a diagrammatic approach to factorizing f-matrices in xxz

A diagrammatic approach to

factorizing F -matrices in XXZ

and XXX spin chains

Stephen Gerard McAteer

Submitted in total fulfilment of the requirements of the degree of

Doctor of Philosophy

October 2015

School of Mathematics and Statistics

The University of Melbourne

This work by Stephen Gerard McAteer is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 3.0 Unported License (2016).

http://creativecommons.org/licenses/by-nc-sa/3.0/

Abstract

The aim of this thesis is a better understanding of certain mathemat-

ical structures which arise in integrable spin chains. Specifically, we

are concerned with XXZ and XXX Heisenberg spin-12chains and their

generalizations.

The mathematical structures in question are the F -matrix (a sym-

metrising, change-of-basis operator) and the Bethe eigenvectors (the

eigenvectors of the transfer matrix of integrable spin chains). A dia-

grammatic tensor notation represents these operators in a way which is

intuitive and allows easy manipulation of the relations involving them.

The sun F -matrix is a representation of a Drinfel’d twist of the R-

matrix of the quantum algebra Uq(sun) and its associated Yangian

Y(sun). The F -matrices of these algebras have proven useful in the

calculation of scalar products and domain wall partition functions in

the spin-12XXZ model. In this thesis we present a factorized dia-

grammatic expression for the su2 F -matrix equivalent to the algebraic

expression of Maillet and Sanchez de Santos. Next we present a fully

factorized expression for the sun F -matrix which is of a similar form

to that of Maillet and Sanchez de Santos [18] for the su2 F -matrix and

equivalent to the unfactorized expression of Albert, Boos, Flume and

Ruhlig [2] for the sun F -matrix.

Using a diagrammatic description of the nested algebraic Bethe Ansatz,

we present an expression for the eigenvectors of the sun transfer matrix

as components of appropriately selected sun F -matrices. Finally, we

present expressions for the sun elementary matrices (and therefore the

local spin operators in the case of su2) in terms of components of the

sun monodromy matrix.

iii

Declaration

This is to certify that:

i) the thesis comprises only my original work towards the PhD ex-

cept where indicated in the Preface,

ii) due acknowledgement has been made in the text to all other ma-

terial used,

iii) the thesis is fewer than 100,000 words in length, exclusive of ta-

bles, maps, bibliographies and appendices.

Stephen Gerard McAteer

v

Preface

The description of diagrammatic tensor notation in Chapter 2 is a

review of the notation developed in [22].

The work on the diagrammatic representation of the su2 factoriz-

ing F -matrix in Chapter 3 and the expression for the sun elementary

matrices in Section 5.6 (Chapter 5) are adaptations and expansions of

work carried out in [19] (S. G. McAteer and M. Wheeler. Factorizing

F -matrices and the XXZ spin-12chain: A diagrammatic perspective.

Nuclear Physics B, 851(2):346379, 2011).

The work on the factorized expression of the sun F -matrix in Chap-

ter 4 and the expression for the sun Bethe eigenvectors in Section 5.5

(Chapter 5) are adaptations and expansions of work carried out in [20]

(S. G. McAteer and M. Wheeler, On factorizing F -matrices in Y(sln)

and Uq(sln) spin chains. Journal of Statistical Mechanics: Theory and

Experiment, 2012(04):P04016, 2012).

The exposition of the nested algebraic Bethe Ansatz in Chapter 5

is a review of [15] and [4] adapted in this thesis to the diagrammatic

tensor notation.

The results presented in this thesis from [19] and [20] are the original

work towards the PhD of the author, Stephen Gerard McAteer.

vii

Acknowledgements

I would like to thank my supervisor Omar Foda for his patience,

enthusiasm and generosity throughout my candidature. I am grateful

for the freedom I was afforded to pursue my ideas and the guidance

and wisdom I was offered when I strayed too far! I would also like to

acknowledge Michael Wheeler as my co-supervisor in all but title. Our

collaboration during my candidature was not just productive but also

thoroughly enjoyable. No two people in the world better deserve the

appellation scholar and gentleman than Omar and Michael. I would

also like to acknowledge the support offered to me by the members of

my supervisory panel: Richard Brak and Jan de Gier.

It was an honour to be part of mathematical physics community at

The University of Melbourne. The following people made up that com-

munity and made my candidature enjoyable and worthwhile: Wendy

Baratta, Nick Beaton, Michael Couch, Richard Hughes, Anita Pon-

saing, Arun Ram, Gus Schrader, Mark Sorrell, Tharatorn Supasiti,

Maria Tsarenko and Matthew Zuparic. The following people have all

had an impact on me during my candidature, whether it was a friendly

face or a helping hand, thank you: Justin Beck, Kostya Borokov, Ellie

Button, Jonathan Budd, Jeff Briffa, Sue Ann Chen, Maurice Chiodo,

Paul Chircop, Sandy Clarke, Sophie Dickson, Yolanda Harbinson, Paul

Keeler, Deb King, Paula King, Daniel Ladiges, Steve Lane, Heather

Lonsdale, Jason Nassios, Michael Neeson, Tracy Nguyen, Aleks Owcza-

rek, Michael Patterson, Sebastian Pucilowski and Peter Taylor.

Words cannot express the strength and encouragement I have re-

ceived from my family and friends. Cherry, Sutee, Nahmmi, Big Mar-

tin, Theresa, Little Martin, Persia, Owen, Faye, Ben, Tess, Donna and

Milesy; thank you, guys!

Finally, I would like to acknowledge the Australian Research Coun-

cil and The University of Melbourne for the funding I received during

my candidature.

ix

I dedicate this thesis to the Australian people: you provided me with

my education and furnished my needs throughout, I would not be

where I am today withour your outrageous generosity. I appreciate

the blood, sweat and tears of every individual and I pledge to repay

the faith shown in me to the full extent of my capacity.

Christopher Robin came down from the Forest to the bridge,

feeling all sunny and careless, and just as if twice nineteen

didn’t matter a bit, as it didn’t on such a happy afternoon,

and he thought that if he stood on the bottom rail of the

bridge, and leant over, and watched the river slipping slowly

away beneath him, then he would suddenly know everything

that there was to be known, and he would be able to tell

Pooh, who wasn’t quite sure about some of it.

— A. A. Milne, The House at Pooh Corner, 1928.

xi

Contents

Abstract iii

Declaration v

Preface vii

Acknowledgements ix

Contents xiii

List of Figures xvii

1 Introduction 1

1.1 The quantum electron . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Quantum spin chains . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The 6-vertex model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Generalization to sun . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Factorizing F -matrices . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 An expression for the elementary matrices . . . . . . . . . . . . . . 10

2 On notation for tensors 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 A vector identity . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16


2.4 Diagrammatic tensor notation . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xiii

Contents

2.4.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 The su2 factorizing F -matrix 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 The R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Identities involving the elementary matrices . . . . . . . . . 36

3.2.3 Unitarity and the Yang-Baxter Relation . . . . . . . . . . . 37

3.3 The bipartite matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Monodromy matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 The rank-(2, 2) F -matrix . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 The partial F -matrices . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 The F -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7.2 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.8 Proof of the factorizing property . . . . . . . . . . . . . . . . . . . . 56

3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 The sun factorizing F -matrix 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 The R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2 Identities involving the colour-s and related matrices . . . . 66

4.2.3 Unitarity and the Yang-Baxter relation . . . . . . . . . . . . 68

4.3 The tier-s R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.2 Tier-s unitarity and the Yang-Baxter relation . . . . . . . . 71

4.4 The bipartite matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.5 Construction of the F -matrix . . . . . . . . . . . . . . . . . . . . . 74

4.5.1 The tier-s partial F -matrix . . . . . . . . . . . . . . . . . . 74

4.5.2 The tier-s F -matrix . . . . . . . . . . . . . . . . . . . . . . . 75

4.5.3 The F -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6 Proof of the factorizing property . . . . . . . . . . . . . . . . . . . . 77

xiv

Contents

4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7.1 The su2 F -matrix . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7.2 The su3 F -matrix . . . . . . . . . . . . . . . . . . . . . . . . 89

4.7.3 Generalisation to sun . . . . . . . . . . . . . . . . . . . . . . 92

4.8 Further properties of the F -matrix . . . . . . . . . . . . . . . . . . 93

4.8.1 Lower triangularity and invertibility . . . . . . . . . . . . . . 93

4.8.2 Construction of the inverse F -matrix . . . . . . . . . . . . . 98

4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 On the nested algebraic Bethe Ansatz, sun Bethe eigenvectors and

elementary matrices 105

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.1 Multi-index notation . . . . . . . . . . . . . . . . . . . . . . 106

5.2.2 A notation for an identity with switching rapidity . . . . . . 107

5.2.3 A summation convention . . . . . . . . . . . . . . . . . . . . 108

5.2.4 The transfer matrix . . . . . . . . . . . . . . . . . . . . . . . 109

5.3 The algebraic Bethe Ansatz (the su2 case) . . . . . . . . . . . . . . 110

5.4 The nested algebraic Bethe Ansatz (the general sun case) . . . . . . 118

5.5 The sun Bethe eigenvectors as components of F -matrices . . . . . . 129

5.6 An expression for the elementary matrices in terms monodromy

matrix components . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6 Conclusion 137

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

References 139

Appendices 143

Appendix A Properties of spin-chain Hamiltonians 145

A.1 The Hamiltonian in terms of the transfer matrix . . . . . . . . . . . 145

A.2 The commutation of the Hamiltonian with the transfer matrix . . . 150

Appendix B Proof of the sun Yang-Baxter relation 153

xv

Contents

Appendix C Notational conventions 159

C.1 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

C.2 Notation for matrix operations . . . . . . . . . . . . . . . . . . . . . 160

Appendix D Notes on presentation and typesetting 165

Index 167

xvi

List of Figures

1.1 Interdependence of topics . . . . . . . . . . . . . . . . . . . . . . . . 2

xvii

Chapter 1

Introduction

The aim of this thesis is a better understanding of certain mathematical struc-

tures which arise in integrable one-dimensional spin chains. Specifically, we are

concerned with XXZ and XXX Heisenberg spin-12chains and their higher-rank

generalizations.

The mathematical structures in question are the F -matrix (a symmetrising,

change-of-basis operator) and the Bethe eigenvectors (the eigenvectors of the trans-

fer matrix of integrable spin chains). The aim of the rest of this chapter is to

explain the preceding sentences.

In Chapter 2 we introduce a diagrammatic notation for the representation of

tensors and compare it to standard notations. This notation was first introduced

in [22].

Under this scheme a tensor is represented as a shape with attached lines; the

shape represents the symbol of the tensor and the lines represent the upper and

lower indices of the tensor. Diagrammatic tensor notation allows an intuitive

treatment of tensors, and relations involving them such as tensor products and

trace. For example taking the trace of a tensor corresponds in diagrammatic

notation to joining two lines of a diagram. In this notation, many relations which

appear complicated in standard notation become obvious or even trivial.

Whilst the use of diagrammatic tensor notation is well established in the liter-

ature, we treat it quite formally. In this thesis the diagrams are not sketches of the

equations, rather they are the equations (and therefore do not appear in figures).

The use of this notation was crucial in the development of the results presented in

Chapters 3 to 5.

In the Section 1.3 we describe the 6-vertex model using this notation without

1

Chapter 1. Introduction

formally defining it†.

In Section 1.1 we provide a brief history of important milestones in the evo-

lution of quantum models of the electron. In Section 1.2, we introduce quantum

spin chains models as generalizations of the Ising model. The 6-vertex model is

introduced in Section 1.3. In this section we provide an overview of the construc-

tion of the R-matrix, the transfer matrix and spin chain Hamiltonians. We also

provide a brief description of the algebraic Bethe Ansatz. In Section 1.4, quan-

tum spin chains are generalized to spin chains with n emphcolours (we use the

term “colour” for the state of a variable in higher-rank spin chains since these spin

chains have more than two states). In Sections 1.5 and 1.6 we describe two im-

portant operators which are examined in this thesis, the factorizing F -matrix and

the elementary matrices. For clarity, remarks on the typesetting and presentation

of the material are made in Appendix D.

An overview of the interdependence of the topics in this thesis is given in Figure

1.1.

Figure 1.1: Interdependence of topics

†A detailed definition is provided in Chapter 2.

2

1.1. The quantum electron

1.1 The quantum electron

In 1927 W. Pauli wrote Zur Quantenmechanik des magnetischen Elektrons (On

the quantum mechanics of magnetic electrons) [21] in which a model describing

an electron with so-called “spin-12symmetry” (this symmetry is described below)

was introduced. The term spin here refers to the dipole moment generated by a

spinning charged particle. A particle with spin-12possesses a physical symmetry

under which it must be rotated through 4π before returning to its initial state.

This model agreed with then contemporary experimental results by allowing pairs

of electrons with different spins to occupy otherwise identical quantum states,

as well as with the results of the Stern-Gerlach experiment, according to which

electrons have spin.

The following standard definitions underlie the remainder of the discussion in

this section.

Definition 1.1. The special unitary group of order n, SU(n) is the group of

all (n×n) complex matrices under matrix multiplication, U such that det(U) = 1

(special) and UU∗ = U∗U = I where U∗ is the conjugate transpose of U (unitary).

The special orthogonal group of order n, SO(n) is the group of all (n× n)

real matrices under matrix multiplication, R such that det(R) = 1 (special) and

UUT = UTU = I where UT is the transpose of U (orthogonal).

The special unitary Lie algebra of order n over C, sun(C) = sun is the

Lie algebra of (n× n) complex matrices with zero trace (special) with Lie bracket

[X, Y ] = XY − Y X . ♦

Remark 1.2. sun is usually taken to be the real vector space of (n × n) anti-

hermitian matrices with trace zero, i.e. sun(R). However, its complexification

sun(C) is equivalent to sln(C). In the literature sln(C) is often the preferred

nomenclature for this Lie algebra, but we use sun(C) because it more clearly

relates to SU(n) from which it derives. ♦

1.2 Quantum spin chains

The following introduction is not meant to be rigorous, but rather to provide an

overview of the basic topics discussed in the thesis and their relationship. In the

rest of the thesis we provide a self-contained and through treatment with references

throughout.

3


Consider a one-dimensional lattice with each site labelled (sequentially) by an

integer, p. Each site possesses a state corresponding to a point in a two-dimensional

vector space {(a, b)|a, b ∈ C} with basis elements corresponding to the two discrete

states, (+)p = (1, 0)p, (−)p = (0, 1)p.

We now discuss the quantization of the spin chain. First we replace the discrete

state-space at a site p, {(+)p, (−)p} = {+1,−1} with a two-dimensional complex

vector space {(a, b)|a, b ∈ C} = C2. This new state-space can be thought of as

complex linear combinations of the discrete state-space elements now represented

by (+)p = (1, 0)p, (−)p = (0, 1)p.

Under this scheme, a Hamiltonian is an operator on the tensor product of the

state-spaces at each site, C2 ⊗ · · · ⊗ C2 = (C2)⊗N where N is the number of spin

sites.

The Hamiltonian defining this system is given by

H(Jx, Jy, Jz) = −1

2

N∑

p=1

(Jxσ

xpσ

xp+1 + Jyσ

ypσ

yp+1 + Jzσ

zpσ

zp+1

)(1.1)

where Jx, Jy and Jz are parameters characterizing the strength of the nearest-

neighbour interaction (for the x, y and z components of spin), σ∗N+1 = σ∗

1 (periodic

boundary conditions), the Pauli matrices† σx, σy, σz are given by

σxp =

[0 1

1 0

]

p

σyp =

[0 −i

i 0

]

p

σzp =

[1 0

0 −1

]

p

(1.2)

and the products of the matrices in the summand are tensor products. This model

is called the Heisenberg spin chain and involves interaction of adjacent spin sites

on a one-dimensional lattice.

We can write the general summand of H as a matrix over the tensor product

of the p and (p + 1) state-space basis vectors‡. In this notation, its action on the

tensor product of the states at sites p and (p + 1), ap ⊗ bp+1 where ap = (a1, a2)p

†Note that the Pauli matrices form a basis for the Hermitian matrices as a vector space.‡That is, the vector space with basis {(+)p⊗(+)(p+1), (+)p⊗(−)(p+1), (−)p⊗(+)(p+1), (−)p⊗

(−)(p+1)}. We call this the flattened representation of a tensor.

4

1.3. The 6-vertex model

and bp+1 = (b1, b2)p+1 becomes

−1

2

Jz 0 0 Jx − Jy

0 −Jz Jx + Jy 0

0 Jx + Jy −Jz 0

Jx − Jy 0 0 Jz

p(p+1)

a1b1

a1b2

a2b1

a2b2

p(p+1)

. (1.3)

To form the overall Hamiltonian, these summands may be tensor multiplied with

the remaining identity matrices (not written here and corresponding to the spaces

i /∈ {p, (p+ 1)}) then summed.

In this most general form, this is the Heisenberg XYZ spin chain. Specializing

to Jx = Jy = 1 and Jz = J we obtain the Heisenberg XXZ model, a model in

which the spin interaction is rotationally symmetric in the x − y plane. Further

specializing to Jx = Jy = Jz = 1 we obtain the Heisenberg XXX model, a model

in which the spin interaction is isotropic. The XXZ model is associated with the

algebra Uq(su2) and the XXX model with Y(su2) (the Yangian) [29, 16].

The first step towards “solving” a quantum mechanical system with Hamilto-

nian H is to solve the eigenvalue problem of the system’s Hamiltonian. That is,

to find the set of all |vp〉 such that H |vp〉 = λp |vp〉 for some complex number λp.

If the eigenvectors |vp〉 form a basis, then we can express any state, |ϕ(0)〉 in the

form

|ϕ(0)〉 =2N∑

p=1

bp |vp〉 (1.4)

where bp ∈ C.

In this thesis, we focus on the calculation of the eigenvalues and eivenvectors,

and in particular on better understanding the mathematical objects that arise in

the course of these calculations.

1.3 The 6-vertex model

In this section we sketch the construction of the transfer matrix, t(λa) of the 6-

vertex model and outline the algebraic Bethe Ansatz method. In Chapters 3–5, we

provide the details and generalize to the sun case. Note that the 6-vertex model

is an example of a broader family of integrable models which can be solved using

the algebraic Bethe Ansatz. Standard references for this material are [26, 15].

An important observation is that the transfer matrix of the 6-vertex model

commutes with the Hamiltonian of the corresponding Heisenberg spin chain. It

5


follows that by solving the transfer matrix eigenvector problem we solve the eigen-

vector problem for the Hamiltonian system.

The fundamental building-block of t(λa) is the R-matrix†. The R-matrix (act-

ing on the 1 and 2-spaces, each a copy of C2) is a 4× 4 matrix which depends on

two rapidities λ1, λ2. The entries of the R-matrix are

R12(λ1, λ2) =

a(λ1 − λ2) 0 0 0

0 b(λ1 − λ2) c(λ1 − λ2) 0

0 c(λ1 − λ2) b(λ1 − λ2) 0

0 0 0 a(λ1 − λ2)

12

(1.5)

which is written as a matrix over the tensor product of the 1 and 2-space basis

vectors as in (1.3).

The entries, called weight functions, are complex valued functions given by

a(λ) = 1, b(λ) =[λ]

[λ+ η], c(λ) =

[η]

[λ+ η](1.6)

where η ∈ C is a model parameter. For the XXX model, [λ] = λ and for the XXZ

model, [λ] = sinhλ. For brevity we write f(λ1 − λ2) = f12, for f = a, b, c. Depen-

dencies on the rapidities are omitted where there is no possibility of ambiguity.

We represent the R-matrix diagrammatically as a pair of intersecting lines—a

vertex—where the correspondence between the indices and lines is given by

(R12)j1j2i1i2

= . (1.7)

When an index takes the value 1 or 2 we represent this diagrammatically as an

upward or downward pointing arrow, respectively. The six nonzero vertex config-

urations are

= = a12 = = b12 = = c12. (1.8)

It is these 6 vertices that give the model its name. Note that each vertex has spin

conservation, that is, the set of spins entering the base of the diagram is equivalent

to the set of spins exiting at the top.

The next object we define is the monodromy matrix, Ra|1···N which is a con-

†The su2 R-matrix and its diagrammatic treatment is provided in detail in Chapter 3.

6

1.3. The 6-vertex model

traction between N R-matrices given by

Ra|1···N(λa|λ1, . . . , λN) = RaN (λa, λN)Ra(N−1)(λa, λ(N−1)) · · ·Ra2(λa, λ2)Ra1(λa, λ1).

(1.9)

Diagrammatically this is given by

(Ra|1···N

)jaj1···jNiai1···iN

= (1.10)

where contraction between the R-matrices is represented here by joining the lines

of the appropriate vertices. By specifying the ia, ja indices of the monodromy

matrix we obtain the following operators

= A1···N(λa|λ1 · · ·λN) (1.11)

= B1···N(λa|λ1 · · ·λN) (1.12)

= C1···N(λa|λ1 · · ·λN) (1.13)

= D1···N(λa|λ1 · · ·λN). (1.14)

Finally, the transfer matrix is defined as

t(λa|λ1 · · ·λN) = A1···N(λa|λ1 · · ·λN) +D1···N(λa|λ1 · · ·λN) (1.15)

which is the trace of the monodromy matrix over the a-space. The Hamiltonians

of the XXZ and XXX models may be written in terms of the transfer matrix as

follows

H(J) = −1

2

N∑

i=1

(σxj σ

xj+1 + σy

jσyj+1 + Jσz

jσzj+1

)

= −√J + 1

d

dλa

log t(λa|λ1, . . . , λN)

∣∣∣∣λ1···λN=λa

(1.16)

where J = 1 corresponds to the XXX model and t(λa) is the scaled transfer matrix,

that is, the transfer matrix made of R-matrices which have been appropriately

scaled. Because the Hamiltonian has this form, it can be shown to commute with

7


the scaled transfer matrix. This is important because it means that they share

a set of eigenvectors (provided that they are diagonalizable which is the generic

case). It follows that solving the transfer matrix eigenvector problem is equivalent

to solving the Hamiltonian system. Proof of Equation (1.16) and the commutation

of the Hamiltonian with the transfer matrix are provided in Appendix A.

Remark 1.3. Although the right hand side of Equation (1.16) appears to depend

on λa, this dependence elegantly vanishes in the specified limit. ♦

We now provide an Ansatz for the eigenvectors of t(λa) and a very brief overview

of the method by which this is established—this method is known as the algebraic

Bethe Ansatz†. First we define the pseudovacuum as

|0〉 =N⊗

p=1

(1, 0)p ! . (1.17)

Now, suppose we act on the pseudovacuum state with M B-operators (where

M is allowed to trivially be 0). Furthermore, suppose the rapidities of the quan-

tum spin states, λ1, . . . , λN and the rapidities of the B-operators λN+1, . . . , λN+M‡

satisfy a set of relations called the Bethe equations, then the eigenvectors of t(λa)

are of the form

B1···N (λN+1)B1···N(λN+2) · · ·B1···N(λN+M) |0〉

! (1.18)

where in the M = 0 case, the claim is that the pseudovacuum is an eigenvector.

The fact that these are indeed eigenvectors of the transfer matrix is estab-

lished by commuting its components A1···N and D1···N through the B-operators

to the pseudovacuum on which they act trivially. Each time an A or D-operator

commutes with a B-operator, two terms are produced: a good term and a bad

†The details of the algebraic Bethe Ansatz are provided in Chapter 5.‡In this thesis the same symbol is used for the spin-state (quantum) rapidities and the B-

operator (auxiliary) rapidities; only the indexes are distinguishing. This convention proves usefullater on. In many texts a different symbol is used for each set.

8

1.4. Generalization to sun

term†. When a certain set of relations between the rapidities are satisfied, the bad

terms arising from the commutation of the A-operator cancel with those of the

D-operator. This set of relations are the Bethe equations.

The commutations used in this process are generated by considering compo-

nents of the relation

RabRb|1···N (λb)Ra|1···N(λa) = Ra|1···N (λa)Rb|1···N(λb)Rab

! = . (1.19)

Equation (1.19) is established by repeated application of the Yang-Baxter relation

R12R13R23 = R23R13R12 ! = (1.20)

and the spin-conservation of the R-matrices‡. This relation is discussed in detail

in the following chapters. For example, the ia = ib = 1, ja = jb = 2 component

implies that the B-operators mutually commute without producing any bad terms.

1.4 Generalization to sun

The Heisenberg model is concerned with chains of electrons which possess an

internal property described by a complex linear combination of two states, referred

to as spins. This can be generalized to systems of particles which possess an

internal property described by a complex linear combination of n states—that is

they transform as elements of a representation of SU(n). In these more general

systems the states are referred to as colours.

The construction of the sun R-matrices (where n is arbitrary) is more com-

plicated, but most of the properties are retained. These details are provided in

Chapter 4.

†In the good term, the B-operator still depends on its original rapidity, and in the bad termit swaps its dependence with the A or D-operator.

‡The Yang-Baxter equation is discussed further in Chapter 3.

9


This thesis does not deal with the issue of generalizing Heisenberg spin chains

themselves, rather it is concerned with the generalization of the transfer matrix

problem (and related problems) from that possessing su2 symmetry to the general

case possessing sun. Following [4], the generalized transfer matrix problem is solved

in Chapter 5 using the nested algebraic Bethe Ansatz which is a generalization of

the algebraic Bethe Ansatz.

1.5 Factorizing F -matrices

Of particular interest in the field of quantum mechanical lattice models is the

study of form factors and correlation functions which are in some cases related

to experimentally measurable quantities. A form factor may be thought of as a

component of an operator, O in some appropriately selected basis. For example

let 〈u| , |v〉 be basis vectors which are acted on by O, then the function, 〈u|O |v〉is a form factor [25]. If the form factors are known for a given operator, they can

be used to calculate correlation functions, that is the expectation of a given state

in a statistical ensemble.

The action of the factorizing F -matrices is equivalent to a change of basis

on the state-space under which certain operators such as the transfer matrix are

simplified [7, 8, 6].

While this thesis focusses on the F -matrices associated with sun, other models

are also of interest. The F -martix for the XYZ spin chain was derived in [1] and

generalized further in [28]. The F -matrix associated with Uq(gl(2|1)) (relating to

the quantum supersymmetric t-J model) was given in [30] and then generalized to

Uq(gl(m|n)) in [31].

In Chapters 3 and 4 diagrammatic expressions for the F -matrix are presented.

In the sun case, a new factorized expression for the sun F matrix is presented.

An expression for the sun Bethe eigenvectors as components of the F -matrix is

presented in Chapter 5. F -matrices have proved useful in the calculation of form

factors and correlation functions and it is hoped that the new expressions may

provide a degree of insight into the higher-dimensional models.

1.6 An expression for the elementary matrices

An sun elementary matrix is an (n × n) matrix with a 0 in all but a single entry

which takes the value 1. These matrices are important since they form a basis for

10

1.6. An expression for the elementary matrices

the vector space of (n× n) matrices. As such, any matrix may be expressed as a

linear combination of the elementary matrices. Specifically in su2, this representa-

tion allows the local spin operators (S+1 = E

(12)1 , S−

1 = E(21)1 , Sz

1 = 12(E

(11)1 −E

(22)1 ))

to be written in terms of monodromy matrix elements and has proved useful in

the calculation of correlation functions [12, 14].

In Chapter 5 we present an expression for the elementary matrices in terms

of components of the sun monodromy matrix. In the case of su2, this result was

provided in [13] and relied on use of the F -matrix. In [9] the su2 result was

provided without use of the F -matrix. We provide an easy proof of the expression

for the elementary matrices in the general case of sun using diagrammatic tensor

notation.

11

Chapter 2

On notation for tensors

2.1 Introduction

In this chapter we review some standard notation for tensors and introduce a

diagrammatic tensor notation originally developed in [22].

The purpose of diagrammatic tensor notation is to allow an intuitive handling

of tensors and relations between them such as tensor products, traces and con-

tractions. In this notation, many relations which appear complicated in standard

notation become obvious or even trivial. The use of diagrammatic tensor notation

was vital to the development of the results presented in the following chapters.

We are interested in tensors as objects which act as linear transformations of

some vector space via left-action and on the dual of this vector space via right-

action. The vector spaces (and dual spaces) that we are interested in are of the

form

Cn ⊗ · · · ⊗ C

n = (Cn)⊗N (2.1)

where Cn ⊗ Cn is the vector space spanned by {b1 ⊗ b2|b1, b2 ∈ B} where B is the

standard basis for Cn. Tensors may possess more structure than matrices since

they need not only possess an action on (Cn)⊗N , they may also possess various

actions on Cn,Cn⊗C

n, . . . , (Cn)⊗N−1. In this thesis we use tensors to describe the

operators on states and dual states that arise in the study of statistical mechanical

lattice models.

In Section 2.2 we briefly describe standard matrix notation. In Section 2.3

we describe standard index notation employing the Einstein summation conven-

tion. In Section 2.4 we describe diagrammatic tensor notation. In each section we

provide a proof of the following standard identity to illustrate the strengths and

weaknesses of each notation.

13

Chapter 2. On notation for tensors

Identity 2.1. Let 〈a| be a 3-component dual vector and let |b〉 and |c〉 be 3-

component vectors, then we have

〈a| ×(|b〉 × |c〉

)=(〈a|c〉

)|b〉 −

(〈a|b〉

)|c〉 (2.2)

♦

In Section 2.5 we discuss diagrammatic tensor notation as it relates to dia-

grammatic representations of statistical mechanical lattice configurations in the

literature. Throughout this chapter, n is the dimension of the underlying vector

space.

2.2 Matrix notation

We regard matrices as linear transformations. They possess left action on the

vector space V ≃ Cn and right action on the dual vector space V ∗ ≃ C

n.

For comparison to index and diagrammatic tensor notation we consider stan-

dard matrix notation and highlight some shortcomings†. We then provide a proof

of Identity 2.1 using this notation.

Matrix notation is powerful, it is sufficient for a full description of linear alge-

bra. Furthermore, any tensorial operation may be treated in this notation with

the appropriate choice of basis. This corresponds to flattening a tensor which

is described later. However it has two major drawbacks: it is difficult to use in

practice in all but the simplest cases and more importantly it obfuscates some of

the extra structure that a tensor may possess. This is illustrated in the following

sections.

2.2.1 A vector identity

We now use matrix notation to establish Identity 2.1. In the following sections we

revisit the proof of this identity under index then diagrammatic notation.

Proof (of Identity 2.1 in matrix notation). On the left hand side of Identity 2.1,

we have

〈a| × (|b〉 × |c〉) =[a1 a2 a3

]×[b2c3 − b3c2 b3c1 − b1c3 b1c2 − b2c1

]

†Although the matrix notation used in this thesis is standard, in order to be completelyunambiguous, we describe it in Appendix C in detail.

14

2.3. Index notation

=

a2(b1c2 − b2c1)− a3(b3c1 − b1c3)

a3(b2c3 − b3c2)− a1(b1c2 − b2c1)

a1(b3c1 − b1c3)− a2(b2c3 − b3c2)

(2.3)

and on the right hand side we have

〈a|c〉 |b〉 − 〈a|b〉 |c〉 = (a1c1 + a2c2 + a3c3) |b〉 − (a1b1 + a2b2 + a3b3) |c〉

=

(a1c1 + a2c2 + a3c3)b1 − (a1b1 + a2b2 + a3b3)c1

(a1c1 + a2c2 + a3c3)b2 − (a1b1 + a2b2 + a3b3)c2

(a1c1 + a2c2 + a3c3)b3 − (a1b1 + a2b2 + a3b3)c3

(2.4)

as required. �

The point of this exercise is that little intuition can be drawn from this proof.

In the following section, the cross product has a more natural interpretation in

terms of Levi-Civita tensors.

2.3 Index notation

In this section we describe the index notation for tensors under the Einstein sum-

mation convention. In this scheme, we do not attempt to write a vector, matrix or

higher rank tensor as an array of elements. Rather, we write general components

of the tensors.

2.3.1 Notation

Definition 2.2 (tensor, state, dual state, scalar). Aj1···jNi1···iM

be elements of C

where 1 ≤ i ≤ n for all indices i. Then a rank-(M,N) tensor, A has components

given by

(A)j1···jNi1···iM= Aj1···jN

i1···iM. (2.5)

If N = 0 then A is a rank-M state and may be written as a ket, A = |a〉. If

M = 0 then A is a rank-M dual state and may be written as a bra, A = 〈a|.If M = N = 0 then A is a scalar. ♦

Remark 2.3. These definitions generalize those given in the previous section. In

terms of matrix notation, the rank-1 states and dual states correspond to the

vectors and dual vectors respectively and the rank-(1, 1) tensors correspond to

matrices. ♦

15


2.3.2 Operators

We now provide definitions for the tensor product, the tensor trace and contrac-

tion. These operations are related back to the matrix notation operations where

applicable.

Throughout this section we use the Einstein summation convention, where

possible by interpreting repeated indices as implicitly summed over. For example,

the trace of a matrix A with components Aji is written as

tr(A) = Akk =

n∑

k=1

Akk (2.6)

where the summation is implied in the right hand side by the repeated index k.

In the definitions below, we write

Aji

∣∣i=j=k

= Akk (2.7)

which is useful notation in cases where many indices are identified.

Definition 2.4 (tensor product, scalar product). Let A be a rank-(M,N)

tensor with components

Aj1···jNi1···iM

and let B be a rank-(M ′, N ′) tensor with components Bj(N+1)···j(N+N′)

i(M+1)···i(M+M′). Then

the tensor product of A and B is the rank-(M +M ′, N +N ′) tensor with com-

ponents given by

(A⊗B)j1···j(N+N′)

i1···i(M+M′)= Aj1···jN

i1···iMB

j(N+1)···j(N+N′)

i(M+1)···i(M+M′). (2.8)

♦

Definition 2.5 (tensor trace). LetA be a rank-(M,N) tensor with components

Aj1···jNi1···iM

, let P = {p1, . . . , pr}, Q = {q1, . . . , qr} be ordered sets of integers such that

1 ≤ p1, . . . , pr ≤ M and 1 ≤ q1, . . . , qr ≤ N . Then the (P,Q)-trace of A is the

rank-(M − r,N − r) tensor trPQA, with components given by

(trPQA)j1···jq1 ···jqr ···jN

i1···ip1 ···ipr ···iM= Aj1···jN

i1···iM

∣∣ip1=jq1=k1,...,ipr=jqr=kr

(2.9)

where the circumflex indicates an omitted index and the expression on the right

hand side is written using the Einstein summation convention. ♦

16

2.3. Index notation

Example 2.6 (tensor trace). If A is a rank-(3, 4) tensor with components Aj1j2j3j4i1i2i3

,

P = {1, 3} and Q = {4, 3}, then

(trPQA)i1i2i2

= Aj1j2j3j4i1i2i3

∣∣i1=j4=k1,i3=j3=k2

= Aj1j2k2k1k1i2k2

(2.10)

a rank-(1, 2) tensor, where summation is implied by the repeated indices. ♦

Definition 2.7 (contraction, left and right action, inner product). Let A

be a rank-(M,N) tensor, let B be a rank-(M ′, N ′) tensor and let P = {p1, . . . , pr},Q = {q1, . . . , qr} be ordered sets of integers such that 1 ≤ p1, . . . , pr ≤ M ′ and

1 ≤ q1, . . . , qr ≤ N . Then the (P,Q)-contraction of A and B is the rank-

(M +M ′ − r,N +N ′ − r)-tensor with components given by

(A ·PQ B)j1···j(N+N′

−r)

i1···i(M+M′−r)

=(tr(P+M)Q (A⊗B)

)j1···j(N+N′−r)

i1···i(M+M′−r)

(2.11)

where P +M = (p1 +M, . . . , pr +M).

In the special case, N = M ′, we write

AB = A ·PQ B (2.12)

where P = Q = {1, . . . , N}.If M = N = M ′ and N ′ = 0, then A is a rank-(N,N) tensor and B = |b〉 is a

rank-N state. In this case A |b〉 is the result of the left action of A on |b〉.If N = M ′ = N ′ and M = 0, then A = 〈a| is a rank-N dual state and B is a

rank-(N,N) tensor. In this case 〈a|B is the result of the right action of B on

〈a|.If N = M ′ and M = N ′ = 0, then A = 〈a| is a rank-N dual state and B = |b〉

is a rank-N state. In this case 〈a|b〉 is the inner product of 〈a| and |b〉. ♦

Remark 2.8. This definition of contraction is not the most general possibility as

we can only contract the upper indices of one tensor with the lower indices of

another, this definition is however general enough for the purposes of this thesis.♦

Example 2.9 (contraction). If A is a rank-(1, 2) tensor with components Aj1j2i1

, B

is a rank-(3, 2) tensor with components Bj3j4i2i3i4

, P = {2, 3} and Q = {2, 1}, then

(A ·PQ B)j3j4i1i2

=(tr(P+1)Q (A⊗B)

)j3j4i1i2

= Aj1j2i1

Bj3j4i2i3i4

∣∣i3=j2=k1,i4=j1=k2

= Ak2k1i1

Bj3j4i2k1k2

(2.13)

17


a rank-(2, 2) tensor. ♦

Example 2.10 (contraction – compact notation). If A is a rank-(1, 2) tensor with

components Aj1j2i1

and B is a rank-(2, 2) tensor with components Bj3j4i2i3

, then N =

M ′ = 2 and

(AB)j3j4i1= (A ·PQ B)j3j4

i1= Aj1j2

i1Bj3j4

i2i3

∣∣i2=j1=k1,i3=j2=k2

= Ak1k2i1

Bj3j4k1k2

(2.14)

where P = Q = {1, 2}, AB is a rank-(1, 2) tensor. ♦

Example 2.11 (inner product). If 〈a| is a rank-3 dual state with components aj1j2j3

and |b〉 is a rank-3 state with components bi1i2i3, then N = M ′ = 3, M = N ′ = 0

and

〈a|b〉 = 〈a| ·PQ |b〉 = aj1j2j3bi1i2i3∣∣i1=j1=k1,i2=j2=k2,i3=j3=k3

= ak1k2k3bk1k2k3 (2.15)

where P = Q = {1, 2, 3}, 〈a|b〉 is a scalar. ♦

Remark 2.12. Operators from the previous section can be extracted as special

cases of this definition. By setting all free ranks to 1, the states, dual states and

tensors become vectors, dual vectors and matrices respectively as required. ♦

Definition 2.13 (identity tensor). The identity tensor is the rank-(N,N)

tensor with components given by

(I)j1···jNi1···iN=

N∏

p=1

δjpip. (2.16)

♦

Definition 2.14 (state and dual state space). Let Vi = V ∗i = Cn for all i, let

B be the standard basis for Cn. Then the rank-N state space is the vector space

VN ≃ V1 ⊗ · · · ⊗ VN (2.17)

with basis {b1 ⊗ · · · ⊗ bN |bi ∈ B}. The rank-N dual state space is the vector

space

V∗N ≃ V ∗

1 ⊗ · · · ⊗ V ∗N (2.18)

with basis {b1 ⊗ · · · ⊗ bN |bi ∈ B}. ♦

18

2.3. Index notation

Remark 2.15. The rank-N states and dual states may be recognized as elements

of VN and V∗N respectively which are vector spaces of dimension nN . The rank-

(M,N) tensors may be regarded as linear transformations from VN to VM via their

left action on rank-N states or as linear transformations from V∗M to V∗

N via their

right action on the rank-M dual states. As such, they may be viewed as nN × nM

matrices. In the case M = N , the rank-(N,N) tensors may be interpreted as

elements of End(VN) or End(V∗N ).

The interpretation of the states and dual states as vectors and dual vectors

respectively and of the tensors as matrices is referred to in this thesis as the

flattened representation. Occasionally we wish to draw a result from linear algebra

and the flattened representation allows us to do this.

In the literature, objects with a tensor structure are referred to asmatrices. For

example in the following chapters we discuss the R-matrix, which can be viewed

as either a rank-(2, 2) tensor or as an (n2 × n2) matrix. ♦

Remark 2.16. Together with the identity tensor, contraction endows the rank-

(N,N) tensors with a monoid structure. Element-wise addition which trivially

has a group structure distributes across contraction and so the rank-(N,N) tensors

possess a ring structure. This allows us to view the rank-N state and dual state

spaces as modules over the ring of rank-(N,N) tensors. As in the previous section,

we identify these structures for convenience so that we may appeal to them later

as required. ♦


We now return to the Identity 2.1, this time providing a proof in terms of index

notation. We provide a definition of the Levi-Civita tensor and the identity matrix

and establish an identity involving them. The cross product is given in terms of

Levi-Civita tensors as in [24] defined below.

Definition 2.17 (Levi-Civita tensor). We call the rank-(n, 0) tensor (state)

with components given by

εi1···in = sgn

(n−1∏

p=1

n∏

q=p+1

(iq − ip)

)(2.19)

the rank-n Levi-Civita tensor, where sgn is the sign function and n is the

dimension of the underlying vector spaces Vi. We call the rank-(0, n) tensor (dual

19


state) with components given by ǫi1...in = εi1···in the rank-n dual Levi-Civita

tensor. ♦

Remark 2.18. In the case of n = 3, the components of ε and ǫ are

εi1i2i3 = ǫi1i2i3 = sgn ((i1 − i2)(i2 − i3)(i3 − i1)) . (2.20)

♦

In index notation, the cross product can be written in terms of the rank-3

Levi-Civita tensors. Let n = 3 and let |a〉 and |b〉 be vectors (rank-1 states), then

their cross product is given by

(|a〉 × |b〉)j = ǫk1k2jak1bk2 (2.21)

a dual vector (rank-1 dual state). Alternatively, let 〈a| and 〈b| be dual vectors,

then their cross product is given by

(〈a| × 〈b|)i = εik1k2ak1bk2 (2.22)

a vector. This expression may be confirmed simply by checking that the three

components of each agree with standard formulations of the cross product.

Remark 2.19. The identity matrix acts on the tensors in the expected way. For

example if A is a rank-(2, 2) tensor with components Aj1j2i1i2

, P = {2} and Q = {1},then the (P,Q)-contraction between A and I is given by

(A ·PQ I)j1j2i1i2

= Aj1ki1i2

δ j2k = Aj1j2

i1i2(2.23)

as expected. ♦

The following identity is used to establish Identity 2.1.

Identity 2.20. The (3, 3)-contraction between the rank-3 Levi-Civita tensor and

the rank-3 dual Levi-Civita tensor may be expressed in the following way

εi1i2kǫj1j2k = δ j1

i1δ j2i2− δ j2

i1δ j1i2

(2.24)

a rank-(2, 2) tensor. ♦

20

2.3. Index notation

Proof. From the definition of the Levi-Civita tensors we have

εi1i2kǫj1j2k = sgn

((i1 − i2)(j1 − j2)

3∑

k=1

(k − i1)(k − i2)(k − j1)(k − j2)

)(2.25)

so all the non-zero components of the tensor have i1 6= i2, j1 6= j2 and there

must also be some k not equal to i1, i2, j1 or j2. This implies that for the non-

zero components, i1, i2, j1 and j2 take exactly two values with the only non-zero

summand corresponding to k taking the one remaining value.

There are two possible values for the non-zero terms. The first possibility is

the case in which we have i1 = j1 = p1 in {1, 2, 3}, i2 = j2 = p2 in {1, 2, 3}\{p1}and k = p3 in {1, 2, 3}\{p1, p2}, in this case the one surviving term is

εi1i2kǫj1j2k = sgn ((p1 − p2)(p1 − p2)(p3 − p1)(p3 − p2)(p3 − p1)(p3 − p2))

= sgn((p1 − p2)

2(p3 − p1)2(p3 − p2)

2)= 1 (2.26)

as required. The remaining possibility is the case where i1 = j2 = p1 in {1, 2, 3},i2 = j1 = p2 in {1, 2, 3}\{p1} and k = p3 in {1, 2, 3}\{p1, p2}. In this case we have

εi1i2kǫj1j2k = sgn ((p1 − p2)(p2 − p1)(p3 − p1)(p3 − p2)(p3 − p1)(p3 − p2))

= sgn(−(p1 − p2)

2(p3 − p1)2(p3 − p2)

2)= −1 (2.27)

as required. �

Proof (of Identity 2.1 in index notation). The components of the left hand side of

Identity 2.1 are

εik1k2ak1(ǫk3k4k2bk3ck4) = (δ k3

i δ k4k1

− δ k4i δ k3

k1)ak1bk3ck4

= δ k3i δ k4

k1ak1bk3ck4 − δ k4

i δ k3k1ak1bk3ck4 = (ak1ck1)bi − (ak1bk1)ci (2.28)

which are the components of the right hand side as required. We are free to

commute the terms here because at the level of components, we are only dealing

with multiplication in C which is indeed commutative. �

It is not easy to visually keep track of the indices in Equation (2.28), in such

cases diagrammatic tensor notation simplifies things.

The proof using index notation depends crucially on a 3-tensor, so it is not

easily accessible in matrix notation. The rank-n Levi-Civita tensors obey a more

21


general relation

εi1···inǫj1···jn =

∑

σ∈Sn

sgn(σ)n⊗

k=1

δjσ(k)

ik(2.29)

where Sn is the symmetric group with n elements and sgn(σ) is the sign of the

permutation σ. This relation suggests many generalizations of the original identity.

Equation (2.29) may be established using the following properties of the Levi-

Civita tensors

1. ε1···n = 1,

2. if there exists p 6= q such that ip = iq, then εi1···in = 0, and

3. εi1···ipi(p+1)...in = −εi1···i(p+1)ip...in for all 1 ≤ p < n.

It is immediate that Equation (2.19) satisfies these properties.

2.4 Diagrammatic tensor notation

In this section we describe the diagrammatic tensor notation† proposed for use

with tensors in [22]‡ . Under this scheme a rank-(M,N) tensor is represented as

a shape with M upward lines and N downward lines coming out of it; the shape

represents the symbol for the tensor and the lines represent the upper and lower

indices of the tensor. Diagrammatic tensor notation allows an intuitive treatment

of tensors, this was crucial in the development of the results presented in Chapters

3 to 5.

2.4.1 Notation

Example 2.21 (scalar). The simplest tensor is the rank-(0, 0) tensor, the scalar. In

diagrammatic tensor notation, this is presented as a shape without trailing lines

as in

α = (2.30)

where the shape corresponds to the symbol. Scalars are not usually boxed in this

thesis. ♦

†The earliest known use of a diagrammatic notation dates back to 1879 in Begriffsschrift

(Concept Notation) by F. L. G. Frege. Begriffsschrift is a foundational text on modern logic butthe diagrammatic notation used therein was not well received at the time [27]. Ominously, Fregedied penniless, ruined by typesetting costs [5].

‡In [22] the fact that the dimension of a tensor n, plays no role at the level of the correspondingdiagram to make the generalization to negative dimensional tensors. This aspect of the notationis not of concern in this thesis.

22

2.4. Diagrammatic tensor notation

Example 2.22 (vector and dual vector). The next simplest object is the rank-(1, 0)

and rank-(0, 1) tensors (vectors and the dual vectors). Vectors are represented as

shapes with one downward line and dual vectors are represented as shapes with

one upward line. We have

ai = (2.31)

and

bj = (2.32)

where the shapes correspond to the symbols a and b and the lines correspond their

respective indices. ♦

Remark 2.23. Once the indices have been associated with the lines, we no longer

need to write them on the diagram, the position of the line relative to the diagram

stands for the index. Equations which only involve diagrams are index-free with

indices only appearing when we wish to actually define the connection between

the diagrammatic and index notation. ♦

Example 2.24 (matrix). Next there is the rank-(1, 1) tensor, the matrix. Matrices

are represented as shapes with one upward and downward line, as in

Aji = . (2.33)

♦

Example 2.25 (general tensor). In general, a rank(M,N) tensor is represented as

a shape with M downward lines and N upward lines, as in

Aj1j2···j(N−1)jNi1i2···i(M−1)iM

= . (2.34)

♦

We formalize the notation characterized by these diagrams in the following

definition.

23


Definition 2.26 (tensor). LetA be a rank-(M,N) tensor with components Aj1···jNi1···iM

.

We represent the components of A diagrammatically as a shape with M down-

ward lines and N upward lines. The shape represents the symbol A and the

(M +N) lines represent the indices. ♦

2.4.2 Operations

In the following examples, let the components of the rank-(M,N) tensor A be

represented diagrammatically as

Aj1j2···j(N−1)jNi1i2···i(M−1)iM

= (2.35)

and the components of the rank-(M ′, N ′) tensor B be represented as

Bj1j2···j(N′

−1)jN′

i1i2···i(M′−1)iM′

= . (2.36)

Example 2.27 (tensor product). As in index notation, tensor products are repre-

sented in diagrammatic tensor notation by juxtaposition. Let A be a rank-(2, 2)

tensor and let B be a rank-(1, 2) tensor. Then

(A⊗B)j1j2j3j4i1i2i3= (2.37)

which is a rank-(3, 4) tensor. ♦

Example 2.28 (tensor trace). The tensor trace corresponds to joining two of the

lines of a diagram. Let A be a rank-(3, 4) tensor, let P = {1, 3} and let Q = {4, 3}.Then

(trPQ(A))j1j2i2

= (2.38)

which is a rank-(1, 2) tensor. This equation corresponds to Equation (2.10). ♦

Remark 2.29. In the above diagram, there is an upper line joined to a lower line,

this implies summation over the corresponding index. (Recall that in index nota-

tion, summation is implied when one symbol is used for both an upper and lower

24


index.) Note that the numbers inside the shape do not play a role in the notation,

they have just been added to clarify the connection between the diagram and the

ordered sets P and Q. Note also that the intersections between lines in this chapter

play no role here whereas in later chapters they take on a particular meaning. ♦

Example 2.30 (contraction). Contraction is achieved by juxtaposition and then

joining pairs of lines. Let A be a rank-(1, 2) tensor, let B be a rank-(2, 2) tensor

and let P = {2, 3} and Q = {2, 1}. Then

(A ·PQ B)j3j4i1i2

= (2.39)

which is a rank-(2, 2) tensor. This equation corresponds to Equation (2.13). ♦

Example 2.31 (compact notation for contraction). Let A be a rank-(1, 2) tensor

and let B be a rank-(2, 2) tensor. Then

(AB)j3j4i1= (2.40)

a rank-(1, 2) tensor. This equation corresponds to Equation (2.14). ♦

Example 2.32 (inner product). Let the components of the rank-3 state |b〉 be givendiagrammatically as

(|b〉)i1i2i3 = (2.41)

and let the components of the rank-3 dual state 〈a| be given diagrammatically as

(〈a|)j1j2j3 = (2.42)

then the inner product 〈a|b〉 is given by

〈a|b〉 = (2.43)

a scalar. This equation corresponds to Equation (2.15). ♦

25


Definition 2.33 (tensor product, contraction, trace, inner product). The

juxtaposition of the diagrams for a pair of tensors is the diagram for their tensor

product.

If in addition to this, some of the upper lines of one diagram are joined to lower

lines of the other we call this a contraction.

If a diagram has some of its own upper lines connected to its lower lines, we

call this a tensor trace.

If all the lines of a state and a dual state are joined in order, we call this their

inner product. ♦


Definition 2.34. We represent the components of the identity matrix diagram-

matically as a detached line

δ ji = (2.44)

where the δ ji is the Kronecker symbol. ♦

Remark 2.35. Diagrammatically the action of the identity matrix on any tensor is

trivial, we simply extend one of the lines on the diagram. ♦

Definition 2.36. Following [23] we represent the components of the Levi-Civita

tensor diagrammatically as a bar with n downward lines, as follows

εi1i2···i(n−1)in = . (2.45)

The components of the dual Levi-Civita tensor are represented as

ǫj1j2···j(n−1)jn = . (2.46)

♦

We can rewrite Equation (2.29) as

=∑

σ∈Sn

sgn(σ)Iσ (2.47)

where Sn is the symmetric group with n elements, sgn(σ) is the sign of the permuta-

tion σ and the diagram for the components of Iσ is a bipartite graph corresponding

26


to σ. For example if σ {1, 2, 3, 4, 5} = {2, 4, 3, 5, 1}, then

(Iσ)j1···j5i1···i5= (2.48)

such that i1 is connected to jσ(1) = j2 by an identity matrix, i2 to jσ(2) = j4 and so

forth. We reiterate that the intersections here have no special meaning, however

in later chapters they are interpreted differently.

In the case of n = 3 Equation (2.47) specializes to

= + + − − − (2.49)

and it follows that

= + + − − − = − (2.50)

since

= δ kk = n = 3 (2.51)

and

= δ ki δ

jk = δ j

i = . (2.52)

Equation (2.50) corresponds to Identity 2.20. Whenever we come across a copy of

the diagram on the left hand side of Equation (2.50) within a diagram, it is legal

to replace it with the right hand side of Equation (2.50) (and vice versa).

We now wish to express the cross product diagrammatically. Let n = 3 and

let the components of two vectors, |b〉 , |c〉 be given diagrammatically as

and (2.53)

then the components of their cross product are given by

(×

)= (2.54)

a dual vector. Let the components of two dual vectors 〈a| , 〈b| be given diagram-

matically as

and (2.55)

27


then the components of their cross product are given by

(×

)= (2.56)

a vector. With the above definitions the proof of Identity 2.1 is immediate.

Proof (of Identity 2.1 in diagrammatic tensor notation). In diagrammatic tensor

notation, the components of the left hand side of Identity 2.1 are

= − = − (2.57)

as required, where we have used Equation (2.50). �

Remark 2.37. In this equation we have used the diagrammatic equivalent of the

distribution in Equation (2.28) in the proof by index notation. ♦

Remark 2.38. In matrix notation the Levi-Civita tensor did not explicitly play a

role and in the index notation its structure was hidden to a degree. By contrast

diagrammatic tensor notation is simultaneously detailed and immediately compre-

hensible. A consequence of this is that diagrammatic tensor notation allows for

intuitive generalization of relations such as the Identity 2.1.

If we increase the dimension of the underlying vector spaces in Equation (2.57)

to n = 4, we may generalize in a number of ways. For example we obtain

= 2(〈a|c〉

)|b〉 − 2

(〈a|b〉

)|c〉 (2.58)

and

=(〈a|d〉〈b|e〉 − 〈a|e〉〈b|d〉

)|c〉+

(〈b|c〉〈a|e〉 − 〈a|c〉〈b|e〉

)|d〉

+(〈a|c〉〈b|d〉 − 〈a|d〉〈b|c〉

)|e〉 . (2.59)

Indeed formulae may be derived for general n in a number of different ways. ♦

2.5 Conclusion

The use of diagrams is common practice in the literature of statistical mechanical

lattice models, for example Korepin et al. [15] and Baxter [3]. A famous example

is Baxter’s exposition of the Yang-Baxter Equation (Figure 11.4 in [3]).

28

2.5. Conclusion

With the formalism set out above, there is no need to treat the diagrams any

differently from any other expressions for tensors. In this thesis the diagrams do

not appear in figures, rather they appear in sentences like any other equation. As

such they move from being a sketch of the relations to a part of the equations in

their own right.

It is often possible to dispense with the algebraic notation altogether, however

for clarity and to allow comparison to other works in the literature we usually

write the equations out in diagrammatic and algebraic forms. When we do so, for

brevity we write A (an algebraic expression) ! B (a diagrammatic expression)

as shorthand for the phrase: A may be expressed in diagrammatic tensor notation

as B.

29

Chapter 3

The su2 factorizing F -matrix

3.1 Introduction

The idea of twisting in quantum groups [7, 8, 6] was applied in [18] in the context

of the algebraic Bethe Ansatz. In [18] the Drinfel’d twists associated with the

quantum algebra Uq(su2) and its associated Yangian Y(su2)† were represented by

an F -matrix F1···N , a rank-(N,N) tensor which satisfies the following theorem.

Theorem 3.1 (factorizing property). Let N > 2 and let σ be a permutation

of {1, . . . , N} then

Fσ(1)···σ(N)Rσ1···N = F1···N (3.1)

where Rσ is a product of R-matrices associated with σ (to be defined later in this

chapter). ♦

F1···N is invertible as will be shown in Chapter 4 in the more general context

of Uq(sun) and we have

Rσ1···N =

(Fσ(1)···σ(N)

)−1F1···N (3.2)

hence F1···N is referred to as a factorizing F -matrix‡. One of the main results of

[18] was an explicit formula for F1···N when specializing to representations of the

quantum affine algebra Uq(su2). This expression for the F -matrix is entirely in

terms of products of the corresponding trigonometric R-matrix.

In this chapter we reproduce several results from [18], using diagrammatic ten-

sor notation as described in Chapter 2. In particular we give a diagrammatic

†For a discussion of the quantum algebra and its associated Yangian, see [10].‡Equation (3.2) corresponds to Equation (11) in [18].

31

Chapter 3. The su2 factorizing F -matrix

representation of the F -matrix F1···N using conventions from the six-vertex model

to draw it as a lattice of vertices. The representation allows us to establish iden-

tities such as Equation (3.1) virtually by inspection.

The role of Drinfel’d twists in the algebraic Bethe Ansatz was further consid-

ered in [13]. There it was noticed that the calculation of certain objects in the

XXZ spin-12chain, such as the domain wall partition function and scalar prod-

uct, is significantly simplified when the objects in question have been transformed

appropriately using F -matrices.

In Section 3.2 we recall the six-vertex notation for the R-matrix corresponding

to Uq(su2) and establish some standard identities relating to it. In Section 3.4

we introduce the monodromy matrix which appears in the algebraic Bethe Ansatz

and establish a number of lemmas related to it. In Section 3.3 we introduce the

bipartite matrix which is a product of R-matrices parametrized by permutations.

In this section we also establish some properties of permutations which are used in

the proof of the main result. In Sections 3.6 and 3.7 we construct the F -matrix. We

first define the partial F -matrix and then the F -matrix as a product of the partial

F -matrices. Several lemmas are established in these sections which culminate in

the proof of Theorem 3.1 in Section 3.8. Diagrammatic tensor notation is used

throughout. Often we also provide the algebraic version of the relations for easy

comparison to [18].

3.2 The R-matrix

In this section we build up to the definition of the R-matrix through the following

sequence; we define the basis vectors and dual vectors, we then define the elemen-

tary matrices as tensor products of the basis vectors and dual vectors; the identity

and permutation matrices are defined as tensor products of the elementary matri-

ces; and finally the R-matrix is defined in terms of the identity and permutation

matrices. We then provide some identities involving the R-matrices.

3.2.1 Definitions

We define the colour-s vector and dual vector which form a basis for the underlying

2 dimensional vector spaces. Tensor products of these may be used to construct a

basis for the state and dual state spaces and thereby a basis for the (vector space

of) tensors in general via the elementary matrices.

32

3.2. The R-matrix

Definition 3.2 (colour-s vector and dual vector). Let s be in {1, 2} and let

δ ji be the Kronecker symbol. Then the colour-s basis vector is given by

(e(s))i= δ s

i (3.3)

and the colour-s dual basis vector is given by

(e∗(s)

)j= δ j

s. (3.4)

♦

We now define the elementary matrices as products of tensor products of the

basis and dual basis vectors.

Definition 3.3 (elementary matrix). Let s, r be in {1, 2}. Then we call the

matrix

E(sr) = e(s) ⊗ e∗(r) (3.5)

an elementary matrix. The components of E(11) are given diagrammatically by

(E(11)

)ji= (3.6)

and the components of E(22) are given diagrammatically by

(E(22)

)ji= . (3.7)

♦

Remark 3.4. In Defintions 3.2 and 3.3 the objects only act on one space, so space

labelling was not required. When labelling is required, subscripts are used, for

example we write E(sr)1 to represent an elementary matrix acting on the 1-space.♦

Remark 3.5. Although not represented diagrammatically in this thesis, E(12) and

E(21) are also important operators. In a lattice they fix the incoming and outgoing

spins to be opposite to one another. ♦

The identity matrix and the permutation matrix are now defined as sums of

tensor products between the elementary matrices.

33


Definition 3.6. [identity matrix] We call the rank-(2, 2) tensor

I12 =

2∑

s,r=1

E(ss)1 ⊗ E

(rr)2 (3.8)

the identity matrix. The components of the identity matrix are given diagram-

matically by a pair of non-intersecting lines which overlap as follows

(I12)j1j2i1i2

= . (3.9)

♦

Definition 3.7. [permutation matrix] We call the rank-(2, 2) tensor

P12 =

2∑

s,r=1

E(sr)1 ⊗ E

(rs)2 (3.10)

the permutation matrix. The components of the permutation matrix are given

diagrammatically by a pair of non-intersecting lines which do not cross as follows

(P12)j1j2i1i2

= . (3.11)

♦

Remark 3.8. From this point on everything can be defined purely in terms of

diagrams. Algebraic notation is maintained mainly for comparison to works in the

literature. ♦

An essential object in the quantum inverse scattering and the algebraic Bethe

Ansatz scheme is the quantum R-matrix. The R-matrix associated to the XXZ

spin-12chain is given by the following definitions.

Definition 3.9. [weight functions] Let λ ∈ C be a free variable and let η ∈ C be

the fixed crossing parameter of the model in question. Then the weight functions

are given by

a(λ) = 1 (3.12)

b(λ) =[λ]

[λ+ η](3.13)

34

3.2. The R-matrix

c(λ) =[η]

[λ+ η]. (3.14)

For the representations of Y(su2), [λ] = λ and for the representations of Uq(su2),

[λ] = sinh λ. ♦

Remark 3.10. Throughout this chapter we treat these models in parallel and refer

to them simply as su2 models. We only explicitly deal with the Uq(su2) model but

lose no generality since Y(su2) may be realised as a small λ, η limit of this model.

For brevity we write f12 = f(λ1 − λ2) where f is a, b or c±. ♦

Definition 3.11. [R-matrix] Let λ1, λ2 be the rapidities associated with the vec-

tor spaces V1, V2 respectively. Then the R-matrix is given by

R12(λ1, λ2) = a12

(E

(11)1 E

(11)2 + E

(22)1 E

(22)2

)I12 + b12

(E

(11)1 E

(22)2 + E

(22)1 E

(11)2

)I12

+ c12

(E

(11)1 E

(22)2 + E

(22)1 E

(11)2

)P12 (3.15)

! = a12

(+

)+ b12

(+

)+ c12

(+

).

(3.16)

The components of the R-matrix are given diagrammatically by a pair of inter-

secting lines

(R12)j1j2i1i2

= . (3.17)

♦

Remark 3.12. i) In any given equation involving diagrams the endpoints of the

lines, in relation to the diagram, are fixed at all times. For example, in Equa-

tion (3.17) the index j1 will always be the rightmost upper index. Therefore

the index labels are redundant data and may be omitted. Whenever there is

no room for confusion, indices will be omitted for clarity of exposition.

ii) In the diagrammatic definition of the R-matrix we could have alternatively

used P -matrices for the a-weighted vertices since the necessary components

of I12 and P12 are equivalent.

iii) Each R-matrix in a diagram has two lines which ultimately exit at the top of

the diagram and two that exit at the bottom; this fixes the orientation to be

up the page.

35


iv) The six diagrams in the definition of the R-matrix correspond to the vertices

of the six-vertex model. ♦

Remark 3.13. In this thesis, each tensor which is constructed from R-matrices has

a corresponding tensor constructed from identity matrices. Diagrammatically, this

corresponds to replacing all of the intersections with non-intersections. To avoid

repetitive definitions, we use the following convention: any symbol involving an I

is taken to be the equivalent symbol involving an R in which all of the R-matrices

have been replaced with identity matrices. ♦

Remark 3.14. Algebraically, the identity matrix acts trivially and commutes with

everything. Diagrammatically, this is expressed as a line which does not interact

with the rest of the diagram; one or more non-intersections. Due to their non-

interaction, such lines may be drawn overlapping the diagram in any convenient

way. When we use this fact to redraw a diagram, we refer to it as a trivial re-

arrangement. ♦

3.2.2 Identities involving the elementary matrices

In this subsection we generate a number of identities which will be used throughout

the remainder of the chapter. The following identity allows us to reduce products

of pairs of elementary matrices.

Identity 3.15 (products of elementary matrices). The elementary matrices

satisfy

E(pq)1 E

(rs)1 = δ r

qE(ps)1 . (3.18)

If we restrict to p = q and r = s we may represent these relations diagrammatically

as

= and = (3.19)

where p = r, and

= = 0 (3.20)

where p 6= r. ♦

Proof. This identity follows from the definition of the basis and dual basis vec-

tors. �

36

3.2. The R-matrix

In the following identity we establish that certain tensor products of elementary

matrices project R-matrices onto identity matrices.

Identity 3.16 (R-matrix normalization). The R-matrices and the identity ma-

trices satisfy

E(11)1 E

(11)2 R12 = E

(11)1 E

(11)2 I12 ! = (3.21)

and

E(22)1 E

(22)2 R12 = E

(22)1 E

(22)2 I12 ! = . (3.22)

♦

Remark 3.17. When a contraction between two tensors is written without speci-

fying which indices are to be paired up, we take the contractions to be between

spaces with matching labels. The indices need not be of concern because the

diagrams unambiguously specify the contractions in all cases. ♦

Proof. This relation is a direct consequence of Identity 3.15 and the definition of

the R-matrix. �

Remark 3.18. These relations arise only when the R-matrix is normalized to have

the weight function a12 = 1 as it is in this case. Selection of a different normal-

ization would have led, in the case of some relations, to the requirement to keep

track of a large number of constants. ♦

3.2.3 Unitarity and the Yang-Baxter Relation

In this subsection we establish unitarity and the Yang-Baxter relation, two of the

most fundamental and important identities involving R-matrices.

Lemma 3.19 (unitarity). The R-matrices and the identity matrices satisfy

R21R12 = I21I12 ! = = . (3.23)

♦

37


Lemma 3.20 (Yang-Baxter relation). The R-matrices satisfy

R12R13R23 = R23R13R12 ! = . (3.24)

♦

Remark 3.21. In the su2 case, Lemma 3.19 the unitarity relation reduces to a 4×4

matrix identity. We have

1 0 0 0

0 b21 c21 0

0 c21 b21 0

0 0 0 1

1 0 0 0

0 b12 c12 0

0 c12 b12 0

0 0 0 1

=

1 0 0 0

0 b21b12 + c12c21 b21c12 + c21b12 0

0 c21b12 + b21c12 c21c12 + b12b21 0

0 0 0 1

.

(3.25)

Therefore we require

b21b12 + c12c21 = c21c12 + b12b21 = 1 (3.26)

and

b21c12 + c21b12 = c21b12 + b21c12 = 0. (3.27)

Multiplying through by [λ1 − λ2 + η][λ2 − λ1 + η] and applying the definitions of

the weight functions, these relations reduce to

[η][η] + [λ1 − λ2][λ2 − λ1] = [λ1 − λ2 + η][λ2 − λ1 + η] (3.28)

and

[η][λ1 − λ2] + [η][λ2 − λ2] = 0 (3.29)

which are established with the following standard identities on the hyperbolic sine

[A+B][A− B] = [A][A] + [B][−B], and [−A] = −[A]. (3.30)

♦

Remark 3.22. The proofs of unitarity and the Yang-Baxter relation will be given

in more detail in the more general case of sun in Chapter 4, Subsection 4.2.3

(unitarity) and Appendix B (Yang-Baxter). ♦

38

3.3. The bipartite matrix

3.3 The bipartite matrix

In this section we define the bipartite matrix as a product of R-matrices parametr-

ized by a permutation σ. We do this via a bipartite graph for σ, interpreting the

intersections in the graph to be R-matrices.

Definition 3.23 (bipartite graph). Let σ be a permutation of the ordered set

{1, . . . , N} for some positive integer N . Then a bipartite graph is a diagram

such that

i) the upper indices are ordered j1, . . . , jN and the lower indices are ordered

iσ(1), . . . , iσ(N) from right to left,

ii) jp is connected to ip by a monotonically downward line for all 1 ≤ p ≤ N ,

iii) no three lines intersect at any point.

Interpreting the intersections of this diagram as R-matrices, this diagram specifies

the components of a rank-(N,N) tensor. ♦

Bipartite graphs for any given permutation (for N > 2) are non-unique. We

establish that if two bipartite graphs correspond to the same permutation, then

their associated rank-(N,N) tensors are equivalent.

Lemma 3.24. The tensors corresponding to two bipartite graphs are equal if the

bipartite graphs correspond to the same permutation. ♦

Proof. The R-matrices may be interpreted as representations of generators from a

standard presentation† of the permutation group. Under this representation, uni-

tarity and the Yang-Baxter relation correspond to the relations of the presentation.

It follows that there is exactly one tensor corresponding to each equivalence class

of bipartite graphs under unitarity and the Yang-Baxter relation, that is for each

permutation. �

Remark 3.25. Lemma 3.24 is used extensively in this thesis since it effectively al-

lows us to use the unitarity and Yang-Baxter relation an arbitrary number of times

in a single move It allows us to perform complicated calculations by manipulating

lines on a page and to leverage visual intuition to discover hidden relations. ♦

†A presentation of a group is a set of generators together with a set of relations betweenthem such that the group is the set of words in the generators modulo the relations. Note thatthis presentation has a third relation which is trivial in terms of R-matrices: R12R34 = R34R12.See [11] for details of this presentation.

39


Lemma 3.24 allows us to make the following definition.

Definition 3.26 (bipartite matrix). Let σ be a permutation of the ordered set

{1, . . . , N} for some positive integer N . We call the rank-(N,N) tensor Rσ1···N

corresponding to a permutation σ, via a bipartite graph, the bipartite matrix.

The bipartite matrix is well defined by Lemma 3.24. ♦

Example 3.27 (bipartite matrix). Let σ be the permutation given by

σ{1, 2, 3, 4, 5} = {3, 5, 2, 1, 4}. (3.31)

The components of the bipartite matrix corresponding to σ are given diagrammat-

ically by

(Rσ1···5)

j1···j5i1···i5

= . (3.32)

This may be expanded into a product of R-matrices as follows

Rσ1···5 = R25R15R12R13R23R45. (3.33)

♦

Remark 3.28. The bipartite matrix is the central object of interest of this chapter.

Recall that Theorem 3.1 concerns the factorization of Rσ1···N . ♦

In order to establish Theorem 3.1, we decompose arbitrary permutations into

products involving two permutations; the cyclic permutation and site-swap permu-

tation. We establish that these two permutations form a generating set by relating

them back to a standard generating set.

Definition 3.29. [cyclic and site-swap permutations] Let N ≥ 2, then the cyclic

permutation, σc is the permutation such that

σc{b, a1, . . . , aN} = {a1, . . . , aN , b}. (3.34)

and the site-swap permutation, σs is the permutation such that

σs{b1, b2, a1, . . . , aN} = {b2, b1, a1, . . . , aN}. (3.35)

40


♦

Lemma 3.30. Any permutation may be written as a product of the cyclic and

site-swap permutations. ♦

Proof. Let σi be the permutation that swaps sites i and (i+ 1), that is

σi{a1, . . . , aN} = {a1, . . . , a(i+1), ai, . . . , aN} (3.36)

for all 1 ≤ i ≤ N − 1. Then (as mentioned above) the σi form a generating

set from a standard presentation for the permutation group (along with relations

which correspond to unitarity and the Yang-Baxter relation) so any permutation

may be written as a product of them. We have

σcσi = σi−1σc (3.37)

which implies that

σi−1c σi = σ1σ

i−1c . (3.38)

Noting that σs = σ1, we express the standard generators in terms of the cyclic and

site-swap permutations as follows. We have

σi = σNc σi = σN−i+1

c σsσi−1c . (3.39)

since σNc is the identity permutation. So the cyclic and site-swap permutations

also form a generating set for the permutations as required. �

Example 3.31 (Lemma 3.30). Let N = 5 and let i = 2. We have

σ2 = σ5cσ2 = σ4

cσ1σc (3.40)

since σ5c is the identity permutation. In terms of the bipartite matrix Rσ2

1···5 we may

rewrite this diagrammatically as

= =

(3.41)

41


where we have equality by Lemma 3.24 since the bipartite graphs in all the dia-

grams correspond to the same permutation; that is the top to bottom connectivity

is the same in each diagram. Note that the line emerging at the right hand side

at the top and bottom of the diagram† could be disengaged from the rest of the

diagram at any stage using unitarity and the Yang-Baxter relation. ♦

3.4 Monodromy matrix

We define the monodromy matrix which is an important tensor in the algebraic

Bethe Ansatz. There is often a distinction made between the quantum spaces and

the auxiliary spaces (corresponding to the index a below) however the distinction

is not important in this chapter.

Definition 3.32. [monodromy matrices] Let N be a positive integer, then the

monodromy matrix is the rank-(N + 1, N + 1) tensor given by

R1···N |a(λa;λ1, . . . , λN)

= R1a(λ1, λa)R2a(λ2, λa) · · ·R(N−1)a(λN−1, λa)RNa(λN , λa). (3.42)

The components of the monodromy matrix are given diagrammatically by

(R1···N |a

)j1···jN ja

i1···iN ia= . (3.43)

The alternate monodromy matrix is the rank-(N + 1, N + 1) tensor given by

Ra|1···N(λa;λ1, . . . , λN)

= RaN (λa, λN)Ra(N−1)(λa, λN−1) · · ·Ra2(λa, λ2)Ra1(λa, λ1). (3.44)

The components of the alternate monodromy matrix are given diagrammatically

by

(Ra|1···N


= . (3.45)

♦

†In this and following diagrams, where available, colour is used to highlight lines of particularinterest.

42

3.4. Monodromy matrix

Remark 3.33. The term alternate is used here only to distinguish the two types

of monodromy matrix, it is not intended to imply that one is more important

than the other. They are simply the right-to-left and left-to-right versions of each

other. ♦

Remark 3.34. The monodromy and alternate monodromy matrices may be defined

in terms of permutations. For example we have R1|2···N = Rσc where σc is the cyclic

permutation. ♦

In the following lemma, we establish a relation involving a product of a mon-

odromy and alternate monodromy matrix.

Lemma 3.35. The monodromy matrices satisfy

R1···N |a(λa)Ra|1···N(λa) = I (3.46)

and

Ra|1···N(λa)R1···N |a(λa) = I (3.47)

where I is the identity tensor. ♦

Proof. Diagrammatically Equations (3.46) and (3.47) may be written as

= (3.48)

and

= . (3.49)

Observe that both of the tensors in both Equations (3.48) and (3.49) correspond

to Rσ where σ is the identity permutation. Therefore the equalities are established

by Lemma 3.24. The unitarity relation is the mechanism by which the equalities

are satisfied. �

We now establish what is referred to in the literature as the intertwining re-

lation, it should be noted however that this is not the usual intertwining relation

from the algebraic Bethe Ansatz - the usual intertwining relation does not involve

both types of monodromy matrix.

43


Lemma 3.36. The monodromy matrices satisfy the relation

R1···N |b(λb)Ra|1···Nb(λa) = Ra|1···N(λa)Ra1...N |b(λb). (3.50)

♦

Proof. Diagrammatically Equation (3.50) may be written as

= . (3.51)

Observe that both of the tensors in Equation (3.51) correspond to Rσ where

σ{a, 1, . . . , N, b} = {b, 1, . . . , N, a}. Therefore the equality is established by Lemma

3.24. The Yang-Baxter relation is the origin of this equality. �

3.5 The rank-(2, 2) F -matrix

We now examine a simple case of the factorizing property. We write two alternate

expressions for the rank-(2, 2) F -matrix, show that these expressions are equivalent

and provide a diagrammatic proof of the factorizing property.

Definition 3.37 (F -matrix). The (rank-(2, 2)) F -matrix is the rank-(2, 2) ten-

sor given by†

F12(λ1, λ2) = E(11)2 R12(λ1, λ2) + E

(22)2 I12. (3.52)

We represent the components of F12 diagrammatically by

(F12)j1j2i1i2

= (3.53)

and we may reproduce the definition diagrammatically as

= + . (3.54)

The (rank-(2, 2)) F ′-matrix is the rank-(2, 2) tensor given by

F ′12(λ1, λ2) = E

(11)1 I12 + E

(22)1 R12(λ1, λ2) (3.55)

†Equation (3.52) corresponds to Equation (91) in [18].

44

3.5. The rank-(2, 2) F -matrix

We represent the components of F ′12 diagrammatically by

(F ′12)

j1j2i1i2

= (3.56)


= + . (3.57)

♦

Remark 3.38. In the case of both F12 and F ′12, when the incoming spin agrees with

the direction of the hollow arrow, the vertex becomes an R-matrix, alternatively

when the incoming spin disagrees with the direction of the hollow arrow, the vertex

becomes an identity matrix. ♦

In the following lemma, we establish that the F -matrix and the F ′-matrix are

equivalent. This is a diagrammatic treatment of the result of [18].

Lemma 3.39. The F -matrix and the F ′-matrix are equivalent†, F12 = F ′12. ♦

Proof. We have

= + = + + +

= a12 + b12 + c12 + + (3.58)

where the first equality is due to the definition of the F -matrix, the second equality

is due to expanding the terms into components and the last equation is due to

Identity 3.15 and the definition of the R-matrix‡. Similarly we have

= + = + + +

= a12 + b12 + c12 + + . (3.59)

The desired result is obtained by observing that a12 = 1. �

†This can be regarded as a special case of Equation (24) in [18].‡The second and third terms (that is, those with coefficients b12 and c12) in the final expres-

sion both arise from the second term in the previous expression.

45


Remark 3.40. Flattening the F -matrix according to the procedure set out in Re-

mark 2.15 we have

F12 = F ′12 =

1

1

c12 b12

1

12

(3.60)

where only the non-zero entries are indicated. Observe that F12 is lower triangular

and invertible provided b12 6= 0 which is true provided λ1 6= λ2. ♦

Example 3.41 (factorization). When N = 2, Theorem 3.1 specializes† to

F21R12 = I21F12 (3.61)

which we represent diagrammatically as

= . (3.62)

We establish this example diagrammatically by decomposing it into two cases. In

the first case we have

= = = (3.63)

where the first and third equalities follow from the definition of F12 and F ′12 re-

spectively and the second is an application of the unitarity relation. In the second

case we have

= = (3.64)

which is true by definition. ♦

Remark 3.42. This method of proof is reused for many of the following lemmas.

The key point is that by multiplying certain tensorial quantities involving partial

F -matrices by the elementary matrices E(ss)1 , we reduce the problem into several

cases which involve only R-matrices and identity matrices. Provided the cases

†This corresponds to Equation (89) in [18].

46

3.6. The partial F -matrices

considered correspond to a basis for the vector space in question, we are able to

establish the relations that we need. ♦

3.6 The partial F -matrices

3.6.1 Definitions

The partial F -matrices which we use to construct the F -matrices are defined in

a similar way to the rank-(2, 2) F -matrix. Indeed, the rank-(2, 2) F -matrix is a

special case of the partial F -matrix.

Definition 3.43 (partial F -matrices). Let N be a positive integer. The par-

tial F -matrix is the rank-(N + 1, N + 1) tensor given by†

F1···N |a(λa;λ1, . . . , λN) = E(11)a R1···N |a(λa;λ1, . . . , λN) + E(22)

a I1···N |a. (3.65)

We represent the components of F1···N |a diagrammatically as

(F1···N |a

)j1···jN ja

i1···iN ia= (3.66)


= + . (3.67)

The (alternate) partial F -matrix is the rank-(N + 1, N + 1) tensor given by

Fa|1···N(λa;λ1, . . . , λN) = E(11)a Ia|1···N + E(22)

a Ra|1···N(λa;λ1, . . . , λN). (3.68)

We represent the components of Fa|1···N diagrammatically as

(Fa|1···N


= (3.69)

†Equation (3.65) corresponds to Equations (96, 97) in [18].

47



= + . (3.70)

♦

Remark 3.44. Whilst we may regard F12 (F ′12) as the N = 2 case of F1|2···N

(F1···(N−1)|N ), F1|2···N is not in equal to F1···(N−1)|N for general N . ♦

3.6.2 Lemmas

We provide several lemmas which establish how the R-matrix, the alternate mon-

odromy matrix and the alternate partial F -matrix commute through the partial

F -matrices. Due to its connection with the Lie bialgebra co-commutator mapping,

the latter commutation relation is referred to as the cocycle relation [7].

The following lemma describes the commutation of the partial F -matrix with

the R-matrix. It may be thought of as the Yang-Baxter relation except that

the R-matrix passes through a partial F -matrix instead of a (short) monodromy

matrix.

Lemma 3.45. The partial F -matrices satisfy the following relation†

F213···N |a(λa)R12(λ1, λ2) = R12(λ1, λ2)F1···N |a(λa)

! = . (3.71)

♦

Proof. Equation (3.71) is equivalent to the two relations given by

E(ss)a F213···N |a(λa)R12(λ1, λ2) = E(ss)

a R12(λ1, λ2)F1···N |a(λa). (3.72)

for s in {1, 2}. We consider the two cases diagrammatically. Case 1 (s = 1). We

have

= (3.73)

†This corresponds to the first equation on Page 13 in [18].

48

3.6. The partial F -matrices

where the equality is due to the Yang-Baxter relation. Case 2 (s = 2). We have

= (3.74)

where the equality is due to a trivial redrawing of the diagram. �

The following relation is similar to the unitarity of monodromy matrices except

that it involves a partial F -matrix instead of a second monodromy matrix.

Lemma 3.46. The partial F -matrices satisfy the following relation†

F1···N |a(λa)Ra|1···N(λa) = I1···N |aFa|1···N(λa)

! = . (3.75)

♦

Proof. Equation (3.75) is equivalent to the two relations given by

E(ss)a F1···N |a(λa)Ra|1···N (λa) = E(ss)

a I1···N |aFa|1···N (λa). (3.76)

for s in {1, 2}. We consider the two cases diagrammatically. Case 1 (s = 1).. We

have

= . (3.77)

Here the equality is due to Lemma 3.35. Case 2 (s = 2). Here both sides of the

equation take the same diagrammatic form

(3.78)

as required. �

The following relation is similar to the intertwining relation except that it

involves partial F -matrices. It is referred to as the cocycle relation in [18].

†This corresponds to the second equation on Page 13 in [18].

49


Lemma 3.47 (cocycle relation). The partial F -matrices satisfy the following

relation†

F1···N |b(λb)Fa|1···Nb(λa) = Fa|1···N(λa)Fa1...N |b(λb)

! = . (3.79)

♦

Remark 3.48. Here and subsequently, each row of dots is associated with its own

hollow arrow. For example on the left hand side of Equation (3.79) the top row of

dots is sensitive to the spin at the downward hollow arrow and the bottom row of

dots is sensitive to the spin at the upward hollow arrow. ♦

Proof. Equation (3.79) is equivalent to

E(ss)a E

(rr)b F1···N |b(λb)Fa|1···Nb(λa) = E(ss)

a E(rr)b Fa|1···N (λa)Fa1...N |b(λb). (3.80)

for r, s in {1, 2}. We consider the four cases diagrammatically. Case 1 (s = 1, r =

1). We have

= = .

(3.81)

Here the first equality is a trivial redrawing of the diagram and the second equality

is an application of Identity 3.16. Case 2 (s = 1, r = 2). We have

= . (3.82)

Here the equality is a trivial redrawing of the diagram; there is no interaction at

the vertices. Case 3 (s = 2, r = 1). We have

= . (3.83)

†This corresponds to Equation (55) in [18].

50

3.7. The F -matrix

Here the equality is due to Lemma 3.36. Case 4 (s = 2, r = 2). We have

= = .

(3.84)

Here the first equality is an application of Identity 3.16 and the second equality is

a trivial redrawing of the diagram. �

3.7 The F -matrix

In this section the F -matrix is defined as a product of partial F -matrices. We

establish a number of lemmas before restating and proving Theorem 3.1.

3.7.1 Definition

Definition 3.49 (F -matrix). Let N ≥ 2, then the F -matrix is the rank-(N,N)

tensor given by†

F1···N(λ1, . . . , λN) = F12(λ1, λ2)F12|3(λ3) · · ·F1···(N−1)|N (λN). (3.85)

We may alternatively define the F -matrix diagrammatically by

(F1···N(λ1, . . . , λN))j1···jNi1···iN

= . (3.86)

♦

Remark 3.50. We adhere to our previous convention whereby each row of dots is

associated with a single open-faced triangle. For example the triangle on the far

right is associated with the uppermost row of dots. ♦

†Equation (3.85) corresponds to Equation (21) in [18].

51


We can also express the F -matrix recursively as

F1···N = F1···(N−1)F1···(N−1)|N (3.87)

which follows from the definition. This form is suggestive of the inductive methods

used in this section.

3.7.2 Lemmas

In Lemma 3.30 it was established that the cyclic and site-swap permutations form

a generating set for the permutation. The following two lemmas correspond to

Theorem 3.1 in the special cases where the permutation is set to be the cyclic or

site-swap permutations. The factorizing property is an almost immediate result of

these lemmas in combination.

Lemma 3.51. Recall that σs is the site-swap permutation. Then Rσs

12 = R12,

Iσs

12 = I21 and we have the following relation

F213···N(λ2, λ1, λ3, . . . λN)R12(λ1, λ2) = I21F1···N(λ1, . . . , λN)

!

= . (3.88)

♦

52

3.7. The F -matrix

Proof. We proceed by induction on N . The base case of the induction (N = 2)

is provided by Lemma 3.46 (with N = 1 in that lemma). The inductive step is

achieved by a single application of Lemma 3.45 to the top part of the diagram to

produce

(3.89)

as required. �

The cyclic permutation case involves the use of the following lemma which is

similar to the cocycle relation except that it involves an F -matrix instead of a

partial F -matrix.

Lemma 3.52. The F -matrices and the alternate partial F -matrices satisfy the

following relation

F2···N (λ2, . . . , λN)F1|2···N(λ1) = F1···N(λ1, . . . , λN)

!

= . (3.90)

♦

53


Proof. We proceed by induction on N . In the base case (N = 3) we have

= = (3.91)

where the first equality is given by the cocycle relation (Lemma 3.47 with N = 1

in that lemma) and the second equality follows from Lemma 3.39. Applying the

cocycle relation to the top of the diagram in Equation (3.90), we obtain

(3.92)

which provides the required inductive step. �

The following lemma describes the commutation of an alternate monodromy

matrix with the F -matrix.

Lemma 3.53. Since Rσc

1···N = R1|2···N and Iσc

1···N = IN ···2|1 where σc is the cyclic

permutation, we have

F2···N1(λ2, . . . , λN , λ1)R1|2···N(λ1) = IN ···2|1F1···N(λ1, . . . , λN).

!

54

3.7. The F -matrix

= . (3.93)

♦

Proof. We apply Lemma 3.46 to the upper part of the diagram on the left hand

side of Equation (3.93) to obtain

. (3.94)

We then trivially redraw the diagram to obtain

. (3.95)

The result then follows from Lemma 3.52. �

55


3.8 Proof of the factorizing property

We are now in a position to prove that

Fσ(1)···σ(N)Rσ1···N = F1···N (3.96)

(that is, Equation 3.1 from Theorem 3.1).

Proof (Theorem 3.1). First we make an observation that allows us to pass a de-

composed bipartite matrix through the F -matrix one component at a time. Let

σ and τ be two permutations of the set {1, . . . , N}. Recall that σ = ρσρ where ρ

is the permutation given by ρ{1, . . . , N} = {N, . . . , 1}. Then it follows from the

definition of Rσ1···N that

Rτ1···NR

σ1···N = Rτσ

1···N (3.97)

and

Iτ1···NIσ1···N = I

(τσ)1···N (3.98)

since (τ )(σ) = ρτρρσρ = ρτσρ = (τσ). So we have

Fτσ(1)···τσ(N)Rτσ1···N = Iτ1···NFσ(1)···σ(N)R

σ1···N = I

(τσ)1···NF1···N (3.99)

as required.

Now since the cyclic and site-swap permutations form a basis for the permu-

tations (Lemma 3.30), the fact that the cyclic and site-swap permutations pass

through the F -matrix in the expected way is sufficient to establish the factorizing

property of the F -matrix in general. �

3.9 Conclusion

Following the algebraic methods of [18], we have outlined a diagrammatic treat-

ment of the factorizing F -matrices. The main feature of our work is the diagram-

matic depiction of the partial F -matrices Fa|1···N and F1···N |a in Section 3.6, which

parallel the standard representation of the XXZ monodromy matrix. In Section

3.6 we also gave diagrammatic proofs of a number of identities involving partial

F -matrices. These proofs are quite transparent in our notation, which allows the

components of all tensors to be extracted automatically. In Section 3.7 we built

the full F -matrix F1···N by stacking partial F -matrices together and proved the

factorizing equation (3.1) in the sufficient cases σ = σc and σ = σs. Our proofs are

56

3.9. Conclusion

inductive in nature, since they only require iterations of the more basic identities.

We used diagrammatic tensor notation to simplify the proofs presented in this

chapter by laying bare the structure of relations. In the next chapter, the results

presented here are generalized to sun and a different approach is used. Specifically,

we decompose permutations into a different generating set; site-swaps (permuta-

tions which switch adjacent spin-sites) rather than σc and σp. This leads to a

differently structured proof of the factorizing property, and although it could have

been used in this chapter, we retain the present approach for consistency with

[18, 19].

57

Chapter 4

The sun factorizing F -matrix

4.1 Introduction

In the previous chapter, we considered quantum spin chains associated with inter-

acting objects with two spin-states — up and down. We now turn our attention

to a generalization of these quantum spin chains in which the associated objects

have n states (in this context, the states are called colours). These spin chains

are associated with the quantum algebra Uq(sun) and its Yangian Y(sun)†. Ex-

tending the partial F -matrix formalism of [18] which was discussed in Chapter 3,

we derive a completely factorized expression for the sun F -matrix of such models.

We prove the equivalence of this expression to the expression obtained by Albert,

Boos, Flume and Ruhlig [2]. We also provide a diagrammatic proof that the sun

F -matrix is invertible and construct a factorized expression for the inverse of the

sun F -matrix.

Since the results of the previous chapter are a special case of the results of this

chapter, there is necessarily some repetition of material. We err on the side of this

chapter being self-contained at the risk of being repetitive.

The main result of this chapter is that the sun F -matrix, a rank-(N,N) tensor

satisfies the following theorem.

Theorem 4.1 (the factorizing property). Let N ≥ 2, let σ be a permutation

of {1, . . . , N} and let σ = ρσρ where ρ is the permutation such that

ρ{1, . . . , N} =

{N, . . . , 1}, if n is even

{1, . . . , N}, if n is odd(4.1)

†For a discussion of the quantum algebra and its Yangian, see [10].

59

Chapter 4. The sun factorizing F -matrix

then

Fσ(1)···σ(N)Rσ1···N = Iσρ(1)···ρ(N)F1···N = F1···N (4.2)

where Rσ and Iσ (specified below) are products ofR-matrices and identity matrices

respectively. ♦

Remark 4.2. Algebraically Iσ plays no role and was omitted from the notation

used in [18]; we retain it because diagrammatically it plays the role of maintaining

the proper position of each line on the page which is important to us. ♦

As is established in Subsection 4.8.1, F1···N is invertible, so we have

Rσ1···N =

(Fσ(1)···σ(N)

)−1Iσρ(1)···ρ(N)F1···N =

(Fσ(1)···σ(N)

)−1F1···N (4.3)

hence F1···N is referred to as a factorizing F -matrix. One of the main results of

[18] was an explicit expression for F1···N when specializing to representations of

the quantum affine algebra Uq(su2). This expression for the F -matrix is entirely

in terms of products of the corresponding trigonometric R-matrix.

In [2], a sum expression for F1···N satisfying Theorem 4.1 was obtained in the

case where the R-matrix corresponds to Y(sun), and went on to study the Bethe

eigenvectors of the higher rank XXX spin chains under the change of basis induced

by the F -matrix. What is absent in [2] is a construction of F1···N using the partial

F -matrix approach developed in [18]. Moreover it is natural to expect that the

sun factorizing problem can be solved by a reduction, in (n − 2) steps, to the

known solution to the su2 factorizing problem as discussed in Chapter 3. Such

a method would be in keeping with the structure of the nested Bethe Ansatz

discussed in [17, 4] and Chapter 5, which is used to construct the eigenvectors of

these models. Indeed, in the nested Bethe Ansatz approach to the sun spin chains,

the eigenvectors of the transfer matrix are obtained via (n− 2) reductions to the

su2 problem, the solution to which is known from the algebraic Bethe Ansatz as

discussed in [15].

The purpose of this chapter is to settle the question raised in [2]; can the

sun F -matrix be represented in a factorized form for general n as it was for n =

2 in Chapter 3 and [18]? The main result of this chapter is a new expression

for the F -matrix F1···N for N -fold tensor products of the fundamental evaluation

representations of Y(sun) and Uq(sun). We show that the F -matrix admits the

60

4.1. Introduction

completely factorized expression

F1···N =

{F 21···NF

3N ···1 · · ·F n

1···N , n even,

F 2N ···1F

31···N · · ·F n

1···N , n odd,(4.4)

where each F s1···N has an analogous form to the F -matrices of [18], and is composed

of a product of partial F -matrices F s1···(i−1)|i as follows

F s1···N = F s

1|2Fs12|3 · · ·F s

1···(N−1)|N . (4.5)

A key feature of our work is the notion of tiers. Throughout this chapter we say

that each F s1···N is situated at tier-s in reference to the fact that it depends on the

interaction of only s state variables, or in other words, has sus type behaviour. To

prove that the F -matrix given in Equation (4.4) satisfies the factorizing property

we proceed by induction on n, with the base case corresponding to the factorizing

property of the su2 F -matrix discussed in Chapter 3. We show that the sum

expression for the sun F -matrix given in Equation (4.4) – despite its appearance

– is equivalent to the result obtained in [2].

In Chapter 3 we presented a review of [18], working in terms of a new dia-

grammatic notation motivated by the six-vertex model. The cornerstone of our

approach was a diagrammatic representation of the partial F -matrices used in

[18]. In this chapter we generalize the notation used in Chapter 3, to allow a

diagrammatic description of the sun F -matrix given in Equation (4.4). For clarity

we present algebraic and diagrammatic versions of most relations.

In Section 4.2 we recall the six vertex notation for the sun R-matrix correspond-

ing to Uq(sun) and establish some standard identities relating to it. In Section 4.3

we define the tier-s R-matrix which is an sus reduction of the sun R-matrix. In

Section 4.4 we recall the definition of the bipartite matrix from Chapter 3 which is

a product of R-matrices parametrized by the permutations. In this section we also

establish a new property of the bipartite matrix which is used later to establish

the equivalence between the factorized and sum expressions for the F -matrix. In

Section 4.5 we construct the F -matrix. We first define the tier-s partial F -matrix.

We then define the tier-s F -matrix as a product of the tier-s partial F -matrices.

Finally the F -matrix is defined as a product of tier-s F -matrices. In Section 4.6

several lemmas are established which culminate in the proof of Theorem 4.1. In

Section 4.7 we provide examples of the F -matrix in the special cases of n = 2

and n = 3 highlighting the sorting property of the F -matrix which characterizes

61


the sum expression of [2]. We establish the equivalence between the factorized

and sum expressions for the F -matrix. In Section 4.8 we provide a diagrammatic

proof of the lower triangularity and invertibility of the F -matrix. We then provide

an explicit construction of the inverse of the F -Matrix. Finally in Section 4.9 we

summarize the chapter. We use the diagrammatic tensor notation described in

Chapter 2 throughout. This chapter closely follows [20].

4.2 The R-matrix

In this section we build up to the definition of the R-matrix through the follow-

ing sequence; we define the basis vectors and dual vectors, we then define the

elementary and related matrices as tensor products of the basis vectors and dual

vectors; the identity and permutation matrices are defined as tensor products of

the elementary and related matrices; and finally the R-matrix is defined in terms

of the identity and permutation matrices. We provide some identities involving

the elementary and related matrices and the R-matrices.

4.2.1 Definitions

We define the colour-s vector and dual vector which form a basis for the n dimen-

sional underlying vector spaces. Tensor products of these may be used to construct

a basis for the state and dual state spaces and thereby a basis for the (vector space

of) tensors in general via the elementary matrices.

Definition 4.3. Let s be in {1, . . . , n} and let δ ji be the Kronecker symbol. Then

the colour-s basis vector is the vector with components given by

(e(s)1

)i= δ s

i (4.6)

and the colour-s dual basis vector is the dual vector with components given by

(e∗(s)1

)j= δ j

s. (4.7)

♦

The elementary matrices are defined as tensor products of the basis and dual

basis vectors.

62

4.2. The R-matrix

Definition 4.4 (elementary and related matrices). Let s, r be in {1, . . . , n},then the colour-(sr) elementary matrix is given by

E(sr)1 = e

(s)1 ⊗ e

∗(r)1 . (4.8)

Specializing to r = s, we call matrix E(s)1 = E

(ss)1 the colour-s matrix. The

non-colour-s matrix is given by

N(s)1 =

n∑

p=1,p 6=s

E(p)1 . (4.9)

The sub-colour-s matrix is given by

S(s)1 =

s∑

p=1

E(p)1 . (4.10)

The space-1 identity matrix is given by

I1 =

n∑

p=1

E(p)1 . (4.11)

The components of the colour-s, non-colour-s and sub-colour-s matrices are given

diagrammatically by

(E

(s)1

)ji= (4.12)

(N

(s)1

)ji= (4.13)

(S(s)1

)ji= (4.14)

respectively. The space-1 identity matrix is given diagrammatically by a simple

line. ♦

Remark 4.5. In the case of n = 2, N(1)1 = E

(2)1 , N

(2)1 = E

(1)1 , S

(1)1 = E

(1)1 and

S(2)1 = I1, the identity matrix for the vector space V1 so the definition of the N

(s)1

and S(s)1 is redundant. Note also that any rank-(N,N) tensor acting on the state-

space V1 ⊗ · · · ⊗ VN and dual state-space V ∗1 ⊗ · · · ⊗ V ∗

N where Vk = V ∗k ≃ Cn can

be written as a complex linear combination of tensor products of the elementary

matrices. ♦

The identity matrix and the permutation matrix are now defined as sums of

63


tensor products between the elementary matrices.

Definition 4.6. [identity matrix] The identity matrix is the rank-(2, 2) tensor

given by

I12 =n∑

s,r=1

E(ss)1 ⊗E

(rr)2 . (4.15)

The components of the identity matrix are given diagrammatically by a pair of

non-intersecting lines which cross each other as follows

(I12)j1j2i1i2

= . (4.16)

♦

Definition 4.7. [permutation matrix] The permutation matrix is the rank-

(2, 2) tensor given by

P12 =

n∑

s,r=1

E(sr)1 ⊗E

(rs)2 . (4.17)

The components of the permutation matrix are given diagrammatically by a pair

of non-intersecting lines which do not cross each other as follows

(P12)j1j2i1i2

= . (4.18)

♦

We now present the definition of the sun R-matrix – the fundamental building

block of the quantum mechanical lattice models we are concerned with in this

thesis. The nonzero entries of the R-matrix are given by the following definition.

Definition 4.8. [weight functions] Let λ be a free variable and let η be the fixed

crossing parameter of the model in question. Then the weight functions are

given by

a(λ) = 1 (4.19)

b(λ) =[λ]

[λ+ η](4.20)

c±(λ) = e±θλ [η]

[λ+ η]. (4.21)

64

4.2. The R-matrix

For the representations of Y(sun), [λ] = λ and θ = 0 and for the representations

of Uq(sun), [λ] = sinh λ and θ = 1. Throughout this chapter we treat these models

in parallel and refer to them simply as sun models. We only explicitly deal with

the Uq(sun) model but lose no generality since Y(sun) may be realised as a small

λ, η limit of this model. For brevity we write f12 = f(λ1 − λ2) where f is a, b or

c±. ♦

Definition 4.9. [R-matrix] Let λ1, λ2 be the rapidities associated with the vec-

tor spaces V1, V2 respectively and let

f(rs)12 (λ1, λ2) =

a(λ1 − λ2), if r = s

b(λ1 − λ2), if r 6= s(4.22)

and

g(rs)12 (λ1, λ2) =

0, if r = s

c+(λ1 − λ2), if r < s

c−(λ1 − λ2), if r > s.

(4.23)

Then the R-matrix is the rank-(2, 2) tensor is given by

R12(λ1, λ2) =

n∑

r,s=1

f(rs)12 (λ1, λ2)E

(r)1 E

(s)2 I12 + g

(rs)12 (λ1, λ2)E

(r)1 R

(s)2 P12

! =n∑

r,s=1

f(rs)12 + g

(rs)12 (4.24)

here we represented the components of the R-matrix diagrammatically by a pair

of intersecting lines

(R12)j1j2i1i2

= . (4.25)

♦

Remark 4.10. i) The definition of the R-matrix via the functions f(rs)12 and g

(rs)12

is somewhat unconventional, but the simplicity of the diagrammatic definition

allows for relatively compact proofs of unitarity and the Yang-Baxter relation

(these proofs are given in Subsection 4.2.3).

ii) We could have alternatively let f(ss)12 = 0 and g

(ss)12 = a(λ1 − λ2) without

affecting the definition of the R-matrix.

65


iii) Each R-matrix in a diagram has two lines which ultimately exit at the top of

the diagram and two which exit at the bottom; this fixes the orientation to

be up the page.

iv) For clarity we omit dependencies of objects on rapidities and indices through-

out where there is no room for confusion.

v) Since the R-matrix is defined in terms of the identity and permutation ma-

trices, it inherits colour conservation from them; for all nonzero components,

the set of colours at the base of the R-matrix must match the set of colours

at the top. ♦

Remark 4.11. In this chapter we use the following notation: any symbol involving

an I is taken to be the equivalent symbol involving an R evaluated at η = 0.

That is, it is the equivalent object where all intersections are replaced with non-

intersections. We use the term trivial re-arrangement in the same sense as in

Chapter 3 for re-arrangements of non-interacting lines arising from identity matri-

ces. ♦

4.2.2 Identities involving the colour-s and related matrices

We establish a number of technical identities which are used later. In the case of

su2 we had relatively few of these relations.

Identity 4.12 (sums and products of the colour-s and related matrices).

We provide several identities which involve sums and products of colour, non-colour

and sub-colour matrices. The diagrammatic equivalent of the relations is also pro-

vided where it is of use later.

i) The colour-s are projective

E(s)1 E

(r)1 =

E

(s)1 , if s = r

0, if s 6= r(4.26)

a corollary of this relation is that

n∑

r=1

E(s)1 E

(r)1 = E

(s)1 !

n∑

r=1

= . (4.27)

66

4.2. The R-matrix

ii) The product of a colour-s and a non-colour-r matrix is given by

E(s)1 N

(r)1 = N

(r)1 E

(s)1 =

0, if s = r

E(s)1 , if s 6= r

(4.28)

a corollary of this relation is that

n∑

s=1

E(s)1 N

(r)1 =

n∑

s=1

N(r)1 E

(s)1 =

n∑

s=1,s 6=r

E(s)1 = N

(r)1

!

n∑

s=1

=n∑

s=1

=n∑

s=1,s 6=r

= . (4.29)

iii) The product of a colour-s matrix and a sub-colour-s matrix is given by

E(s)1 S

(r)1 =

E

(s)1 , if s ≤ r

0, if s > r.(4.30)

In the case of s ≤ r diagrammatically this is

= . (4.31)

iv) The product of a non-colour-s matrix and a sub-colour-s matrix is given by

N(s)1 S

(s)1 = N

(s)1 S

(s−1)1 = S(s−1) ! = = . (4.32)

♦

Proof. These identities follow from the definition of the basis and dual basis vec-

tors. �

In the following identity we establish that certain tensor products of elementary

matrices project R-matrices onto identity matrices.

Identity 4.13 (R-matrix normalization). Let 1 ≤ s ≤ n, then

E(s)1 E

(s)2 R12 = E

(s)1 E

(s)2 I12 ! = . (4.33)

♦

67


Remark 4.14. In the above equations and in this chapter in general when a con-

traction between two tensors is written without specifying which indices are to

be paired up, we take the contractions to be between spaces with matching labels

such that left-to-right in the algebraic expressions corresponds to bottom-to-top in

the diagrammatic notation. The indices are not of concern because the diagrams

unambiguously specify the contractions in all cases. ♦

Proof. This relation is a direct consequence of Identity 4.12. Using the definition

of the R-matrix the right hand side of Equation (4.33) becomes

n∑

p,q=1

f

(pq)12 + g

(pq)12

= f

(ss)12 + g

(ss)12 = (4.34)

where the first equality is due to Identity 4.12 (i) and the second equality is es-

tablished by observing that f(ss)12 = a12 = 1 and g

(ss)12 = 0. �

Remark 4.15. These relations arise only when the R-matrix is normalized to have

the weight function a12 = 1 as it is in this case. Selection of a different normal-

ization would have led, in the case of some relations, to the requirement to keep

track of a large number of constants. ♦

4.2.3 Unitarity and the Yang-Baxter relation

In this subsection we establish unitarity and the Yang-Baxter relation, two of the

most fundamental and important identities involving R-matrices.

Lemma 4.16 (unitarity). We have the following relation

R21R12 = I21I12. ! = = . (4.35)

♦

Proof. Unitarity is a consequence of Identity 4.12 and standard identities on the

hyperbolic functions. Expanding the left hand side of 4.35 we have

n∑

r,s,p,q=1

f

(rs)12 f

(pq)21 + f

(rs)12 g

(pq)21 + g

(rs)12 f

(pq)21 + g

(rs)12 g

(pq)21

68

4.2. The R-matrix

=

n∑

p,q=1

f

(qp)12 f

(pq)21 + f

(pq)12 g

(pq)21 + g

(qp)12 f

(pq)21 + g

(pq)12 g

(pq)21

=n∑

p,q=1

((f(qp)12 f

(pq)21 + g

(pq)12 g

(pq)21

)+(f(pq)12 g

(pq)21 + g

(qp)12 f

(pq)21

) )

=n∑

p,q=1

= (4.36)

where the first equality is due to Identity 4.12 (i) and the second equality is due

to a trivial re-arrangement of non-interacting lines and gathering common factors.

Using the additional notation [λ]∗ = cosh(λ), we have [A][B] = 12([A+B]∗ − [A−

B]∗) and it follows that

b12b21 + c±12c±21 =

[λ1 − λ2][λ2 − λ1] + e±(λ1−λ2)[η]e±(λ2−λ1)[η]

[λ1 − λ2 + η][λ2 − λ1 + η]

=[0]∗ − [2λ1 − 2λ2]

∗ + [2η]∗ − [0]∗

[2η]∗ − [2λ1 − 2λ2]∗= 1 (4.37)

and

b12c±21 + c∓12b21 =

[λ1 − λ2]e±(λ2−λ1)[η] + e∓(λ1−λ2)[η][λ2 − λ1]

[λ1 − λ2 + η][λ2 − λ1 + η]

=[λ1 − λ2]e

∓(λ1−λ2)[η]− e∓(λ1−λ2)[η][λ1 − λ2]

[λ1 − λ2 + η][λ2 − λ1 + η]= 0. (4.38)

Since we also have a12 = 1, the third equality is established by considering the

following three cases. Case 1 (p = q). In this case have

f(qp)12 f

(pq)21 + g

(pq)12 g

(pq)21 = a12a21 + 0× 0 = 1 (4.39)

and

f(pq)12 g

(pq)21 + g

(pq)12 f

(pq)21 = a12 × 0 + 0× a21 = 0. (4.40)

Case 2 (p < q). In this case have

f(qp)12 f

(pq)21 + g

(pq)12 g

(pq)21 = b12b21 + c−12c

−21 = 1 (4.41)

and

f(pq)12 g

(pq)21 + g

(pq)12 f

(pq)21 = b12c

−21 + c+12b21 = 0. (4.42)

69


Case 3 (p > q). In this case have

f(qp)12 f

(pq)21 + g

(pq)12 g

(pq)21 = b12b21 + c+12c

+21 = 1 (4.43)

and

f(pq)12 g

(pq)21 + g

(pq)12 f

(pq)21 = b12c

+21 + c−12b21 = 0. (4.44)

The final equality follows from the definition of the space-1 and 2 identity matrices.

This establishes the lemma. �

Lemma 4.17 (Yang-Baxter relation). The R-matrices satisfy

R12R13R23 = R23R13R12 ! = . (4.45)

♦

Proof. As in the case of unitarity, the Yang-Baxter relation is a consequence of

Identity 4.12 (i) and standard identities on the hyperbolic functions. The details

are provided in Appendix B. �

4.3 The tier-s R-matrix

The tier-s R-matrix is a reduced version of the sun R-matrix which behaves as

a sus R-matrix in the absence of any colours greater than s and as an identity

matrix otherwise. We establish that the tier-s R-matrix possesses many of the

important properties of the R-matrix. In particular, they satisfy unitarity and

the Yang-Baxter relation. These relations are retained since the tier-s R-matrix is

effectively an sus restriction of the sun R-matrix.

4.3.1 Definition

Definition 4.18. [tier-s R-matrix] Let 1 ≤ s ≤ n. Then the tier-s R-matrix is

the rank-(2, 2) tensor given by

Rs12 = S

(s)1 S

(s)2 (R12 − I12)+ I12 ! = − + (4.46)

70

4.3. The tier-s R-matrix

here we represented the components of the tier-s F -matrix diagrammatically by

(Rs12)

j1j2i1i2

= . (4.47)

♦

The following identity comes into play in the final steps of the proof of the

factorizing property.

Identity 4.19 (Rn

12and R

1

12). For s = n the tier-s R-matrix specializes to the

R-matrix

Rn12 = R12 (4.48)

and for s = 1 it specializes to the identity matrix

R112 = I12. (4.49)

♦

Proof. Equation (4.48) follows from the fact that S(n)1 = I1. Equation (4.49)

follows from the fact that S(1)1 = E

(1)1 combined with Identity 4.13. �

The following Lemma is a tier-s version of Identity 4.13.

Identity 4.20 (tier-s R-matrix normalization). Let s, r be in {1, . . . , n}, thenwe have the following relation

E(r)1 E

(r)2 Rs

12 = E(r)1 E

(r)2 I12 ! = . (4.50)

♦

Proof. This result follows from the definition of the tier-s R-matrix, Identity 4.13

and Identity 4.12 (iii). �

4.3.2 Tier-s unitarity and the Yang-Baxter relation

The following two lemmas are tier-s versions of unitarity and the Yang-Baxter

relation.

71


Lemma 4.21 (tier-s Yang-Baxter relation). The tier-s R-matrices satisfy

Rs12R

s13R

s23 = Rs

23Rs13R

s12 ! = . (4.51)

♦

Proof. The Yang-Baxter relation is established by considering two cases. Case 1

(all colours i1, i2, i3 ≤ s). In this case the tier-s R-matrices become R-matrices and

the relation holds by Lemma 4.17. Case 2 (one or more of i1, i2, i3 are greater than

s). In this case, at least one of the lines in the diagram only interacts by identity

matrices and so the relation is true by a trivial re-arrangement of the diagram. �

Example 4.22 (tier-s Yang-Baxter equation). Let r > s. Then we have

E(r)3 Rs

12Rs13R

s23 = E

(r)3 Rs

12I13I23 = E(r)3 I23I13R

s12 = E

(r)3 Rs

23Rs13R

s12

! = = = (4.52)

where the first and third equalities are due to the definition of the tier-s R-matrix

and Identity 4.12 (ii) and the second equality is a trivial re-arrangement of the

non-interacting line. The cases in which we multiply by E(r)1 or E

(r)2 are similar.♦

Lemma 4.23. [tier-s unitarity] The tier-s R-matrices satisfy

Rs21R

s12 = I21I12 ! = = . (4.53)

♦

Proof. Unitarity is established by considering cases similar to those for the Yang-

Baxter relation. �

4.4 The bipartite matrix

In Section 3.3 of Chapter 3 the bipartite matrix was defined and its properties were

established without reference to the dimension of the underlying vector spaces, all

72


that was required was that unitarity and the Yang-Baxter relation were satisfied.

Since we have unitarity and the Yang-Baxter relation for the sun tier-s R-matrices,

the definitions and results established in that section for the su2 R-matrices case

carry over to the sun tier-s R-matrices.

The following lemma which does not appear in Chapter 3 is used in Section 4.7

to show the equivalence between the factorized expression for the sun F -matrix

and the sum expression for the sun F -matrix of [2].

Lemma 4.24. Let {i1, . . . , iN} be an ordered set of integers taking values in

{1, . . . , n} and let σ, τ be any two permutations of {1, . . . , N} which satisfy

iσ(1) ≤ · · · ≤ iσ(N), and iτ(1) ≤ · · · ≤ iτ(N). (4.54)

Then the components of the graphs Rσ1···N and Rτ

1···N , satisfy

(Rσ1···N)

j1···jNi1···iN

= (Rτ1···N)


. (4.55)

♦

Proof. Up to applications of the unitarity and Yang-Baxter equation, the graphs

corresponding to σ and τ only differ from one another in the ordering of the spaces

at the base of the diagram. Furthermore, as a consequence of the assumptions

given by Inequalities (4.54), they only differ within groups of consecutive identical

colours. The lemma is established by applying Identity 4.13 to consecutive pairs

of spaces with identical colour in the graph for σ as required, until the graph for

τ is produced. �

Example 4.25 (equivalence of components of bipartite matrices). Let N = 5 and

{i1, i2, i3, i4, i5} = {2, 1, 2, 1, 1} (4.56)

then two permutations which achieve the ordering required by Inequalities (4.54)

are given by σ such that

σ{1, 2, 3, 4, 5} = {5, 2, 4, 1, 3} (4.57)

and τ such that

τ{1, 2, 3, 4, 5} = {2, 4, 5, 1, 3}. (4.58)

73


We then find that

(Rσ12345)

j1j2j3j4j52 1 2 1 1 = = =

= (Rτ12345)

j1j2j3j4j52 1 2 1 1 . (4.59)

Note that the indices have been retained in the diagrams for purely for clarity; once

an index has been specified, it plays no role. Indeed the tensor in this equation

is a rank-(0, 5) tensor, or in other words a rank-5 dual state. The second equality

uses the relation E(11)2 E

(11)5 R25 = E

(11)2 E

(11)5 I25, then E

(11)4 E

(11)5 R45 = E

(11)4 E

(11)5 I45.

The third equality is simply a trivial re-arrangement of the non-intersecting lines.♦

4.5 Construction of the F -matrix

In this section we construct the sun F -matrix via the tier-s partial F -matrix and

the tier-s F -matrix.

4.5.1 The tier-s partial F -matrix

We define the tier-s partial F -matrix which is used to construct the tier-s F -matrix.

It is defined in a similar way to the su2 partial F -matrix, and the diagrammatic

notation has been chosen to reflect this similarity. The tier-s partial F -matrix is

defined in terms of the tier-s monodromy matrix and a chain of identity matrices.

Definition 4.26. [tier-s monodromy matrix] Let N be a positive integer, then

the tier-s monodromy matrix is the rank-(N + 1, N + 1) tensor given by

(Rs

1···N |a(λa;λ1, . . . , λN))j1···jN ja

i1···iN ia

=(Rs

1a(λa, λ1)Rs2a(λa, λ2) · · ·Rs

Na(λa, λN))j1···jN ja

i1···iN ia= .

(4.60)

♦

74

4.5. Construction of the F -matrix

Definition 4.27 (tier-s partial F -matrix). Let N be a positive integer, then

the tier-s partial F -matrix is the rank-(N + 1, N + 1) tensor given by

F s1···N |a(λa;λ1, . . . , λN) = N (s)

a Rs1···N |a(λa;λ1, . . . , λ2) + E(s)

a I1···N |a

! = +

(4.61)

here we represented the components of F1···N |a diagrammatically by

(F s1···N |a

)j1···jN ja

i1···iN ia= . (4.62)

♦

Remark 4.28. As in the case of the su2 partial F -matrix, the diagrammatic nota-

tion is chosen so that if the arrow corresponding to the colour-s or non-colour-s

matrix

i) agrees with the hollow arrow then the diagram becomes a monodromy matrix

and if it

ii) disagrees with the hollow arrow then the diagram becomes a chain of identity

matrices

(provided all arrows have the same label s). ♦

4.5.2 The tier-s F -matrix

In this subsection the tier-s F -matrix is defined as a product of tier-s partial

F -matrices.

Definition 4.29. Let N ≥ 2, then the F -matrix is the rank-(N,N) tensor given

by

(F s1···N (λ1, . . . , λN))


=(F s1|2(λ2;λ1)F

s12|3(λ3;λ1, λ2) · · ·F s

1···(N−1)|N (λN ;λ1, . . . , λN−1))j1···jNi1···iN

75


= . (4.63)

♦

Remark 4.30. We adhere to our previous convention whereby each row of dots is

associated with a single hollow arrow. For example the hollow arrow on the far

right is associated with the uppermost row of dots. ♦

4.5.3 The F -matrix

In this subsection we define the F -matrix as a product of tier-s F -matrices.

Definition 4.31. Let N ≥ 2, then the F -matrix is the rank-(N,N) tensor given

by

(F1···N)j1···jNi1···iN

=

(F 2

1···NF3N ···1 · · ·F n

1···N)j1···jNi1···iN

, if n is even

(F 2N ···1F

31···N · · ·F n

1···N)j1···jNi1···iN

, if n is odd.

= . (4.64)

♦

76

4.6. Proof of the factorizing property

Remark 4.32. i) The order of indices at the base of the diagrammatic represen-

tation of the F -matrix depends on the parity of n.

ii) For n = 2 we have only one tier, and the diagram for the sun F -matrix

specializes to the su2 F -matrix obtained in Chapter 3. ♦

From its definition, a tier-s F -matrix admits non-trivial interaction only be-

tween the colours {1, . . . , s}. Any line bearing a colour greater than s is simply

peeled away from this part of the lattice by the definition of the tier-s R-matrix.

Hence we say that tier-s has a reduced, sus type behaviour. This decomposition

of the F -matrix into structures which are reduced iteratively, is reminiscent of

the nested Bethe Ansatz approach to the sun spin chains [17, 4]. We review the

nested Bethe Ansatz in Chapter 5 and make this correspondence more concrete,

by showing that the F -matrices and the Bethe eigenvectors of these models are

explicitly linked.

Recall that the su2 F -matrix obeys the recursion relation

F 21···N = F 2

1···(N−1)F21···(N−1)|N (4.65)

in which all action in the quantum space VN comes from the partial F -matrix

F 21···(N−1)|N . This recursion allows an inductive proof of results for the twisted

monodromy matrix operators† in the Y(su2) and Uq(su2) models as in [18]. A

similar recursion relation does not appear to exist in the cases n ≥ 3. As discussed

in [2], this means that methods which involve the recursive form of the F -matrix

may not translate into the higher rank models. This shortcoming is of no concern

for the purposes of this thesis.

4.6 Proof of the factorizing property

In this section we establish a number of lemmas which lead to a proof of the

factorizing property of the sun F -matrix. First we provide two lemmas that provide

a method of commuting tier-s R-matrices through the tier-s partial F -matrices. In

the first lemma the R-matrix passes through unchanged and in the second lemma

the R-matrix moves down a tier to become a tier-(s−1) R-matrix. When the tier-s

R-matrix is not in the leftmost position, it commutes through the tier-s partial

F -matrices according to the following lemma.

†The twist of the operator O1···N is defined to be F1···NO1···NF−11···N .

77


Remark 4.33. As opposed to the proof presented in Chapter 3 for the su2 case,

we do not require a version of the cocycle relation. Indeed the lemmas given in

this chapter may be specialized to n = 2 in order to make the cocycle relation

unnecessarily even in the case of su2 the F -matrix. ♦

The following lemma establishes a commutation relation between a tier-s par-

tial F -matrix and a tier-s R-matrix, Rsi(i+1) where i < N (that is, when the tier-s

R-matrix is not in the leftmost position on the diagram). The remaining case is

dealt with below.

Lemma 4.34. Let 1 ≤ i ≤ N − 2, then the tier-s partial F -matrix satisfies

F s1···(i+1)i···N |a(λa)R

si(i+1)(λi, λi+1) = Rs

i(i+1)(λi, λi+1)Fs1···N |a(λa).

! = .

(4.66)

♦

Remark 4.35. In this case, the spaces Vi and Vi+1 are reversed in the state space on

which the partial F -matrix acts but not in the state-space on which the R-matrix

acts. ♦

Proof. Using the definition of the tier-s partial-F matrix Equation (4.66) decom-

poses into the following two relations. Case 1 (N(s)a component). In this case we

have

= (4.67)

where the equality is due to the tier-s Yang-Baxter relation. Case 2 (E(s)a compo-

nent). In this case we have

= (4.68)

where the equality is due to a trivial re-arrangement of the diagram. �

Remark 4.36. In the special case of s = 2, i = 1, the above lemma reduces to

Lemma 3.45 of Chapter 3. ♦

78


When the tier-s R-matrix is in the leftmost position, it commutes through

tier-s partial F -matrices according to the following lemma.

Lemma 4.37. This lemma describes the commutation of two tier-s partial F -

matrices with a tier-s R-matrix. The tier-s partial F -matrix satisfies

F s1···N |b(λb)F

s1···Nb|a(λa)R

sab(λa, λb) = R

(s−1)ba (λb, λa)F

s1···N |a(λa)F

s1···Na|b(λb)

! = . (4.69)

♦

Here the dotted line is used to demarcate the tiers, this device is used whenever

more than one tier occurs in a diagram†.

Remark 4.38. In this case, not only are the a and b-spaces reversed in the state

space on which the partial F -matrices act, but also in the spate space on which

the R-matrix acts. ♦

Proof. Applying the definition of the tier-s partial-F matrix, Equation (4.69) de-

composes into the following four relations. Case 1 (N(s)a N

(s)b component). In this

case we have

=

= = (4.70)

where the first equality is due to Lemma 3.24 and the second equality is due to

the following observation

†In this and following diagrams, where available, colour is used to highlight lines of particularinterest.

79


= − + = − + =

(4.71)

which follows from Identity 4.12 (iv). The third equality in Equation (4.70) is

due to the colour conservation of the tier-(s − 1) R-matrix. Case 2 (N(s)a E

(s)b

component). In this case we have

=

= =

= (4.72)

where the first equality is due to the tier-k unitarity relation, the second and third

equalities are due to trivial re-arrangement of the diagram. The final equality is

established by the following observation

= − + = (4.73)

which is a consequence of Identity 4.12 (iii). Case 3 (E(s)a N

(s)b component). In this

case we have

=

80


= = (4.74)

where the first and second equalities are due to trivial redrawing of the diagrams.

The final equality is established similarly to Case 2. Case 4 (E(s)a E

(s)b component).

In this case we have

=

= =

= (4.75)

Where the first and third equalities are due to trivial redrawing of the diagrams and

the second equality which is due to Lemma 4.20. The final equality is established

similarly to Cases 2 and 3. �

The following lemma allows us to commute a tier-s R-matrix through a tier-s

F -matrix.

Lemma 4.39. [commutation of a tier-s F -matrix with a tier-s R-matrix] Let

N ≥ 2 and let 1 ≤ i ≤ N − 1. Then we have

F s1···(i+1)i···N (λ1, . . . , λi+1, λi, . . . , λN)R

si(i+1)(λi, λi+1)

= R(s−1)(i+1)i(λi+1, λi)F

s1···N(λ1, . . . , λN)

♦

Proof. Diagrammatically, Equation (4.39) is

81


= . (4.76)

We apply Lemma 4.34, N − 1− i times, to obtain

F sσ(1)|σ(2)F

sσ(1)σ(2)|σ(3) · · ·F s

σ(1)···σ(N−1)|σ(N)Ri(i+1)

= F sσ(1)|σ(2)F

sσ(1)σ(2)|σ(3) · · ·F s

σ(1)···σ(i)|σ(i+1)Rsi(i+1)F

s1···(i+1)|(i+2) · · ·F s

1···(N−1)|N (4.77)

where σ is the permutation such that

σ{1, . . . , N} = {1, . . . , (i+ 1), i, . . . , N}. (4.78)

Diagrammatically, the right hand side of Equation (4.77) is

. (4.79)

The result is then established by Lemma 4.37. �

The following lemma makes the factorizing property of the F -matrix almost

82


immediate.

Lemma 4.40 (commutation of an F -matrix with an R-matrix). Let N ≥2 and let 1 ≤ i ≤ N−1. Then the sun R-matrices commute through the F -matrix

according to the relation†

F1···(i+1)i···N (λ1, . . . , λ(i+1), λi, . . . , λN)Ri(i+1)(λi, λi+1)

=

I(i+1)iF1···N(λ1, · · · , λN), if n is even

Ii(i+1)F1···N(λ1, · · · , λN), if n is odd.(4.80)

♦

Proof. Diagrammatically, Equation (4.80) is

†The colours in the diagram (where colour display is available) correspond to the case inwhich n is odd. In the case where n is even, the colours in the lower part of the diagram wouldbe reversed. The case in which n is even has been omitted for brevity.

83


= . (4.81)

We recall that by Identity 4.19 Rn12 = R12 and R1

12 = I12, then proceed by induction

on n. The base case which corresponds to n = 2 is established by Lemma 4.39

(with s = 2 in that lemma). The inductive step is given by applying Lemma 4.39

to Equation (4.80) to obtain

F 21···(i+1)i···N · · ·F n−1

N ···i(i+1)···1Fn1···(i+1)i···NRi(i+1)

= F 21···(i+1)i···N · · ·F n−1

N ···i(i+1)···1R(i+1)iFn1···N (4.82)

when n is even and

F 2N ···i(i+1)···1 · · ·F n−1

N ···i(i+1)···1Fn1···(i+1)i···NRi(i+1)

= F 2N ···i(i+1)···1 · · ·F n−1

N ···i(i+1)···1R(i+1)iFn1···N (4.83)

when n is odd. Diagrammatically the right hand side is

84


(4.84)

as required. �

We are now in a position to prove Theorem 4.1. Recall that this theorem

requires us to establish Equation (4.2) which we restate here for convenience

Fσ(1)···σ(N)Rσ1···N = Iσρ(1)···ρ(N)F1···N = F1···N (4.85)

where ρ is given by Equation (4.1).

Proof (Proof of Theorem 4.1). First we make an observation that allows us to pass

a decomposed bipartite matrix through the F -matrix one R-matrix at a time. Let

σ and τ be two permutations of the set {1, . . . , N}. Then it follows from the

definition of Rσ1···N that

Rτσ(1)···σ(N)R

σ1···N = Rτσ

1···N (4.86)

and

Iτσ(1)···σ(N)Iσ1···N = I

(τσ)1···N (4.87)

since (τ )(σ) = ρn−1τρn−1ρn−1σρn−1 = ρn−1τσρn−1 = (τσ). So if we have

Fτ(1)···τ(N)Rτ1···N = Iτρ(1)···ρ(N)F1···N (4.88)

and

Fσ(1)···σ(N)Rσ1···N = Iσρ(1)···ρ(N)F1···N (4.89)

85


then

Fτσ(1)···τσ(N)Rτσ1···N = Iτρσ(1)···ρσ(N)Fσ(1)···σ(N)R

σ1···N = I

(τσ)ρ(1)···ρ(N)F1···N (4.90)

as required. We also observe that Ri(i+1) = Rσi for σi such that

σi{1, . . . , N} = {1, . . . , (i+ 1), i, . . . , N} (4.91)

and that the σi form the generating set from a standard presentation of the sym-

metric group (with relations corresponding to unitarity and the Yang-Baxter rela-

tion). These observations show that Lemma 4.40 is sufficient to establish Theorem

4.1 by passing the Rσi through the F -matrix one at a time. �

4.7 Examples

In this section we present some examples which clarify the structure of the fac-

torized expression for the F -matrix given in Equation (4.64). We firstly consider

the su2 case of the F -matrix in which it reduces to the expression discussed in

[18] and Chapter 3. We then study in more detail the next simplest case, the su3

specialization of the F -matrix. Finally, we provide the sum expression for the sun

F -matrix of [2] and relate it to the factorized expression for the F -matrix.

4.7.1 The su2 F -matrix

We specialize the F -matrix to n = 2 to obtain

F1···N = F 21···N = F 2

1|2F212|3 · · ·F 2

1···(N−1)|N . (4.92)

Recall that

F 21···(i−1)|i = E

(2)i I1···(i−1)|i +N

(2)i R2

1···(i−1)|i (4.93)

86

4.7. Examples

for all 2 ≤ i ≤ N . The expression for the su2 F -matrix given in [18] and Chapter

3 is recovered here since Rn12 = R2

12 = R12. Diagrammatically we have

(F1···N)j1···jNi1···iN

= (4.94)

where the labelling of the tier is now redundant. We can realize the diagrams for

the su2 elementary matrices E(11)1 and E

(22)1 by omitting the label 2 from the dia-

grams for N(2)1 and E

(2)1 . Under these caveats, the present diagrammatic notation

corresponds precisely to that of 3.

The diagram in Equation (4.94) represents a general component of the su2 F -

matrix. In the following example we specialize the lower colours {i1, . . . , iN} and

redraw the diagram to demonstrate the sorting property of the su2 F -matrix. This

sorting property can be taken as the defining quality of the expression presented

in [2].

By applying the definition of the tier-2 partial F -matrix to a component of the

su3 F -matrix, we move the 2-colours to the right of the 1-colours. This provides

an algorithm for recognizing each component of the su3 F -matrix as a component

of Rσ where σ is a permutation which sorts the specified colours.

Example 4.41 (ordering property on the su2 F -matrix). Let n = 3 and let N = 7.

We consider the component of the su2 F -matrix with

{i1, i2, i3, i4, i5, i6, i7} = {1, 2, 2, 1, 2, 1, 1}. (4.95)

Note that this component is a rank-(0, 7) tensor (indeed a rank-7 dual state) since

only free indices are the jk for 1 ≤ k ≤ 7, therefore rearranging the lines at the

base of the diagram has no effect. The indices are retained in the diagrams for

clarity. Diagrammatically we have

87


=

= (4.96)

where the first equality is due to the definition of the tier-2 partial F -matrix and

the second equality is a trivial re-arrangement of the non-interacting lines. ♦

Remark 4.42. This simple procedure of stripping-off the non-interacting lines pro-

duces a bipartite matrix which sorts the specified colours into monotonically de-

creasing order from left to right. ♦

If n = 2, {i1, . . . , iN} is an ordered set of integers taking values in {1, 2} and σ

is a permutation of {1, . . . , N} such that iσ(1) ≤ · · · ≤ iσ(N), then

(F1···N)j1···jNi1···iN

= (Rσ1···N)


(4.97)

by the definition of the tier-2 partial F -matrix. Note that there is some degree of

freedom in the choice of σ due to Lemma 4.24.

88

4.7. Examples

4.7.2 The su3 F -matrix

We specialize the F -matrix to n = 3 to obtain

F1···N = F 21···NF

31···N = (F 2

1|2F212|3 · · ·F 2

1···(N−1)|N )(F31|2F

312|3 · · ·F 3

1···(N−1)|N ). (4.98)

Recall that

F 21···(i−1)|i = E

(2)i I1···(i−1)|i +N

(2)i R2

1···(i−1)|i (4.99)

F 31···(i−1)|i = E

(3)i I1···(i−1)|i +N

(3)i R3

1···(i−1)|i (4.100)

for all 2 ≤ i ≤ N .

As in the case of the su2 F -matrix above, we specialize Equation (4.98) by

fixing the lower colours {i1, . . . , iN} and demonstrate the sorting property of the

su3 F -matrix using the diagrammatic notation.

By applying the definition of the tier-2 partial F -matrix to a component of

the su3 F -matrix, we move the 2-colours to the right of the 1-colours, as occurred

in the previous example. Then by applying the definition of the tier-2 R-matrix

and the tier-3 partial F -matrix, we move the 3-colours to the right of the 1 and

2-colours. This provides an algorithm for recognizing each component of the su3

F -matrix as a component of Rσ where σ is a permutation which sorts the specified

colours.

Example 4.43 (ordering property of the su3 F -matrix). Let n = 3 and let N = 7.

We consider the component of the su3 F -matrix with

{i1, i2, i3, i4, i5, i6, i7} = {2, 3, 1, 1, 2, 3, 2}. (4.101)

Note that this component is a rank-(0, 7) tensor since the only free indices are the

jk for 1 ≤ k ≤ 7. Diagrammatically we have

89


=

90

4.7. Examples

=

=

= (4.102)

where the first equality is due to the definition of the tier-2 partial F -matrix,

the second equality is a trivial re-arrangement of the non-interacting lines, the

third equality is due to the definition of the tier-2 R-matrix and the tier-3 partial

F -matrix and the final equality is again due to a trivial re-arrangement of the

lines. ♦

Remark 4.44. As in the previous case, this simple procedure of stripping-off the

91


non-interacting lines produces a bipartite matrix which sorts the specified colours

into monotonically decreasing order from left to right. ♦

If n = 3, {i1, . . . , iN} is an ordered set of integers taking values in {1, 2, 3} and

σ is a permutation of {1, . . . , N} such that iσ(1) ≤ · · · ≤ iσ(N) then

(F1···N)j1···jNi1···iN

= (Rσ1···N)


. (4.103)

This equality is established by the definition of the tier-2 partial F -matrix, the

tier-2 R-matrix and the tier-3 partial F -matrix. This idea extends to the general

case naturally; in the following subsection we recognise the F -matrix is an object

which sorts the lower colours.

4.7.3 Generalisation to sun

We establish that the sun F -matrix is equivalent to the sum expression given by

Albert et al. in [2].

Theorem 4.45. [equivalence to the expression for the sun F -matrix of [2]] Let

SN be the symmetric group of order N , then the factorized expression for the sun

F -matrix is equivalent to the expression of [2].

F1···N =∑

σ∈SN

∑∗

α1,...,αN

N∏

i=1

(E

(αi)σ(i)

)Rσ

1···N (4.104)

where the sum∑∗ is over all sequences of integers α1, . . . , αN ∈ {1, . . . , N} satis-

fying the conditions

αi ≤ αi+1, if σ(i) < σ(i+ 1), (4.105)

αi > αi+1, if σ(i) > σ(i+ 1). (4.106)

♦

Proof. We isolate each component of the F -matrix by fixing the ik for 1 ≤ k ≤ N .

On the left hand side, we apply the algorithm discussed in the previous sub-

sections to the factorized expression of the F -matrix. For each 2 ≤ s ≤ n, we

consider the lines with index ik fixed to be colour-s. We then apply the definition

of the tier-r R-matrix for 2 ≤ r ≤ s − 1 and the tier-s partial F -matrix to those

lines and trivially rearrange the diagram so that the bases of those lines are to the

far left of the others.

92

4.8. Further properties of the F -matrix

This algorithm produces the result that

(F1···N)j1···jNi1···iN

= (Rσ1···N)


(4.107)

where σ is the unique permutation such that iσ(1) ≤ · · · ≤ iσ(N) with the further

conditions that σ(k) < σ(k + 1) if ik = ik+1 = s, and (n− s) is even, and σ(k) >

σ(k + 1) if ik = ik+1 = s, and (n− s) is odd.

On the right hand side, since the colour-s matrices E(s)1 are projections, the

only summand which survives the specialization of the ik is

(N∏

k=1

(E

(αk)τ(k)

)Rτ

1···N

)j1···jN

i1···iN

= (Rτ1···N)


(4.108)

where αk = iτ(k) and τ is the unique permutation such that iτ(1) ≤ · · · ≤ iτ(N) and

τ(k) < τ(k + 1) when αk = αk+1. Now since the permutations σ and τ only differ

in groups of equal colours, Lemma 4.24 asserts that

(Rσ1···N)


= (Rτ1···N)


(4.109)

as required. �

4.8 Further properties of the F -matrix

In this section we establish that the sun F -matrix is lower triangular and invertible.

We then provide a diagrammatic construction of the inverse.

4.8.1 Lower triangularity and invertibility

Definition 4.46. For convenience, we define the following rank-(2, 2) tensor

(∆12(λ1, λ2))j1j2i1i2

=

a(λ1 − λ2), if i1 = i2, i1 = j1, i2 = j2

b(λ1 − λ2), if i1 > i2, i1 = j1, i2 = j2

b(λ2 − λ1), if i1 < i2, i1 = j1, i2 = j2

0, otherwise.

(4.110)

♦

93


Remark 4.47. Recall that in our present normalization, the weight a(λ1, λ2) = 1.

Note that provided λi 6= λj for all i 6= j, the flattened representation of ∆12 is a

diagonal matrix with nonzero components on the diagonal so is invertible with its

inverse is given by

(∆−1

12 (λ1, λ2))j1j2i1i2

=

1a(λ1−λ2)

, if i1 = i2, i1 = j1, i2 = j2

1b(λ1−λ2)

, if i1 > i2, i1 = j1, i2 = j2

1b(λ2−λ1)

, if i1 < i2, i1 = j1, i2 = j2

0, otherwise.

(4.111)

♦

The following lemma provides the inductive step for the proof of Theorem 4.49

which follows.

Lemma 4.48. The flattened representation of the tensor

E(s)1

N∏

k=2

(S(s)k

)R1|2···N (4.112)

is lower triangular with diagonal components given by

(E

(s)1

N∏

k=2

(S(s)k

)R1|2···N

)j1···jN

i1···iN

=

(N∏

k=2

∆1k

)j1···jN

i1···iN

(4.113)

where ik = jk for all 1 ≤ k ≤ N . ♦

Proof. We have

E(s)1

N∏

k=2

(S(s)k

)R1|2···N = E

(s)1

N∏

k=2

(S(s)k

)R1|2···NS

(s)1

= E(s)1

N∏

k=2

(S(s)k

)R1|2···NS

(s−1)1 + E

(s)1

N∏

k=2

(S(s)k

)R1|2···NE

(s)1 (4.114)

! =

= + (4.115)

94


where the first equality is by the colour conservation of the R-matrices and the

second is simply a decomposition of the tensor into two components.

The tensor represented on the right hand side of Equation (4.115) is decom-

posed as follows. The first term corresponds to the strictly lower blocks and the

second to the diagonal blocks. The second term corresponds to the diagonal blocks.

If N = 2, then i2 = j2 by colour conservation and the diagonal subblocks are them-

selves diagonal. If N > 2, then we consider two cases. Let the colour connecting

the rightmost pair of R-matrices be k (so that the rightmost R-matrix becomes

(R12)j1j2ki2

). Case 1 (k < s). In this case, i2 = s and j2 = k by colour conservation

which implies i2 > j2 and we have all nonzero components in the strictly lower

subblocks. Case 2 (k = s). In this case, since k = j1 we have i2 = j2 by colour

conservarion, (note that these components correspond to the diagonal subblocks

of the diagonal blocks). Since k = s, removing the rightmost R-matrix leaves us

with a tensor of the same form as the second term in Equation (4.115) but with

N reduced by 1. Repeating this argument until N = 2, gives the desired result.

The value of the diagonal components is obtained by observing that by colour

conservation, when i1 = j1 the colour i1 must propagate along the space V1 all

the way to the top of the diagram. Colour conservation then forces ik = jk for all

2 ≤ k ≤ N , so the vertices are all either a or b weighted and the result follows. �

The following theorem establishes the lower triangularity of the flattened rep-

resentation of the F -matrix.

Theorem 4.49 (lower triangularity of the F -matrix). The flattened repre-

sentation of the sun F -matrix is lower triangular and, has nonzero diagonal com-

ponents provided λi 6= λj for all i 6= j. These diagonal components corresponding

to ik = jk for all 1 ≤ k ≤ N are given by

(F1···N )j1···jNi1···iN

=

(N−1∏

k=1

mk−1∏

l=1

∆kσk(l)

)j1···jN

i1···iN

(4.116)

where σ is a permutation of {1, . . . , N} such that iσ(1) ≤ · · · ≤ iσ(N), σk is

the restriction of σ to a permutation of the set {k, . . . , N} and mk is such that

σk(mk) = k. ♦

Proof. For any given {i1, . . . , iN} taking values in {1, . . . , n}, by Lemma 4.45 dia-

95


grammatically we have

(F1···N)j1···jNi1···iN

= (Rσ1···N)


= (4.117)

where σ is any permutation satisfying iσ(1) ≤ · · · ≤ iσ(N) and the rectangle con-

taining the symbol Rσ stands for a bipartite matrix corresponding to σ.

To establish the lower triangularity property, we proceed by induction on N .

In the base case for which N = 2, there are two possible permutations σ. Case

1 (σ{1, 2} = {1, 2}). In this case, there is no interaction between the spaces and

Rσ = I12 which is diagonal. Case 2 (σ{1, 2} = {2, 1}). In this case, Theorem

4.49 corresponds to Lemma 4.48 (with N = 2 in that lemma). So the base case is

established.

We now consider the line in the bipartite graph corresponding to the space V1,

which connects the indices i1 and j1. Since by Lemma 3.24 we are free to rearrange

the lines in Rσ, we can move the line corresponding to the space V1 to the bottom

of the diagram to produce

(F1···N)j1···jNi1···iN

=(Rσ(m)|σ(m−1)···σ(1)R

σ2···N

)j1···jNi1···iN

=

(4.118)

where m is such that σ(m) = 1 and σ is the permutation of {2, . . . , N} which

results from removing the line corresponding to the space V1 from the bipartite

graph for σ.

Since we have iσ(l) ≤ iσ(m) for all l < m, Rσ(m)|σ(m−1)···σ(1) is lower triangular by

Lemma 4.48. By the inductive assumption the flattened representation of Rσ2···N

is lower triangular, then since the flattened representation of Rσ1···N is the product

of two lower triangular matrices, it is itself lower triangular as required.

The value of the diagonal components of the F -matrix follows from iterative

application of Lemma 4.48. �

Example 4.50 (lower triangularity of the F -matrix). In the case of the final dia-

gram in Equation (4.102) the decomposition which appears in the inductive step

96


becomes

= (4.119)

Colour conservation ensures that the diagram has weight zero for all for j1 > i1 = 2,

these weights correspond to the strictly upper blocks in the flattened representation

of R1|3475. The diagonal blocks of R1|3475 for which j1 = i1 = 2 are themselves

diagonal by colour conservation. The vertices are therefore forced to be a or

b weighted, specifically, the line connecting i1 to j1 may be removed from the

diagram and replaced with the factor

a(λ1 − λ5)a(λ1 − λ7)b(λ1 − λ4)b(λ1 − λ3) = b(λ1 − λ4)b(λ1 − λ3). (4.120)

It follows that this particular diagonal component is nonzero provided that λ1 6= λ4

and λ1 6= λ3.

Iterating this procedure over the lines connecting ik to jk for 2 ≤ k ≤ 6

decomposes the entire tensor into a product of tensors which are individually lower

block diagonal with diagonal blocks on the diagonal when written in flattened

form. ♦

A corollary of the lower triangularity of the F -matrix is that it is invertible,

the following corollary establishes this fact.

Corollary 4.51 (F -matrix invertibility). It is an immediate consequence of

Theorem 4.49 that the sun F -matrix is invertible provided that λi 6= λj for all

i 6= j. ♦

97


4.8.2 Construction of the inverse F -matrix

We present a diagrammatic construction of the inverse of the sun F -matrix. This

construction parallels that of the F -matrix itself. The inverse is constructed via an

intermediate object, the F ∗-matrix. The flattened representation of the product

of the F -matrix and the F ∗-matrix is diagonal; the inverse of the F -matrix is then

obtained by multiplying the F ∗-matrix by the inverse of this diagonal matrix.

The construction of the F ∗-matrix parallels that of the F -matrix. First we

define the tier-s partial F ∗ matrix. Then the tier-s F ∗-matrix is defined as a

product of tier-s partial F ∗-matrices. Finally the F ∗-matrix is defined as a product

of the tier-s F ∗-matrices.

The diagrammatic representation of the tier-s partial F ∗-matrix, the tier-s

F ∗-matrix and the F ∗-matrix is essentially the 180 degree rotation of the diagram-

matic representation of the corresponding non-starred objects. Indeed with some

refinement this could be taken as their definition.

Definition 4.52 (tier-s partial F ∗-matrix). Let N be a positive integer, then

the tier-s partial F ∗-matrix is the rank-(N + 1, N + 1) tensor given by

F ∗s1···N |a(λa;λ1, · · · , λN) = I1···N |aE

(s)a +Rs

1···N |a(λa;λ1, · · · , λN)N(s)a

! = +

(4.121)

here we represented the components of the tier-s partial F ∗-matrix diagrammati-

cally by

(F ∗s1···N |a

)j1···jN ja

i1···iN ia= . (4.122)

♦

Remark 4.53. The convention for the tier-s partial F -matrices is maintained here:

when an arrow corresponding to a colour-s of non-colour-s matrix agrees with the

hollow arrow the partial F -matrix becomes a monodromy matrix and when they

disagree, it becomes a chain of identity matrices. ♦

Definition 4.54 (tier-s F -matrix). LetN ≥ 2, then the F -matrix is the rank-

(N,N) tensor given by

(F ∗s1···N)


98


=(F ∗s(N−1)···1|N(λN ;λ1, · · · , λN−1) · · ·F ∗s

21|3(λ3;λ1, λ2)F∗s1|2(λ2;λ1)


= . (4.123)

♦

Definition 4.55 (the F∗-matrix). Let N ≥ 2, then the F ∗-matrix is the rank-

(N,N) tensor given by

(F ∗1···N)


=

(F ∗n

N ···1 · · ·F ∗31···NF

∗2N ···1)


, if n is even

(F ∗nN ···1 · · ·F ∗3

N ···1F∗21···N)


, if n is odd.

= . (4.124)

Where the order of indices at the top of the diagram depends on the parity of n.♦

Since the F ∗-matrix is closely related to the F -matrix, all of the results relating

to the F -matrix carry over to the F ∗-matrix. Of particular interest here is the lower

triangularity and the value of the diagonal components. Consistent with the 180

degree rotation, the F ∗-matrix has the property that it sorts the upper colours in

99


weakly increasing order from left to right (recall that the F -matrix sorted the lower

colours in decreasing order from left to right). We establish two results which lead

to an explicit form for the inverse of the F -matrix.

Theorem 4.56 (lower triangularity of the F∗-matrix). The flattened repre-

sentation of the sun F ∗-matrix is lower triangular and, has nonzero diagonal com-

ponents provided λi 6= λj for all i 6= j. These diagonal components corresponding

to ik = jk for all 1 ≤ k ≤ N are given by

(F ∗1···N)


=

(N−1∏

k=1

nk−1∏

l=1

∆kτk(l)

)j1···jN

i1···iN

(4.125)

where τ is a permutation of {1, . . . , N} such that jτ(1) ≥ · · · ≥ jτ(N), τk is the

restriction of τ to a permutation of the set {k, . . . , N} and nk is such that τk(nk) =

k. ♦

Proof. The proof of this theorem is essentially the same to that of Theorem 4.49

rotated 180 degrees. One thing to note is that in the intermediate steps when

fixing the jk for all 1 ≤ k ≤ N , we write

(F ∗1···N)


=(Rτ (−1)

τ(1)···τ(N)


(4.126)

and that the condition on τ is to sort the indices jk in increasing order from left

to right as opposed to decreasing order in the case of the F -matrix. These points

are simply bookkeeping consistent with the 180 degree rotations involved. �

In the following lemma we establish the form of the product of the product of

the F -matrix with the F ∗-matrix. A corollary of this is the explicit construction

of the inverse of the F -matrix.

Lemma 4.57. F1···NF∗1···N is diagonal and given by

F1···NF∗1···N =

∏

1≤k<l≤N

∆kl. (4.127)

♦

Proof. Diagrammatically we have

(F1···NF∗1···N )

j1···jNi1···N

=(Rσ

1···NRτ−1

τ(1)···τ(N)


100


= (Rσ1···N)

k1···kNi1···iN

(Rτ−1

τ(1)···τ(N)

)j1···jNk1···kN

= (4.128)

where σ is a permutation of {1, . . . , N} such that iσ(1) ≤ · · · ≤ iσ(N), and τ is a

permutation of {1, . . . , N} such that jτ(1) ≥ · · · ≥ jτ(N). Let σ(m) = τ(l) = a.

Then we can move the line connecting ia to ja to the exterior of the diagram on

either the left or the right in the following way. We have

= =

(4.129)

where σ and τ are the restrictions of σ and τ respectively to the set {1, . . . , a −1, a+ 1, . . . , N}. Recall that the indices are monotonically decreasing from left to

right on the bottom of the diagram and on monotonically increasing from left to

right at the top of the diagram. It follows from colour conservation that for each

nonzero component of F1···NF∗1···N we have ia ≥ ka ≥ ja and ia ≤ ka ≤ ja for all

1 ≤ a ≤ N which implies ia = ja for all 1 ≤ a ≤ N .

By the symmetry between the F and F ∗-matrices, we can choose τ−1 to be

the permutation corresponding to a 180 degree rotation of the bipartite graph

corresponding to σ. We can write this as τ = ρσρ†. We have

F1···NF∗1···N =

(N−1∏

k=1

mk−1∏

l=1

∆kσk(l)

)(N−1∏

k=1

mk−1∏

l=1

∆kτk(l)

)

=

(N−1∏

k=1

mk−1∏

l=1

∆kσk(l)

)(N−1∏

k=1

N−k+1∏

l=mk+1

∆kσk(l)

)

†Recall that ρ is the permutation such that ρ{1, . . . , N} = {N, . . . , 1}.

101


=

N−1∏

k=1

N−k+1∏

l=1,l 6=mk

∆kσk(l) =∏

1≤k<l≤N

∆kl (4.130)

Where the first equality is by Theorems 4.49 and 4.56, the second equality is due

to the relation τ = ρσρ and the last two equalities are simply re-arrangements of

the terms. Note that in Equation (4.130) we are free to commute the terms since

they are all scalar multiples of identity matrices. �

In the following theorem, we explicitly state the inverse of the F -matrix.

Theorem 4.58. The inverse of the F -matrix is given by the F ∗-matrix multiplied

by a diagonal matrix. We have

F−11···N = F ∗

1···N

∏

1≤k<l≤N

∆−1kl . (4.131)

♦

Proof. Theorem 4.58 is an immediate consequence of the Lemma 4.57. �

Finally for completeness we present a diagrammatic representation of the tensor∏1≤k<l≤N ∆−1

kl . If ∆−112 be represented diagrammatically as

(∆−1

12

)j1j2i1i2

= (4.132)

then

( ∏

1≤k<l≤N

∆−1kl

)j1···jN

i1···iN

= . (4.133)

This allows us to write out the factorization of Rσ in full. We have

Rσ1···N = F ∗

σ(1)···σ(N)

( ∏

i≤k<l≤N

∆−1kl

)Iσρ(1)···ρ(N)F1···N . (4.134)

102

4.9. Conclusion

Diagrammatically this is

=

. (4.135)

4.9 Conclusion

In Sections 4.2 to 4.5 we presented a new expression for the F -matrix of quantum

spin chains based on the algebras Y(sun) and Uq(sun). The factorized expression

for the sun F -matrix given by Equation (4.64) is similar to that of [18] for Y(su2)

and Uq(su2), in the sense that it is factorized into a product of partial F -matrices.

The tier-s of the sun F -matrix only exhibits non-trivial interaction between the

colours {1, . . . , s}. Hence the decomposition of the sun F -matrix into tiers is sim-

103


ilar to the nested Bethe Ansatz construction of eigenstates of the transfer matrix.

In Subsection 4.6 we also proved that the factorized expression for the F -matrix

satisfies the factorizing property given by Theorem 4.1. The proof was based on

Lemmas 4.34 and 4.37, which give information about the commutation of a tier-s

R-matrix through tier-s partial F -matrices. Having established these two lemmas,

Theorem 4.1 was almost immediate since the factorized expression for the F -matrix

is a product of tier-s partial F -matrices, with s taking values in {2, . . . , n}.In Section 4.7 we studied the special cases n = 2, 3 of the factorized expression

for the F -matrix. The main observation was that all components of the F -matrix

are given by expressions of the form (F1···N )j1···jNi1···iN

= (Rσ1···N )


, where σ is any

permutation which sorts the lower colours of the diagram into weakly decreasing

order from left to right. This enabled us to show that the factorized expression

for the F -matrix is equivalent to the sum expression for the F -matrix given in

Equation (4.104), as obtained in [2]. In Section 4.8, we started from the sum

expression for the F -matrix and gave diagrammatic proofs of its lower triangularity

and invertibility. We then went on to express the F ∗-matrix F ∗1···N in terms of

partial F ∗-matrices. This was achieved by making an appropriate definition for the

partial F ∗-matrices such that the upper colours are sorted in increasing order from

left to right, whereas F1···N sorts the lower colours in decreasing order from left to

right. The diagrammatic representation of the F ∗-matrix is in close correspondence

to the diagrammatic representation of the F -matrix rotated 180 degrees.

104

Chapter 5

On the nested algebraic Bethe

Ansatz, sun Bethe eigenvectors

and elementary matrices

5.1 Introduction

In this chapter we present new expressions for the eigenvectors of the sun transfer

matrix as components of the sun F -matrix following [20]. In order to facilitate this

we use diagrammatic tensor notation to write the sun eigenvectors and describe

algebraic Bethe Ansatz and the nested algebraic Bethe Ansatz following [15] and

[4] respectively.

The algebraic Bethe Ansatz is a method for calculating the eigenvectors and

eigenvalues of the su2 transfer matrix, t(λa) = traRa|1···N(λa). This is of importance

because the transfer matrix commutes with certain spin chain Hamiltonians and

this implies that they share a common set of eigenvectors. So solving the transfer

matrix eigenvalue problem solves the Hamiltonian system. Specifically

H(J) =J

2[η]

d

dλa

log t(λa|λ1 · · ·λN)

∣∣∣∣λ1···λN=λa

(5.1)

where the Hamiltonian corresponding to Y(su2) is that of the XXX spin-12chain

(with [η] = η) and the Hamiltonian corresponding to the Uq(su2) is that of the XXZ

spin-12chain (with [η] = sinh η). Proof of Equation (5.1) and the commutation of

the Hamiltonian with the transfer matrix are provided in Appendix A.

The nested algebraic Bethe Ansatz is a generalization of the algebraic Bethe

Ansatz to models with sun symmetry. It proceeds by induction on n and takes the

105

Chapter 5. On the nested algebraic Bethe Ansatz, sun Bethe eigenvectors andelementary matrices

algebraic Bethe Ansatz as its base-case.

In this chapter we also present an expression for the elementary matrices (given

by Equation (4.8)) in terms of components of the sun monodromy matrix. In su2,

this representation allows the local spin operators (S+1 = E

(12)1 , S−

1 = E(21)1 , Sz

1 =12(E

(11)1 − E

(22)1 )) to be written in terms of monodromy matrix elements and has

proved useful in the calculation of correlation functions [12, 14]. In the case of

su2, this result was provided in [13] and relied on use of the F -matrix. In [9] the

su2 result was provided without use of the F -matrix. We provide an easy proof of

the expression for the elementary matrices in the more general case of sun using

diagrammatic tensor notation.

In Section 5.2 we build on the diagrammatic notation established in Chapter

2 and make some new definitions required later in the chapter. In Section 5.3

we derive an expression for the su2 eigenvectors using the algebraic Bethe Ansatz

and diagrammatic tensor notation. In Section 5.4 we derive an expression for

the sun eigenvectors using the nested algebraic Bethe Ansatz and diagrammatic

tensor notation. In Section 5.5 we present an expression for the eigenvectors of the

sun transfer matrix as components of the sun F -matrix. In Section 5.6 we derive

an expression for the elementary matrices in terms of the components of the sun

monodromy matrices. Finally in Section 5.7 we summarize the chapter.

5.2 Definitions

5.2.1 Multi-index notation

In the following definition we establish a notation for combining multiple indices

into a single multi-index. This corresponds to recasting the tensor such that some

of the underlying vector spaces are combined into a single vector space spanned

by tensor products of the basis elements of the original vector spaces. This is

of particular use when we wish to simplify a certain type of diagram whilst still

retaining all of the data encoded in the original.

Definition 5.1. This definition concerns permutations which permute groups of

consecutive indices, but do not rearrange the indices within each group.

Let q be a positive integer, then for all p in {1, . . . , q} let Np be a non-negative

integer, let Mp =∑p

k=1Nk and let Mp = {M(p−1) + 1, . . . ,Mp} be an ordered set.

Let σ be a permutation of the set {1, . . . ,Mq} such that for all p in {1, . . . , q}and k in {0, . . . , Np − 1}, σ(Mp − k) = σ(Mp) − k. Let Ip = {ir|r ∈ Mp} and

106

5.2. Definitions

Jp = {jr|r ∈ Mp} be ordered sets for all p in {1, . . . , q}. We call each Ip a lower

multi-index and each Jp an upper multi-index.

A diagram for a bipartite matrix involving a bold line with lower multi-index

Ip and upper multi-index Jp is equivalent to the diagram in which the bold line

is expanded into Np mutually non-intersecting lines with lower and upper indices

being the entries of Ip and Jp respectively each ordered from right to left.

If s is in {1, . . . , n}, Ip = s means i = s for all i in Ip and likewise for the

Jp = s. ♦

Example 5.2 (multi-indices). Using multi-indices, we have

= . (5.2)

♦

Later in this chapter we will use multi-indices to greatly simplify the diagram

for the sun Bethe eigenvectors.

Analogously we will also represent multiple crossing lines of identity matrices

as

= . (5.3)

5.2.2 A notation for an identity with switching rapidity

A feature of the nested Bethe Ansatz is that tensors arise which have a peculiar

property captured by the following definition. The tensors which have this property

ultimately have their coefficients set to zero.

Definition 5.3. Let λp and λq be rapidities (i.e. elements of C) then we represent

107


the components of a switching identity diagrammatically as

(Ip∽q)jqip= . (5.4)

♦

Remark 5.4. Note that the indication of the rapidities λp, λq in Equation (5.4) is

necessary since the indices ip, jq may in some cases be fixed. In such cases it would

otherwise become ambiguous as to which section of line possessed which rapidity.♦

Example 5.5 (rapidity changing alone a line). In the following example we use the

device defined in Equation (5.4) to indicate that the two R-matrices do not depend

on the same rapidity. We have

(R(λ1, λp))j1k

i1ip(R(λ2, λq))

j2jqi2k

= . (5.5)

Note that even if we fix the indices such that ip = 1, jq = 2 (for example), the

rapidities contributing to the R-matrices are still unambiguously specified. ♦

5.2.3 A summation convention

In this subsection, we establish a diagrammatic convention which will be used

extensively in the proof of the nested algebraic Bethe Ansatz.

Definition 5.6. We will use the following notation

=

n∑

k=1

(5.6)

and

=n∑

k=2

(5.7)

108

5.2. Definitions

where the box represents an arbitrary tensor. Note that Equation (5.6) corresponds

to taking the trace over the indicated space. ♦

5.2.4 The transfer matrix

In Chapters 3 and 4 we introduced the monodromy matrix as a product of R-

matrices as follows

(Ra|1···N(λa;λ1, . . . , λN)


=(RaN (λa, λN)Ra(N−1)(λa, λN−1) · · ·Ra1(λa, λ1)


! . (5.8)

We may regard the monodromy matrix as an n× n matrix over the a-space with

entries being tensors over the 1 to N -spaces. In the case of n = 2 we write

Ra|1···N (λa) =

[A1···N(λa) B1···N(λa)

C1···N(λa) D1···N (λa)

]

a

(5.9)

and for general n we write

Ra|1···N(λa) =

A(1)1···N (λa) B

(12)1···N(λa) · · · B

(1n)1···N(λa)

C(21)1···N(λa) A

(2)1···N(λa)

. . ....

.... . .

. . . B((n−1)n)1···N (λa)

C(n1)1···N(λa) · · · C

(n(n−1))1···N (λa) A

(n)1···N (λa)

a

. (5.10)

The following definition describes the transfer matrix, the tensor for which we

seek to solve the eigenvector problem.

Definition 5.7. The transfer matrix is the rank-(N,N) tensor given by taking

the trace over the a-space of the monodromy matrix. We have

t(λa) = tra(Ra|1···N (λ1)) =

A1···N(λa) +D1···N(λa) if n = 2,∑n

k=1A(k)1···N (λa) in general.

(5.11)

109


♦

The goal of the next two sections is to find a basis for the rank-n state and

dual-state spaces made up of eigenvectors of the transfer matrix.

5.3 The algebraic Bethe Ansatz (the su2 case)

In this section we use the algebraic Bethe Ansatz to solve the transfer matrix

eigenvalue problem using diagrammatic tensor notation.

Theorem 5.8 (algebraic Bethe Ansatz). Let N1 and N2 be positive integers

such that N1 ≥ N2, let M1 = N1 and M2 = N1 + N2 and let M1 = {1, . . . ,M1}and M2 = {M1 + 1, . . . ,M2} be ordered sets. Let the pseudo-vacuum be the

rank-M1 state given by

|0〉 =⊗

k∈M1

e(1)k . (5.12)

Then if the rapidities, λ1, . . . , λM2 satisfy

∏

r∈M1

bqr∏

p∈M2p 6=q

bpqbqp

= 1 (5.13)

for all q ∈ M2, then

B1···M1(λM1+1) · · ·B1···M1(λM2) |0〉 (5.14)

is an eigenvector of the su2 transfer matrix, t(λa) with eigenvalue

∏

q∈M2

1

bqa+∏

r∈M1

bar∏

q∈M2

1

baq. (5.15)

♦

Remark 5.9. The relations given by Equation (5.13) are called the Bethe equations.

Expanding the equations using the definition of the b-weights we obtain

∏

p∈M2p 6=q

[λp − λq + η]

[λp − λq]=∏

r∈M1

[λq − λr]

[λq − λr + η]

∏

p∈M2p 6=q

[λq − λp + η]

[λq − λp]. (5.16)

Specializing to Y(su2), solving the Bethe equations corresponds to finding all

110

5.3. The algebraic Bethe Ansatz (the su2 case)

unique sets {λr|r ∈ M2} such that

∏

r∈M1

(λq − λr + η)∏

p∈M2p 6=q

(λp − λq + η) =∏

r∈M1

(λq − λr)∏

p∈M2p 6=q

(λp − λq − η) (5.17)

for all q ∈ M2. Specializing to Uq(su2), if we let Lp = exp (2λp) and E = exp η,

solving the Bethe equations corresponds to finding all unique sets {Lr|r ∈ M2}such that

∏

r∈M1

(LqE2 − Lr)

∏

p∈M2p 6=q

(LpE2 − Lq) =

∏

r∈M1

E(Lq − Lr)∏

p∈M2p 6=q

(Lp − LqE2) (5.18)

for all q ∈ M2.

So the algebraic Bethe Ansatz reduces the XXX and XXZ eigenvalue problem

to that of solving a set of polynomials over {λr|r ∈ M2} and {Lr|r ∈ M2}respectively. ♦

Proof. We write the proposed eigenvector as

(B1···M1(λM1+1) · · ·B1···M1(λM2) |0〉

)i1···iM1

=(B1···M1(λp)B1···M1(λM1+1) · · · B1···M1(λp) · · ·B1···M1(λM2) |0〉

)i1···iM1

!

= (5.19)

where the circumflex indicates an omitted term and we have used the multi-index

111


notation for the 1 to M1-spaces. In the diagrammatic form we have isolated the

tensor B1···M1(λp) for some arbitrary p ∈ M2 by indicating the rapidity λp†. The

equality arises from repeated application of the relation

B(λp)B(λq) = IpqB(λp)B(λq)Iqp = RqpB(λp)B(λq)Rpq = B(λq)B(λp)

! = = =

(5.20)

where the first equality is a trivial redrawing of the diagram, the second equality

is due to the normalization of the R-matrix (Identity 3.16) and the last equality

is due to Lemma 3.24 since both diagrams correspond to the bipartite matrix of

the same permutation, This has the corollary that the proposed eigenvector is be

symmetric in the rapidities {λp|p ∈ M2}.We seek a scalar Λ(λa) such that

t(λa)B(λM1+1) · · ·B(λM2) |0〉 = Λ(λa)B(λM1+1) · · ·B(λM2) |0〉

! = Λ(λa)

(5.21)

where we have indicated the transfer matrix by writing the rapidity λa‡.

We proceed by considering each summand in the transfer matrix individually

and then summing them at the end of the process. Case 1 (the A(λa) component

of t(λa)). In this case we, commute the A-operator through the B-operators to

the top of the diagram. The A-operator may then be removed due to its trivial

†Where colour is available B(λp) is also indicated by a blue line.‡Where colour is available the transfer matrix is also indicated by a red line.

112


action on the pseudo-vacuum. We have

=

=

= bap +cap

(5.22)

where the first equality is due to Lemma 3.24 since both diagrams correspond to

the bipartite matrix of the same permutation, the second equality is due to the

normalization of the R-matrix (Identity 3.16) and the third equality is due to a

trivial redrawing of the diagram and the definition of the su2 R-matrix. Note

that in the second term on the right hand side we have indicated the change in

rapidity along two of the lines. This feature is of no consequence in the case

of su2 (since there are no R-matrices on the diagram above the point where the

rapidity changes), but we write it here because in the nested algebraic Bethe

Ansatz discussed in the next section, this data becomes crucial.

113


Rearranging Equation (5.22), we obtain

=1

bap−capbap

(5.23)

where the term on the left hand side has been trivially redrawn and the second

term on the right hand side has been simplified by trivially removing the switching

identities. Note that later in the general sun case, the switching identities will not

always be removable in this way. In algebraic notation this relation is

A(λa)B(λp)B(λM1+1) · · · B(λp) · · ·B(λM2) |0〉

=1

bapB(λp)A(λa)B(λM1+1) · · · B(λp) · · ·B(λM2) |0〉

− capbap

B(λa)A(λp)B(λM1+1) · · · B(λp) · · ·B(λM2) |0〉 . (5.24)

Recursively repeating this commutation process a further N2 − 1 times, we obtain

2N2 terms. Each of these terms will have an A-operator at the top of the diagram

which depends on one of the rapidities λa or {λp|p ∈ M2}, and N2 B-operators

which depend on the remaining rapidities. Gathering these terms according to the

dependency of the B-operators and using the symmetry of the B-operators in the

rapidities, {λp|p ∈ M2} we obtain

114


=∏

q∈M2

1

bqa

−∑

p∈M2

cpabpa

∏

q∈M2q 6=p

1

bqp

=∏

q∈M2

1

bqa

−∑

p∈M2

cpabpa

∏

q∈M2q 6=p

1

bqp(5.25)

115


where the second equality is due to colour conservation and removal of a-weighted

vertices.

Case 2 (the D(λa) component of t(λa)). In this case, following a similar pro-

cedure to that used in Equation (5.22), we obtain

=∏

q∈M2

1

baq

−∑

p∈M2

capbap

∏

q∈M2q 6=p

1

bpq

=∏

r∈M1

bar∏

q∈M2

1

baq

116


−∑

p∈M2

∏

r∈M1

bprcapbap

∏

q∈M2q 6=p

1

bpq(5.26)

where the second equality is due to colour conservation and replacement of the

b-weighted vertices with the specified product of b-weights.

Combining these terms we obtain

=

( ∏

q∈M2

1

bqa+∏

r∈M1

bar∏

q∈M2

1

baq

)

−∑

p∈M2

cpabpa

∏

q∈M2q 6=p

1

bqp−∏

r∈M1

bpr∏

q∈M2q 6=p

1

bpq

(5.27)

as required. �

Remark 5.10. Note that setting η = 0 also corresponds to a solution since all c-

117


weights become zero. We do not consider this solution however because at η = 0

the R-matrices become identity matrices and system becomes trivial. Indeed at

this evaluation, A(λa)+D(λa) = 2I1···M1 and since all the b-weights take the value

1, the eigenvalue becomes 2 as expected. ♦

5.4 The nested algebraic Bethe Ansatz (the gen-

eral sun case)

In this section we describe the nested algebraic Bethe Ansatz using diagrammatic

tensor notation.

Theorem 5.11 (nested algebraic Bethe Ansatz). LetNk be positive integers

for all k in {1, . . . , n} such that Nk ≥ N(k+1) for all k in {1, . . . , n − 1}. Let

Mk =∑k

p=1Np for all k in {0, . . . , n}. Let Mk be the ordered set {M(k−1) +

1,M(k−1) + 2, . . . ,Mk}.Let σ be the permutation given by

σ{Mn,M(n−1), . . . ,M1} = {M1M2 . . . ,Mn, } (5.28)

and let the sun pseudo-vacuum be the rank-M1 state given by

|Bn〉i1···iM1= (Rσ)

(n)Nn ···(2)N2 (1)N1

i1···iM1(1)N2 ···(n−1)Nn

(5.29)

where the indices on the right hand side use the notation established in Equation

(5.50), i.e. I(k+1) = Jk = k for all k in {1, . . . , n − 1} and Jn = n with I1 left

unspecified. Then if for all t ∈ {2, . . . , n} and all p ∈ Mt

∏

r∈M(t−1)

bpr∏

q∈Mt

q 6=p

bqpbpq

∏

s∈M(t+1)

1

bsp= 1 (5.30)

where we take M(n+1) = ∅, then |Bn〉 is an eigenvector of the sun transfer matrix

with eigenvalue Λn(λa) given recursively by Λ1(λa) = 1 and

Λm(λa) =∏

q∈M(n−m+2)

1

bqa+ Λm−1(λa)

∏

r∈M(n−m+1)

bar∏

q∈M(n−m+2)

1

baq(5.31)

for all m in {2, . . . , n}. ♦

118

5.4. The nested algebraic Bethe Ansatz (the general sun case)

Remark 5.12. Each sun Bethe eigenvector is a linear combination of rank-N1 states

made up of a tensor product of (N(m+1)−Nm) colour-m vectors for m ∈ {1, . . . , n−1} and Nn colour-n vectors. ♦

Remark 5.13. The sun Bethe eigenvectors are also sum Bethe eigenvectors for all

m > n. Indeed the sun Bethe eigenvectors are precisely the subset of the sum

Bethe eigenvectors such that N(n+1) = · · · = Nm = 0. ♦

Proof. Diagrammatically we may represent Rσ using multi-index notation (indi-

cated by bold lines) as

(Rσ)Jn···J2J1In···I2I1= . (5.32)

Note that without the use of multi index notation, this diagram would be much

more complicated. Specifying the components of Rσ according to Equation (5.29)

we obtain

|Bn〉i1···iM1=

= G = G (5.33)

119


where the second equality is due to colour conservation and replacement of b-

weighted vertices with the appropriate product of b-weights and G given by

G =∏

p∈M1

∏

q∈M3∪···∪Mn

bqp. (5.34)

Note that G is nonzero provided that there are no repeated rapidities. In the

final term, the state |Bn−1〉 corresponds to the part of the diagram omitted in the

third equality and is of the same form as |Bn〉 under a trivial mapping of colours

{1, . . . , n− 1} 7→ {2, . . . , n}. We seek a scalar Λn(λa) such that

t(λa) |Bn〉 = Λn(λa) |Bn〉 ! = Λn(λa) (5.35)

where the factor of G has been omitted in the diagrammatic representation and

the line corresponding to the a-space is indicated by the rapidity λa.

We proceed by induction on n. The base case, in which n = 2, corresponds to

the algebraic Bethe Ansatz given in Section 5.3†. We assume the n = m− 1 case

and seek to establish the n = m case. In terms of the n = m case, the inductive

hypothesis may be written diagrammatically as

= Λm−1(λa) (5.36)

where the double-struck line (in contrast to the triple-struck line in Equation

(5.35)) refers to summation over {2, . . . , m}.

Expanding the M2 multi-index on the left hand side of Equation (5.35) we

obtain

†Indeed, we may take the base case to be n = 1 by noting that |B0〉 =⊗

k∈M2e(2)k

andΛ1(λa) = 1.

120


=

= (5.37)

where the second equality is due to the commutation of the B-operators as in

Section 5.3, Equation (5.19) and the line corresponding to the p-space is indicated

by the rapidity λp.

As in the case of su2 we will decompose the transfer matrix into two compo-

nents; the A(1)(λa) component and the∑m

k=2A(k)(λa) component. Case 1 (the

A(1)(λa) component of t(λa)). We have

=

121


=

= bpa +c+pa

(5.38)

where the first equality is due to Lemma 3.24 since both diagrams correspond

to the bipartite matrix of the same permutation, the second equality is due to

the normalization of the R-matrix (Identity 4.13) and the third equality is due

to a trivial redrawing of the diagram and the definition of the sum R-matrix.

Rearranging Equation (5.38) we obtain

122


=1

bpa−c+pabpa

(5.39)

where the diagrams have been trivially redrawn. Note that there are now two lines

at the top of the diagram associated with the rapidity λp and that – in contrast to

the su2 case – one of the switching identities can not be removed. As in the case of

su2, we apply this commutation procedure recursively to both terms, gather terms

according to their dependencies and use the symmetry of the expression in the λp

(for p ∈ M2) to obtain

=∏

q∈M2

1

bqa

123


−∑

p∈M2

c+pabpa

∏

q∈M2q 6=p

1

bqp

=∏

q∈M2

1

bqa

−∑

p∈M2

c+pabpa

∏

q∈M2q 6=p

1

bqp(5.40)

where the second equality is due to colour conservation and remonval of a-weighted

vertices.

Case 2 (the∑n

k=2A(k)(λa) component of t(λa)). Here we use the notational

device provided by Equation (5.7) to obtain a diagrammatic representation of the

remaining components of the transfer matrix. We have

=

124


= bap +c−ap

(5.41)

where the first equality is due to Lemma 3.24 since both diagrams correspond to

the bipartite matrix of the same permutation, and the second equality is due to

the definition of the sun R-matrix.

Rearranging Equation (5.41), we obtain

=1

bap−c−apbap

(5.42)

where the diagram in the second term on the right hand side has been simplified by

moving the point on the line at which the rapidity changes. This move is allowed

since there are no R-matrix dependencies affected by the change.

Applying this commutation process recursively and using the symmetry as in

125


the previous case, we obtain

=∏

q∈M2

1

baq

−∑

p∈M2

c−apbap

∏

q∈M2q 6=p

1

bpq

=∏

q∈M2

1

baq

126


−∑

p∈M2

c−apbap

∏

q∈M2q 6=p

1

bpq

= Λm−1(λa)∏

r∈M1

bar∏

q∈M2

1

baq

−∑

p∈M2

Λm−1(λp)∏

r∈M1

bprc−apbap

∏

q∈M2q 6=p

1

bpq(5.43)

where the second equality is due to

R12(λ1, λ2)|λ1=λ2= P12 (5.44)

to the second term. This relation arises because the b-weights become zero and

the c-weights become one when λ1 = λ2. The third equality is due to colour con-

servation, replacement of b-vertices with appropriate b-weights and the inductive

hypothesis which is given by Equation (5.36).

Combining Equations (5.40) and (5.43), we obtain

127


=

( ∏

q∈M2

1

bqa+ Λm−1(λa)

∏

r∈M1

bar∏

q∈M2

1

baq

)

−∑

p∈M2

c+pabpa

∏

q∈M2q 6=p

1

bqp− Λm−1(λp)

∏

r∈M1

bpr∏

q∈M2q 6=p

1

bpq

. (5.45)

We observe that

Λm−1(λp) =∏

s∈M3

1

bsp+ Λm−2(λp)

∏

r∈M2

bpr∏

s∈M3

1

bas=∏

s∈M3

1

bsp(5.46)

since p ∈ M2, the second term in the eigenvalue contains bpp = 0. Using Equation

(5.46) allows us to avoid the need to write a recursive expression for the Bethe

equations. Finally undoing the trivial remapping of rapidities gives us the desired

result. �

Remark 5.14. The nested algebraic Bethe Ansatz may also be used to obtain the

sun dual-eigenvectors. Indeed the statements and proofs of the theorems may

be obtained from those for the eigenvectors almost exactly by rotating all of the

128

5.5. The sun Bethe eigenvectors as components of F -matrices

diagrams 180 degrees. The only algebraic change induced by this rotation is that

the c+pa in Equation (5.38) becomes a c−pa which is of no consequence to the final

result since this term only contributes to trivial solutions of the eigenvalue problem

(see Remark 5.10). The dual-eigenvector is given by

〈Cn|j1···jM1 ! (5.47)

where 〈Cn|j1···jM1 like |Bn〉i1···iM1is a specialization of Rσ (Equation (5.32)). In this

case we fix Ik = Jk+1 = k for all k in {1, . . . , n− 1} and In = n. The eigenvalues

and Bethe equations corresponding to the dual-eigenvectors are identical to those

corresponding to the eigenvectors. This is expected from the symmetry between

the non-dual and dual objects. ♦

5.5 The sun Bethe eigenvectors as components of

F -matrices

In this section we make the observation that the sun Bethe eigenvectors and

dual-eigenvectors may be written as components of appropriately selected sun F -

matrices. In the above notation, the result is almost immediate.

Theorem 5.15. The sun Bethe eigenvectors and dual-eigenvectors may be writ-

ten as components of appropriately selected sun F ∗ and F -matrices, respectively.

We have (F ∗1···Mn

)(1)N1 (2)N2 ···(n)Nn

i1···iM1(1)N2 ···(n−1)Nn

= |Bn〉i1···iM1(5.48)

and

(F1···Mn)j1···jM1

(1)N2 ···(n−1)Nn

(1)N1 (2)N2 ···(n)Nn= 〈Cn|j1···jM1 (5.49)

where we have used the following notation to specify the indices

(a)b =

b copies︷︸︸︷a · · · a . (5.50)

129


♦

Proof. This result is a consequence of Theorem 4.45, Lemma 3.24 and Lemma

4.24. (An illustrative example is provided below.) �

Example 5.16 (Bethe eigenvector as a component of an F ∗-matrix). Let n = 3,N1 =

4, N2 = 3 and N3 = 1. Then specializing the sun F ∗ matrix, we obtain

=

130

5.5. The sun Bethe eigenvectors as components of F -matrices

=

=

= (5.51)

where the first equality is due to the sorting property of the F ∗-matrix (Theorem

4.45), the second equality is due to the equivalence of bipartite matrices (Lemma

3.24) and the third equality is due to the equivalence of a bipartite matrix under

rearrangement the rearrangement of lines with equal colour (Lemma 4.24). The

final equality is a trivial redrawing of the diagram.

The right hand side of Equation (5.51) may be rewritten to highlight the nested

structure of the Bethe eigenvectors, B3 and B2. We have

131


= G (5.52)

where G is given by Equation (5.34). ♦

5.6 An expression for the elementary matrices

in terms monodromy matrix components

In this section we present an expression for the elementary matrices (Equation

(3.5) in terms of components of the sun monodromy matrix. This result is a

generalization of [13, 9] to sun.

Theorem 5.17. Let p ∈ {1, . . . , N} and let λa = λp, then

Pap =

(p−1∏

q=1

t(λq)

)Ra|1···N(λa|λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.53)

♦

Remark 5.18. In the case of su2, Equation (5.53) specialises to the following four

cases. Case 1 (ia = 1, ja = 1).

E(11)p =

(p−1∏

q=1

t(λq)

)A1···N(λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.54)

Case 2 (ia = 2, ja = 1).

E(21)p =

(p−1∏

q=1

t(λq)

)B1···N (λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.55)

132

5.6. An expression for the elementary matrices in terms monodromy matrixcomponents

Case 3 (ia = 1, ja = 2).

E(12)p =

(p−1∏

q=1

t(λq)

)C1···N(λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.56)

Case 4 (ia = 2, ja = 2).

E(22)p =

(p−1∏

q=1

t(λq)

)D1···N(λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.57)

Theorem 5.17 is not restricted to su2 and for sun in general, the components

of Equation (5.53) may be written as follows. Case 1 (ja = ia).

E(jaia)p =

(p−1∏

q=1

t(λq)

)A

(ja)1···N(λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.58)

Case 2 (ja > ia).

E(jaia)p =

(p−1∏

q=1

t(λq)

)B

(iaja)1···N (λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.59)

Case 3 (ja < ia).

E(jaia)p =

(p−1∏

q=1

t(λq)

)C

(iaja)1···N (λ1, . . . , λN)

(N∏

q=p+1

t(λq)

). (5.60)

♦

The following Lemma establishes that the transfer matrix evaluates to a mon-

odromy matrix when its (auxiliary) rapidity is set equal to one of the other (quan-

tum) rapidities,

Lemma 5.19. Let p ∈ {1, . . . , N} and let λa = λp then

t(λa) = Rp|(p+1)···(N)(1)···(p−1). (5.61)

♦

Proof. Diagrammatically, Equation (5.61) may be written

133


=

= = (5.62)

where in the first and second equalities are due to the specialization of the R-

matrix to a P -matrix where λp = λa and the third equality is due to a trivial

rearrangement of the indices. �

Remark 5.20. All indices are included in Equation (5.62) so that the rearrange-

ment in the final equality may be understood and index ka is included on the right

hand side clarity. ♦

Remark 5.21. This section includes expressions involving multiple transfer matri-

ces so in order to keep track of identified lines we retain the repeated indices to

complement the notation established in Equation (5.6). ♦

Proof. (of Theorem 5.6) Diagrammatically, Equation (5.53) may be written

134

5.7. Conclusion

=

=

=

= (5.63)

where the first equality is due to repeated application of Lemma 5.19 (omitting

the final rearrangement step in the proof of the lemma), the second equality is

due to Lemma 3.24 since both diagrams correspond to bipartite matrices of the

same permutation, the third equality is due to a trivial rearrangement and the

final equality simply involves recognizing the P -matrix. �

Remark 5.22. On the left hand side of Equation (5.53), we omit the identity ma-

trices which appear in the diagrammatic form at the right hand side of Equation

(5.63). We have

Pap =

(p−1⊗

q=1

Iq

)⊗ Pap ⊗

(N⊗

q=p+1

Iq

)(5.64)

♦

5.7 Conclusion

In Sections 5.3 and 5.4 diagrammatic methods are used to describe the algebraic

Bethe Ansatz and the nested algebraic Bethe Ansatz. It is hoped that readers find

this diagrammatic treatment transparent and intuitive.

A key result of this chapter, found in Section 5.5, is the new expressions

presented in Equations (5.48) and (5.49). These are expressions for the sun

135


eigenvectors and dual-eigenvectors as components of sun F -matrices of length

N1 + N2 + · · · + Nn with N1 ≥ · · · ≥ Nn where N1 is the rank of the eigen-

vector or dual eigenvector. The remaining the Ni are the number B-operators and

N(m+1) − Nm (for m ∈ {1, . . . , n − 1} is the number of m-colours and Nn is the

number of n-colours in each component of the eigenvector.

The second key result, found in Section 5.6, is the expressions for the elementary

matrices in terms of components of the sun monodromy matrices. This allows us

to construct a basis for the trace-free matrices which are related to local spin

operators.

We hope that the expressions presented in Equations (5.48) and (5.49) for the

Bethe eigenvectors and dual-eigenvectors and Equation (5.53) the expression for

the elementary matrices combined with their diagrammatic justification of those

expressions provide a new approach for the study of scalar products and correlation

functions in the su(n) models, but this is beyond the scope of this thesis.

136

Chapter 6

Conclusion

6.1 Summary

Chapter 1 is an introduction to the topics discussed in this thesis. In Chapter 2

we describe diagrammatic tensor notation in detail. This notation allows us to

manipulate complicated tensorial relations intuitively.

In Chapter 3, following the algebraic methods of [18], we present a diagram-

matic treatment of the factorizing F -matrices of [18]. The main feature here is the

diagrammatic representation of the partial F -matrices Fa|1···N and F1···N |a which

parallels the standard representation of the XXZ monodromy matrix. The full

F -matrix F1···N is constructed as a product partial F -matrices, diagrammatically

this corresponds to stacking the diagrams for the partial F -matrices.

In this chapter we also establish the factorizing property of the F -matrix (The-

orem 3.1) using diagrammatic techniques and by decomposing the bipartite matrix

into generating components. The use of diagrammatic tensor notation simplifies

the proofs presented in this chapter by laying bare the structure of the underlying

operators. Indeed some of the preliminary results stated in [18] are not required

here.

In Chapter 4 we generalise the results presented of the previous chapter from

models with su2 symmetry to models with general sun symmetry. These general-

izations to sun are established by induction on n where the base case corresponds

to su2. This inductive procedure is consistent with the procedure used in the

nested algebraic Bethe-Ansatz.

Specifically, the sun F -matrix is factorized into a product of n − 1 tier -s F -

matrices. Each tier-s F -matrix has a similar form to the su2 F -matrix of the previ-

ous chapter, but only exhibits nontrivial interaction between the colours {1, . . . , s}.

137

Chapter 6. Conclusion

Diagrammatically, the sun F -matrix is constructed by stacking tier-s F -matrices

for s in {2, . . . , n}.We then establish the factorizing property of the sun F -matrix by induction

on n. Finally we establish the equivalence of our expression to the sum expression

for the sun F -matrix of [2]. This is done by observing that the factorized form of

the F -matrix possesses the sorting property which is central to the definition of

the sum expression. The F -matrix is sorting in the sense that the nontrivial part

of it acts on the incoming colours sorted in decreasing order from left-to-right.

In Chapter 5, we establish the algebraic Bethe Ansatz and the nested alge-

braic Bethe Ansatz using diagrammatic methods following [15] and [4] respectively.

These results establish expressions for the eigenvectors of the su2 and sun mod-

els and therefore solve the related quantum lattice models. The nested algebraic

Bethe Ansatz is established by induction on n and the algebraic Bethe Ansatz is

used as the base case.

A key result of this chapter is the expression for the sun eigenvectors as compo-

nents of appropriately selected sun F -matrices. Another key result is the expression

for the elementary matrices (and therefore the local spin operators) in terms of

components of sun monodromy matrices.

6.2 Future work

In the field of statistical lattice models the generalization of results to models

with sun symmetry is generally nontrivial. Since diagrammatic tensor notation is

insensitive to the dimension of the underlying vector spaces and allows relatively

complicated tensors to be manipulated easily, there may be hope for progress in

these generalizations. Examples of problems to which this methodology may be

applied include the calculation of scalar products, form factors and correlation

functions in models with sun symmetry.

138

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141

Appendices

143

Appendix A

Properties of spin-chain

Hamiltonians

In this Appendix, we establish two results concerning the Hamiltonian of the XXZ

spin chain model. The first concerns the expression of the Hamiltonian in terms of

the transfer matrix, and the second concerns the commutation of the Hamiltonian

with the transfer matrix. A standard reference on this material is [15].

Note that this appendix relies on definitions made in Chapters 2–5.

A.1 The Hamiltonian in terms of the transfer

matrix

Definition A.1. In the case of the XXZ model, let

ϕ(λ1, λ2) = sinh(λ1 − λ2 + η) exp

(−λ1

cosh η + 1

sinh η

)(A.1)

and in the case of the XXX model, let

ϕ(λ1, λ2) = (λ1 − λ2 + η) exp

(−2λ1

η

)(A.2)

then the scaled R-matrix is given by

R12(λ1, λ2) = R12(λ1, λ2)ϕ(λ1, λ2) (A.3)

145

Appendix A. Properties of spin-chain Hamiltonians

and the scaled transfer matrix is given by

t(λa|λ1, . . . , λN) = tra(RaN (λa, λN)RaN (λa, λN−1) · · · RaN (λa, λ1)). (A.4)

♦

Remark A.2. The scaling factors ϕ(λ1, λ2) are chosen such that

d

dλ1

(a(λ1 − λ2)ϕ(λ1, λ2))

∣∣∣∣λ1=λ2

= −1 (A.5)

andd

dλ1(b(λ1 − λ2)ϕ(λ1, λ2))

∣∣∣∣λ1=λ2

= 1 (A.6)

where a, b are the weight functions of the appropriate model. ♦

Remark A.3. The scaling of the R-matrix and the transfer matrix do not affect

their properties as discussed in the remainder of the thesis (up to accounting for

constants). ♦

Remark A.4. In contrast to the remainder of this thesis, the intersecting lines in

the diagrams in this chapter represent the scaled R-matrices. ♦

Theorem A.5. Let η be a solution of cosh η = −(J+2)/J , then the Hamiltonian

of the spin chain models may be written as

H(J) = −1

2

N∑

i=1

(σxj σ

xj+1 + σy

jσyj+1 + Jσz

jσzj+1

)

= −√J + 1

d

dλa

log t(λa|λ1, . . . , λN)

∣∣∣∣λ1···λN=λa

(A.7)

where J = 1 in the case of the XXX model. ♦

Definition A.6. The shift operator is given by

S = ϕ(λa, λa)N(δj(N−1)

iNδj(N−2)

i(N−1)· · · δj1i2 δ

jNi1

)(A.8)

where the scaling factor ϕ(λa, λa)N has been added to simplify later expressions.♦

Lemma A.7. The scaled transfer matrix evaluated at λ1 = · · · = λN = λa is

equal to the shift operator. ♦

146

A.1. The Hamiltonian in terms of the transfer matrix

Proof. We have

t∣∣λ1···λN=λa

= tra(RaN (λa, λN)RaN (λa, λN−1) · · · RaN (λa, λ1))∣∣∣λ1···λN=λa

=

(N∏

p=1

ϕ(λa, λp)tra(RaNRa(N−1) · · ·Ra1

))∣∣∣∣∣

λ1···λN=λa

= ϕ(λa, λa)Ntra

(PaNPa(N−1) · · ·Pa1

)

= ϕ(λa, λa)N(δj(N−1)

iNδj(N−2)

i(N−1)· · · δj1i2 δ

jNi1

)= S (A.9)

where the first equality is due to the definition of the scaled transfer matrix, the

second is due to the evaluation of the scaled R-matrices and the final equality is

due to the definition of the P -matrices. Diagrammatically, this may be expressed

as

∣∣∣∣∣∣∣λ1···λN=λa

= ϕ(λa, λa)N

= ϕ(λa, λa)N (A.10)

where the second equality is a trivial rearrangement of the diagram†¡++¿. This

establishes the result. �

Lemma A.8. The shift operator commutes with the derivative of the scaled trans-

fer matrix evaluated at λ1, . . . , λN = λa. That is

[S,

dt

dλa

∣∣∣∣λ1···λN=λa

]= 0. (A.11)

♦

Proof. We have

Sdt

dλa

∣∣∣∣λ1···λN=λa

= Sd

dλa

tra

(RaN · · · Ra1

)∣∣∣∣λ1···λN=λa

= SN∑

j=1

tra

(RaN · · · Ra(j+1)

dRaj

dλa

Ra(j−1) · · · Ra1

)∣∣∣∣∣λ1···λN=λa

†The triple struck line (Definition 5.6) indicates contraction over the indicated index—kahere.

147


= Sϕ(λa, λa)N−1

N∑

j=1

tra

(PaN · · ·Pa(j+1)

dRaj

dλa

∣∣∣∣∣λ1···λN=λa

Pa(j−1) · · ·Pa1

)

= ϕ(λa, λa)N−1

N∑

j=1

tra

(PaN · · ·Paj

dRa(j−1)

dλa

∣∣∣∣∣λ1···λN=λa

Pa(j−2) · · ·Pa1

)S

=dt

dλa

∣∣∣∣λ1···λN=λa

S (A.12)

where the first equality is due to the definition of the scaled transfer matrix, the

second equality is an application of the product rule and the third equality is due

to applying the evaluation. The fourth equality can be understood by observing

that there is only one nontrivial operator (the differentiated and evaluated scaled

R-matrix) in each summand and the effect P -matrices is simply to shift the spaces

on which the nontrivial matrix acts. The final equality is due to the observation

that shifting the spaces on which the nontrivial matrix acts has no effect under

the sum.

Diagrammatically this may be expressed as

∣∣∣∣∣∣∣∣∣λ1···λN=λa

= ϕ(λa, λa)2N−1

= ϕ(λa, λa)2N−1

=

∣∣∣∣∣∣∣∣∣λ1···λN=λa

(A.13)

where the derivative of the scaled R-matrix is represented by the square labelled

R′. �

Lemma A.9. We have the following expression for the logarithmic derivative of

148

A.1. The Hamiltonian in terms of the transfer matrix

the scaled transfer matrix

(d

dλa

log(t)

)∣∣∣∣λ1···λN=λa

= S−1 dt

dλa

∣∣∣∣λ1···λN=λa

. (A.14)

♦

Proof. Writing the logarithm of transfer matrix as a series about I, the left hand

side of Equation A.14 becomes

−(

d

dλa

∞∑

n=1

(I − t

)n

n

)∣∣∣∣∣λ1···λN=λa

=

∞∑

n=1

1

n

n∑

m=1

((I − t

)m−1 dt

dλa

(I − t

)n−m

)∣∣∣∣∣λ1···λN=λa

=

∞∑

n=1

1

n

n∑

m=1

(I − S)m−1 dt

dλa

∣∣∣∣λ1···λN=λa

(I − S)n−m

=

∞∑

n=1

(I − S)n−1 dt

dλa

∣∣∣∣λ1···λN=λa

= S−1 dt

dλa

∣∣∣∣λ1···λN=λa

(A.15)

where the first equality is due to the product rule, the second equality is due to

the Lemma A.7 and the third equality is due to Lemma A.8 The final equality

may be confirmed by multiplication of the series on the left hand side by the shift

operator. Explicitly, let I − S = T , then we have

S

∞∑

n=1

(I − S)n−1 = (I − T )

∞∑

n=1

T n−1 = I (A.16)

as required. �

Proof. (of Theorem A.5) We observe that the right hand side of Equation (A.15)

may be expressed as

S−1 dt

dλa

∣∣∣∣λ1···λN=λa

=N∑

p=1

dR(p−1)p(λa, λp)

dλa

∣∣∣∣∣λ1···λN=λa

(A.17)

where we identify the index 0 with the index N . Diagrammatically this relation is

immediate

149


N∑

p=1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣λ1···λN=λa

=

N∑

p=1

=

N∑

p=1

(A.18)

Differentiating the scaled R-matrix entry-by-entry and evaluating, we obtain

S−1 dt

dλa

∣∣∣∣λ1···λN=λa

= [η]−1

N∑

i=1

(1

2(cosh η + 1)

(σxi+1σ

xi + σy

i+1σyi

)− σz

i+1σzi

)

(A.19)

where the scaling factors have cancelled in all but one term in which only the

exponential part of the scaling factor cancels. Finally, setting cosh η = −(J+2)/J

and therefore [η] = −2√J + 1/J establishes the result. �

Remark A.10. The XXX case is established along similar lines. ♦

A.2 The commutation of the Hamiltonian with

the transfer matrix

Theorem A.11. The spin chain Hamiltonian commutes with the transfer ma-

trix. ♦

Remark A.12. In this section, we do not specify whether we are concerned with

the XXZ or XXX case—the arguments hold in both cases. ♦

Lemma A.13. Transfer matrices commute. ♦

Proof. We have

t(λa)t(λb) = tra(Ra|1···N

)trb(Rb|1···N

)= tra

(trb(RbaRabRa|1···NRb|1···N

))

= tra(trb(RabRa|1···NRb|1···NRba

))= tra

(trb(Rb|1···NRa|1···NRabRba

))

150

A.2. The commutation of the Hamiltonian with the transfer matrix

= tra(trb(Rb|1···NRa|1···N

))= t(λb)t(λa) (A.20)

where the first and last equalities are due to the definition of the transfer ma-

trix, the second, fourth and fifth are due to Lemma 3.24 since both sides of those

equalities correspond to the same permutations and the third equality is a triv-

ial rearrangement due to presence of the traces. Diagrammatically this may be

represented as

=

= =

= (A.21)

where the second equality is a trivial rearrangement (all that has been done is

that the point the contraction is indicated on the diagram has been moved). This

establishes the required result. �

Proof. (of Theorem A.11) An immediate consequence of Lemma A.13 is that that

the logarithm of a transfer matrix commutes with a transfer matrix. That is

[log t(λa), t(λb)] = 0. (A.22)

Differentiating with respect to λa provides the desired result. �

151

Appendix B

Proof of the sun Yang-Baxter

relation

We restate the sun Yang-Baxter relation and provide a proof.

Lemma B.1 (Yang-Baxter relation). We have the following relation

R12R13R23 = R23R13R12. ! = . (B.1)

♦

Proof. As in the case of unitarity, the Yang-Baxter relation is a consequence of

Identity 4.12 (i) and standard identities on the hyperbolic functions. Recall the

notation [λ] = sinh(λ). We will also use the notation [λ]∗ = cosh(λ). In particular

we will use the following standard identities

2[A][B] = [A+B]∗ − [A− B]∗ (B.2)

and

4[A][B][C] = [A +B + C]− [A− B + C]− [A+B − C] + [A−B − C] (B.3)

Expanding the R-matrices in the right hand side of Equation (B.1) we obtain

153

Appendix B. Proof of the sun Yang-Baxter relation

n∑p,q,r,s,

t,u=1

f(pq)23 f

(ut)13 f

(rs)12 + g

(pq)23 f

(ut)13 f

(rs)12 + f

(pq)23 g

(rt)13 f

(rs)12

+g(pq)23 g

(ut)13 f

(rs)12 + f

(pq)23 f

(ut)13 g

(rs)12 + g

(pq)23 f

(ut)13 g

(rs)12

+f(pq)23 g

(ut)13 g

(rs)12 + g

(pq)23 g

(ut)13 g

(rs)12

(B.4)

=

n∑p,q,r,s,

t,u=1

f(pq)12 f

(ru)13 f

(st)23 + g

(pq)12 f

(ru)13 f

(st)23 + f

(pq)12 g

(ru)13 f

(st)23

+g(pq)12 g

(ru)13 f

(st)23 + f

(pq)12 f

(ru)13 g

(st)23 + g

(pq)12 f

(ru)13 g

(st)23

+f(pq)12 g

(ru)13 g

(st)23 + g

(pq)12 g

(ru)13 g

(st)23

(B.5)

applying Identity 4.12 (i) we obtain

n∑

r,s,t=1

f(st)23 f

(rt)13 f

(rs)12 + g

(st)23 f

(rt)13 f

(rs)12 + f

(sr)23 g

(rt)13 f

(rs)12

154

+g(sr)23 g

(rt)13 f

(rs)12 + f

(rt)23 f

(st)13 g

(rs)12 + g

(rt)23 f

(st)13 g

(rs)12

+f(rs)23 g

(st)13 g

(rs)12 + g

(rs)23 g

(st)13 g

(rs)12

(B.6)

=n∑

p,q,r,s,

t,u=1

f(rs)12 f

(rt)13 f

(st)23 + g

(rs)12 f

(rt)13 f

(st)23 + f

(ts)12 g

(rt)13 f

(st)23

+g(ts)12 g

(rt)13 f

(st)23 + f

(rt)12 f

(rs)13 g

(st)23 + g

(rt)12 f

(rs)13 g

(st)23

+f(st)12 g

(rs)13 g

(st)23 + g

(st)12 g

(rs)13 g

(st)23

. (B.7)

We now consider several cases over the colours r, s, t. Case 1 (r = s = t). In this

case all the terms involving a g-weight are zero and we trivially require

f(ss)23 f

(ss)13 f

(ss)12 = f

(ss)12 f

(ss)13 f

(ss)23 . (B.8)

Case 2 (r = s, t 6= r, s). In this case we have three nonzero components

f(st)23 f

(st)13 f

(ss)12 = f

(ss)12 f

(st)13 f

(st)23 (B.9)

f(ss)23 g

(st)13 f

(ss)12 = f

(ts)12 g

(st)13 f

(st)23 + g

(st)12 f

(ss)13 g

(st)23 (B.10)

g(st)23 f

(st)13 f

(ss)12 = g

(ts)12 g

(st)13 f

(st)23 + f

(st)12 f

(ss)13 g

(st)23 . (B.11)

The first relation is immediate and applying the definition of the f and g-weights

155


the other relations become

c±13 = b12c±13b23 + c±12c

±23 (B.12)

c±23b13 = c∓12c±13b23 + b12c

±23 (B.13)

which are established by standard identities on the hyperbolic functions. In the

case of Equation (B.12), multiplying both sides by

[λ2 − λ3 + η][λ1 − λ3 + η][λ1 − λ2 + η]

[η](B.14)

and using the definition of the a, b and c-weights, we obtain

[λ2 − λ3 + η]e±(λ1−λ3)[λ1 − λ2 + η]

= [λ1 − λ2]e±(λ1−λ3)[λ2 − λ3] + e±(λ1−λ2)[λ1 − λ3 + η]e±(λ2−λ3)[η] (B.15)

observing that

e±(λ1−λ2)e±(λ2−λ3) = e±(λ1−λ3) (B.16)

and applying Equation (B.2), we obtain

[λ1 − λ3 + 2η]∗ − [−λ1 + 2λ2 − λ3]∗

= [λ1 − λ3]∗ − [λ1 − 2λ2 + λ3]

∗ + [λ1 − λ3 + 2η]∗ − [λ1 − λ3]∗ (B.17)

which is indeed true. The other relations are established along similar lines. Case

3 (s = t, r 6= s, t). This case is true by a symmetry of the Yang-Baxter relation;

consider the mirror image of the diagrams involved. Case 4 (r = t, s 6= r, t). In

this case we again have three nonzero components

f(st)23 f

(tt)13 f

(ts)12 + g

(ts)23 g

(st)13 g

(ts)12 = f

(ts)12 f

(tt)13 f

(st)23 + g

(st)12 g

(ts)13 g

(st)23 (B.18)

f(tt)23 f

(st)13 g

(ts)12 = g

(ts)12 f

(tt)13 f

(st)23 + f

(st)12 g

(ts)13 g

(st)23 (B.19)

g(st)23 f

(tt)13 f

(ts)12 + f

(ts)23 g

(st)13 g

(ts)12 = f

(tt)12 f

(ts)13 g

(st)23 . (B.20)

Applying the definition of the f and g-weights we obtain

b23b12 + c∓23c±13c

∓12 = b12b23 + c±12c

∓13c

±23 (B.21)

b13c∓12 = c∓12b23 + b12c

∓13c

±23 (B.22)

c±23b12 + b23c±13c

∓12 = b13c

±23 (B.23)

156

which are established by standard identities on the hyperbolic functions as as in

the previous cases. Case 5 (r, s, t distinct). In this case all six components are

nonzero. We have

f(st)23 f

(rt)13 f

(rs)12 = f

(rs)12 f

(rt)13 f

(st)23 (B.24)

g(st)23 f

(rt)13 f

(rs)12 = f

(rt)12 f

(rs)13 g

(st)23 (B.25)

f(sr)23 g

(rt)13 f

(rs)12 + g

(rs)23 g

(st)13 g

(rs)12 = f

(ts)12 g

(rt)13 f

(st)23 + g

(st)12 g

(rs)13 g

(st)23 (B.26)

g(sr)23 g

(rt)13 f

(rs)12 + f

(rs)23 g

(st)13 g

(rs)12 = g

(rt)12 f

(rs)13 g

(st)23 (B.27)

where the first two relations are immediate. Note that we need only consider these

four relations since the remaining two may be obtained from Equations (B.25)

and (B.27) by symmetry; once again consider the mirror image of the diagrams.

Applying the definition of the f and g-weights and simplifying we obtain

c〈rs〉23 c

〈st〉13 c

〈rs〉12 = c

〈st〉12 c

〈rs〉13 c

〈st〉23 (B.28)

c〈sr〉23 c

〈rt〉13 b12 + b23c

〈st〉13 c

〈rs〉12 = c

〈rt〉12 b13c

〈st〉23 (B.29)

where

〈rs〉 =

+, if r < s

−, if r > s.(B.30)

In the case of Equation (B.28), multiplying both sides by

[λ2 − λ3 + η][λ1 − λ3 + η][λ1 − λ2 + η]

[η]3(B.31)

and using the definition of the c-weights, the right hand side of Equation (B.28)

becomes

e〈rs〉(λ2−λ3)e〈st〉(λ1−λ3)e〈rs〉(λ1−λ2) = e(〈rs〉+〈st〉)(λ1−λ3)

= e〈st〉(λ1−λ2)e〈rs〉(λ1−λ3)e〈st〉(λ2−λ3) (B.32)

as required. In the case of Equation (B.29), multiplying both sides by

[λ2 − λ3 + η][λ1 − λ3 + η][λ1 − λ2 + η]

[η]2(B.33)

and using the definition of the b and c-weights, Equation (B.29) becomes

157


e〈sr〉(λ2−λ3)e〈rt〉(λ1−λ3)[λ1 − λ2] + [λ2 − λ3]e〈st〉(λ1−λ3)e〈rs〉(λ1−λ2)

− e〈rt〉(λ1−λ2)[λ1 − λ3]e〈st〉(λ2−λ3) = 0. (B.34)

Multiplying both sides by

e〈rs〉(λ2−λ3)e〈ts〉(λ1−λ3)e〈tr〉(λ1−λ2) (B.35)

we obtain

e〈rt〉(λ2−λ3)e〈ts〉(λ1−λ3)[λ1 − λ2] + [λ2 − λ3]e〈rs〉(λ1−λ3)e〈tr〉(λ1−λ2)

− e〈ts〉(λ1−λ2)[λ1 − λ3]e〈rs〉(λ2−λ3) = 0 (B.36)

so it follows that Equation (B.34) is symmetric under cyclic permutation of the

colours r, s and t. It is therefore sufficient to consider the cases r < s < t and

r > s > t. In these cases the right have side of Equation (B.34) becomes

e∓(λ2−λ3)e±(λ1−λ3)[λ1 − λ2] + [λ2 − λ3]e±(λ1−λ3)e±(λ1−λ2)

− e±(λ1−λ2)[λ1 − λ3]e±(λ2−λ3) (B.37)

= e±(λ1−λ2)(eλ1−λ2 − e−(λ1−λ2)

)+ e±(2λ1−λ2−λ3)

(eλ2−λ3 − e−(λ2−λ3)

)

− e±(λ1−λ3)(eλ1−λ3 − e−(λ1−λ3)

)(B.38)

± e±2(λ1−λ2) ∓ 1± e±2(λ1−λ3) ∓ e±2(λ1−λ2) ∓ e±2(λ1−λ3) ± 1 = 0 (B.39)

as required. �

Note that the additional factor e±λ of c-weights is only required for Equation

(B.29) to hold; all the remaining relations would hold in its absence. This explains

why this factor may be omitted in the case of su2.

158

Appendix C

Notational conventions

Although the matrix notation used in this thesis is standard, in order to be com-

pletely unambiguous, we describe it here in detail.

C.1 Matrix notation

Matrix notation involves writing vectors, dual vectors and matrices as two-dimensional

arrays of elements of C.

Definition C.1 (matrices). Let Aji be elements of C for all i, j in {1, . . . , n}†.

Then we call

A =

A11 · · · An

1...

. . ....

A1n · · · An

n

(C.1)

a matrix. ♦

Definition C.2 (vectors and dual vectors). Let ai, bi be elements of C for all

i in {1, . . . , n}. Then we call

|a〉 =

a1...

an

(C.2)

a vector and we call

〈b| =[b1 · · · bn

](C.3)

a dual vector. ♦

†A more general treatment would allow for n×m matrices, but is not required in this thesis.

159

Appendix C. Notational conventions

Remark C.3. We have used Dirac bra-ket notation to make the distinction between

vectors and dual vectors. When a space is equipped with an inner product there

is actually no distinction between vectors and dual vectors since we may write the

basis for the vectors in terms of the basis for the dual vectors (and visa-versa). ♦

C.2 Notation for matrix operations

We describe scalar multiplication, matrix multiplication and the left and right

actions of the matrices on the vectors and dual vectors respectively. Under matrix

notation these are thought of in terms of manipulation of the arrays involved.

Definition C.4 (scalar product). Let α be a scalar in the field C. Let A be an

matrix, a be a vector and b be a dual vector. Then we call the matrix

αA = Aα = α

A11 · · · An

1...

. . ....

A1n · · · An

n

=

αA11 · · · αAn

1...

. . ....

αA1n · · · αAn

n

(C.4)

the scalar product of α and A, we call the vector

αa = aα = α

a1...

an

=

αa1...

αan

(C.5)

the scalar product of α and a and we call the dual vector

αb = Aα = α[b1 · · · bn

]=[αb1 · · · αbn

](C.6)

the scalar product of α and b. ♦

Definition C.5 (matrix multiplication). Let A and B be matrices. Then we

call the matrix

AB =

A11 · · · An

1...

. . ....

A1n · · · An

n

B11 · · · Bn

1...

. . ....

B1n · · · Bn

n

=

A11B

11 + · · ·+ An

1B1n · · · A1

1Bn1 + · · ·+ An

1Bnn

.... . .

...

A1nB

11 + · · ·+ An

nB1n · · · A1

nBn1 + · · ·+ An

nBnn

. (C.7)

160

C.2. Notation for matrix operations

the matrix product of A and B. ♦

Definition C.6 (left and right action). LetA be a matrix, |a〉 be a vector and〈b| be a dual vector. Then we call

A |a〉 =

A11 · · · An

1...

. . ....

A1n · · · An

n

a1...

an

=

A11a1 + · · ·+ An

1an...

A1na1 + · · ·+ An

nan

(C.8)

the left action of A on a and we call

〈b|A =[b1 · · · bn

]

A11 · · · An

1...

. . ....

A1n · · · An

n

=[b1A1

1 + · · ·+ bnA1n · · · b1An

1 + · · ·+ bnAnn

](C.9)

the right action of A on b. ♦

The identity matrix serves as the identity element in the monoid structure of

the matrices described below.

Definition C.7 (identity matrix). We call

δ ji =

1, if i = j,

0, otherwise(C.10)

the Kronecker symbol and we call the matrix

I =

δ 11 · · · δ n

1...

. . ....

δ 1n · · · δ n

n

(C.11)

the identity matrix. ♦

Definition C.8 (vector and dual vector spaces). Let V = V ∗ = Cn be copies

of the standard complex vector space of dimension n. ♦

Vectors and dual vectors are interpreted as elements of the vector spaces V

and V ∗ respectively. With the left and right actions defined above, matrices may

be interpreted as endomorphisms, End(V ) and End(V ∗) respectively.

161

Appendix C. Notational conventions

Together with the identity matrix, matrix multiplication endows the matrices

with a monoid structure. Element-wise addition which trivially has a group struc-

ture distributes across matrix multiplication and so the matrices posses a ring

structure. This allows to view the vector and dual vector spaces as modules over

the ring of matrices. We identify these structures for convenience so that we may

appeal to them later as required.

For completeness, we now define the inner product between a vector and a

dual vector. Note that this provides us with an inner product between the spaces

provided that we take the dual of a vector |a〉 with elements ai to be the dual

vector 〈a| with elements ai = ai (the complex conjugate of ai).

Definition C.9 (inner product). Let |a〉 be a vector and let 〈b| be a dual vec-

tor. Then we call

〈b|a〉 =[b1 · · · bn

]

a1...

an

= a1b1 + · · ·+ anbn (C.12)

the inner product of |a〉 and 〈b|. ♦

We now provide a definition of the cross product. This definition seems ad

hoc in matrix notation, but the definition in terms of index and tensor notation is

more natural and is discussed in the following sections.

Definition C.10 (cross product). Let n = 3 and let 〈a| and 〈b| be vectors.

Then we call the dual vector

|a〉 × |b〉 =

a1

a2

a3

×

b1

b2

b3

=

[a2b3 − a3b2 a3b1 − a1b3 a1b2 − a2b1

](C.13)

the cross product of 〈a| and 〈b|. Let |c〉 and |d〉 be dual vectors. Then we call

the vector

〈c| × 〈d| =[c1 c2 c3

]×[d1 d2 d3

]=

c2d3 − c3d2

c3d1 − c1d3

c1d2 − c2d1

(C.14)

the cross product of |c〉 and |d〉. ♦

162

C.2. Notation for matrix operations

The idea that the cross product of a pair of vectors is a dual vector (and vise-

versa) may seem odd at first sight. It is in fact consistent with the differential

geometric interpretation of vectors as differential operators and dual vectors as

volume elements. Indeed it is well known that geometrically the cross product of

a pair of vectors is a vector perpendicular to both with length equal to the area of

the parallelogram with the vectors as edges.

163

Appendix D

Notes on presentation and

typesetting

In this thesis there are several theorem-like environments which are detailed below.

It is felt that a closing symbol for all environments is important because it may

not always be clear where it concludes and normal body text resumes. Identities,

lemmas, theorems and corollaries are of the following format and terminated by

the lozenge symbol (♦) as in the following example.

Theorem D.1 (sample). Lorem ipsum dolor sit amet, consectetuer adipiscing

elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. ♦

Note that the term(s) being defined (such as adipiscing elit in the example

above) are printed in bold. Proofs are of the following format and terminated by

the Halmos symbol.

Proof (sample). Curabitur dictum gravida mauris. Nam arcu libero, nonummy

eget, consectetuer id, vulputate a, magna. Donec vehicula augue eu neque. �

Examples and remarks are of the following format and are also terminated by the

lozenge symbol.

Example D.2 (sample). Pellentesque habitant morbi tristique senectus et netus et

malesuada fames ac turpis egestas. Mauris ut leo. Cras viverra metus rhoncus

sem. ♦

The optional term in parenthesis (sample in each of the examples above) is an

optional phase indicating the subject matter of the environment in question.

165

Appendix D. Notes on presentation and typesetting

All items (e.g. equations, theorems, etc.) in this thesis are numbered without

regard to whether or not they are actually referred to. Although this causes a

degree of clutter in the presentation, it is desirable to facilitate the ability to easily

refer to this document later.

The numbered environments in this thesis are all part of the same sequence and

of the form m.n where m is the chapter number and n is a counter. For example,

above we have Example D.2, not Example D.1.

This thesis is produced entirely using free and open source software (FOSS). In

particular, the thesis is typeset using LATEX and the diagrams are produced using

the Inkscape vector graphics editor and the PSTricks set of LATEX macros.

In all cases this software was run on free and open source operating systems –

specifically Fedora and Ubuntu.

This source LATEXfiles of thesis is based on the Cambridge University, Depart-

ment of Engineering (CUED) LATEX thesis template by Harish Bhanderi. The

CUED thesis template is released under the GNU General Public Licence and as

such the source of this derivative work is also available under the same licence

upon request from the author. Table D provides links to some of the software used

in this thesis.

Table D.1: Links to software used in the production of this thesis

Software URLLATEX http://latex-project.org/

Inkscape http://inkscape.org/

PSTricks http://tug.org/PSTricks/

Fedora http://fedoraproject.org/

Ubuntu http://www.ubuntu.org/

CUED thesis template http://www-h.eng.cam.ac.uk/help/ ...

... tpl/textprocessing/ThesisStyle/

166

http://latex-project.org/

http://inkscape.org/

http://tug.org/PSTricks/

http://fedoraproject.org/

http://www.ubuntu.org/

http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ThesisStyle/

http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ThesisStyle/

Index

Symbols

6-vertex model . . . . . . . . . . . . . . . . . . . . . . . . . . 5

A

algebraic Bethe Ansatz . . . . . . . . . . . 5, 8, 110

B

Bethe eigenvectors

dual eigenvectors . . . . . . . . . . . . . . . . . 128

relation to F -matrices . . . . . . . . . . . . 129

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

structure . . . . . . . . . . . . . . . . . . . . .118, 119

Bethe equations

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

bipartite matrix . . . . . . . . . . . . . . . . . . . . . . . . 40

equivalence under consecutive identical

colours . . . . . . . . . . . . . . . . . . . . . . . . 73

bra-ket notation . . . . . . . . . . . . . . . . . . . . . . . 160

C

cocycle relation . . . . . . . . . . . . . . . . . . . . . . . . .50

colour-s vector

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

commutation relations on elementary and

related matrices

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

cyclic and site-swap permutations . . . . . . 40

as spanning generators . . . . . . . . . . . . . 41

D

diagrammatic tensor notation . . . . . . . . 1, 22

identity matrix . . . . . . . . . . . . . . . . . . . . 26

indices represented by lines . . . . . . . . 23

joined lines representing summation 24

operations . . . . . . . . . . . . . . . . . . . . . . . . . 26

tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

E

electron, models of the . . . . . . . . . . . . . . . . . . .3

elementary matrix . . . . . . . . . . . . . . . . . . . . . . 10

in terms of monodromy matrices . . 132

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

F

F -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

commutation relation . . . . . . . . 52–54

equivalence to F ′-matrix . . . . . . . . .45

example . . . . . . . . . . . . . . . . . . . . . . . . . 44

factorizing property . . . . . . . . . . . . . 31

factorizing property example . . . . .46

hollow arrow convention . . 45, 50, 51

sorting property . . . . . . . . . . . . . . . . . 87

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

commutation relation . . . . . . . . . . . . 83

equivalence to expression of Albert et

al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

factorizing property . . . . . . . . . . . . . 59

hollow arrow convention . . . . . . . . . 75

inverse construction . . . . 98–100, 102

invertibility . . . . . . . . . . . . . . . . . . . . . . 97

lower triangularity . . . . . . . . . . . 95, 96

sorting property . . . . . . . . . . . . . . . . . 89

H

Hamiltonian of the spin- 12 chain . . 4, 7, 105

Heisenberg model . . . . . . . . . . . . . . . . . . . . . 5, 9

167

Index

I

identity matrix

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

index notation . . . . . . . . . . . . . . . . . . . . . . . . . .15

contraction, left and right action, inner

product . . . . . . . . . . . . . . . . . . . . . . . 17

state spaces . . . . . . . . . . . . . . . . . . . . . . . . 18

tensor product, scalar product . . . . . 16

tensor trace . . . . . . . . . . . . . . . . . . . . . . . . 16

tensor, state, dual state, scalar . . . . . 15

intertwining relation . . . . . . . . . . . . . . . . . . . . 44

L

Levi-Civita tensor . . . . . . . . . . . . . . . . . . . . . . 19

diagrammatic representation . . . . . . . 26

M

matrix notation . . . . . . . . . . . . . . . . . . . 14, 159

cross product . . . . . . . . . . . . . . . . . . . . . 162

identity matrix . . . . . . . . . . . . . . . . . . . 161

inner product . . . . . . . . . . . . . . . . . . . . . 162

left and right action of a matrix on a

vector and dual vector . . . . . . . 161

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

matrix multiplication . . . . . . . . . . . . . 160

scalar product . . . . . . . . . . . . . . . . . . . . 160

monodromy matrix

as a locus of a transfer matrix . . . . 133

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 42

commutation relation . . . . . . . . . . . . 49

unitarity-like identity . . . . . . . . . . . . 43

sun (tier-s) . . . . . . . . . . . . . . . . . . . . . . . . 74

multi-index notation . . . . . . . . . . . . . . . . . . 106

N

nested algebraic Bethe Ansatz . . . . . . . . . 118

normalization of the R-matrix

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

sun (tier-s) . . . . . . . . . . . . . . . . . . . . . . . . 71

P

partial F -matrix (su2) . . . . . . . . . . . . . . . . . . 47

commutation relation . . . 48–50, 53, 54

permutation matrix

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Q

quantum algebra . . . . . . . . . . . . . . . . . . . . . . . 59

quantum spin chains . . . . . . . . . . . . . . . . . . . . .3

R

R-matrix

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 35

su2 commutation relation . . . . . . 48, 52

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

sun commutation relation . . . . . . . . . . 83

S

six-vertex model . . . . . . . . . . . . . . . . . . . . . . . . . 5

special orthogonal group . . . . . . . . . . . . . . . . . 3

special unitary group . . . . . . . . . . . . . . . . . . . . 3

special unitary Lie algebra . . . . . . . . . . . . . . . 3

state spaces as vector spaces . . . . . . . . . . . . 19

summation convention, diagrammatic . 108

switching identity . . . . . . . . . . . . . . . . . . . . . 107

T

tier-s F -matrix (sun) . . . . . . . . . . . . . . . . . . . 75

commutation relation . . . . . . . . . . . . . . 81

tier-s partial F -matrix (sun) . . . . . . . . . . . .75



tier-s R-matrix (sun) . . . . . . . . . . . . . . . . . . . 70

commutation relation . . . . . . . 78, 79, 81

transfer matrix . . . . . . . . . . . . . . . . . . . . . . . . 109

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

U

unitarity

proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

sun (tier-s) . . . . . . . . . . . . . . . . . . . . . . . . 72

V

vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 161

W

weight functions

168

Index

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

X

XYZ, XXZ and XXX model . . . . . . . . . . . . . 5

Y

Yang-Baxter relation . . . . . . . . . . . . . . . . . . 153

proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

su2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

sun (tier-s) . . . . . . . . . . . . . . . . . . . . . . . . 72

Yangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

169

Minerva Access is the Institutional Repository of The University of Melbourne

Author/s:

McAteer, Stephen Gerard

Title:

A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains

Date:

2015

Persistent Link:

http://hdl.handle.net/11343/58978

File Description:

A diagrammatic approach to factorizing F-matrices in XXZ and XXX spin chains

a diagrammatic approach to factorizing f-matrices in xxz

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