evident analogy of (3.24),
(3.28)Vλ¯=V(b1b2b3b4)¯+λV(b4b1b2b3)¯+λ2V(b3b4b1b2)¯+λ3V(b2b3b4b1)¯.
One obtains, for distinct indices, the same set of eigenvalues as in (3.26). For some equal indices, but not all, the preceding results can be carried over essentially. For all equal indices one obtains the following spectrum of eigenvalues (grouping together the even and the odd subspaces):
(3.29)e4mii(+)θ(1,1),[3-times]:e2(mii(+)+mii(-))θ(1,ei(2π/4),ei2⋅(2π/4),ei3⋅(2π/4)),e4mii(-)θ(1,-1),
(In fact these are the same as in (A.13) of [7], since the distinguishing index for odd dimensions signalled in (2.14) and (2.15) is not involved.) We will not present here the 16 orthogonal eigenstates corresponding to (3.29). But a new feature, as compared to (3.22), should be pointed out. For r=3, considering together even and odd spaces, one has for identical indices the eigenvalues
(3.30)e3mii(+)θ(1,1),[2-times]:e(mii(+)+2mii(-))θ(1,ei(2π/3),ei2⋅(2π/3)).
In fact, for each r one obtains, for ϵ=1, ermii(+)θ(1,1) giving
(3.31)Tr⁡(T(r))=2∑i=1nermii(ϵ)θ.
All other multiplets have each one zero sum. Concerning the zero sum multiplets, comparing (3.30) and (3.29), we note that(1)
forr=3, one has only triplets;(2)
forr=4, one has quadruplets and also a doublet (1,-1). This last enters due to factorizability of r (4=2×2).Such a feature is worth signalling since it generalizes.
Whenr is a prime number, the zero-sum multiplets appear only as r-plets (with rth roots of unity). How they consistently cover the whole base space (along with the doublets giving the trace as in (3.31)) has been amply discussed in our previous papers [7, 9] under the heading “An encounter with a theorem of Fermat.” When r has many factors the multiplicity of submultiplets corresponding to each one is difficult to formulate giving, say, an explicit general prescription.(v)
r
≥
5
Another problem concerning systematic, complete enumeration one encounters already forr=5. Starting with (3.9) the generalizations of (3.23)–(3.26) are quite evident. We will not present them here. But as compared to (3.29) one has now for the 32-dimensional subspace (for i1=i2=⋯=i5)
(3.32)e5mii(+)θ(1,1),[n2-times]:e(3mii(+)+2mii(-))θ(1,ei(2π/5),ei2⋅(2π/5),ei3⋅(2π/5),ei4⋅(2π/5)),[n4-times]:e(mii(+)+4mii(-))θ(1,ei(2π/5),ei2⋅(2π/5),ei3⋅(2π/5),ei4⋅(2π/5)),
where n2+n4=6. At this stage it is not difficult to present complete construction. But as r increases, for prime numbers, one has
(3.33)ermii(+)θ(1,1),[n2-times]:e((r-2)mii(+)+2mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),[n4-times]:e((r-4)mii(+)+4mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),⋮⋮[nr-1-times]:e(mii(+)+(r-1)mii(-))θ(1,ei(2π/r),…,ei(r-1)⋅(2π/r)),
the sum of the multiplicities of such r-plets satisfying
(3.34)n2+n4+⋯+nr-1=1r(2r-2).
(That the right-hand side is an integer for r a prime number is guaranteed by a theorem of Fermat, as has been discussed before.) A general explicit prescription for the sequence (n2,n4,…,nr-1) is beyond the scope of this paper.The situation is as above for prime numbersr. For factorizable r the presence, in addition, of submultiplets has already been pointed out. For r=4, only such submultiplets were (1,-1). For r=6, one can have (1,-1), (1,ei(2π/3),ei2·(2π/3)) corresponding to the factors 6=2×3, respectively. For r=30=2×3×5 one can have also submultiplets corresponding to 5. And so on.But apart from the above-mentioned limitations concerning multiplicities of zero-sum (“roots of unity”) multiplets and submultiplets we can claim to have elucidated the spectrum of eigenvalues for all(N,r). For the generic case with distinct indices for any (N,r), one has eigenvalues
(3.35)λke(mi1i2(ϵ12)+mi2i3(ϵ23)+⋯+mir-1ir(ϵr-1,r)+mi1ir(ϵ1r))θ,
where (1+ϵ12ϵ23⋯ϵr-1,rϵ1,r)=2 and λk=eik·(2π/r)(k=0,1,2,…,r-1). Here each mikil(ϵkl) is a free parameter. We solve only sets of linear equations with quite simple constant coefficients to obtain the eigenstates and the eigenvalues.


## 3.2. Odd Dimensions

We refer back to (2.14)–(2.16). When the special index n (n=n¯) for N=2n-1 is not present in the basis state |b1b2⋯br〉 and hence in the subspace it generates via cyclic permutations as in the forgoing examples, the foregoing constructions can be taken over wholesale. When n is present, the modifications are not difficult to take into account. The trace is now Tr(T(r))=2∑i=1n-1‍ermii(ϵ)θ+1, since the central state |n〉⊗|n〉⊗⋯|n〉⊗≡|nn⋯n〉 contributes with our normalization 1 to the trace. Various aspects have been studied in considerable detail in our previous papers [6, 7] on odd N. Here we just refer to them.


## 4. Spin Chain Hamiltonians

Spin chains corresponding to our braid matrices have already been studied in our previous papers [7, 8]. Here we formulate a unified approach for all dimensions (N=2n-1,2n).The basic formula (see sources cited in [7, 8]) is
(4.1)H=∑k=1rI⊗⋯⊗I⊗R^˙k,k+1(0)⊗I⊗⋯⊗I,
where for circular boundary conditions, k+1=r+1≈1. For even N (N=2n), from (2.4),(4.2)R^˙(0)=ddθR^(θ)|θ=0=∑ϵ,i,jmij(ϵ)(Pij(ϵ)+Pij¯(ϵ))
the projectors being given by (2.1) and the remark below (2.1). For N odd (N=2n-1; n=2,3,…) we introduce a modified overall normalization factor to start with. Such a factor is trivial concerning the braid equation, but not for the Hamiltonian (since a derivative is involved) if the factor is θ-dependent. Multiply (2.16) by emθ and redefine
(4.3)((m+mij(ϵ)),(m+mni(ϵ)),(m+min(ϵ)))⟶(mij(ϵ),mni(ϵ),min(ϵ))
since (mij(ϵ),…) are arbitrary to start with. Now for odd N (with the ranges of (i,j), (i-,j-) of (2.16)),
(4.4)R^˙(0)=mPnn+∑ϵ,i(mni(ϵ)Pni(ϵ)+min(ϵ)Pin(ϵ))+∑ϵ,i,jmij(ϵ)(Pij(ϵ)+Pij¯(ϵ)).
This extends the m=0 case by including the presence of Pnn.InH, R^˙(0)k,k+1 acts on the basis |V〉(k)⊗|V〉(k+1). One can denote, using standard ordering of spin components for each k,
(4.5)|V〉(k)=||n-1/2〉k⋮|1/2〉k|-1/2〉k⋮|-n+1/2〉k〉,|V〉(k)=||n-1〉k⋮|1〉k|0〉k|-1〉k⋮|-n+1〉k〉,
for N=2n,2n-1, respectively. Without being restricted to spin, one can consider more generally any system with N orthogonal states. We will continue, however, to use the terminology of spin. At each site one can consider a superposition at each level such as (∑l‍cil(k)|l〉k). To keep the notation tractable, we will just denote
(4.6)|V〉(k)=||1〉k|2〉k⋮|2-〉k|1-〉k〉,
and keep possible significances of |i〉k in mind. Each index k is acted upon twice by H, namely, by R^˙k-1,k(0), R^˙k,k+1(0), and, for closed chains, the index 1 is also thus involved in R^˙12(0) and R^˙r1(0). Some simple examples are(i)
N
=
2
(4.7)
R
^
˙
(
0
)
=
|
a
^
+
0
0
a
^
-
0
a
^
+
a
^
-
0
0
a
^
-
a
^
+
0
a
^
-
0
0
a
^
+
|
,
where a^±=(1/2)(m11(+)±m11(-)) and hence in evident notations
(4.8)R^˙(0)|V〉(k)⊗|V〉(k+1)=|a^+|11〉+a^-|1-1-〉a^+|11-〉+a^-|1-1〉a^-|11-〉+a^+|1-1〉a^-|11〉+a^+|1-1-〉〉(k,k+1),(ii)
N
=
4
In the notation of Section 7 of [9]
(4.9)R^(θ)=|D^1100A^11-0D^22A^22-00A^2-2D^2-2-0A^1-100D^1-1-|,
where
(4.10)D^11=D^1-1-=(a^+0000b^+0000b^+0000a^+),D^22=D^2-2-=(c^+0000d^+0000d^+0000c^+),A^11-=A^1-1=(000a^-00b^-00b^-00a^-000),A^22-=A^2-2=(000c^-00d^-00d^-00c^-000),a^±=12(m11(+)±m11(-)),b^±=12(m12(+)±m12(-)),c^±=12(m21(+)±m21(-)),d^±=12(m22(+)±m22(-)).
Hence,
(4.11)R^˙(0)|V〉(k)⊗|V〉(k+1)=||1〉⊗(D^11|V〉)+|1-〉⊗(A^11-|V〉)|2〉⊗(D^22|V〉)+|2-〉⊗(A^22-|V〉)|2〉⊗(A^22-|V〉)+|2-〉⊗(D^22|V〉)|1〉⊗(A^11-|V〉)+|1-〉⊗(D^11|V〉)〉(k,k+1),
where for each k, |V〉=||1〉|2〉|2-〉|1-〉〉(
iii
)
N
=
3
Adapting the results of Section 1 of [7] and Section 11 of [8] to notations analogous to the cases above, one can write
(4.12)R^(θ)=|D0A0C0A0D|,
where
(4.13)D=(a^+000b^+000a^+),A=(00a^-0b^-0a^-00),C=(c^+0c^-0m0c^-0c^+),
the central element m corresponding to mPnn of (4.4) and
(4.14)a^±=12(m11(+)±m11(-)),b^±=12(m12(+)±m12(-)),c^±=12(m21(+)±m21(-)).Denote the basis factors for eachk as, |V〉(k)=||1〉|2〉|1-〉〉(k) and
(4.15)|V〉(k)⊗|V〉(k+1)=||1〉⊗|V〉|2〉⊗|V〉|1-〉⊗|V〉〉(k,k+1).
One obtains
(4.16)R^˙(0)|V〉(k)⊗|V〉(k+1)=|D0A0C0A0D|||1〉⊗|V〉|2〉⊗|V〉|1-〉⊗|V〉〉(k,k+1)=||1〉⊗(D|V〉)+|1-〉⊗(A|V〉)|2〉⊗(C|V〉)|1〉⊗(A|V〉)+|1-〉⊗(D|V〉)〉(k,k+1).
For N=5,6,…, one can generate such results systematically. They, considering the action of all the terms of (4.1), furnish the transition matrix elements and expectation values for possible states of the chain and permit a study of correlations.One can also consider higher-order conserved quantities (Section 1.5 of [10]) given by
(4.17)Hl=dldθllog⁡T(r)(θ)|θ=0.
For l=2, (4.1) is generalized by the appearance of factors of the type (apart from non-overlapping derivatives)
(4.18)R^˙(0)k-1,k⊗R^˙(0)k,k+1,R^¨(0)k,k+1.
For l>2 this generalizes in an evident fashion. A study of spin chains for the “exotic” SØ3 can be found in [11]. It is interesting to compare it with the 4×4 (for N=2) case briefly presented above by exploring the latter case in comparable detail.One may note that for our class of braid matrices, for evenN as compared to (4.2), are
(4.19)dldθlR^(θ)|θ=0=∑ϵ∑i,j‍(mij(ϵ))l(Pij(ϵ)+Pij¯(ϵ)),
and for odd N, as compared to (4.4),
(4.20)dldθlR^(θ)|θ=0=mlPnn+∑ϵ,i((mni(ϵ))lPni(ϵ)+(min(ϵ))lPin(ϵ))+∑ϵ,i,j(mij(ϵ))l(Pij(ϵ)+Pij¯(ϵ)).


## 5. Potentials for FactorizableS-Matrices

Such potentials can be obtained as inverse Cayley transforms of the Yang-Baxter matrices of appropriate dimensions (see, for example, Section 3 of [12] and Section 1 of [2]). Starting with the Yang-Baxter matrix R(θ)=PR^(θ), the required potential V(θ) is given by
(5.1)-iV(θ)=(R(θ)-λ(θ)I)-1(R(θ)+λ(θ)I).
We have emphasized in our previous studies (Section 5 of [7], Section 8 of [8]) that λ(θ) cannot be arbitrary, a set of values must be excluded for the inverse (5.1) to be well defined. Here we generalize our previous results to all N. As compared to other well-known studies [13] of fields corresponding to factorizable scatterings, here our construction starts with the braid matrices (2.4) and (2.16).Define(5.2)X(θ)=(R(θ)-λ(θ)I)-1
when
(5.3)-iV(θ)=I+2λ(θ)X(θ).
We give below immediately the general solution and then explain the notations more precisely. The solution was obtained via formal series expansions. But the final closed form can be verified directly. The solution is
(5.4)X(θ)=-12∑ϵ=±∑a,b=1N1λ2(θ)-e(mab(ϵ)+mba(ϵ))θ{λ(θ)[(aa)⊗(bb)+ϵ(aa¯)⊗(bb¯)]+emba(ϵ)θ[(ab)⊗(ba)+ϵ(ab¯)⊗(ba¯)]},
where
(5.5)λ(θ)≠±e(1/2)(mab(ϵ)+mba(ϵ))θ.
For N=2n, (2.9) is implicit in this result. For N=2n-1, when n¯=n, the conventions (2.16) (or (4.4)) are to be implemented when a or b or both (a,b) are n. From (5.3) and (5.4),
(5.6)V(θ)=-i2∑ϵ=±∑a,b=1N{λ2(θ)+e(mab(ϵ)+mba(ϵ))θλ2(θ)-e[mab(ϵ)+mba(ϵ)]θ[(aa)⊗(bb)+ϵ(aa¯)⊗(bb¯)]+emab(ϵ)θλ2(θ)-e(mab(ϵ)+mba(ϵ))θ[(ba)⊗(ab)+ϵ(ba¯)⊗(ab¯)]}≡∑ab,cdV(θ)(ab,cd)(ab)⊗(cd).
Now one can write down the Lagrangians for scalar and spinor fields. For the spinor case, for example,
(5.7)L=∫dx[iψ¯aγν∂νψa-g(ψ¯aγνψc)Vab,cd(ψ¯bγνψd)].
The simpler scalar case can be written analogously. The scalar Lagrangian has an interaction term of the form (ϕ¯aϕ¯c)Vab,cd(ϕbϕd). For N=2n, one obtains the nonvanishing elements of V as
(5.8)Vbb,dd=-i2∑ϵλ2(θ)+e(mbd(ϵ)+mdb(ϵ))θλ2(θ)-e(mbd(ϵ)+mdb(ϵ))θ,Vb¯b,d¯d=-i2∑ϵϵλ2(θ)+e(mbd(ϵ)+mdb(ϵ))θλ2(θ)-e(mbd(ϵ)+mdb(ϵ))θ,Vdb,bd=-i2∑ϵembd(ϵ)θλ2(θ)-e(mbd(ϵ)+mdb(ϵ))θ,Vd¯b,b¯d=-i2∑ϵϵembd(ϵ)θλ2(θ)-e(mbd(ϵ)+mdb(ϵ))θ,
where a,b∈{1,…,N} and (a,b)≠(n,n) if N=2n-1. When N is odd (N=2n-1), one has, for the special index n,
(5.9)Vnn,nn=-i(λ2(θ)+2λ2(θ)-1),
where mnn(ϵ) are taken to be zero. For our models, the scattering process can be schematically, presented as (see in Figure 1)Figure 1
Scattering process.


## 6. Remarks

We have obtained explicit eigenvalues of the transfer matrixT(r) corresponding to our class of N2×N2 braid matrices for all (r,N). Starting with our nested sequence of projectors, we obtained a very specific structure of T(r). Exploiting this structure fully, we obtained the eigenvalues and eigenstates. The zero-sum multiplets parametrized by sets of roots of unity arose from the circular permutations of the indices in the tensor products of the basis states under the action of T(r). The same structure led to systematic constructions of multiparameter Hamiltonians of spin chains related to our class of braid matrices for all N. Finally we constructed the inverse Cayley transformation for the general case (any N) giving potentials compatible with factorizability of S-matrices. The contents of such matrices (generalizing Section 5 of [7]) will be further studied elsewhere. In the treatment of each aspect we emphasized the role of a remarkable feature of our formalism—the presence of free parameters whose number increase as N2 with N. A single class of constraints (θ≥0, mab(+)>mab(-), a,b∈{1,…,N}) assures nonnegative Boltzmann weights in the statistical models. But our constructions also furnish directly some important properties of such models. Thus Tr(T(r))=2∑i=1n‍ermii(+)θ (N=2n) and Tr(T(r))=2∑i=1n-1‍ermii(+)θ+1 (N=2n-1) (for the normalization (2.16)). The eigenvalues can be ordered in magnitude by choosing the order of values of the parameters. Thus choosing m11(+)>m22(+)>⋯, the largest eigenvalue of T(r) is for θ>0, say erm11(+)θ, the next largest is erm22(+)θ and so on. We intend to study elsewhere the properties of our models more thoroughly.In previous papers [8, 14] we pointed out that for purely imaginary parameters (imab(±) with mab(±) real) our braid matrices are all unitary, providing an entire class with free parameters for all N. Here we have not repeated this discussion. But the fact that they generate parametrized entangled states is indeed of interest.Unitary matrices provide valid transformations of a basis (of corresponding dimension) of quantum states. One may ask the following question. If such a matrix, apart from being unitary, also satisfies the braid equation what consequence might be implied? Link between quantum and topological entanglements has been discussed by several authors [15, 16] cited in our previous papers [8, 14]. For the braid property to be relevant, a triple tensor product (V⊗V⊗V) of basis space is essential. We hope to explore elsewhere our multiparameter unitary matrices in such a context.In a following paper we will present a quite different aspect of our multiparameter braid matrices. It will be shown how for imaginary parameters (when the matrices are unitary) they can be implemented to generate quantum entanglements. Topological and quantum entanglements will thus be brought together in the two papers.Note Added
Professor J. H. H. Perk has kindly pointed out that a class of multiparameter generalization of the 6-vertex model is provided bysl(m∣n) ones [17, 18]. As the Perk-Schultz models they have been studied by many authors and have led to various important applications. (Relevant references can be easily found via arXiv.) In this class the sources of parameters are multi-component rapidities. The study of eigenvectors and eigenvalues was pioneered in [18] the most recent followup being [19], where other references can be found. Such studies may be compared to our systematic explicit constructions for all (r,N).



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