six-vertex model from the Drinfeld doubles of dihedral group algebras. ..... where x, y â C. Given a parameter dependent solution, R(z), constant solutions will be.
The Drinfeld Double of Dihedral Groups and Integrable Systems Peter Edward Finch
A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in November 2009
School of Mathematics and Physics
Declaration by author
This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the General Award Rules of The University of Queensland, immediately made available for research and study in accordance with the Copyright Act 1968. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material.
iii
Statement of Contributions to Jointly Authored Works Contained in the Thesis
Integrable boundary conditions for a non-Abelian anyon chain with D(D3 ) symmetry [25] was a jointly authored paper by K.A. Dancer, P.S. Isaac, J. Links and myself. I was responsible for the majority of calculations and the writing of the body. K.A. Dancer, P.S. Isaac and J. Links were primarily responsible for verification of calculations, writing the introduction, general interpretation of results and editing. I also have a draft manuscript Solutions of the Yang–Baxter equation: descendants of the six-vertex model from the Drinfeld doubles of dihedral group algebras. It is jointly authored with K.A. Dancer, P.S. Isaac and J. Links, and will soon be submitted to an appropriate journal. I completed the majority of the calculations and writing of the body, although these have changed significantly in the process of editing by the other authors. The other authors have also added significantly to the introduction and discussion of results.
Statement of Contributions by Others to the Thesis as a Whole
Each of my supervisors, K.A. Dancer, P.S. Isaac and J. Links, played a significant role in the development my thesis. Their contributions range from the general design of the project to providing me with background and technical explanations. They also gave input to approaches and techniques that were useful for calculations, informed me of accepted conventions and pointed out grammatical or spelling errors they uncovered during their reviewing of my thesis.
Statement of Parts of the Thesis Submitted to Qualify for the Award of Another Degree
None. iv
Published Works by the Author Incorporated into the Thesis
None.
Additional Published Works by the Author Relevant to the Thesis but not Forming Part of it
The article Universal Baxterization for Z-graded Hopf algebras [24] was jointly authored by K.A. Dancer, P.S. Isaac and myself. Although the article was based primarily on work done during my honours year it was completed and published in the first year of my PhD candidature.
v
Acknowledgements I would first like to thank my supervisors Karen Dancer, Phil Isaac and Jon Links for their support and advice. Most of all I want to thank them for helping me enjoy my time as a PhD student and for never making me dread a meeting. For their discussions with me and my supervisors concerning the connection between the Fateev–Zamolodchikov and D(D3 ) models, I would like to thank H. Au-Yang, V. V. Bazhanov, V. V. Mangazeev and J. H. H. Perk. I would like to thank all my fellow PhD students who understand the frustrations that go hand in hand with doing a PhD and have helped me deal with it. In particular I would like to thank Tristan for the many games of lawn bowls, Chris for drinking with me at the wonder that was Tingl Tangl, Geoff for the games of Hallway tennis, Tom for making the office fun and Nick for the many drinks, conversations and games of pool at the uni bar. Lastly I thank my friends and family for the support they gave me. To my mum Beth, thanks for all conversations we had and for your continued belief in me. To my sister and brother-in-law, Kate and Jason, thanks for always being available to help me.
vii
Abstract A little over 20 years ago Drinfeld presented the quantum (or Drinfeld) double construction. This construction takes any Hopf algebra and embeds it in a larger quasi-triangular Hopf algebra, which contains an algebraic solution to the constant Yang–Baxter equation. One such class of algebras consists of the Drinfeld doubles of finite groups, which are currently of interest due to their connections with non-Abelian anyons. The smallest non-commutative Drinfeld double of a finite group algebra is the Drinfeld double of D3 , the dihedral group of order six, which was recently used to construct solutions to the Yang–Baxter equation corresponding to 2-state and 3-state integrable spin chains with periodic boundary conditions. In this thesis we construct R-matrices from the Drinfeld double of dihedral group algebras, D(Dn ) and consider their associated integrable systems. The 3-state spin chain from D(D3 ) is generalised to include open boundaries and it is also shown that there exists a more general R-matrix for this algebra. For general D(Dn ) an R-matrix is constructed as a descendant of the zero-field six-vertex model.
Keywords: Yang–Baxter equation, descendant, Drinfeld double, quantum double, finite group algebra, dihedral group, integrable systems, reflection equations, R-matrices
Australian and New Zealand Standard Research Classifications (ANZSRC): 010501 Algebraic Structures in Mathematical Physics 50% 010502 Integrable Systems (Classical and Quantum) 50%
ix
Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
1 Introduction
1
2 Background Knowledge
7
2.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
The Yang–Baxter equation and integrability . . . . . . . . . . . . . . . . .
8
2.3
Drinfeld doubles of finite groups . . . . . . . . . . . . . . . . . . . . . . . .
14
2.4
Representations of D(Dn ), R-matrices and projection operators when n is odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.5
Representations of D(D2n ), R-matrices and projection operators . . . . . .
19
2.6
Review of the periodic spin chain associated with D(D3 ) . . . . . . . . . .
23
3 Integrable Models Associated with D(D3 ) 3.1
3.2
Integrable boundary conditions . . . . . . . . . . . . . . . . . . . . . . . .
28
3.1.1
Boundary Quantum Inverse Scattering Method (BQISM) . . . . . .
28
3.1.2
Reflection matrices for an R-matrix associated with D(D3 ) . . . . .
33
3.1.3
An integrable Hamiltonian with open boundary conditions . . . . .
41
A general R-matrix associated with D(D3 ) . . . . . . . . . . . . . . . . . .
42
3.2.1
3.3
A general descendant of the zero-field six-vertex model with D(D3 ) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.2.2
Peculiar cases of the general R-matrix from D(D3 ) . . . . . . . . .
46
3.2.3
General periodic spin chain for D(D3 ) . . . . . . . . . . . . . . . .
47
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4 Descendants with D(Dn ) Symmetry 4.1
27
51
Constructing descendants using D(Dn ) when n is odd . . . . . . . . . . . .
52
4.1.1
52
L-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
4.2
4.3
4.4
4.1.2
Descendants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.1.3
Local Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Constructing descendants using D(D2n ) . . . . . . . . . . . . . . . . . . . .
70
4.2.1
L-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.2.2
Descendants when n is even . . . . . . . . . . . . . . . . . . . . . .
70
4.2.3
Descendants when n is odd
. . . . . . . . . . . . . . . . . . . . . .
76
Other descendants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.3.1
Using other initial representations . . . . . . . . . . . . . . . . . . .
77
4.3.2
More R-matrices for models with few states . . . . . . . . . . . . .
78
4.3.3
Self-adjoint descendants with free parameter when n even . . . . . .
82
4.3.4
General descendants with free parameter when n even . . . . . . . .
87
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5 Conclusion
91
A From the STR to the YBE
101
A.1 Converting weights to R-matrices . . . . . . . . . . . . . . . . . . . . . . . 101 A.2 Proving the R-matrix satisfies the YBE . . . . . . . . . . . . . . . . . . . . 103 B The Fateev–Zamolodchikov (FZ) Model
109
B.1 The connection between the 3-state D(D3 ) and FZ model . . . . . . . . . . 109 B.2 Properties of the FZ model . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.3 The connection between the general n-state models . . . . . . . . . . . . . 114 C Calculations
123
C.1 Calculations for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 C.2 Calculations for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 C.3 Calculations for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xii
List of Tables
2.1
The representations of D(Dn ), n is odd. . . . . . . . . . . . . . . . . . . .
16
2.2
The ordered pairs labelling the irreps for D(Dn ), n odd.
. . . . . . . . . .
18
2.3
The representations of D(D2n ). . . . . . . . . . . . . . . . . . . . . . . . .
19
2.4
The ordered pairs labeling the irreps for D(D2n ), n odd. . . . . . . . . . .
21
2.5
The ordered pairs labeling the irreps for D(D2n ), n even, case (i ). . . . . .
22
2.6
The ordered pairs labeling the irreps for D(D2n ), n even, case (ii ). . . . . .
22
xiii
List of Figures
2.1
Yang–Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Braid equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Action of σ and τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.1
Reflection equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4.1
Tensor product diagram for odd n . . . . . . . . . . . . . . . . . . . . . . .
55
4.2
Tensor product diagram for even n . . . . . . . . . . . . . . . . . . . . . .
71
A.1 Representation of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.2 Star-triangle relation (i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.3 Star-triangle relation (ii ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.4 R-matrix from weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.5 STR to YBE: proof part (i) . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A.6 STR to YBE: proof part (ii ) . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A.7 STR to YBE: proof part (iii ) . . . . . . . . . . . . . . . . . . . . . . . . . 105 A.8 STR to YBE: proof part (iv ) . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A.9 STR to YBE: proof part (v ) . . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.1 FZ weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B.2 Reversal of arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.3 Properties of FZ weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.4 FZ R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.5 Inverse relation for FZ R-matrix . . . . . . . . . . . . . . . . . . . . . . . . 113 B.6 Relation for Rt1 (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 B.7 Relation for R21 (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xiv
List of Abbreviations
BQISM
Boundary quantum inverse scattering method
FZ
Fateev–Zamolodchikov
QISM
Quantum inverse scattering method
STR
Star-triangle relation
YBE
Yang–Baxter equation
xv
Chapter 1
Introduction The Yang–Baxter equation (YBE) has been of significant interest ever since its connection to integrable systems was discovered [67, 68, 88]. Although the YBE appeared in the mid 1960’s, other forms and related equations were considered earlier [76]. One of the related equations is the star-triangle relation, which is an equivalence condition for weights on a lattice in statistical mechanics. This relation was first observed in spin models by Onsager [75], but it was Baxter who brought it to prominence after considering a variety of exactly solved models [10]. When considered without spectral parameters the YBE is equivalent to the braid equation, a defining relation of the braid group. Representations of the braid group allow the construction of link invariants [2, 87]. Thus the YBE is one connection, amongst others, between knot and link invariants and theoretical physics [41, 52, 73]. The YBE in its current form was put forth by McGuire [67, 68] who introduced the equation as a consistency condition for a many-body S-matrix to be factorisable into two-body S-matrices. Yang also used this condition to solve the one dimensional spin- 12 fermionic Bose gas [88]. There are also generalisations of the YBE including the dynamical YBE [34, 38, 42], which first appeared in the quantisation of Liouville theory, and the tetrahedron equations of both the Zamolodchikov [11, 89] and Frenkel–Moore type [40], which are three dimensional analogues of the YBE. The quantum inverse scattering method (QISM) is a highly successful technique that uses solutions to the YBE, referred to as R-matrices, to construct integrable models [83]. This method relies on the construction of a transfer matrix, which contains a spectral parameter and also commutes with the Hamiltonian of the system. The Hamiltonian and transfer matrix are able to be simultaneously diagonalised; furthermore, the resulting eigenvectors are independent of the spectral parameter. The diagonalisation of the Hamiltonian is commonly accomplished though the use of the algebraic Bethe ansatz [36]. Some of the more prominent integrable systems are the six-vertex [62, 63, 84], eight-vertex [9], chiral Potts [8], Kashiwara–Miwa [50] and the Fateev–Zamolodchikov [37] models. 1
2
Chapter 1. Introduction
These models are relevant to this thesis as each is constructed from a solution of the YBE. Connections arise between some of these models when one is viewed as a descendant of another with fewer states (an explicit definition using L-operators [55] is given in the Section 2.2). The most notable cases of descendants were found by Bazhanov and Stroganov [15], who showed that the chiral Potts model is a descendant of the six-vertex model, and Hasegawa and Yamada [45], who showed that the Kashiwara–Miwa model is a descendant of the zero-field eight-vertex model. It is also of interest that the six-vertex and the zerofield eight-vertex models intersect at the zero-field six-vertex model, whose descendant is the Fateev–Zamolodchikov model, which is precisely the intersection between the chiral Potts and Kashiwara-Miwa models [12]. Quasi-triangular Hopf algebras provide an appropriate algebraic framework to study solutions of the YBE [20, 51, 61, 66]. Hopf algebras themselves have been objects of interest for many years [1, 85] and can roughly be described as unital associative algebras whose duals are also unital and associative. They are also required to have a certain antihomomorphism. Hopf algebras are called quasi-triangular if they have an associated canonical element, referred to as a universal R-matrix, which amongst other things intertwines the coproduct and is an algebraic solution of the constant YBE. The first quasi-triangular Hopf algebras appeared in the works of Drinfeld [33] and Jimbo [47] in the context of quantum groups, which are deformations of universal enveloping algebras of Lie algebras. A great feat of Drinfeld was the development of a construction which allows any Hopf algebra to be embedded into a quasi-triangular Hopf algebra [33]. This technique is referred to as the Drinfeld or quantum double construction. As stated earlier, the original quasi-triangular Hopf algebras studied were deformations of universal enveloping algebras of Lie algebras [33, 47]. These algebras, termed quantum groups, have been used to model a variety of quantum mechanical systems. Affine quantum groups are extensions of these algebras which allow the introduction of a spectral parameter into the universal R-matrix [92]. Quantum groups can also be generalised to the Z2 -graded quantum supergroups, which can be used to model systems with both bosonic and fermionic degrees of freedom [17, 19, 29]. As with quantum groups, quantum supergroups can be extended to contain an affine structure [30, 57, 80]. Over time, with use of the Drinfeld double construction, many other quasi-triangular Hopf algebras were found [18, 21, 32, 43, 77]. Within these quasi-triangular Hopf algebras lie a class known as the Drinfeld doubles of finite group algebras. The representation the-
3 ory of the Drinfeld doubles of finite group algebras has been well studied [32, 43]. These articles contain within them techniques which allow the explicit construction of all irreducible representations. For the case of the Drinfeld double of the dihedral group algebra the representations have been explicitly given [26]. It has been stated that the Drinfeld doubles of finite group algebras provide appropriate algebraic descriptions of non-Abelian anyons. The conjugacy classes and irreducible representations of the centraliser subgroups of the finite group label generalised notions of the magnetic and electric charges [28, 53, 54]. Anyons themselves are particles which are generalisations of bosons and fermions. When two anyons interchange the wave function is altered by more than the usual phase of ±1 associated with bosons and fermions. Anyonic models have recently been paid a large amount of attention, with two prominent areas of study being the fractional quantum Hall effect [5, 35, 72] and topological quantum computing [54, 71, 74]. Quantum computation has been of major interest since it became apparent that certain algorithms could be computed much faster than known algorithms for classical computers. A major challenge of quantum computing is overcoming decoherence, a phenomenon which leads to information loss. One proposed solution is the so-called topological quantum computer, which uses anyons to store information in non-local topological properties. Despite the success of the Drinfeld double construction in producing solutions to the constant YBE, there is no systematic technique which allows the introduction of a parameter into these solutions. This does not imply there are not techniques which work in specific cases. Indeed there are many such techniques, commonly referred to as Baxterisations [48, 49], including algebraic [49, 58, 59] and representation dependent [31, 46, 47, 48, 90] methods. In the case of algebraic techniques often a property of the algebra is utilised, for example the previously mentioned affine structure [92] or a grading over the integers [24]. Alternately for representation dependent techniques a property of the matrix solution may be used; for example the Baxterisation ansatz presented in [22, 91] requires that the constant R-matrix multiplied by the permutation operator has precisely three distinct eigenvalues. One Baxterisation technique, which we shall adapt for our purpose, is the tensor product graph method. This technique has successfully been applied to twisted and untwisted affine quantum (super)algebras to construct trigonometric [31, 44, 90] and rational [65] R-matrices. The technique relies on the tensor product decomposition of two irreducible representations into irreducible representations without multiplicities. The R-matrices
4
Chapter 1. Introduction
which are constructed are all expressed in terms of the projection operators associated with the tensor product decomposition. For integrable systems with boundaries a generalisation of the QISM is required. This problem was first addressed by Sklyanin [81]. In Sklyanin’s paper the XXZ and XYZ models were considered along with the non-linear Schrodinger equation and Toda chain. This method of introducing open boundaries required several conditions being imposed on the solutions of the YBE, including P -symmetry, T -symmetry and crossing symmetry. Given a suitable R-matrix two reflection equations were defined, each of which has an associated operator referred to as a reflection matrix. Subsequently these conditions were weakened and then removed. The first generalisation was the combining of P -symmetry and T symmetry into a single P T -symmetry [69]; later P T -symmetry was removed altogether [64]. Similarly the requirement of crossing symmetry was weakened to crossing unitarity [70] and then removed completely [16, 93]. For a general overview of the Yang–Baxter and reflection equation see [23, 76]. The goal of this thesis is to generalise the work of Dancer, Isaac and Links [26]. In their paper they consider the Drinfeld double of the Dihedral group of order six, D(D3 ). For this algebraic structure they used the work of [32, 43] to explicitly construct all the irreducible representations. These representations allowed the construction of parameter dependent R-matrices which yield 2- and 3-state periodic spin chains. In this thesis we generalise their results in three ways. The first of these generalisations is the breaking of the periodic conditions in the spin chain and allowing boundary terms. Secondly a more general parameter dependent solution of the YBE is found using D(D3 ). Lastly, we construct parameter dependent R-matrices using the Drinfeld double of dihedral groups of arbitrary order, D(Dn ). The structure of the thesis is outlined below. In Chapter 2 we provide the necessary background information. This includes the appropriate variants of the YBE as well as the definition of descendants. The general algebraic structure of the Drinfeld double of finite group algebras is presented followed by the defining relations for the dihedral group Dn . The representations of the double of Dn are given along with some projection operators that are required for later chapters. In Section 3.1 we modify the spin chain to allow boundary conditions. To do this we consider a general formulation of the BQISM, similar to that of [16, 93]. We first reduce the problem of solving the two reflection equations by reducing them each to a special case
5 in which they coincide. We then are able to find the most general solution to this equation and from this we are able to find the most general forms for the two reflection matrices. Next in Section 3.2 we provide a general 3-state R-matrix whose underlying algebraic structure is given by D(D3 ). We find this R-matrix by considering a more general ansatz, specifically that the R-matrix is a descendant. We also discuss the observations of [14] in which it was shown that these R-matrices can be considered special cases of the well-known Fateev–Zamolodchikov model. Lastly, in Chapter 4 we construct a family of R-matrices for models with an arbitrary number of states. We first split the construction into two cases using either D(Dn ) for odd n or D(D2n ) as our algebraic structure. Using this structure, we recall the R-matrix associated with the zero-field six-vertex model and construct an associated L-operator. Given the L-operator, it is a simple procedure to construct an operator R(z) which satisfies the intertwining relation with the L-operator. Difficulty arises when we enforce that R(z) satisfies the YBE. We are able to impose certain conditions which yield a family of operators which are conjectured to satisfy the YBE. For lower order cases it has been computationally verified that R(z) satisfies the YBE. The local Hamiltonian is constructed for models with an odd number of states. Finally the conditions we imposed on R(z) are relaxed to obtain more R-matrices. Throughout we discuss the connection between these R-matrices and those arising from the Fateev–Zamolodchikov model. For the completeness of the thesis we have also included appendices. In Appendix A we have included the construction of an R-matrix from weights satisfying the star-triangle relation. The validity of the construction is proved graphically based upon the work of [45]. Appendix B contains the observations of [14] that connect the D(D3 ) R-matrix and the Fateev–Zamolodchikov model, as well as calculations of our own about the relationship between the D(Dn ) R-matrix and the Fateev–Zamolodchikov model. Appendix C contains details of several calculations which the reader might find helpful.
Chapter 2
Background Knowledge This chapter is devoted to giving the reader the required background information for this thesis. We first present any non-standard notation which we use. In particular we utilise notations and conventions which allow us to avoid tedious technical statements involving special cases. Two Kronecker delta functions are defined and we present notation for elementary matrices. Also provided is our convention for the summation and product symbols. We provide the Yang–Baxter equation in three forms: the constant and parameter dependent forms, and a variant of the parameter dependent form which reduces to the braid equation in some special cases. We also define the intertwining relation, which allows us to introduce the concept of descendants. Following that we provide the general construction for the Hamiltonian of an integrable periodic spin chain. For a general group we give the maps and definitions required for it to become a Hopf algebra. Subsequently the dual and Drinfeld double of this Hopf algebra are given. The general dihedral group is defined, along with its presentation as a subgroup of the group of permutations. Finally we review the work of [26]. We provide the method the authors of [26] used to construct solutions of the Yang–Baxter equation and the solutions themselves. We also reproduce the 3-state local Hamiltonian associated with one of the solutions.
2.1
Notation
Throughout this thesis we often work modulo n. We define a map as follows: given a ∈ Z then 0 ≤ a ≤ n − 1 is the integer which satisfies a ≡ a (mod n). It should be noted that the overline does not represent the conjugacy class as it often does in number theory.
7
8
Chapter 2. Background Knowledge
The following two delta functions are used: ( 1, i = j, δij = and 0, i 6= j,
( δ¯ij =
1,
i ≡ j (mod n),
0,
i 6≡ j (mod n).
We use ei,j to denote an elementary matrix in Mn×n (C) whose indices are consider modulo n. The matrix ei,j has a one in the ith row and jth column and zeros elsewhere. Here we adopt the convention that e0,0 when expressed in the usual matrix form will correspond to the nth row and nth column entry. Thus we start at the 1st row rather than the 0th and similarly with the columns. These matrices obey the relation ei,j ek,l = δ¯jk ei,l . We will also use I to denote the identity matrix. For the product symbol we use the convention: ( k Y 1, k < j, ai = aj aj+1 ...ak , k ≥ j. i=j A similar convention is also used for summation: ( k X 0, k < j, ai = aj + aj+1 + ... + ak , k ≥ j. i=j
2.2
The Yang–Baxter equation and integrability
Throughout this dissertation the Yang–Baxter equation will appear in a variety of forms. The variations of the YBE which appear arise from both necessity and convenience. The YBE is a non-linear matrix equation and throughout this thesis it will usually be considered as an equation in End(V ⊗ V ⊗ V ) for some vector space V , with End(V ) being the space of endomorphisms on V . There is also a brief mention of the algebraic YBE, where the equation is in A ⊗ A ⊗ A for some algebra A. For more information on the algebraic YBE see [20, 66]. The first form we consider is the constant YBE, given by R12 R13 R23 = R23 R13 R12 ,
9
2.2 The Yang–Baxter equation and integrability
where the subscripts of R refer to which vector spaces the operator is acting upon. That is, given R=
X
ai ⊗ bi ∈ End(V ⊗ V ),
i∈S
we have R12 =
X
ai ⊗ bi ⊗ id,
R13 =
i∈S
X
ai ⊗ id ⊗ bi ,
etc.,
i∈S
where id is the identity map and S is an indexing set. Pictorially we can represent the YBE equation by the following diagram: Figure 2.1: Yang–Baxter equation 1 2
2 3 3
1
In this diagram the lines represent particles (each labeled by a number) and the crossings represent interactions. We equate Rij with a crossing between the ith and jth particle and the order of the Rs is determined from the vertical order of the crossings. The second form is known as the parameter dependent YBE and is given by R12 (x)R13 (xy)R23 (y) = R23 (y)R13 (xy)R12 (x),
(2.1)
where x, y ∈ C. Given a parameter dependent solution, R(z), constant solutions will be recovered when we take the limits z → 0, 1 and ∞. In this thesis solutions of either equation are referred to as R-matrices and generally will only be considered if invertible. This second form is the most prevalent within this thesis and hereafter we shall refer to it simply as the YBE equation. The third variant of the YBE used is ˇ 12 (x)R ˇ 23 (xy)R ˇ 12 (y) = R ˇ 23 (y)R ˇ 12 (xy)R ˇ 23 (x). R
(2.2)
Given a matrix solution R(z) to Equation (2.1), a solution to this form of the YBE is given by ˇ R(z) = P R(z), where P ∈ End(V ⊗ V ) is the usual permutation operator defined by P (v ⊗ w) = w ⊗ v,
v, w ∈ V.
10
Chapter 2. Background Knowledge
Alternately the permutation operator can be expressed as
P =
dim(V X )
ei,j ⊗ ej,i .
i,j=1
It is known that in the constant limits, i.e. when the parameters go to 0, 1 or ∞, this variant reduces to the braid equation. The pictorial representation of the braid equation is given below where each line represents a strand of string. Figure 2.2: Braid equation 1
2
3
1
2
3
In this diagram the numbers label the vertical columns rather than the individual strings. ˇ i (i+1) (0) with the string in the ith column crossing over the string in the We equate R (i + 1)th column, with order determined by the order of the crossings. Alternately we ˇ i (i+1) (1) or R ˇ i (i+1) (∞) instead of R ˇ i (i+1) (0). could have used R We now consider other associated equations.
Suppose we have the R-matrix
R(z) ∈ End(V ⊗ V ) and an invertible operator L(z) ∈ End(V ⊗ W ). We define the intertwining relation on End(V ⊗ V ⊗ W ) to be R12 (xy −1 )L13 (x)L23 (y) = L23 (y)L13 (x)R12 (xy −1 ).
(2.3)
Similarly for L(z) ∈ End(W ⊗ V ) we have L12 (x)L13 (y)R23 (x−1 y) = R23 (x−1 y)L13 (y)L12 (x). We refer to any invertible L(z) which satisfies either of these equations as an L-operator. Lastly consider R-matrices r(z) ∈ End(V ⊗ V ) and R(z) ∈ End(W ⊗ W ) where dim(V ) < dim(W ). We call R(z) a descendant of r(z) if and only if there exists an nontrivial invertible L(z) ∈ End(V ⊗ W ) which satisfies the intertwining relation with both
11
2.2 The Yang–Baxter equation and integrability r(z) and R(z). That is, together these operators satisfy the four relations: r12 (x)r13 (xy)r23 (y) = r23 (y)r13 (xy)r12 (x), r12 (x)L13 (xy)L23 (y) = L23 (y)L13 (xy)r12 (x),
(2.4)
L12 (x)L13 (xy)R23 (y) = R23 (y)L13 (xy)L12 (x),
(2.5)
R12 (x)R13 (xy)R23 (y) = R23 (y)R13 (xy)R12 (x).
To describe integrability we first introduce transfer matrices. Transfer matrices are a family of commuting matrices characterised by a parameter. That is, there is t(x) such that [t(x), t(y)] = 0,
∀x, y ∈ C.
This implies that t(x) is diagonalisable independent of x. It is also possible to express t(x) in the series t(x) =
∞ X
Ci x i ,
i=−∞
with each Ci and Cj commuting, [Ci , Cj ] = 0,
∀i, j ∈ Z.
If there is a Hamiltonian H which is expressible in terms of the operators Ci then it follows that [H, Ci ] = 0,
∀i ∈ Z,
which implies Ci are conserved quantities and furthermore the transformation which diagonalises t(x) will also diagonalise H. In classical systems a model is said to be integrable if the number of independent conserved quantities is equal to the number of degrees of freedom. For quantum mechanical systems there is no clear and agreed upon definition of integrability within the mathematical physics community. However, we shall adopt Sklyanin’s definition [82] which states that a quantum system is integrable provided it is possible to calculate exactly some quantities of physical interest, which for the purposes of this thesis is the set commuting quantum integrals of motion. These are the conserved quantities Ci mentioned above. We now move forward and consider the well known construction, known as the quantum inverse scattering method (QISM), of an integrable Hamiltonian for a periodic spin chain
12
Chapter 2. Background Knowledge
from a solution of the YBE equation [56, 60, 83]. Suppose we have an R-matrix R(z) ∈ End(V ⊗ V ) which is intertwined by an L-operator L(z) ∈ End(V ⊗ W ). Using this we construct the monodromy matrix T (z) = L0N (z)...L02 (z)L01 (z) ∈ End(V ⊗ W N ). This monodromy matrix will lead to an N -site model. As a consequence of the intertwining relation it is easily shown that R12 (xy −1 )T13 (x)T23 (y) = T23 (y)T13 (x)R12 (xy −1 ). Thus the monodromy matrix, T (z), is an L-operator. We then construct the transfer matrix by taking the trace over the space V of T (z), that is, t(z) = tr0 [L0N (z)...L02 (z)L01 (z)] , where tri is the trace over the ith space. It is then a straightforward procedure to show that the transfer matrices commute: t(x)t(y) = (tr1 ⊗ tr2 ) [T13 (x)T23 (y)] −1 −1 = (tr1 ⊗ tr2 ) R12 (xy )T23 (y)T13 (x)R12 (xy −1 ) = (tr1 ⊗ tr2 ) [T23 (y)T13 (x)] = t(y)t(x). We now suppose that V = W and R(z) = L(z). Furthermore we suppose the R(z) obeys regularity, i.e. R(1) = P , where P is the permutation operator. Using the transfer matrix we present the well-known integrable Hamiltonian for a spin chain with periodic boundary conditions: Theorem 2.2.1. The integrable Hamiltonian for a periodic spin chain, d ln(t(z)) H=c dz z=1
is given by H = HN 1 +
N −1 X
Hi(i+1) ,
i=1
is the local Hamiltonian and c ∈ C.
where
d Hij = cPij Rij (z) dz z=1
13
2.2 The Yang–Baxter equation and integrability Proof. We calculate that t(1) = tr0 [R0N (1)...R02 (1)R01 (1)] = tr0 [P0N ...P02 P01 ] = tr0 [P0N ...P03 P01 ] P12 = tr0 [P0N ...P01 ] P13 P12 = tr0 [P01 ] P1N ...P13 P12 = P1N ...P13 P12 . Similarly we have N X d 0 0 t(z) = tr0 [P0N ...P02 R01 (1)] + tr0 P0N ...P0(i+1) R0i (1)P0(i−1) ...P01 dz z=1 i=2 =
0 tr0 [R01 (1)P0N ...P02 ]
+
N X
0 tr0 P0N ...P0(i+1) P0(i−1) R(i−1)i (1)...P01
i=2 N X
0 = tr0 [P0N ...P02 ] RN 1 (1) +
0 tr0 P0N ...P0(i+1) P0(i−1) ...P01 R(i−1)i (1)
i=2 0 = tr0 [P02 ] P2N ...P23 RN 1 (1) +
N X
0 tr0 [P01 ] P1N ...P1(i+1) P1(i−1) ...P12 R(i−1)i (1)
i=2 0 = P2N ...P23 RN 1 (1) +
N X
0 P1N ...P1(i+1) P1(i−1) ...P12 R(i−1)i (1).
i=2
Lastly d ln(t(z)) H = c dz z=1 d = ct−1 (1) t(z) dz z=1 (
0 = c P12 ...P1N P2N ...P23 RN 1 (1) +
N X
) 0 P12 ...P1N P1N ...P1(i+1) P1(i−1) ...P12 R(i−1)i (1)
i=2
( 0 = c P23 ...P2(N −1) P12 P1N P2N ...P23 RN 1 (1) +
N X
( 0 = c P23 ...P2(N −1) P1N PN 2 P2N ...P23 RN 1 (1) +
( = c
0 P23 ...P2(N −1) P1N P2(N −1) ...P23 RN 1 (1)
+
0 = c P1N RN 1 (1) +
N X i=2
) 0 P(i−1)i R(i−1)i (1) ,
0 P12 ...P1(i−1) P1i P1(i−1) ...P12 R(i−1)i (1)
i=2 N X i=2 N X i=2
(
)
) 0 P12 ...P1(i−1) P1(i−1) ...P12 P(i−1)i R(i−1)i (1)
) 0 P(i−1)i R(i−1)i (1)
14
Chapter 2. Background Knowledge
as required.
2.3
Drinfeld doubles of finite groups
To construct an integrable model using the QISM a parameter dependent R-matrix is required. Drinfeld’s double construction gives rise to algebraic solutions to the constant YBE through the embedding of a Hopf algebras into a quasi-triangular Hopf algebras; Baxterisation of these solutions are then sought. Here we review Drinfeld’s construction as applied to finite group algebras [32, 43]. Given a group G with identity e, we consider the algebra CG, whose basis vectors are the elements of the group. The multiplication and unit of CG are inherited from the group in the natural way. We equip CG with a coproduct, counit and antipode defined respectively by ∆(g) = g ⊗ g,
(g) = 1
and
γ(g) = g −1 ,
∀g ∈ G.
With these maps CG becomes a Hopf algebra. We next consider the dual space of CG, (CG)o = C{g ∗ |g ∈ G}. The multiplication and unit are given, respectively, by g ∗ h∗ = δgh
and
u(1) =
X
g∗.
g∈G
The costructure and antipode are defined by X ∆(g ∗ ) = (hg)∗ ⊗ (h−1 )∗ , (g ∗ ) = δge
and
γ(g ∗ ) = (g −1 )∗ ,
∀g ∈ G.
h∈G
Under these maps (CG)o is also a Hopf algebra. Using the dual and the original Hopf algebra we can construct the Drinfeld double, D(G) = C{gh∗ |g, h ∈ G}. We impose the relation h∗ g = g(g −1 hg)∗ and adopt the required Hopf structure from CG and (CG)o . It is known that D(G) is a quasi-triangular Hopf algebra, containing the universal R-matrix X R= g ⊗ g∗. g∈G
15
2.3 Drinfeld doubles of finite groups This element satisfies the following relations: R∆(a) = ∆T (a)R, ∀a ∈ D(G), (∆ ⊗ id)R = R13 R23 , (id ⊗ ∆)R = R13 R12 , where ∆T (a) = T ◦ ∆(a)
and
T (a ⊗ b) = b ⊗ a,
∀a, b ∈ D(G).
As a consequence of these relations R is a solution of the constant YBE equation, as shown below. R12 R13 R23 = R12 (∆ ⊗ id)R = (∆T ⊗ id)R R12 = [(T ⊗ id)(R13 R23 )] R12 = R23 R13 R12 For more information on the doubles of finite groups see [32, 43]. The groups that we are interested in are known as the dihedral groups. These groups are based upon the symmetries of regular polygons. For a regular polygon with n vertices we have the dihedral group given by Dn = {σ, τ |σ n = τ 2 = e, στ σ = τ }. The order of this group is 2n. For example D3 is defined on the symmetry of a triangle where σ and τ are represented by the following actions: Figure 2.3: Action of σ and τ σ
τ and
When n > 3 it is useful to consider the dihedral group Dn as subgroup of Sn , the group of permutations of on n objects. For details on the definition and notation for Sn see [6, 7, 39]. Thus we can consider, using the usual notation for Sn , that the generators of the dihedral group are given by
σ = [1, 2, 3, ..., n]
and
τ=
n−1 bY 2 c
k=1
[k, n − k],
16
Chapter 2. Background Knowledge
where bac ∈ Z is the floor of a. We have not included D2 in this presentation as it is Abelian and we do not take any interest in it. Using this presentation for Dn we consider each element, γ ∈ Dn , as a map, γ : Zn → Zn , where Zn are the integers reduced modulo n. For example for σ ∈ D3 we have σ(2) = [1, 2, 3](2) = 3.
For ease of calculation we will investigate the dihedral groups in a few different general cases. The division of these cases will be based on the order of the group.
2.4
Representations of D(Dn), R-matrices and projection operators when n is odd
We first consider D(Dn ) for the case where n is odd. As described in [26, 43] the irreducible representations (irreps) of the double of a group are naturally partitioned by the conjugacy classes of the group. For these representations we consider w ∈ C to be a primitive nth root of unity. The irreducible representations are given in the table below:
Table 2.1: The representations of D(Dn ), n is odd. Irrep π
Constraints
π(σ)
π(τ )
π(g ∗ ), g ∈ Dn
1
±1
δge
π1± (0,k)
π2
(l,k)
π2
1≤k≤ 1≤l≤
n−1 2
n−1 , 2
0≤k ≤n−1
(a,b)
0
0
πn±
Here π1± , π2
w
k
wk
w
−k
0 −k
0 w Pn i=1 ei+1,i
0 1
1 0 0 1
1 0 Pn ± i=1 ei,2−i
δge
0
0
δge
δgσ
l
0 δgσ
2j τ
0 −l δgσ
ej+1,j+1
and πn± have dimensions 1, 2 and n respectively, and their associated mod(a,b)
ules are denoted V1± , V2
and Vn± .
Also associated with these representations are the solutions of the YBE equation arising
2.4 Representations of D(Dn ), R-matrices and projection operators when n is odd 17 from the canonical element. If we apply a two-dimensional irrep we find the R-matrix wkl 0 0 0 0 w−kl 0 0 (l,k) (l,k) . (π2 ⊗ π2 )R = −kl 0 w 0 0 kl 0 0 0 w It was shown in [26] that this constant solution leads to the parameter dependent solution w2kl z −1 − w−2kl z 0 0 0 −1 2kl −2kl 0 z − z w − w 0 . r(z) = (2.6) 2kl −2kl −1 0 w − w z − z 0 0 0 0 w2kl z −1 − w−2kl z This R-matrix corresponds to the six-vertex model with zero-field. For a review on the explicit construction of r(z) in the case of D(D3 ), see Section 2.6. Also of interest is the representation of the canonical element using the n-dimensional irreps, n X
(πn± ⊗ πn± )R = ±
ei+j,i−j ⊗ ei,i .
i,j=1
We also require the projection operators from the tensor product decomposition of two irreps onto the irreps in its decomposition. These are necessary for the construction of descendants in Chapter 4. The projection operators can be calculated in a systematic manner using the structure of the doubles of finite group algebras, D(G). It is known that, by Sweedler’s generalisation [85] of Maschkes Theorem, that for any finite group, D(G) is semi-simple. This, combined with the Artin–Wedderburn theorem, implies that D(G) is decomposable into the direct sum of ideals of D(G), that is D(G) =
M
Aα ,
α
where α are irreducible representations and Aα are ideals of D(G), each isomorphic to a matrix algebra. Furthermore there exist projection operators, Eα , satisfying Eα D(G) = Aα
and
Eα Eβ = δαβ Eα .
These projections operators are given by [43] Eα =
d[α] X χα (h∗ g −1 )gh∗ , |G| g,h∈G
18
Chapter 2. Background Knowledge
where d[α] denotes the dimension of α and χα : D(Dn ) → C is the group character defined by χα (a) = tr πα (a), ∀a ∈ D(Dn ). From these operators we consider pα = (πn± ⊗ πn± )∆(Eα ).
(2.7)
Here we have that pα is a projection operator from πn± ⊗ πn± on to all copies of the representation α in the decomposition of πn± ⊗ πn± . Note that we consider πn+ ⊗ πn+ and πn− ⊗ πn− together since (πn+ ⊗ πn+ )∆(a) = (πn− ⊗ πn− )∆(a),
∀a ∈ D(Dn ).
We calculate1 pα explicitly using the expression given in Equation (2.7), pα =
n d[α] X X χα ((σ 2j )∗ g −1 )πn± (g)ei−j,i−j ⊗ πn± (g)ei,i . 2n g∈D i,j=1 n
From this we see that the only irreps which have non-zero projection operators are associated with the conjugacy classes {e} and {σ i , σ −i } for 1 ≤ i ≤
n−1 . 2
This implies that
πn± ⊗ πn± will only decompose into one and two-dimensional irreps. For convenience, we slightly modify our notation for the irreps. Instead of using π1+ and (l,k)
π2
, we use ordered pairs corresponding only to irreps that appear in the direct sum
decomposition of πn± ⊗ πn± . The correspondence is summarised in Table 2.2. Table 2.2: The ordered pairs labelling the irreps for D(Dn ), n odd. α
Irrep
(0, 0)
π1+ (0,2b)
(0, b) (0, b)
Constraint on a
1 ≤ b ≤ n−1 4 n+3 ≤ b ≤ n−1 4 2
π2
(0,n−2b)
π2
(2a,2b)
(a, b)
Constraint on b
1 ≤ a ≤ n−1 4 n+3 ≤ a ≤ n−1 4 2
π2
(n−2a,n−2b)
(a, b) π2
0≤b≤n−1 0≤b≤n−1
For α = (0, 0), we find2 the projection operator is determined by n 1 X p = ei,j ⊗ ei,j . n i,j=1 α
1 2
See Calculation C.1.1 in Appendix C.1 for details. See Calculation C.1.2 in Appendix C.1 for details.
2.5 Representations of D(D2n ), R-matrices and projection operators
19
For α = (0, b), b 6= 0, the projection operator is given by pα =
n 1 X 2bj (w + w−2bj )ei+j,i ⊗ ei+j,i . n i,j=1
Lastly for α = (a, b), a 6= 0, we find the projection operator n 1 X 2bj [w ei+a+j,i+a ⊗ ei+j,i + w−2bj ei−a+j,i−a ⊗ ei+j,i ]. p = n i,j=1 α
We have calculated non-zero projection operators for (n2 − 1)/2, 2-dimensional irreps and one 1-dimensional irrep. Using a counting argument, it is clear this is the complete decomposition of πn± ⊗ πn± for n odd.
2.5
Representations of D(D2n), R-matrices and projection operators
We similarly catalogue the irreps of D(D2n ). Throughout this section, w ∈ C is a primitive 2nth root of unity. Table 2.3: The representations of D(D2n ). Constraints
π(σ)
π(τ )
π(g ∗ ), g ∈ D2n
(a,b)
a, b ∈ {0, 1}
(−1)b
(−1)a
δge
(a,b)
a, b ∈ {0, 1}
(−1)b
(−1)a
δgσ
Irrep π π1,e
π1,σn
(0,k)
π2
1≤k 2, along with l, k such that gcd(l, n) = gcd(k, n) = 1 and w2 a primitive nth root of unity, then for all b ∈ Z, we have n Y
z+w
2l((2j−1)k+b)
j=1
=
n Y
1 + zw2l((2j−1)k+b) .
j=1
Proof. Let h1 (z) =
n Y
z+w
2l((2j−1)k+b)
and
h2 (z) =
j=1
n Y
1 + zw2l((2j−1)k+b) ,
j=1
for some b ∈ Z. As gcd(k, n) = 1 then ∃k −1 ∈ Z such that kk −1 = 1. Using this we see that h2 (z) =
n Y
1 + zw2l((2j−1)k+b)
j=1
=
n Y
z + w−2l((2j−1)k+b)
z + w2l((−2j+1)k−b)
j=1
=
n Y j=1
= = =
n Y
z + w2l((2j+1)k−b)
j=1 n Y j=1 n Y
z + w2l((2(j+bk
−1 −1)+1)k−b)
z + w2l((2j−1)k+b) ,
j=1
as required. The previous lemma can also be considered in a more general setting (see Calculation C.3.6 in Appendix C.3). When done in the more general manner we find that given gcd(l, n) = 1 we must have gcd(k, n) = 1 for odd n if h1 (z) = h2 (z). In the case n is even only a slight difference arises.
4.1 Constructing descendants using D(Dn ) when n is odd
57
Corollary 4.1.4. Given n > 2, along with l, k such that gcd(l, n) = gcd(k, n) = 1 and w2 a primitive nth root of unity, then for all b ∈ Z, we have Y a a Y z + w2l((2j−1)k+b) z + w2l((2j−1)k+b) = , 1 + zw2l((2j−1)k+b) 1 + zw2l((2j−1)k+b) j=1 j=1 for all a ∈ N, b ∈ Z Proof. Follows directly from Proposition 4.1.3. Proposition 4.1.5. Let S 0 = {(a, b)|a ≥ 0, b ∈ Z}, and consider a set of functions {f(a,b) |(a, b) ∈ S 0 }. If the functions satisfy the relations z + w2((a+l)k+bl) f(a+l,b+k) (z) = f(a,b) (z), 1 + zw2((a+l)k+bl) f(0,b) (z) = f(0,−b) (z), f(a,b) (z) = f(a,b+n) (z), f(a,b) (z) = f(a+n,b) (z), f(a,b) (z) = f(n−a,−b) (z), (a, b) ∈ S 0 . then the functions also satisfy the constraints given in Proposition 4.1.2 for all (a, b) ∈ S. Moreover, every set of functions satisfying the conditions of Proposition 4.1.2 can be extended in a unique way to a set of functions defined on S 0 satisfying the above conditions. Hence the two sets of constraints are equivalent. Proof. The first condition is directly required by Proposition 4.1.2. The remaining conditions ensure that the extension from S to S 0 is unique and that the relations in Proposition 4.1.2 are satisfied. Lemma 4.1.6. The set of functions f(a,b) (z) =
−1 al Y
j=1
−1
z + w2l((2j−1)k+b−akl ) 1 + zw2l((2j−1)k+b−akl−1 )
! f(0,b−akl−1 ) (z),
satisfy the conditions in Proposition 4.1.5 given f(0,b) (z) = f(0,−b) (z), for b ∈ Z. Proof. Inductively we are able to show that a Y z + w2l((2j−1)k+b) f(al,ak+b) (z) = f(0,b) (z). 1 + zw2l((2j−1)k+b) j=1
(a, b) ∈ S 0
58
Chapter 4. Descendants with D(Dn ) Symmetry
Using Corollary 4.1.4 we are able to rewrite this as
f(a,b) (z) =
−1 al Y
j=1
−1
z + w2l((2j−1)k+b−akl ) 1 + zw2l((2j−1)k+b−akl−1 )
! f(0,b−akl−1 ) (z).
Thus the first condition in Proposition 4.1.5 is met, the next three are trivial satisfied by definition of f(0,b) (z) and w. The last condition is verified by Calculation C.3.7 in Appendix C.3. We now have an appropriate set of functions which we will use herein. These functions, coupled with their associated projection operators, give an operator which satisfies Equation (4.1). Combining these results we recover the operator −1 n−1 al X Y 1 2bj ˇ R(z) = w n i,j,a,b=0 p=1
−1
z + w2l((2p−1)k+b−akl ) 1 + zw2l((2p−1)k+b−akl−1 )
! f(0,b−akl−1 ) (z)ei+a+j,i+a ⊗ ei+j,i ,
which satisfies Equation (4.1). Note that in these calculations the only property that we have used is that w2 is a primitive nth root of unity, irrespective of whether n is odd or even. This allows us to use these calculations later on without alteration. ˇ It is still not known whether R(z) will satisfy the YBE. For the moment we will set l = k = 1, and consider the more general case in Section 4.3.1. a Y z + w2(2j−1+b) f(a,a+b) (z) = f(0,b) (z). 2(2j−1+b) 1 + zw j=1
(4.9)
This gives the operator n−1 a 2 Y X 2(2j−1) z+w (a,a) ˇ p R(z) = f(0,0) (z) 1 + zw2(2j−1) a=0 j=1 n−1 n−1 " a 2 2 X X Y z + w2(2j−1+b) (0,b) + f(0,b) (z) p + p(a,a+b) 2(2j−1+b) 1 + zw a=1 j=1 b=1 #) a Y z + w2(2j−1−b) + p(a,a−b) , 2(2j−1−b) 1 + zw j=1
(4.10)
or equivalently ˇ R(z) =
n−1 X i,j,a=0
"
# n−1 a 2(2p−1+b−a) X Y 1 z+w w2bj f(0,b−a) (z) ei+a+j,i+a ⊗ ei+j,i . n b=0 1 + zw2(2p−1+b−a) p=1
(4.11)
4.1 Constructing descendants using D(Dn ) when n is odd ˇ In R(z) there remains
n+1 2
59
arbitrary functions. To make this operator inherit properties
from the canonical element we enforce two additional conditions. We enforce that R(z) is self-adjoint, which is equivalent to † ˇ 12 (z) = R ˇ 21 R (z)
z ∈ R,
where † is the adjoint operator. We also enforce that R(z) obeys a limiting condition, lim R(z) = ±(πn± ⊗ πn± )R.
z→0
We now proceed to prove various properties of R(z) along with what our limiting and self-adjointness conditions imply. The first propositions we prove involve limits of R(z) namely z → 0 and z → 1. To compute this limit we consider the following proposition. Proposition 4.1.7. R(z) obeys the limiting condition lim R(z) = ±(πn± ⊗ πn± )R,
z→0
if and only if f(0,b) (0) = 1 for all 0 ≤ b ≤
n−1 . 2
Proof. Due to the linear independence of the projection operators it suffices to show that f(0,b) (0) = 1, 0 ≤ b ≤
n−1 2
implies
lim R(z) = ±(πn± ⊗ πn± )R.
z→0
We calculate "
# n−1 a X Y 1 ˇ P R(0) = P w2bj w2(2p−1+b−a) ei+a+j,i+a ⊗ ei+j,i n p=1 i,j,a=0 b=0 # " n−1 n−1 X 1 X 2b(j+a) w ei+a+j,i+a ⊗ ei+j,i = P n i,j,a=0 b=0 n−1 X
= P
n−1 X
0 δ¯a+j ei+a+j,i+a ⊗ ei+j,i
i,j,a=0
= P =
as required.
n−1 X
ei,i−j ⊗ ei+j,i
i,j,a=0 ±(πn± ⊗
πn± )R,
60
Chapter 4. Descendants with D(Dn ) Symmetry
The second proposition involving limits is concerned with regularity. Although regularity is not essential, it is useful and as such it has been utilised in all the constructions of integrable models which have appeared in this thesis so far. We produce the following proposition to determine when R(z) will satisfy regularity. ˇ ˇ Proposition 4.1.8. The operator R(z) given in (4.11) satisfies regularity, i.e. R(1) = I ⊗I (equivalently R(1) = P ), if and only if f(0,b) (1) = 1 for 0 ≤ b ≤
n−1 . 2
Proof. We first note that n−1
I ⊗I =
2 X
n−1
p(0,b) +
b=0
n−1 2 X X
p(a,b) ,
a=1 b=0
and that the projection operators p(a,b) are linearly independent. By definition we have n−1 n−1 n−1 2 2 2 X X h i X ˇ R(1) = f(0,0) (1) p(a,a) + f(0,b) (1) p(0,b) + p(a,a+b) + p(a,a−b) . a=0
b=1
a=1
Matching coefficients of projection operators proves the proposition. We now determine what conditions R(z) being self-adjoint imposes. † ˇ ˇ 12 (z) = R ˇ 21 Proposition 4.1.9. The R(z) of Equation (4.11) satisfies R (z) if and only if
the coefficient functions defined in Equation (4.9) satisfy (f(a,b) (z))∗ = f(a,−b) (z), for (a, b) ∈ S 0 , where
∗
denotes complex conjugation. This condition is equivalent to (f(0,b) (z))∗ = f(0,b) (z) = f(0,b+2c) (z),
∀z ∈ R and b, c ∈ Z. Proof. Looking at the general form of the projection operators we find that † (a,b) (a,n−b) p˜21 = p˜12 . ˇ Recalling the form of R(z) given in Equation (4.2), we find that self-adjointness implies (f(a,b) (z))∗ = f(a,−b) (z).
4.1 Constructing descendants using D(Dn ) when n is odd
61
Equation (4.9) gives us f(1,1+b) (z) = and
f(1,−1−b) (z) =
z + w2(b+1) 1 + zw2(b+1)
z + w−2(b+1) 1 + zw−2(b+1)
f(0,b) (z),
f(0,−b−2) (z).
We see that ∗
(f(1,−1−b) (z)) =
z + w2(b+1) 1 + zw2(b+1)
∗
(f(0,−b−2) (z)) =
z + w2(b+1) 1 + zw2(b+1)
f(0,b+2) (z),
which implies that f(0,b) (z) = f(0,b+2) (z). Combining this with the constraint of f(0,b) (z) we find (f(0,b) (z))∗ = f(0,b) (z) = f(0,b+2c) (z), for b, c ∈ Z. Conversely, suppose (f(0,b) (z))∗ = f(0,b+2c) (z), for all b, c ∈ Z. We are able to see that a Y z + w2(2j−1+b−a) f(0,b−a) (z). f(a,b) (z) = 1 + zw2(2j−1+b−a) j=1 We can now show that ∗
(f(a,b) (z))
= = = =
!∗ a Y z + w2(2j−1+b−a) f(0,b−a) (z) 1 + zw2(2j−1+b−a) j=1 a Y ∗ z + w−2(2j−1+b−a) f (z) (0,b−a) 1 + zw−2(2j−1+b−a) j=1 a Y z + w2(2(−j)+1−b+a) f(0,−b−a) (z) 1 + zw2(2(−j)+1−b+a) j=1 a Y z + w2(2j−1−b−a) f(0,−b−a) (z) 1 + zw2(2j−1−b−a) j=1
= f(a,−b) (z). This implies from previous calculations that R(z) must be self-adjoint if we have that (f(0,b) (z))∗ = f(0,b+2c) (z), for all b, c ∈ Z.
62
Chapter 4. Descendants with D(Dn ) Symmetry
As the second index of the function can be considered modulo n and n is odd it implies that there is only one arbitrary function left. Without loss of generality we consider it to be f(0,0) (z). This can be seen as an overall scalar multiple of our operator thus we are able to set it to a constant. This constant is determined by the limiting condition, and thus by Proposition 4.1.7 we set f(0,0) (z) = 1. This gives us the functions a Y z + w2(2j−1+b−a) f(a,b) (z) = , 2(2j−1+b−a) 1 + zw j=1 which produces the operator n−1
n−1
n−1 Y a 2 X X z + w2(2j−1+b−a) (0,b) ˇ p + p(a,b) , R(z) = 2(2j−1+b−a) 1 + zw a=1 b=0 j=1 b=0 2 X
or equivalently ˇ R(z) =
n−1 X i,j,a=0
"
# n−1 a 1 X 2bj Y z + w2(2p−1+b−a) w ei+a+j,i+a ⊗ ei+j,i . 2(2p−1+b−a) n b=0 1 + zw p=1
As we have set f(0,b) (z) = 1 for b ∈ Z by Proposition 4.1.8 we have that R(z) satisfies regularity. To simplify the operator R(z) we define the functions n−1 n−1 a 1 X 2bj 1 X 2bj Y z + w2(2p−1+b−a) g(a,j) (z) = w f(a,b) (z) = w , n b=0 n b=0 1 + zw2(2p−1+b−a) p=1
(4.12)
where a ∈ N and j ∈ Z. In terms of these new functions our operator becomes ˇ R(z) =
n−1 X
g(a,j) (z)ei+a+j,i+a ⊗ ei+j,i ,
i,j,a=0
or R(z) =
n−1 X
g(a,j) (z)ei+j,i+a ⊗ ei+a+j,i
(4.13)
i,j,a=0
We have found the required conditions for R(z) to be self-adjoint and limiting appropriately. We now wish to determine other properties of R(z). We first show that when z ∈ R then R(z) only contains real entries. Proposition 4.1.10. The R(z) given in Equation (4.13) satisfies R(z)∗ = R(z),
∀z ∈ R.
4.1 Constructing descendants using D(Dn ) when n is odd
63
Proof. We first restrict z to the reals and 0 ≤ a, j ≤ n − 1. We then proceed to calculate #∗ n−1 a 1 X 2bj Y z + w2(2p−1+b−a) w n b=0 1 + zw2(2p−1+b−a) p=1 n−1 a 1 X −2bj Y z + w−2(2p−1+b−a) = w n b=0 1 + zw−2(2p−1+b−a) p=1 n−1 a 1 X 2bj Y z + w2(−2p+1+b+a) = w n b=0 1 + zw2(−2p+1+b+a) p=1 n−1 a 1 X 2bj Y z + w2(2p−1+b−a) = w n b=0 1 + zw2(2p−1+b−a) p=1 "
∗ g(a,j) (z) =
= g(a,j) (z). As the only entries of R(z) are g(a,j) (z) we have proved the proposition. We calculate the inverse of R(z). Proposition 4.1.11. The R(z) given in Equation (4.13) satisfies R2 (z) = I ⊗ I. Proof. We first note that R(z) squaring to the identity is equivalent to ˇ 21 (z)R ˇ 12 (z) = I ⊗ I. R We are able to calculate that (a,b)
p˜21
(a,−b)
= p˜12
and f(a,−b) (z) =
=
=
= =
a Y z + w2(2j−1−b−a) 1 + zw2(2j−1−b−a) j=1 " a #−1 Y 1 + zw2(2j−1−b−a) z + w2(2j−1−b−a) j=1 " a #−1 Y z + w2(−2j+1+b+a) 1 + zw2(−2j+1+b+a) j=1 " a #−1 Y z + w2(2j−1+b−a) 1 + zw2(2j−1+b−a) j=1 −1 f(a,b) (z) .
64
Chapter 4. Descendants with D(Dn ) Symmetry
Combining these results leads to n−1
ˇ 21 (z)R ˇ 12 (z) = R
2 X
n−1
f(0,b) (z)f(0,−b) (z)p
(0,b)
n−1 2
=
f(a,−b) (z)f(a,b) (z)p(a,b)
a=1 b=0
b=0
X
+
n−1 2 X X
n−1 2
p
(0,b)
+
n−1 XX
p(a,b)
a=1 b=0
b=0
= I ⊗ I.
We still have not made any assertions to whether R(z) satisfies the YBE. We simplify the matter by providing a list of identities which hold if and only if R(z) satisfies the YBE. Proposition 4.1.12. The operator R(z) =
n−1 X
g(a,j) (z)ei+j,i+a ⊗ ei+a+j,i
i,j,a=0
is a solution of the Yang–Baxter equation if and only if n−1 X
g(a,k−d) (x)g(k,a−b) (xy)g(b,c−k) (y) =
k=0
n−1 X
g(c,k−b) (x)g(k,c−d) (xy)g(d,a−k) (y),
k=0
for for 0 ≤ a, b, c, d ≤ n − 1. Proof. See Calculation C.3.8 in Appendix C.3. We have converted the problem of verifying if R(z) is a solution of the YBE into a problem of verifying relations of functions. It is also note worthy in the proof there is no assumption of the form of g(a,j) (z) which means this will apply to any R-matrix with this general form. The main properties of the conjectured descendant can be summarised by the following corollary which is an amalgamation of previous propositions. Corollary 4.1.13. The operator R(z) of Equation (4.13) has the following properties: (i) R(z)∗ = R(z), ∀z ∈ R (ii) R† (z) = R(z), ∀z ∈ R (iii) R−1 (z) = R(z), ∀z ∈ C
4.1 Constructing descendants using D(Dn ) when n is odd
65
(iv) limz→0 R(z) = ±(πn± ⊗ πn± )R (v) limz→1 R(z) = P (vi) R(z) is a solution of the Yang–Baxter equation if and only if n−1 X
g(a,k−d) (x)g(k,a−b) (xy)g(b,c−k) (y) =
k=0
n−1 X
g(c,k−b) (x)g(k,c−d) (xy)g(d,a−k) (y),
k=0
for 0 ≤ a, b, c, d ≤ n − 1. Proof. This is the collection of Propositions 4.1.7–4.1.12. The matrix R(z) agrees with the R-matrix obtained in [26] for n = 3. It has also been verified computationally for all odd n ≤ 17 that the functions g(a,j) (x) satisfy the property given in (vi), and thus that R(z) is a solution of the YBE. The computations were performed with x and y treated as arbitrary complex numbers and w as an arbitrary primitive nth root of unity. Conjecture 4.1.14. The matrix R(z) given in Equation (4.13) is a solution of the Yang– Baxter equation for all odd n. As with the case of the 3-state model we expect that this general n-state model will again be a special case of the Fateev–Zamolodchikov. In Appendix B.3 it was shown that when n is odd there is a limit of the Fateev–Zamolodchikov model which squares to the identity and limits to a representation of the canonical element from D(Dn ). Furthermore it was shown that for small odd n that the D(Dn ) model is precisely a special case of the Fateev– Zamolodchikov model. It is reasonable to expect that this connection between the models holds for larger n. We remark that should the connection hold as expected then the above conjecture must hold. Earlier it was also noted that there is a more general D(D3 ) R-matrix than that presented in [26]. This more general R-matrix was again seen to be a special case of the Fateev– Zamolodchikov model. Later in Section 4.3.2 we show that we have not found the most general form of the D(Dn ) R-matrix, it is expected that again the more general R-matrix will be a special case of Fateev–Zamolodchikov model.
66
4.1.3
Chapter 4. Descendants with D(Dn ) Symmetry
Local Hamiltonian
Here we construct the local Hamiltonian for the periodic spin chain as presented in Theorem 2.2.1. We start by stating simple results which are useful. Firstly, for λ ∈ C, d z+λ 1−λ = . dz 1 + zλ z=1 1+λ If n ∈ N is odd and λn = 1 then n−1
(1 + λ)−1 =
1X (−λ)i . 2 i=0
We also have i 1h 0 (−1)i + (−1)i+δi = (1 − δi0 )(−1)i , 2 for 0 ≤ i ≤ n − 1. We are able to combine these into a single proposition. Proposition 4.1.15. If n ∈ N is odd and λn = 1 then
d dz
z+λ 1 + zλ
n−1 X
= z=1
(1 − δi0 )(−λ)i
i=0
Proof.
d dz
z+λ 1 + zλ
=
z=1
1−λ 1+λ
n−1
X 1 = (1 − λ) (−λ)i 2 i=0 n−1
1X = (−λ)i + (−λ)i+1 2 i=0 = =
n−1 X 1 i=0 n−1 X
2
h i 0 λi (−1)i + (−1)i+δi
(1 − δi0 )λi (−1)i .
i=0
We can now use this to differentiate the entries of the R-matrix. Proposition 4.1.16. Given g(a,j) (z) as defined in Equation (4.12) for 0 ≤ a, j ≤ n − 1 then
a X d 0 j g(a,j) (z) = (δj − 1)(−1) w−2(2p−1−a)j dz z=1 p=1
4.1 Constructing descendants using D(Dn ) when n is odd
67
Proof.
" d 1 d g(a,j) (z) = dz n dz z=1
!# a 2(2p−1+b−a) Y z + w w2bj 1 + zw2(2p−1+b−a) p=1 b=0 !#z=1 " n−1 a 1 X 2bj d Y z + w2(2p−1+b−a) = w n b=0 dz p=1 1 + zw2(2p−1+b−a) z=1 n−1 a 2(2p−1+b−a) 1 X 2bj X d z+w = w n b=0 dz 1 + zw2(2p−1+b−a) z=1 p=1 n−1 X
We now utilise Proposition 4.1.15 and continue the calculation:
n−1 a n−1 d 1 X 2bj X X g(a,j) (z) = (1 − δi0 )(−1)i w2(2p−1+b−a)i w dz n b=0 z=1 p=1 i=0 n−1 a 1 XX = (1 − δi0 )(−1)i w2(2p−1+b−a)i+2bj n b,i=0 p=1 " n−1 # a n−1 X X X 1 (1 − δi0 )(−1)i w2(2p−1−a)i w2b(i+j) = n b=0 i=0 p=1
= =
n−1 X a X
0 δ¯i+j (1 − δi0 )(−1)i w2(2p−1−a)i
i=0 p=1 a X
0 (1 − δn−j )(−1)n−j w−2(2p−1−a)j
p=1
=
(δj0
j
− 1)(−1)
a X
w−2(2p−1−a)j .
p=1
This gives the required result. These propositions provide the bulk of the calculation required to determine the local Hamiltonian. We come to the final proposition for this section. Proposition 4.1.17. The local Hamiltonian for the D(Dn ) R-matrix is given by H =
w−w
n−1 X n−1 X −1 i=0 a,j=1
" (−1)j
a X
# w2(2p−1−a)j ei+a−j,i+a ⊗ ei−j,i ,
p=1
which is Hermitian. Proof. We start with the definition of the local Hamiltonian for a periodic spin chain as
68
Chapter 4. Descendants with D(Dn ) Symmetry
given in Theorem 2.2.1 (setting c = (w − w−1 )), d −1 ˇ H = w−w R(z) dz z=1 # " a n−1 X X w−2(2p−1−a)j ei+a+j,i+a ⊗ ei+j,i = w − w−1 (δj0 − 1)(−1)j p=1
i,j,a=0
=
w−w
−1
n−1 X n−1 X
−1
n−1 X n−1 X
" (−1)j+1
=
w−w
# w−2(2p−1−a)j ei+a+j,i+a ⊗ ei+j,i
p=1
i=0 a,j=1
a X
" j
(−1)
#
a X
w
2(2p−1−a)j
ei+a−j,i+a ⊗ ei−j,i .
p=1
i=0 a,j=1
This proves the first statement made in the proposition. Next we check if H is Hermitian. " # n−1 X n−1 a X X (−1)j w−2(2p−1−a)j ei+a,i+a−j ⊗ ei,i−j H † = − w − w−1 p=1
i=0 a,j=1
= − w−w
−1
n−1 X n−1 X
" (−1)n−j
=
w−w
−1
# w2(2p−1−a)j ei+a,i+a+j ⊗ ei,i+j
p=1
i=0 a,j=1 n−1 X n−1 X
a X
" (−1)j
a X
# w2(2p−1−a)j ei+a−j,i+a ⊗ ei−j,i
p=1
i=0 a,j=1
This calculation completes the proof of the proposition. We have shown that the R-matrix from D(Dn ) yields a spin chain with a Hermitian local Hamilton given in Proposition 4.1.17. We also wish to illustrate the symmetry of the Hamiltonian inherited from D(Dn ). We consider the Dn as a subgroup of Sn as described in Section 2.3. The generators of Dn are given by n−1
σ = [1, 2, 3, ..., n]
and
τ=
2 Y
[k, n − k].
k=1
Using this notation we can express6 the local Hamiltonian as H = w−w
−1
n−1 X j=1
n−1
(−1)
j
a 2 X X
" w
2(2p−1−a)j
a=1 p=1
n−1 X
# eγ(a−j),γ(a) ⊗ eγ(n−j),γ(n) .
γ∈Dn
This form in which we sum over elements of Dn written as permutations of integers is the a generalisation of the form shown in [26] in the case of n = 3. 6
See Calculation C.3.9 in Appendix C.3 for details.
4.1 Constructing descendants using D(Dn ) when n is odd
69
From its construction the local Hamiltonian inherits symmetries from D(Dn ). Specifically we have [H12 , (πn ⊗ πn )∆(a)] = 0,
∀a ∈ D(Dn ).
Due to the structure of D(Dn ) we have the additional symmetry that [H21 , (πn ⊗ πn )∆(a)] = 0,
∀a ∈ D(Dn ),
even though ∆ 6= ∆T . However, we find that the D(Dn ) symmetry does not appear in the global Hamiltonian for spin chains of finite length. In this case the inherited symmetry is given by the subalgebra, CH = {a ∈ D(Dn )|∆(a) = ∆T (a)}. That is, L
[H, (πn⊗ )∆(L) (a)] = 0,
a ∈ CH
where L is the number of sites and ∆(l) is defined recursively by the relations ∆(l) = (∆ ⊗ id⊗
l−2
)∆(l−1)
and
∆(2) = ∆.
Due to the co-commutative nature of group algebras and the coproduct of e∗ , CH satisfies the condition Dn ⊂ CH ⊂ D(Dn ). In the thermodynamic limit of an infinite site spin chain the D(Dn ) invariance is recovered. It is important to note that although our R-matrix appears to be a special case of the Fateev–Zamolodchikov R-matrix, the Hamiltonian given above is not connected to the usual Fateev–Zamolodchikov Hamiltonian presented [3, 4, 78]. It is often the case that the Hamiltonian presented for the Fateev–Zamolodchikov model is associated with the socalled uniform square lattice [13]. The Hamiltonian which appears in these papers is given by H=−
L N −1 X X j=1 k=1
1 sin
h i † k k + (Z Z ) , X j j+1 j kπ N
where N is the number of states, L is the number of sites and w 0 0 0 1 0 0 w2 0 . X= and Z = 0 0 1 0 0 1 1 0 0 The periodic spin chain with local Hamiltonian given in Proposition 4.1.17 has a different eigenspectrum to the above Hamiltonian.
70
Chapter 4. Descendants with D(Dn ) Symmetry
4.2
Constructing descendants using D(D2n)
In this section we construct a family of solutions of the Yang–Baxter equation using D(D2n ). We first construct the L-operator in a similar manner as before. Then to determine the solutions of the YBE we will split this case further, again depending on the parity of n. It turns out that the case when n is odd is in fact very similar to the case of the previous section. From this point on we will consider w to be a primitive 2nth root of unity.
4.2.1
L-operator
In this subsection we construct an operator L(z) ∈ End(V2 ⊗ Vn ) using a similar approach (l,k)
as before in Section 4.1.1. We use r(z) associated with π2
given in Section 2.5 and
assume that the form of the L-operator is going to be (l,k)
L(z) = (π2 (0,b)
⊗ πn )[R + h(z)(RT )−1 ],
(0,b)
where πn is either πn,τ or πn,στ with b ∈ {0, 1}. Applying these representations7 and an appropriate basis transformation on the two-dimensional space we find L(z) =
n−1 X
(w2ik e1,2 + w−2ik e2,1 ) ⊗ ei,i + h(z) [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ] ,
i=0
We note that the basis transformation is of the same general form used in Section 4.1.1. Also both r(z) and L(z) are of the same form as Section 4.1.1. Thus it follows that our L-operator is of the form L(z) =
n−1 X
(w2ik e1,2 + w−2ik e2,1 ) ⊗ ei,i + z [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ] .
i=0
4.2.2
Descendants when n is even
Here we consider the case D(D2n ) where n is even. We use the predicted form of the descendant ˇ P R(z) = R(z) =
X
fα (z)pα ,
α∈S
where pα are the projection operators previously calculated, fα (z) are continuous functions and S is the set of ordered pairs which correspond to non-zero projection operators. We ˇ require R(z) to satisfy Equation (4.1). We define the operators 7
See Calculations C.3.10 and C.3.11 in Appendix C.3 for details.
4.2 Constructing descendants using D(D2n ) (a,b)
p˜
n X
=
71
[w2jb ei+a+j,i+a ⊗ ei+j,i + w−2bj ei−a+j,i−a ⊗ ei+j,i ]
i,j=1
for (a, b) ∈ S. These operator have the property that p˜α is a scalar multiple of pα for all α ∈ S. We notice that the form of these operators and the L-operator are equivalent to those found in previous sections. This implies that four constraints of Proposition 4.1.2 apply in this section. As done in Section 4.1.2, we draw the tensor product diagram, Figure 4.2, which displays the required relations between the projection operators for the case k = l = 1. Figure 4.2: Tensor product diagram for even n ) (0, n−2 2
(0, 1)
(0, 0)
) (0, n 2
(1, 1)
(1, 0)
(1, 2)
(1, n+4 ) 2
(1, n ) 2
(1, n+2 ) 2
, n−2 ) ( n−2 2 2
( n−2 , n−4 ) 2 2
( n−2 , n+4 ) 2 2
( n−2 , 0) 2
( n−2 , n − 2) 2
( n−2 , n − 1) 2
(n , n−2 ) 2 2
(n , n) 2 2
(n , 1) 2
(n , 0) 2
Again we define the set S 0 = {(a, b)|a ∈ N, b ∈ Z}, which leads to a natural extension for the functions fα (z). This extension is analogous to that provided in Proposition 4.1.5. We use calculations from previous sections, while again requiring gcd(k, n) = gcd(l, n) = 1, and determine that for R(z) to be a descendant we must have
f(a,akl−1 +b) (z) =
−1 al Y
j=1
z + w2l((2j−1)k+b) 1 + zw2l((2j−1)k+b)
f(0,b) (z),
72
Chapter 4. Descendants with D(Dn ) Symmetry
where the functions f(0,b) (z) satisfy f(0,b) (z) = f(0,b+n) (z), f(0,b) (z) = f(0,−b) (z),
(a, b) ∈ S 0 .
We now choose to set l = k = 1; this gives the functions a Y z + w2(2j−1+b) f(a,a+b) (z) = f(0,b) (z). 1 + zw2(2j−1+b) j=1 This produces the operator n a 2 Y X 2(2j−1) z+w (a,a) ˇ R(z) = f(0,0) (z) p 1 + zw2(2j−1) a=0 j=1 n a 2 Y X 2(2j−1+ n ) 2 z+w +a) (a, n 2 +f(0, n2 ) (z) p n a=0 1 + zw2(2j−1+ 2 ) j=1 n n −1 −1 " a 2 2 X X Y z + w2(2j−1+b) (0,b) + + f(0,b) (z) p p(a,a+b) 2(2j−1+b) 1 + zw a=1 j=1 b=1 # ) a a 2(2j−1+ n ) Y Y 2 n z + w2(2j−1−b) z + w + p(a,a−b) + p(a,a+ 2 ) 2(2j−1+ m ) 2(2j−1−b) 2 1 + zw 1 + zw j=1 j=1 or equivalently
R(z) =
n−1 X i,j,a=0
"
# n−1 a 1 X 2bj Y z + w2(2p−1+b−a) w f(0,b−a) (z) ei+j,i+a ⊗ ei+a+j,i . n b=0 1 + zw2(2p−1+b−a) p=1
As before we start by determining if our R-matrix is able to satisfy regularity and if so under what conditions. Proposition 4.2.1. We have the relation c+ n 2
Y z + w2(2p−1+b−a) a+b+ n 2, = (−1) 2(2p−1+b−a) 1 + zw p=c+1 for c ∈ N, w a primitive 2nth root of unity and n > 2 even.
4.2 Constructing descendants using D(D2n )
73
Proof. It is a straight forward calculation: n c+ n 2 Y2 z + w2(2p−1+b−a) Y z + w2(2p−1+b−a) = 1 + zw2(2p−1+b−a) 1 + zw2(2p−1+b−a) p=c+1 p=1 n
=
2 Y
w
−2(2p−1+b−a)
w
−2(2p−1+b−a)
w
−2(2p−1+b−a)
z + w2(2p−1+b−a) z + w−2(2p−1+b−a)
z + w2(2p−1+b−a) z + w2(2p−1−b−a)
z + w2(2p−1+b−a) z + w2(2p−1+b−a)
p=1
n
=
2 Y
p=1 n
=
2 Y
p=1 n
=
2 Y
w−2(2p−1+b−a)
p=1 n
= (−1)a+b+ 2 .
ˇ Proposition 4.2.2. For R(z) as define above we have ˇ lim R(z) 6= cI ⊗ I
z→1
for all non-zero c ∈ C. Proof. We have n
I ⊗I =
2 X
n
p(0,b) + p
,b) (n 2
+
−1 n−1 2 X X
p(a,b) ,
a=1 b=0
b=0
where the projection operators are linearly independent. We also have n
n
ˇ R(z) =
2 X
f(0,b) (z)p
(0,b)
+ f( n2 ,b) (z)p
,b) (n 2
+
f(a,b) (z)p(a,b) .
a=1 b=0
b=0
We recall that
−1 n−1 2 X X
a Y z + w2(2j−1+b−a) f(0,b−a) (z). f(a,b) (z) = 1 + zw2(2j−1+b−a) j=1
using Proposition 4.2.1 we obtain f( n2 ,b) (z) = (−1)b f(0,b− n2 ) (z). This leads to n
ˇ R(z) =
2 X
n
f(0,b) (z) p(0,b) + (−1)
b=0
This proves the proposition.
n +b 2
p
(n , n −b) 2 2
+
−1 n−1 2 X X a=1 b=0
f(a,b) (z)p(a,b) .
74
Chapter 4. Descendants with D(Dn ) Symmetry
The previous propositions states that irrespective of the choice of functions, f(0,b) (z), the R-matrix obtained will not satisfy regularity. Here we again enforce that our operator is self-adjoint. We are able to use the previous calculations and recall that (f(0,b) (z))∗ = f(0,b) (z) = f(0,b+2c) (z), ∀z ∈ R and b, c ∈ N. The second index of the function can be considered modulo n and as n is even the functions are partitioned into two sets. We see that we have only two functions in which we have any freedom left; without loss of generality we consider them to be f(0,0) (z) and f(0,1) (z). As there are two functions we find that we cannot consider them as an overall scalar multiple, so we need to enforce additional conditions. We enforce that our descendant is unitary; this also implies the condition that our descendant squares to the identity. We enforce this because the n-dimensional representations of the canonical element along with our descendant in the case where n was odd have this property. From this it follows that f(0,0) (z) = ±1
and
f(0,1) (z) = ±1.
Enforcing the limiting condition we find that we must have f(0,0) (z) = f(0,1) (z) = 1. This yields the operator " n−1 # n−1 a 2(2p−1+b−a) X X Y 1 z + w ˇ R(z) = w2bj ei+a+j,i+a ⊗ ei+j,i . 2(2p−1+b−a) n 1 + zw p=1 i,j,a=0 b=0
(4.14)
To simplify this we define the functions n−1 n−1 a 1 X 2bj Y 1 + zw2(a−b+1−2k) 1 X 2bj w f(a,b) (z) = w , g(a,j) (z) = 2(a−b+1−2k) n b=0 n b=0 z + w k=1 where a ∈ N and j ∈ Z. With the use of these new functions our operator becomes R(z) =
n−1 X
g(a,j) (z)ei+j,i+a ⊗ ei+a+j,i .
(4.15)
i,j,a=0
This operator is of the same form given in Proposition 4.1.12, which implies that the operator of Equation (4.15) is a solution of the YBE if and only if the following constraint is satisfied: n−1 X k=0
g(a,k−d) (x)g(k,a−b) (xy)g(b,c−k) (y) =
n−1 X k=0
g(c,k−b) (x)g(k,c−d) (xy)g(d,a−k) (y),
4.2 Constructing descendants using D(D2n )
75
for 0 ≤ a, b, c, d ≤ n − 1. This has been computationally shown to be true for even n up to 16 and we conjecture that these possible descendants are actual descendants for all even n greater than 16. ˇ We now investigate a few properties of R(z) given in Equation (4.14) and the functions used in its construction. We first observe that again we have the relation (f(a,b) (z))∗ = f(a,−b) (z) = [f(a,b) (z)]−1 , for z ∈ R, which is a consequence of imposing that R(z) is self-adjoint and unitary. Again we find that R(z) squares to the identity. The above relations also imply that g(a,j) (z) ∈ R for all z ∈ R, as n−1
(g(a,j) (z))∗ =
n−1
1 X −2bj 1 X −2bj w (f(a,b) (z))∗ = w f(a,−b) (z) = g(a,j) (z). n b=0 n b=0
ˇ Thus R(z) is a real matrix when z is real. We are also able to show that f(a,b) (z −1 ) = [f(a,b) (z)]−1 , ˇ from which it follows that R(z) obeys the unitarity relation, ˇ −1 )R(z) ˇ R(z = I ⊗ I. We also recall that we required a limiting condition ˇ lim R(z) = (−1)b P (πn ⊗ πn )R,
z→0 (0,b)
(0,b)
where πn = π(0,τ ) or πn = π(0,στ ) for b ∈ {0, 1}. To summarise, we have found a possible descendant of a zero-field six-vertex model for each even n ∈ N. We applied additional constraints to the possible descendant and found the operator
R(z) =
n−1 X i,j,a=0
"
# n−1 a 1 X 2bj Y z + w2(2p−1+b−a) w ei+j,i+a ⊗ ei+a+j,i . n b=0 1 + zw2(2p−1+b−a) p=1
(4.16)
This operator is of the same form as our conjectured descendant from D(Dn ) for odd n. We have found that R(z) satisfies the YBE for even n up to 16. We conjecture that R(z)
76
Chapter 4. Descendants with D(Dn ) Symmetry
satisfies the YBE for n ≥ 18. This conjectured descendant shares many properties with the conjectured descendant for n odd. We have that R(z) is self-adjoint, unitary and squares to the identity. We also have that R(z) limits to a representation of the canonical element as z goes to zero. Lastly R(z) obeys the unitarity condition but not the regularity condition. We can also compare this R-matrix to the Fateev–Zamolodchikov R-matrix. In Corollary B.3.9 in Appendix B.3 it was shown that any R-matrix derived from the Fateev– Zamolodchikov model which squares to the identity has the same eigenvalues (and multiplicities) as the permutation operator. It is known that the permutation operator’s eigenvalues are 1 and -1 which have multiplicities
n(n+1) 2
and
n(n−1) 2
while the eigenvalues for
the R-matrix given by Equation (4.16) are 1 and -1 which have multiplicities n( n2 + 1) and n( n2 − 1). We deduce that any connection between the Fateev–Zamolodchikov model and this R-matrix lies beyond transformations which preserve eigenvalues.
4.2.3
Descendants when n is odd
For completeness, we include the descendants associated to D(D2n ) for odd n despite this yielding an R-matrix which is equivalent to the R-matrix from D(Dn ) when n is odd. We provide the crucial points of the construction and relate the calculations to those done previously. We start by recalling w is a primitive 2nth root of unity. We have already calculated the L-operator, L(z), associated with r(z) in Section 4.2.1. Recalling the projection operators defined in Section 2.5 for odd n and Section 2.4 we find that their form is the same. This implies that as we again use the ansatz ˇ R(z) =
X
fα (z)pα ,
α∈S
where S is the set of non-zero projection operators, that we must obtain the same R-matrix as given in Section 4.1. Furthermore if we apply the same assumptions then we find the operator R(z) =
n−1 X i,j,a=0
"
# n−1 a 1 X 2bj Y z + w2(2p−1+b−a) w ei+j,i+a ⊗ ei+a+j,i . n b=0 1 + zw2(2p−1+b−a) p=1
It also obeys the properties stated in Section 4.1.2. Thus our operator is self-adjoint, unitary and squares to the identity. It also obeys the regularity and unitarity conditions and
77
4.3 Other descendants limits to a representation of the canonical element.
For D(D2n ) where n is odd we have therefore found descendants which are equivalent to the descendant found using D(Dn ) where n is odd. These descendants are therefore valid for odd n up to 17 and conjectured to be valid for greater odd n.
4.3
Other descendants
In this section we investigate some of the choices we made in order to obtain descendants. Specifically, we explore the effects of choosing different 2-dimensional irreps and of removing the self-adjointness, unitary and limiting conditions. Throughout this section for the R-matrix associated with the n-dimensional irrep we will consider w as a primitive nth root of unity. As a consequence most operators will differ slightly in the powers of w compared to their counterparts in previous sections. This is done to unify the cases of both odd and even n.
4.3.1
Using other initial representations
We first show that starting with two-dimensional irreps with certain properties yields equivalent descendants. Here we consider any n ≥ 3, with w a primitive nth root of unity. We recall that we started with the R-matrix wkl z −1 − w−kl z 0 0 0 −1 kl −kl 0 z − z w − w 0 , r(z) = kl −kl −1 0 w − w z − z 0 kl −1 −kl 0 0 0 w z −w z (k,l) where 0 ≤ l ≤ n2 and 0 ≤ k ≤ n − 1. This r(z) is associated with π2 although we note that the w used here differs slightly to that used in Sections 2.4 and 2.5. Using r(z) we found the L-operator L(z) =
n−1 X ik (w e1,2 + w−ik e2,1 ) ⊗ ei,i + z [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ] . i=0
We consider the case where l and k satisfy gcd(l, n) = gcd(k, n) = 1. We obtain the possible descendant −1 n−1 al X Y 1 bjk ˇ R(z) = w n i,j,a,b=0 p=1
−1
z + wlk(2p−1+b−al ) 1 + zwlk(2p−1+b−al−1 )
! f(0,bk−akl−1 ) (z)ei+a+j,i+a ⊗ ei+j,i .
78
Chapter 4. Descendants with D(Dn ) Symmetry
We consider the change of basis on the n-dimensional space which yields ei,j → eil−1 ,jl−1 ,
i ∈ Z.
Under this change of basis we find that our L-operator becomes L(z) =
n−1 X
(wilk e1,2 + w−ilk e2,1 ) ⊗ ei,i + z [e1,1 ⊗ ei−1,i + e2,2 ⊗ ei+1,i ] ,
i=0
while our descendant must be of the form n−1 a Y 1 X z + wlk(2p−1+b−a) bjlk ˇ f(0,(b−a)k) (z)ei+a+j,i+a ⊗ ei+j,i . R(z) = w n i,j,a,b=0 1 + zwlk(2p−1+b−a) p=1 From this we see that these different choices of initial two-dimensional representation provide equivalent descendants. The different choices yield different basis transformations, a permutation on the arbitrary functions and a change of the root of unity, which still must be a primitive nth root of unity. Thus using any 2-dimensional irrep satisfying gcd(l, n) = gcd(k, n) = 1 results in equivalent descendants.
4.3.2
More R-matrices for models with few states
In Section 3.2 we presented an R-matrix with symmetry inherited from the double of a dihedral group that was more general than the one from [26]. Similarly, here we wish to construct more descendants with symmetry inherited from the double of a dihedral group that will not necessarily be self-adjoint and unitary.
The General Form When we constructed our conjectured descendant we imposed that, under an appropriate limiting condition, R(z) was self-adjoint and unitary. If we do not impose these constraints then it is possible to uncover more descendants. We recall that our possible descendant for 3 ≤ n ∈ Z was of the form " n−1 # n−1 a (2p−1+b−a) X X Y 1 z + w ˇ R(z) = wbj f(0,b−a) (z) ei+a+j,i+a ⊗ ei+j,i , (2p−1+b−a) n 1 + zw p=1 i,j,a=0 b=0 where w is a primitive nth root of unity and the functions obey the relations f(0,b) (z) = f(0,b+n) (z) = f(0,−b) (z),
79
4.3 Other descendants for all b ∈ Z. We recall that this is alternately defined by n−1 X
ˇ R(z) =
g(a,j) (z)ei+a+j,i+a ⊗ ei+j,i ,
i,j,a=0
where n−1
1 X bj g(a,j) (z) = w f(a,b) (z) n b=0
and
a Y z + w(2p−1+b−a) f(0,b−a) (z). f(a,b) (z) = (2p−1+b−a) 1 + zw p=1
For convenience we redefine the projection operators n X 1 [wjb ei+a+j,i+a ⊗ ei+j,i + w−bj ei−a+j,i−a ⊗ ei+j,i ], p(a,b) = (1 + δ¯0 δ¯0 )n 2a 2b
i,j=1
for a ∈ N, b ∈ Z. Using these operators we are able to construct more descendants which we shall catalogue with respect to the root of unity used.
n=4 ˇ For n = 4 we have 3 arbitrary functions in our possible descendant. Expressing R(z) in terms of projection operators yields (1,1) z+w ˇ R(z) = f(0,0) (z) p(0,0) + 1+zw p + p(2,2) +f(0,1) (z) p(0,1) + p(1,0) − p(1,2) − p(2,1) (1,3) z−w +f(0,2) (z) p(0,2) + 1−zw p + p(2,0) . We find that if 1 + zw 1 − zw f(0,0) (z) ∈ 1, and f(0,2) (z) ∈ 1, z+w z−w ˇ and f(0,1) (z) remains arbitrary then R(z) satisfies Equation (2.2), and thus R(z) is a descendant of the zero-field six-vertex model. n=5 Here we find examples of other descendants when n = 5. In terms of the projection operators we have n o (z+w) (1,1) (z+w)(z+w3 ) (2,2) ˇ R(z) = f(0,0) (z) p(0,0) + (1+zw) p + (1+zw)(1+zw 3) p n (z+w2 ) (2,1) (z+w2 ) (1,2) +f(0,1) (z) p(0,1) + p(1,0) + (1+zw + (1+zw + 2) p 2) p o n (z+w2 )(z+w4 ) (2,3) (z+w3 ) (1,3) + (1+zw + f(0,2) (z) p(0,2) + (1+zw 2 )(1+zw 4 ) p 3) p o (z+w4 ) (1,4) (z+w3 ) (2,4) + (1+zw + p(2,0) + (1+zw . 4) p 3) p
80
Chapter 4. Descendants with D(Dn ) Symmetry
This can be rewritten as ˇ R(z) =
2 X
f(0,b) (z)p
(0,b)
+
1 X 4 X
f(a,b) (z)p(a,b) ,
a=0 b=0
b=0
where a Y z + w(2p−1+b−a) f(0,b−a) (z) f(a,b) (z) = 1 + zw(2p−1+b−a) p=1
and
f(0,b) (z) = f(0,−b) (z).
We modify this to the equivalent form ˇ R(z) =
"
n−1 X i,j,a=0
# n−1 1 X bj w f(a,b) (z) ei+a+j,i+a ⊗ ei+j,i . n b=0
(4.17)
Our approach here is to set a collection of fα (z) = 1. We note that setting f(a,b) (z) = 1 is equivalent to setting f(0,b−a) (z) =
a Y 1 + zw(2p−1+b−a) z + w(2p−1+b−a)
p=1
.
If we choose f(0,0) (z) = f(1,2) (z) = f(1,3) (z) = 1, then we find a solution of the YBE. Setting f(1,0) (z) = f(1,2) (z) = f(0,2) (z) = 1, also yields a solution. We can also use a brute force method where we make the assumption that f(0,b) (z) =
i 1 Y (1 + zwkb ) i
i=0
(z + wkb )
,
where b ∈ {0, 1, 2} and kbi ∈ {0, 1, 2, 3, 4}. Testing all possibilities provides only two more descendants, which are found by setting f(0,0) =
(1 + zw)2 , (z + w)2
and
f(0,1) (z) = f(0,2) (z) =
(1 + zw2 ) , (z + w2 )
or f(0,0) =
(1 + zw)2 , (z + w)2
f(0,1) (z) =
(1 + zw2 )2 (z + w2 )2
and
f(0,2) (z) = 1.
81
4.3 Other descendants n=6 We look at other descendants for n = 6. o n (z+w) (2,2) (z+w) (1,1) (3,3) (0,0) ˇ − (1+zw) p −p R(z) = f(0,0) (z) p + (1+zw) p o n (z+w2 ) (1,2) (z+w2 ) (2,1) (2,3) (3,2) p + p + p + p +f(0,1) (z) p(0,1) + p(1,0) + (1+zw 2) (1+zw2 ) n o (z−w2 ) (1,5) (z−w2 ) (2,4) (2,0) (3,1) +f(0,2) (z) p(0,2) − p(1,3) + (1−zw p + p − p − p 2) (1−zw2 ) n o (z−w) (1,4) (z−w) (2,5) +f(0,3) (z) p(0,3) + (1−zw) p + (1−zw) p + p(3,0) .
Using the previous approach for finding other descendants we choose our free functions such that other functions associated with projection operators become 1. We find the ˇ following cases in which R(z) satisfies Equation (2.2): f(0,0) (z) = 1, f(0,2) (z) =
(1 − zw2 ) (1 − zw) and f(0,3) (z) = f(0,1) (z) 2 (z − w ) (z − w)
and f(0,0) (z) = 1, f(0,2) (z) = −
(1 − zw2 ) (1 − zw) and f (z) = f(0,1) (z), (0,3) (z − w2 ) (z − w)
where in both cases f(0,1) (z) is still an arbitrary function. n=7 We take a similar approach to that in the case of n = 5 and use the operator as presented in Equation (4.17). We find that the following list provides other descendants f(0,0) (z) = f(0,1) (z) = f(0,2) (z) = f(0,3) (z) = 1, f(0,0) (z) = f(3,4) (z) = f(1,6) (z) = f(2,5) (z) = 1, f(1,1) (z) = f(2,3) (z) = f(0,2) (z) = f(1,4) (z) = 1, f(2,2) (z) = f(1,2) (z) = f(1,3) (z) = f(0,3) (z) = 1
and
f(3,3) (z) = f(0,1) (z) = f(2,4) (z) = f(1,5) (z) = 1.
Connection to Fateev–Zamolodchikov model We stated earlier that some of the descendants we constructed were clearly connected to the Fateev–Zamolodchikov model while others could not be through a transformation which preserved the eigenvalues of the R-matrix. Here we expect that the additional descendants obtained in this section for odd n are special cases of the Fateev–Zamolodchikov model, as is the general descendant found in Section 3.2. However, for even n as we have an arbitrary function we do not expect a connection to the Fateev–Zamolodchikov model.
82
Chapter 4. Descendants with D(Dn ) Symmetry
4.3.3
Self-adjoint descendants with free parameter when n even
Here we return to D(D2n ) where n is even and we construct more descendants by imposing fewer constraints. We use the general form ˇ R(z) =
"
n−1 X i,j,a=0
# n−1 a 1 X bj Y z + w(2p−1+b−a) f(0,b−a) (z) ei+a+j,i+a ⊗ ei+j,i . w (2p−1+b−a) n b=0 1 + zw p=1
We recall that previously we imposed that R(z) is self-adjoint, unitary and obeys a limiting condition. If we ignore the limiting condition but enforce the other two conditions we find that we must have f(0,2b) (z) = f(0,0) (z) = ±1
and
f(0,2b+1) (z) = f(0,1) (z) = ±1,
for all b ∈ Z. Without loss of generality we can set f(0,0) (z) = 1. This gives us two possibilities; the first choice is f(0,1) (z) = 1, which yields the operator "
n−1 X
ˇ + (z) = R
i,j,a=0
# n−1 a 1 X bj Y z + w(2p−1+b−a) w ei+a+j,i+a ⊗ ei+j,i . (2p−1+b−a) n b=0 1 + zw p=1
We find that this operator corresponds to the conjectured descendant given by Equation (4.14). The second choice is f(0,1) (z) = −1, which gives the operator n−1 X
ˇ − (z) = R
"
i,j,a=0
# n−1 a (2p−1+b−a) Y 1X z + w (−1)b−a wbj ei+a+j,i+a ⊗ ei+j,i . n b=0 1 + zw(2p−1+b−a) p=1
Although R+ (z) and R− (z) are different they share many properties, including that they ˇ + (z) and R ˇ − (z) square to the identity and obey the unitarity condition. We now write R in their equivalent forms # a n−1 1 X bj Y z + w(2p−1+b−a) w ei+j,i+a ⊗ ei+a+j,i (2p−1+b−a) n b=0 1 + zw p=1
(4.18)
# n−1 a (2p−1+b−a) Y 1X z + w (−1)b−a wbj ei+j,i+a ⊗ ei+a+j,i . n b=0 1 + zw(2p−1+b−a) p=1
(4.19)
n−1 X
R+ (z) =
i,j,a=0
"
and R− (z) =
n−1 X i,j,a=0
"
We can also show that R+ (z) satisfies the YBE if and only if R− (z) does.
83
4.3 Other descendants Proposition 4.3.1. The operator " # a n−1 (2p−1+b−a) Y X z + w 1 ˇ + (z) = wbj f(0,b−a) (z) ei+a+j,i+a ⊗ ei+j,i R (2p−1+b−a) n i,j,a,b=0 1 + zw p=1 satisfies the Yang–Baxter equation if and only if " # a n−1 (2p−1+b−a) Y X z + w 1 ˇ − (z) = R (−1)a+b wbj f(0,b−a) (z) ei+a+j,i+a ⊗ ei+j,i (2p−1+b−a) n i,j,a,b=0 1 + zw p=1 does. Proof. We first use the following result of Proposition 4.2.1: c+ n 2
Y z + w(2p−1+b−a) n = (−1)a+b+ 2 , (2p−1+b−a) 1 + zw p=c+1 for all c ∈ N. This implies that a+ n n−1 2 (2p−1+b−a) X Y n 1 z+w wbj R− (z) = (−1) 2 f(0,b−a) (z) ei+j,i+a ⊗ ei+a+j,i . (2p−1+b−a) n i,j,a,b=0 1 + zw p=1 We now consider a basis transformation of R(z). We note that for any λ ∈ C there exist a basis transformation which yields ei,j → λi−j ei,j . We can choose an appropriate λ such that our operator becomes (after scaling) a+ n n−1 2 (2p−1+b−a) X Y 1 z+w w(b+ n2 )j R− (z) = f(0,b−a) (z) ei+j,i+a ⊗ ei+a+j,i . (2p−1+b−a) n i,j,a,b=0 1 + zw p=1 We recall the functions a Y z + w(2p−1+b−a) g(a,j) (z) = w f(0,b−a) (z), 1 + zw(2p−1+b−a) p=1 b=0 n−1 X
bj
for a ∈ N, j ∈ Z, which allow us to write +
R (z) =
n−1 X
g(a,j) (z)ei+j,i+a ⊗ ei+a+j,i
i,j,a=0
and −
R (z) =
n−1 X i,j,a=0
g(a+ n2 ,j) (z)ei+j,i+a ⊗ ei+a+j,i .
84
Chapter 4. Descendants with D(Dn ) Symmetry
Hence R− (z) differs from R+ (z) only by a basis transformation and a permutation of the entries. We calculated previously that R+ (z) satisfies the YBE if and only if n−1 X g(a,k−d) (x)g(k,a−b) (xy)g(b,c−k) (y) − g(c,k−b) (x)g(k,c−d) (xy)g(d,a−k) (y) = 0, k=0
ˇ − (z) satisfies the YBE if and only if 0 ≤ a, b, c, d ≤ n − 1. Similarly we have that R n−1 Xh
i g(a+ n2 ,k−d) (x)g(k+ n2 ,a−b) (xy)g(b+ n2 ,c−k) (y) − g(c+ n2 ,k−b) (x)g(k+ n2 ,c−d) (xy)g(d+ n2 ,a−k) (y) = 0,
k=0
0 ≤ a, b, c, d ≤ n − 1. As we can consider each of the indices of the functions modulo n we find that the two above constraints are equivalent, hence the result. We have therefore shown that R− (z) given in Equation (4.19) satisfies the YBE if and only if R+ (z) given in Equation (4.18) does. Thus we have found another family of conjectured descendants. We can explain the existence of R− (z) by considering the values of functions associated with the irreps. To obtain R+ (z) we simply set every function associated with irreps from the conjugacy class {e} equal to one i.e. f(0,b) (z) = 1 for b ∈ Z. To obtain R− (z) we set every function associated with irreps from the conjugacy class {σ n } equal to one i.e. f( n2 ,b) (z) = 1 for b ∈ Z. We recall e and σ n are the only central group elements in D2n . It is also possible for us to ignore the unitary condition and only impose self-adjointness. This gives the constraints f(0,2b) (z) = f(0,0) (z) = (f(0,0) (z))∗
and
f(0,2b+1) (z) = f(0,1) (z) = (f(0,1) (z))∗ ,
for all b ∈ Z and z ∈ R. Using these constraints we set f(0,0) (z) = f (z) + g(z)
and
f(0,1) (z) = f (z) − g(z),
where f (z) and g(z) are arbitrary real functions. This yields the operator ˇ ˇ + (z) + g(z)R ˇ − (z), R(z) = f (z)R or equivalently R(z) = f (z)R+ (z) + g(z)R− (z).
(4.20)
We are able to verify using Maple that R(z) satisfies the YBE for even n up to 12. It is also worth noting that R(z) is invertible provided f (z) 6= ±g(z).
85
4.3 Other descendants
The freedom of these two arbitrary functions can be replaced by a single second parameter. That is, we have the operator R(z, µ) = R+ (z) + µR− (z).
(4.21)
We prove that this new parameter replaces the arbitrary functions in the following proposition. Proposition 4.3.2. Provided the operators R+ (z) and R− (z) have entries which are rational functions of z then R(z, µ) = R+ (z) + µR− (z) satisfies R12 (x, λ)R13 (xy, µ)R23 (y, ν) = R23 (y, ν)R13 (xy, µ)R12 (x, λ),
(4.22)
if and only if R(z) = f (z)R+ (z) + g(z)R− (z) is a solution of the Yang–Baxter equation for arbitrary functions f (z) and g(z). Proof. Firstly, we assume that R(z) given by (4.20) with arbitrary functions f (z) and g(z) satisfies the YBE. As f (z) is arbitrary we set f (z) = 1. We also know that the entries of R+ (z) and R− (z) are rational function so we rescale each of them by the same factor so that they have polynomial entries. This means that every entry in R(z) can now be written as a polynomial of z and g(z). Let Ω = R12 (x)R13 (xy)R23 (y) − R23 (y)R13 (xy)R12 (x). We observe that every entry of Ω is expressible as 1 X
hlijk (x, y)[g(x)]i [g(xy)]j [g(y)]k ,
i,j,k=0
where hlijk (x, y) are polynomials in x and y (and hence continuous functions), and l indexes the entries of Ω. As R(z) satisfies the YBE, i.e. Ω = 0, and g(z) is an arbitrary function we deduce that hlijk (x, y) = 0,
∀x, y ∈ C/{0, 1, ∞} s.t. x 6= y.
Let y0 ∈ C/{0, 1, ∞}. There are at most four x values for which hlijk (x, y0 ) can be non-zero, but hlijk (x, y0 ) is continuous in x. This means that hlijk (x, y0 ) = 0,
∀x ∈ C.
86
Chapter 4. Descendants with D(Dn ) Symmetry
By symmetry hlijk (x, y) = 0,
∀(x, y) ∈ (C × C)/({0, 1, ∞} × {0, 1, ∞}).
Thus there are at most 9 points in which hlijk (x, y) can be non-zero. As hlijk (x, y) is continuous in x and y we must have hlijk (x, y) = 0,
∀x, y ∈ C.
Let R(z, µ) be the operator derived from R(z) in which we have replaced the free function g(z) with µ. We know that every entry must be expressible as a polynomial in terms of z and µ. If we let Ω = R12 (x, λ)R13 (xy, µ)R23 (y, ν) − R23 (y, ν)R13 (xy, µ)R12 (x, λ), x, y, λ, µ, ν ∈ C, then we find every entry of Ω can be written ∞ X
hlijk (x, y)λi µj ν k .
i,j,k=0
As shown previously hlijk (x, y) = 0,
∀x, y ∈ C.
Thus we must have Ω = 0 which implies R(z, µ) satisfies R12 (x, λ)R13 (xy, µ)R23 (y, ν) = R23 (y, ν)R13,µ (xy)R12 (x, λ), as required. Conversely, suppose we have R(z, µ) which satisfies R12 (x, λ)R13 (xy, µ)R23 (y, ν) = R23 (y, ν)R13 (xy, µ)R12 (x, λ). If we consider R(z) = f (z)R(z, g(z)[f (z)]−1 ), where f (z) and g(z) are arbitrary non-zero functions then R(z) must satisfy the YBE. To obtain the R-matrix R(z) with f (z) arbitrary and g(z) = 0 then we set R(z) = f (z)R(z, 0). If we wish to recover the R-matrix R(z) with g(z) arbitrary and f (z) = 0 then we set R(z, µ) . µ→∞ µ
R(z) = g(z) lim
In each of these two cases we can verify that R(z) will satisfy the YBE. This completes the proof.
87
4.3 Other descendants
4.3.4
General descendants with free parameter when n even
Here we again consider the case of even n and use insight from the previous section to introduce an unconstrained parameter into R-matrices which are not necessarily self-adjoint. We use the operators R+ (z) =
"
n−1 X i,j,a=0
# n−1 a 1 X bj Y z + w(2p−1+b−a) f(0,b−a) (z) ei+j,i+a ⊗ ei+a+j,i w n b=0 1 + zw(2p−1+b−a) p=1
(4.23)
and n−1 X
R− (z) =
"
i,j,a=0
# n−1 a (2p−1+b−a) Y 1X z + w f(0,b−a) (z) ei+j,i+a ⊗ ei+a+j,i . (−1)b−a wbj (2p−1+b−a) n b=0 1 + zw p=1 (4.24) +
We have shown previously that for certain selections of functions that R (z) is solution of the YBE. We also have Proposition 4.3.1 which states that both or neither R+ (z) and R− (z) are solutions of the YBE. Now suppose that we have fixed the functions f(0,b) (z) for 0 ≤ b ≤
n 2
so that there are no
−
+
arbitrary functions in R (z) and R (z). We extend the idea introduced in the previous section and consider the operator R(z, µ) = R+ (z) + µR− (z). Our assertion is that if R+ (z) satisfies the YBE then R(z, µ) should satisfy Equation (4.22). We have made this assertion as every R-matrix found earlier in Section 4.3.2 when n was even contained an arbitrary function and by Proposition 4.3.2 it is known these arbitrary functions are equivalent to a second unconstrained parameter. This allows us to revisit the descendants found for n = 4 and n = 6 and rewrite them as R-matrices with two parameters. We can also write R(z, µ) = U (µ)R+ (z), where U (µ) = I ⊗ I + µ
n X
(−1)i+j ei+ n2 ,i ⊗ ej+ n2 ,j .
i,j=1
Thus R(z, µ) can be simply derived from R+ (z), although the relationship between them is not trivial. For instance, in general they do not have the same eigenvalue spectrum. They do, however, share eigenvectors, as R(z, µ) commutes with both R+ (z) and R− (z).
88
4.4
Chapter 4. Descendants with D(Dn ) Symmetry
Summary
This chapter was devoted to the construction of R-matrices from the doubles of dihedral groups using the concept of descendants. Despite differences in the representations of D(Dn ) for odd and even n a single form for R-matrices was found under the assumptions of self-adjointness and a limiting condition. The result can be expressed in the following manner: For n > 2 and w a primitive nth root of unity then wz −1 − w−1 z 0 0 0 0 z −1 − z w − w−1 0 r(z) = 0 w − w−1 z −1 − z 0 −1 0 0 0 wz − w−1 z
satisfies the Yang–Baxter equation and " n−1 # n−1 a X 1 X bj Y z + w(2p−1+b−a) R(z) = w ei+j,i+a ⊗ ei+a+j,i n b=0 1 + zw(2p−1+b−a) p=1 i,j,a=0 satisfies the Yang–Baxter equation for 3 ≤ n ≤ 17. There also exists L(z) =
n−1 X
(wi e1,2 + w−i e2,1 ) ⊗ ei,i + z [e1,1 ⊗ ei−1,i + e2,2 ⊗ ei+1,i ] ,
i=0
which satisfies Equations (2.4) and (2.5) with r(z) and R(z) respectively. With this presentation we see that R(z) is a descendant of the zero-field six-vertex model. We have conjectured that R(z) satisfies YBE for n ≥ 18. We discussed the connection between our descendants and the R-matrices from the Fateev– Zamolodchikov model. We found that in the case of n odd there was a clear connection between the two. Specifically, only a limit of the variables and basis transformation was required. In the case where n was even it was shown that should a connection exist between the two models it is one which does not preserve the eigenvalue spectrum. We also proved other results including those relating to the choice of two dimensional irreps. This leads to the conclusion that when n is an odd prime then R(z) as given above is indeed the only self-adjoint descendant from D(Dn ). We then extended our results and provided examples of models with few states which do not correspond to self-adjoint Rmatrices. The last significant result obtained in this section was that when n is even a second parameter appears in the R-matrix which is not a spectral parameter nor is it a deformation
89
4.4 Summary
parameter like the q which appears in quantum groups. Specifically (for n > 2 even) given " n−1 # a n−1 (2p−1+b−a) X Y X z+w 1 ei+j,i+a ⊗ ei+a+j,i R+ (z) = wbj n b=0 1 + zw(2p−1+b−a) p=1 i,j,a=0 and R− (z) =
n−1 X
"
i,j,a=0
# n−1 a (2p−1+b−a) X Y 1 z+w ei+j,i+a ⊗ ei+a+j,i , (−1)b−a wbj n b=0 1 + zw(2p−1+b−a) p=1
then R(z, µ) = R+ (z) + µR− (z) is proven to satisfy Equation (4.22) for small n and conjectured to satisfy it for larger n. Furthermore it satisfies L12 (x)L13 (y)R23 (x−1 y, µ) = R23 (x−1 y, µ)L13 (y)L12 (x), with L(z) as given earlier.
Chapter 5
Conclusion
The focus of this thesis was to develop integrable models associated with the Drinfeld double of dihedral groups. We chose these algebraic structures as they provide a description of non-Abelian anyons, which are currently of great interest. In this thesis we specifically looked at generalising the work of [26] in which a 3-state R-matrix was constructed from D(D3 ). In Chapter 3 we investigated two different generalisations which retain the algebraic structure of D(D3 ). The first generalisation was the introduction of open boundary conditions for an anyonic spin chain. This was accomplished using a general formalisation of the boundary quantum inverse scattering method that did not require crossing unitarity. A systematic method was then used to explicitly determine all reflection matrices and hence boundary interaction terms. The second generalisation was the construction of a general R-matrix which possesses D(D3 ) symmetry. In Chapter 4 we considered the general family of dihedral groups. The Drinfeld doubles of these groups were used to construct R-matrices for an arbitrary number of states. For each Drinfeld double of a dihedral group we constructed an R-matrix as a descendant of the zero-field six-vertex model. For the R-matrices which correspond to models with an odd number of states we found strong evidence implying that they are special cases of the Fateev–Zamolodchikov model. The models with an even number of states, however, do not appear to correspond to the Fateev–Zamolodchikov model. Of additional interest in the case of an even number of states was the appearance of a second (non-spectral) parameter in the R-matrix. This thesis opens several avenues for future research. For each of the general R-matrices associated with an odd number of states we could use the Bethe ansatz to determine the eigenvalues of the Hamiltonian; these models could also have integrable boundary 91
92
Chapter 5. Conclusion
interaction terms incorporated. For the family of self-adjoint and unitary R-matrices we could remove these requirements and determine the most general R-matrix which has D(Dn ) symmetry. Another interesting problem is how to construct an R-matrix from the Drinfeld double of a general group algebra. Associated with this would be the problem of determining which groups have R-matrices with an unconstrained parameter.
References
[1] E. Abe, Hopf algebras (translated by H. Kinoshita and H. Tanaka), Cambridge University Press, (1980). [2] Y. Akutsu and M. Wadati, Exactly solvable models and new link polynomials. I. N State Vertex Models, J. Phys. Soc. Jpn., 56, 3039–3051, (1987). [3] G. Albertini, Bethe-ansatz type equations for the Fateev–Zamolodchikov spin model, J. Phys. A: Math. Gen., 25, 1799-1813, (1992). [4] G. Albertini, B.M. McCoy, J.H.H. Perk and S. Tang, Excitation spectrum and order parameter for the integrable N -state chiral Potts model, Nucl. Phys. B, 314, 741–763, (1989). [5] E. Ardonne and K. Schoutens, A new class of non-Abelian spin-singlet quantum Hall states, Phys. Rev. Lett., 82, 5096-5099, (1999). [6] M.A. Armstrong, Groups and symmetry, Springer-Verlag, (1988). [7] M. Aschbacher, Finite group theory, Cambridge University Press, (1986). [8] H. Au-Yang, B.M. McCoy, J.H.H. Perk, S. Tang and M. Yan, Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus>1, Phys. Lett. A, 123, 219–223, (1987). [9] R.J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys., 70, 193– 228, (1972). [10] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London, (1982). [11] R.J. Baxter, On Zamolodchikov’s solution of the tetrahedron equations, Commun. Math. Phys., 88, 185–205, (1983). 93
94
References
[12] R.J. Baxter, The six and eight-vertex models revisited, J. Stat. Phys., 116, 43–66, (2004). [13] R.J. Baxter, J.H.H. Perk and H. Au-Yang, New solutions of the star-triangle relations for the chiral Potts model, Phys. Lett. A, 128, 138–142, (1988). [14] V.V. Bazhanov and J.H.H. Perk, Connection of D(D3 ) model and 3-state self-dual Potts model I-II, Private Communications, (2009). [15] V.V. Bazhanov and Y.G. Stroganov, Chiral Potts model as a descendant of the sixvertex model, J. Stat. Phys., 59, 799–817, (1990). [16] A.J. Bracken, X.Y. Ge, Y.Z. Zhang and H.Q. Zhou, An open-boundary integrable model of three coupled XY spin chains, Nucl. Phys. B, 516, 603–622, (1998). [17] A.J. Bracken, M.D. Gould and R.B. Zhang, Quantum supergroups and solutions of the Yang-Baxter equation, Mod. Phys. Lett. A, 5, 831-840, (1990). [18] S. Burciu, A class of Drinfeld doubles that are ribbon algebras, J. Algebra, 320, 2053– 2078, (2008). [19] M. Chaichian and P. Kulish, Quantum Lie superalgebras and q-oscillators, Phys. Lett. B, 234, 72-80, (1990). [20] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, (1994). [21] H.X. Chen, Irreducible representations of a class of quantum doubles, J. Algebra, 225, 391–409, (2000). [22] Y. Cheng, M.L. Ge and K. Xue, Yang–Baxterisation of braid group representations, Commun. Math. Phys., 136, 195–208, (1991). [23] E.V. Damaskinskii and P.P. Kulish, Symmetries connected with the Yang–Baxter and reflection equations, Journal of Mathematical Sciences, 115, 1986-1993, (2003). [24] K.A. Dancer, P.E. Finch and P.S. Isaac, Universal Baxterization for Z-graded Hopf algebras, J. Phys. A: Math. Theor., 40, F1069-F1075, (2007). [25] K.A. Dancer, P.E. Finch, P.S. Isaac and J. Links, Integrable boundary conditions for a non-Abelian anyon chain with D(D3 ) symmetry, Nucl. Phys. B, 812, 456–469, (2009).
References
95
[26] K.A. Dancer, P.S. Isaac and J. Links, Representations of the quantum double of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation, J. Math. Phys., 47, 103511, (2006). [27] K.A. Dancer and J. Links, Universal spectral parameter-dependent Lax operators for the Drinfeld double of the dihedral group D3 , J. Phys. A: Math. Gen., 42, 042002, (2009). [28] M. de Wild Propitius and F.A. Bais, Discrete gauge theories, In G. Semenoff and L. Vinet, (eds.), Particles and Fields, CRM Series in Mathematical Physics (SpringerVerlag, New York), 353–353, (1998). [29] T. Deguchi, A. Fujii and K. Ito, Quantum superalgebra Uq osp(2, 2), Phys. Lett. B, 238, 242-246, (1990). [30] G.W. Delius, M.D. Gould, J. Links and Y.Z. Zhang, On type I quantum affine superalgebras, Int. J. Mod. Phys., 10, 3259–3281, (1995). [31] G.W. Delius, M.D. Gould and Y..Z. Zhang, On the construction of trigonometric solutions of the Yang–Baxter equation, Nucl. Phys. B, 432, 377–403, (1994). [32] R. Dijkgraaf, V. Pasquier and P. Roche, Quasi Hopf algebras, group cohomology and orbifold models, Nucl. Phys. (Proc. Supp.), 18, 60–72, (1990). [33] V. Drinfeld, Quantum groups, ICM Berkeley, (1986). [34] P. Etingof and O. Schiffmann, Lectures on the dynamical Yang–Baxter equations, arXiv:math/9908064v2, Lecture Notes, (2000). [35] Z.F. Ezawa, Quantum hall effects (2nd ed), World Scientific, (2008). [36] L.D. Faddeev, Algebraic aspects of the Bethe ansatz, Int. J. Mod. Phys. A, 10, 1845– 1878, (1995). [37] V.A. Fateev and Zamolodchikov, Self-dual solutions of the star-triangle relation in ZN -Models, Phys. Lett. A, 92, 37–39, (1982). [38] G. Felder, Conformal field theory and integrable systems associated to elliptic curves, Proceedings of the International Congress of Mathematicians, Z¨ urich, Birkh¨auser, 1247–1255, (1995). [39] J.B. Fraleigh, A first course in abstract algebra (3rd ed.), Addison-Wesley, (1982).
96
References
[40] I. Frenkel and G. Moore, Simplex equations and their solutions, Commun. Math. Phys., 138, 259–271, (1991). [41] R. Gambini and J. Pullin, Loops, gauge theories and quantum gravity, Cambridge University Press, (1996). [42] J.L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nucl. Phys. B, 238, 125–141, (1984). [43] M.D. Gould, Quantum double finite group algebras and their representations, Bull. Aust. Math. Soc., 48, 275–301, (1993). [44] M.D. Gould and Y.Z. Zhang, R-matrices and the tensor product graph method, J. Mod. Phys., 16, 2145–2152, (2002). [45] K. Hasegawa and Y. Yamada, Algebraic derivation of the broken ZN -symmetric model, Phys. Lett. A, 146, 275–301, (1990). [46] M. Jimbo, A q-difference analogue of U (g) and the Yang–Baxter equation, Lett. Math. Phys., 10, 63–69, (1985). [47] M. Jimbo, Quantum R-matrix for the generalized toda system, Commun. Math. Phys., 102, 537–547, (1986). [48] V.F.R. Jones, On a certain value of the Kauffman polynomial, Commun. Math. Phys, 125, 459–467, (1989). [49] V.F.R. Jones, Baxterization, Int. J. Mod. Phys. B, 4, 701–713, (1990). [50] M. Kashiwara and T. Miwa, A class of elliptic solutions to the star-triangle relation, Nucl. Phys. B, 275, 121–134, (1986). [51] C. Kassel, Quantum Groups, Springer-Verlag, (1995). [52] L.H. Kauffman, Knots and physics (2nd ed), World Scientific Publishing, (1993). [53] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys., 321, 2–111, (2006). [54] A.Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys., 303, 2-30, (2003).
97
References
[55] I.G. Korepanov, The method of vacuum vectors in the theory of the Yang– Baxter equation,
In Applied Problems in Calculus,
English translation at
arxiv.org/abs/nlin.SI/0010024, Publishing House of the Chelyabinsk Polytechnical Institute, Chelyabinsk, USSR, (1986). [56] V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, (1993). [57] P.P. Kulish and N.Y. Resetikhen, Universal R-matrix of the quantum superalgebra osp(2|1), Lett. Math. Phys., 18, 143–149, (1989). [58] P.P. Kulish and N.Y. Reshetikhin, Quantum linear problem for the sine-Gordon equation and higher representations, J. Soviet Math., 23, 2435–2446, (1983). [59] P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang–Baxter equation and representation theory: I, Lett. Math. Phys., 5, 393–403, (1981). [60] P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method. Recent developments., Integrable quantum field theories, Tv¨arminne, Finland, (1981). [61] L.A. Lambe and D.E. Radford, Introduction to the quantum Yang–Baxter equation and quantum groups : an algebraic approach, Kluwer Academic Publishers, (1997). [62] E.H. Lieb, Exact solution of the F model of an antiferroelectric, Phys. Rev. Lett., 18, 1046–1048, (1967). [63] E.H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett., 18, 692–694, (1967). [64] J. Links and M.D. Gould, Integrable systems on open chains with quantum supersymmetry, Int. J. Mod. Phys. B, 25, 3461-3480, (1996). [65] N.J. MacKay, Rational R-matrices in irreducible representations, J. Phys. A, 24, 4017– 4026, (1991). [66] S. Majid, Foundations of quantum group theory, Cambrigde University Press, (1995). [67] J.B. McGuire, Interacting fermions in one dimension. I. Repulsive potential, J. Math. Phys., 6, 432, (1965). [68] J.B. McGuire, Interacting fermions in one dimension. II. Attractive potential, J. Math. Phys., 7, 123, (1966).
98
References
[69] L. Mezincescu and R.I. Nepomechie, Integrable open spin chains with non-symmetric R-matrices, J. Phys. A: Math. Gen., 24, L15-L23, (1991). [70] L. Mezincescu and R.I. Nepomechie, Addendum: Integrability of open chains with quantum algebra symmetry, Int. J. Mod. Phys. A, 7, 5657-5659, (1992). [71] C. Mochon, Anyon computers with smaller groups, Physical Review A, 69, 032306, (2004). [72] G. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B, 360, 362–396, (1991). [73] K. Murasugi, Knot theory and its applications, Birkh¨auser, (1996). [74] C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian Anyons and topological quantum computation, Rev. Mod. Phys., 80, 1083, (2008). [75] L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev., 65, 117–149, (1944). [76] J.H.H. Perk and H. Au-Yang, The Yang–Baxter equations, math-ph/0606053, (2006). [77] D.E. Radford, On oriented quantum algebras derived from representations of the quantum double of a finite-dimensional Hopf algebra, J. Algebra, 270, 670-695, (2003). [78] S. Ray and J. Shamanna, A Bethe ansatz study of free energy and excitation spectrum for even spin Fateev–Zamolodchikov model, J. Math. Phys., 46, 043301, (2005). [79] N.Y. Reshetikhin and M.A. Semenov-Tian-Shansky, Central extensions of quantum current groups, Lett. Math. Phys., 19, 133–142, (1990). [80] H. Saleur, Quantum osp(2|1) and solutions of the graded Yang–Baxter equation, Nucl. Phys. B, 336, 363–376, (1990). [81] E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen., 21, 2375-2389, (1988). [82] E.K. Sklyanin, Quantum group and quantum integrable systems, Nankai Lectures in Mathematical Physics, ed. Mo-Lin Ge, World Scientific, 63–97, (1992). [83] E.K. Sklyanin, L.A. Takhtadzhyan and L.D. Faddeev, Quantum inverse problem method. 1, Theor. Math. Phys., 40, 688-706, (1979).
References
99
[84] B. Sutherland, Exact solution of a two-Dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett., 19, 103–104, (1967). [85] M.E. Sweedler, Hopf algebras, Benjamin, New York, (1969). [86] M.C. Takizawa, The ladder operator approach to constructing conserved operators in integrable one-dimensional lattice models, Doctoral Thesis, The University of Queensland, (2004). [87] V.G. Turaev, The Yang–Baxter equation and invariants of links, Invent. math., 92, 527–553, (1988). [88] C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett., 19, 1312–1315, (1967). [89] A.B. Zamolodchikov, Tetrahedron equations and the relativistic S-matrix of straightstrings in 2 + 1-Dimensions, Commun. Math. Phys., 79, 489–505, (1981). [90] R.B. Zhang, M.D. Gould and A.J. Bracken, From representations of the braid group to solutions of the Yang–Baxter equation, Nucl. Phys. B, 354, 625–652, (1991). [91] Y. Zhang, L.H. Kauffman and M.L. Ge, Yang–Baxterisation, universal quantum gates and Hamiltonians, Quantum Inf. Process, 4, 159–197, (2005). [92] Y.Z. Zhang and M.D. Gould, Quantum affine algebras and universal R-matrix with spectral parameter, Lett. Math. Phys., 31, 101–110, (1994). [93] H.Q. Zhou, Integrable open-boundary conditions for the one-dimensional Bariev chain, Phys. Rev. B, 53, 5098–5100, (1996). [94] Y.K. Zhou, Fateev–Zamolodchikov and Kashiwara-Miwa models: boundary startriangle relations and surface critical properties, Nucl. Phys. B, 487, 779–794, (1997).
Appendix A
From the STR to the YBE
It has long been known that solutions of the star-triangle relation give rise to solutions of the Yang–Baxter equation. Here we present the construction of an R-matrix from weights; we closely follow the proof of [45], while including additional details.
A.1
Converting weights to R-matrices
We will present how to use weights satisfying the star-triangle relations (STR) to construct a solution of the YBE. We define two weights: Figure A.1: Representation of weights p a
a
b
W pq (a, b) =
and
b
Wpq (a, b) = p
q
q
These weights must satisfy the two star-triangle relations: Figure A.2: Star-triangle relation (i ) b
b
d
c
∝
r
c
p
p
r a
a
q
and 101
q
102
Appendix A. From the STR to the YBE Figure A.3: Star-triangle relation (ii ) a
c
a
∝
d
c
p
r r
p
b
q
b
q
In these diagrams we assume summation over internal nodes. This corresponds to nodes with two or more W (red arrows) connected to it. We also require that the proportionality is only dependent upon p, q and r. These diagrams are equivalent to requiring: N X
W pq (a, d)Wpr (d, c)W qr (d, b) = Rpqr Wpq (b, c)W pr (a, b)Wqr (a, c)
d=1
and N X
¯ pqr Wpq (c, b)W pr (b, a)Wqr (c, a). W pq (d, a)Wpr (c, d)W qr (b, d) = R
d=1
Using these weights it is possible to construct an R-matrix (a solution of the YBE). We define the R-matrix to be R(˜ p, q˜) =
N X
Rab11ba22 (˜ p, q˜)eb1 ,a1 ⊗ eb2 ,a2
a1 ,a2 ,b1 ,b2 =1
where p1 p˜ = , p2
q1 q˜ = q2
and Figure A.4: R-matrix from weights b1
q2
Rab11ba22 (˜ p, q˜) =
b2
a2
q1
p1
a1
p2
A.2 Proving the R-matrix satisfies the YBE
103
or alternately p, q˜) = W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )Wp1 q2 (b2 , b1 ). Rab11ba22 (˜
A.2
Proving the R-matrix satisfies the YBE
We will present the diagrams as proofs but will also include the algebraic interpretation of them. Figure A.5: STR to YBE: proof part (i ) c2
c1
b1 c3
b2
b3 a1
a3
a2 p1
p2
q1
q2
r1
r2
A graphical representation of R12 (˜ p, q˜)R13 (˜ p, r˜)R23 (˜ q , r˜). Rbc11 bc22 (˜ p, q˜)Rab11cb33 (˜ p, r˜)Rab22ba33 (˜ q , r˜) X = W p1 q1 (b1 , c2 )Wp2 q1 (b1 , b2 )W p2 q2 (b2 , c1 )Wp1 q2 (c2 , c1 ) b1 b2 b3
×W p1 r1 (a1 , c3 )Wp2 r1 (a1 , b3 )W p2 r2 (b3 , b1 )Wp1 r2 (c3 , b1 ) ×W q1 r1 (a2 , b3 )Wq2 r1 (a2 , a3 )W q2 r2 (a3 , b2 )Wq1 r2 (b3 , b2 )
=
X Wp2 q1 (b1 , b2 )W p2 r2 (b3 , b1 )Wq1 r2 (b3 , b2 ) W p1 q1 (b1 , c2 ) b1 b2 b3
×W p2 q2 (b2 , c1 )Wp1 q2 (c2 , c1 )W p1 r1 (a1 , c3 )Wp2 r1 (a1 , b3 ) ×Wp1 r2 (c3 , b1 )W q1 r1 (a2 , b3 )Wq2 r1 (a2 , a3 )W q2 r2 (a3 , b2 )
104
Appendix A. From the STR to the YBE
Figure A.6: STR to YBE: proof part (ii )
c2
c1
b1 c3 d b2
b3 a1
a3
a2 p1
p2
q1
q2
r1
r2
We used the STR on the triangle (b1 b2 b3 ) to introduce a new node d.
=
X Wp2 q1 (b1 , b2 )W p2 r2 (b3 , b1 )Wq1 r2 (b3 , b2 ) W p1 q1 (b1 , c2 ) b1 b2 b3
×W p2 q2 (b2 , c1 )Wp1 q2 (c2 , c1 )W p1 r1 (a1 , c3 )Wp2 r1 (a1 , b3 ) ×Wp1 r2 (c3 , b1 )W q1 r1 (a2 , b3 )Wq2 r1 (a2 , a3 )W q2 r2 (a3 , b2 )
∝
X W p2 q1 (b3 , d)Wp2 r2 (d, b2 )W q1 r2 (d, b1 ) W p1 q1 (b1 , c2 ) b1 b2 b3 d
×W p2 q2 (b2 , c1 )Wp1 q2 (c2 , c1 )W p1 r1 (a1 , c3 )Wp2 r1 (a1 , b3 ) ×Wp1 r2 (c3 , b1 )W q1 r1 (a2 , b3 )Wq2 r1 (a2 , a3 )W q2 r2 (a3 , b2 )
=
X W p1 q1 (b1 , c2 )Wp1 r2 (c3 , b1 )W q1 r2 (d, b1 ) Wp1 q2 (c2 , c1 ) b1 b2 b3 d
× W p2 q2 (b2 , c1 )Wp2 r2 (d, b2 )W q2 r2 (a3 , b2 ) W p1 r1 (a1 , c3 ) × W p2 q1 (b3 , d)Wp2 r1 (a1 , b3 )W q1 r1 (a2 , b3 ) Wq2 r1 (a2 , a3 )
A.2 Proving the R-matrix satisfies the YBE
105
Figure A.7: STR to YBE: proof part (iii ) c2
c1
c3
d
a1
a3
a2 p1
p2
q1
q2
r1
r2
We introduced triangles (c3 c2 d), (dc1 a3 ) and (a1 da2 ) by removing b1 , b2 and b3 .
=
X W p1 q1 (b1 , c2 )Wp1 r2 (c3 , b1 )W q1 r2 (d, b1 ) Wp1 q2 (c2 , c1 ) b1 b2 b3 d
× W p2 q2 (b2 , c1 )Wp2 r2 (d, b2 )W q2 r2 (a3 , b2 ) W p1 r1 (a1 , c3 ) × W p2 q1 (b3 , d)Wp2 r1 (a1 , b3 )W q1 r1 (a2 , b3 ) Wq2 r1 (a2 , a3 )
∝
X Wp1 q1 (c3 , d)W p1 r2 (d, c2 )Wq1 r2 (c3 , c2 ) Wp1 q2 (c2 , c1 ) d
× Wp2 q2 (d, a3 )W p2 r2 (a3 , c1 )Wq2 r2 (d, c1 ) W p1 r1 (a1 , c3 ) × Wp2 q1 (a1 , a2 )W p2 r1 (a2 , d)Wq1 r1 (a1 , d) Wq2 r1 (a2 , a3 )
=
X Wp1 q2 (c2 , c1 )W p1 r2 (d, c2 )Wq2 r2 (d, c1 ) Wq1 r2 (c3 , c2 ) d
× Wp1 q1 (c3 , d)W p1 r1 (a1 , c3 )Wq1 r1 (a1 , d) W p2 r2 (a3 , c1 ) × Wp2 q2 (d, a3 )W p2 r1 (a2 , d)Wq2 r1 (a2 , a3 ) Wp2 q1 (a1 , a2 )
106
Appendix A. From the STR to the YBE Figure A.8: STR to YBE: proof part (iv ) c2
c3 c1 b3
b2
d
b2
a3
a1
a2
p1
p2
q1
q2
r1
r2
We removed triangles (a1 c3 d), (dc2 c1 ) and (a2 da3 ) by introducing b1 , b2 and b3 .
=
X Wp1 q2 (c2 , c1 )W p1 r2 (d, c2 )Wq2 r2 (d, c1 ) Wq1 r2 (c3 , c2 ) d
× Wp1 q1 (c3 , d)W p1 r1 (a1 , c3 )Wq1 r1 (a1 , d) W p2 r2 (a3 , c1 ) × Wp2 q2 (d, a3 )W p2 r1 (a2 , d)Wq2 r1 (a2 , a3 ) Wp2 q1 (a1 , a2 )
∝
X W p1 q2 (d, b3 )Wp1 r2 (b3 , c1 )W q2 r2 (b3 , c2 ) Wq1 r2 (c3 , c2 ) b1 b2 b3 d
× W p1 q1 (a1 , b2 )Wp1 r1 (b2 , d)W q1 r1 (b2 , c3 ) W p2 r2 (a3 , c1 ) × W p2 q2 (a2 , b1 )Wp2 r1 (b1 , a3 )W q2 r1 (b1 , d) Wp2 q1 (a1 , a2 )
=
X W p1 q2 (d, b3 )Wp1 r1 (b2 , d)W q2 r1 (b1 , d) W q1 r1 (b2 , c3 ) b1 b2 b3 d
×W q2 r2 (b3 , c2 )Wq1 r2 (c3 , c2 )Wp2 r1 (b1 , a3 )W p2 r2 (a3 , c1 ) ×Wp1 r2 (b3 , c1 )W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )
A.2 Proving the R-matrix satisfies the YBE
107
Figure A.9: STR to YBE: proof part (v ) c2
c3 c1 b3
b2
b1
a3
a1
a2
p1
p2
q1
q2
r1
r2
We introduce the triangle (b1 b2 b3 ) by removing d. The resulting diagram implies that we are left with R23 (˜ q , r˜)R13 (˜ p, r˜)R12 (˜ p, q˜).
=
X W p1 q2 (d, b3 )Wp1 r1 (b2 , d)W q2 r1 (b1 , d) W q1 r1 (b2 , c3 ) b1 b2 b3 d
×W q2 r2 (b3 , c2 )Wq1 r2 (c3 , c2 )Wp2 r1 (b1 , a3 )W p2 r2 (a3 , c1 ) ×Wp1 r2 (b3 , c1 )W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )
∝
X Wp1 q2 (b2 , b1 )W p1 r1 (b1 , b3 )Wq2 r1 (b2 , b3 ) W q1 r1 (b2 , c3 ) b1 b2 b3
×W q2 r2 (b3 , c2 )Wq1 r2 (c3 , c2 )Wp2 r1 (b1 , a3 )W p2 r2 (a3 , c1 ) ×Wp1 r2 (b3 , c1 )W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )
=
X
W q1 r1 (b2 , c3 )Wq2 r1 (b2 , b3 )W q2 r2 (b3 , c2 )Wq1 r2 (c3 , c2 )
b1 b2 b3
×W p1 r1 (b1 , b3 )Wp2 r1 (b1 , a3 )W p2 r2 (a3 , c1 )Wp1 r2 (b3 , c1 ) ×W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )Wp1 q2 (b2 , b1 )
=
X b1 b2 b3
Rbc23 bc33 (˜ q , r˜)Rbc11 ab33 (˜ p, r˜)Rab11ba22 (˜ p, q˜).
108
Appendix A. From the STR to the YBE
Thus we have X
q , r˜) ∝ p, r˜)Rab22ba33 (˜ p, q˜)Rab11cb33 (˜ Rbc11bc22 (˜
X
p, q˜), p, r˜)Rab11ba22 (˜ q , r˜)Rbc11ab33 (˜ Rbc23bc33 (˜
b1 b2 b3
b1 b2 b3
where the proportionality depends only on p˜, q˜ and r˜. This implies R12 (˜ p, q˜)R13 (˜ p, r˜)R23 (˜ q , r˜) ∝ R23 (˜ q , r˜)R13 (˜ p, r˜)R12 (˜ p, q˜). To prove they are equal we consider the trace (over all space) of each side tr (R12 (˜ p, q˜)R13 (˜ p, r˜)R23 (˜ q , r˜)) 3 X X = W p1 q1 (b1 , a2 )Wp2 q1 (b1 , b2 )W p2 q2 (b2 , a1 )Wp1 q2 (a2 , a1 ) i=1 ai bi
×W p1 r1 (a1 , a3 )Wp2 r1 (a1 , b3 )W p2 r2 (b3 , b1 )Wp1 r2 (a3 , b1 ) ×W q1 r1 (a2 , b3 )Wq2 r1 (a2 , a3 )W q2 r2 (a3 , b2 )Wq1 r2 (b3 , b2 )
=
3 X X
W q1 r1 (b2 , a3 )Wq2 r1 (b2 , b3 )W q2 r2 (b3 , a2 )Wq1 r2 (a3 , a2 )
i=1 ai bi
×W p1 r1 (b1 , b3 )Wp2 r1 (b1 , a3 )W p2 r2 (a3 , a1 )Wp1 r2 (b3 , a1 ) ×W p1 q1 (a1 , b2 )Wp2 q1 (a1 , a2 )W p2 q2 (a2 , b1 )Wp1 q2 (b2 , b1 )
= tr (R23 (˜ q , r˜)R13 (˜ p, r˜)R12 (˜ p, q˜)) as required. All this together implies R12 (˜ p, q˜)R13 (˜ p, r˜)R23 (˜ q , r˜) = R23 (˜ q , r˜)R13 (˜ p, r˜)R12 (˜ p, q˜). Thus R(˜ p, q˜) is a solution of the YBE.
Appendix B
The Fateev–Zamolodchikov (FZ) Model
This appendix reviews the observations of V. Bazhanov and J. Perk [14], which relate the 3-state Fateev–Zamolodchikov model to the 3-state D(D3 ) model. Also investigated is the connection between the N -state Fateev–Zamolodchikov model and the n-state D(Dn ) or D(D2n ) model. Along with the communications of V. Bazhanov and J. Perk we acknowledge the influence of discussions with H. Au-Yang and V. Mangazeev concerning the Fateev–Zamolodchikov and D(Dn ) models.
B.1
The connection between the 3-state D(D3) and FZ model
Here we show the relationship between the Fateev–Zamolodchikov model and the D(D3 ) model. The work presented is a modification of the work of Bazhanov and Perk [14] done in a manner more in line with the rest of this thesis. Throughout we will use use uppercase N when referring to the Fateev–Zamolodchikov model and n when referring to models associated with the double of dihedral groups. The N -state Fateev–Zamolodchikov [37] model is defined by weights Wpq (0) = 1,
Wpq (l) =
l Y sin[π(p − q + 2j − 1)/2N ] j=1
W pq (0) = 1
and
W pq (l) =
sin[π(q − p + 2j − 1)/2N ]
l Y sin[π(q − p + 2j − 2)/2N ] j=1
sin[π(p − q + 2j)/2N ]
,
,
for 1 ≤ l ≤ N − 1. We refer to these as the FZ weights. These functions are extended through the relations: Wpq (l) = Wpq (N + l) = Wpq (−l)
and 109
W pq (l) = W pq (N + l) = W pq (−l).
(B.1)
110
Appendix B. The Fateev–Zamolodchikov (FZ) Model
These weights satisfy the star-triangle relation. Thus it is possible to construct an R-matrix using the formula given in the previous section. Henceforth we instead consider the multiplicative analog where λ = eiπ/N ,
x = eiπp/N ,
y = eiπq/N ,
and the weights become Wxy (l) =
l Y λ2j−1 x − y
and
λ2j−1 y − x j=1
W xy (l) =
l Y λ2j−1 y − λx j=1
λ2j x − y
.
We denote the R-matrix arising from this form by R(˜ x, y˜). For N = 3 we set w = λ2 , which is equivalent to λ = −w2 . To recover the D(D3 ) R-matrix we define x1 x2 and ν2 = , y1 y2 and then take the limit as x2 and y2 go to zero (while x1 and y1 remain non-zero). This ν1 =
yields the R-matrix R(ν1 , ν2 ) =
3 X
w
(j−l)2 −(k−i)2
i,j,k,l=1
w2 (ν1 − 1) wν1 − 1
1−δ¯jk
w2 (ν2 − 1) wν2 − 1
1−δ¯li ei,j ⊗ ek,l .
This operator satisfies R12 (ν1 , ν2 )R13 (ν1 µ1 , ν2 µ2 )R23 (µ1 , µ2 ) = R23 (µ1 , µ2 )R13 (ν1 µ1 , ν2 µ2 )R12 (ν1 , ν2 ). To show that this is indeed the same R-matrix from D(D3 ) we first must consider the basis transformation on the 3 dimensional space given by 1 w 1 1 1 w2 1 1 −1 w2 1 1 . √ and S = S=√ 1 1 w 3 3 w 1 1 1 w2 1 Using this basis transformation we have P U R(ν1 , ν2 )U
−1
ν2 + w 2 ν2 + w (1,1) (0,0) = p + p 1 + w 2 ν2 (1 + wν2 ) (ν2 + w2 ) (1,2) ν1 + w (0,1) (1,0) + p +p + p , 1 + wν1 (1 + w2 ν2 )
where pα are the projection operators defined in Section 3.2, U = (S ⊗ S)
and
U −1 = (S −1 ⊗ S −1 ).
111
B.2 Properties of the FZ model We can now set ν2 = z −1 and ν1 = z η which gives η
−1
P U R(z , z )U
(z + w) (z + w2 ) (1,1) (0,0) p + p = (1 + zw) (1 + zw2 ) (z η + w) (z + w) (1,2) (0,1) (1,0) + . p +p + p (1 + z η w) (1 + zw)
−1
This is the R-matrix which appears in Equation (3.11), with projection operators defined in Section 3.2. Thus we have that the D(D3 ) model is a special case of the Fateev– Zamolodchikov model.
B.2
Properties of the FZ model
Here we present results involving the Fateev–Zamolodchikov model which will later help with determining its connection to the double of dihedral groups. We will use the FZ weights, which are given by
Wxy (0) = 1,
Wxy (l) =
W xy (0) = 1
and
W xy (l) =
l Y λ2j−1 x − y
λ2j−1 y − x j=1
,
l Y λ2j−1 y − λx j=1
λ2j x − y
,
(B.2)
where x, y ∈ C and 0 ≤ l ≤ N − 1. We utilise the difference property and introduce the notation W (xy −1 |l) = Wxy (l)
and
W (xy −1 |l) = W xy (l).
(B.3)
We extend the weights as described in the previous section. We use diagrammatic arguments to obtain results with the weights being represented by Figure B.1: FZ weights W (x|b − a) =
a
x
b
and
W (x|b − a) =
a
x
We recall that the weights satisfy W (x|b − a) = W (x|a − b)
and
W (x|b − a) = W (x|a − b),
a, b ∈ Z, which diagrammatically corresponds to the reversal of arrows.
b
112
Appendix B. The Fateev–Zamolodchikov (FZ) Model Figure B.2: Reversal of arrows a
x
b
a
b
x
=
a
and
b
x
a
=
x
b
It is known that the weights satisfy the following relations Figure B.3: Properties of FZ weights a
a
a
b
λx−1
x
b
x
Σb
a
x
= x−1
a
x−1
c
a
b
b
a
=
δac
∝
(
a
c
)
where Σ implies summation of the node. Algebraically, these properties are written as W (λx−1 |b − a) = W (x|b − a), W (x|b − a)W (x−1 |a − b) = 1, N X
W (x|b − a)W (x−1 |c − b) = ρ(x)δac ,
b=1
for a, b, c ∈ Z, x ∈ C and some function ρ(x). For convenience we will adopt the following notation: x1 x˜ = , x2
x2 x˜ = , x1 T
x˜
−1
=
x−1 1 , x−1 2
x˜
−T
−1 x2 = , x−1 1
and
αx1 α˜ x= , αx2
for α, x1 , x2 ∈ C. We will use the R-matrix defined in the previous section, namely Figure B.4: FZ R-matrix a2
Rab11ba22 (˜ x, y˜) =
x2 y2−1
x2 y1−1
a1
b1
x1 y2−1
x1 y1−1
b2
We shall refer to this R-matrix with the FZ weights as the FZ R-matrix.
113
B.2 Properties of the FZ model We first calculate the inverse by considering the following diagram: Figure B.5: Inverse relation for FZ R-matrix a2
x2 y2−1
x2 y1−1
Σb1
x1 y2−1
y2 x−1 2
c2
y2 x−1 1
y1 x−1 2
a2
δac22
(
δac11
(a
c2
∝ c1
1
a1
x1 y1−1
y1 x−1 1
Σb2
) )
c1
This implies that N X
Rab11ba22 (˜ x, y˜)Rbc11bc22 (˜ x−T , y˜−T ) ∝ δac11 δac22
R(˜ x, y˜)R(˜ x−T , y˜−T ) ∝ I ⊗ I.
⇒
b1 ,b2 =1
Thus we know the inverse of R(˜ x, y˜) up to a scalar multiple. We can also calculate Figure B.6: Relation for Rt1 (z) a2
λy1 x−1 2
λy2 x−1 2
a1
λy1 x−1 1
λy2 x−1 1
a2
b1
=
b2
x2 y2−1
x1 y2−1
x2 y1−1
b1
a1
x1 y1−1
b2
This implies Rab11ba22 (λ˜ x−1 , y˜−T ) = Rba11ab22 (˜ x, y˜)
⇒
Rt1 (˜ x, y˜) = R(λ˜ x−1 , y˜−T ),
where t1 is the partial transpose over the first space. Similarly we have Rt2 (˜ x, y˜) = R(λ˜ x−T , y˜−1 ). Next we consider
114
Appendix B. The Fateev–Zamolodchikov (FZ) Model Figure B.7: Relation for R21 (z) a2
x1 y1−1
x2 y1−1
a1
b1
a1
x1 y2−1
x2 y2−1
=
b2
x2 y2−1
x2 y1−1
a2
b2
x1 y2−1
x1 y1−1
b1
This implies x, y˜) y −T , x˜−T ) = Rab22ba11 (˜ Rab11ba22 (˜
⇒
R(˜ y −T , x˜−T ) = P R(˜ x, y˜)P,
where P is the usual permutation operator. The last property we wish to mention is the regularity condition. We find that the weights (as defined by Equations (B.2) and (B.3)) have the property W (1|l) = δl0 , for 0 ≤ l ≤ N − 1, thus we have R(˜ x, x˜) = P.
B.3
The connection between the general n-state models
We now present results which relate the n-state model constructed from the double of a dihedral group to the Fateev–Zamolodchikov model. In the case where n is odd there is strong evidence that the dihedral model is a special case of the Fateev–Zamolodchikov model. When n is even, however, the evidence does not appear to support a connection. To reveal the connection when n is odd we determine when the FZ R-matrix squares to the identity. We then show that in this case, the FZ R-matrix limits to a representation of the canonical element from D(Dn ). Lastly a few cases for small N are checked. FZ R-matrices which square to the identity Here we determine exactly what conditions are placed on the parameters of the FZ Rmatrix if it is to square to the identity, as that is a property of the n-state dihedral model. Furthermore we will see that if an FZ R-matrix squares to the identity then its eigenvalues
B.3 The connection between the general n-state models
115
are the same as the permutation operator’s. If the FZ R-matrix, R(˜ x, y˜), squares to a scalar multiple of the identity then by the previous section we must have R(˜ x, y˜) = γ(˜ x, y˜)R(˜ x−T , y˜−T )
(B.4)
for some function γ(˜ x, y˜). We wish to determine what conditions this places on the variables x˜ and y˜. In terms of the FZ weights, Equation (B.4) is equivalent to −1 −1 −1 γ(˜ x, y˜)W (y2 x−1 2 |b2 − a1 )W (y2 x1 |a2 − a1 )W (y1 x1 |b1 − a2 )W (y1 x2 |b1 − b2 )
= W (x1 y1−1 |b2 − a1 )W (x2 y1−1 |a2 − a1 )W (x2 y2−1 |b1 − a2 )W (x1 y2−1 |b1 − b2 )
(B.5)
for all 0 ≤ a1 , a2 , b1 , b2 ≤ N − 1. We now start the classification by proving various propositions. Firstly we show that γ(˜ x, y˜) = 1. Proposition B.3.1. If R(˜ x, y˜) satisfies (B.4) then γ(˜ x, y˜) = 1. Proof. We consider the special case of Equation (B.5) where a1 = a2 = b1 = b2 and recall that W (x|0) = W (x|0) = 1. This proves the proposition. From this point on we set γ(˜ x, y˜) = 1. The symmetry of Equation (B.5) provides a useful tool. Proposition B.3.2. Any condition on x1 , x2 , y1 and y2 implied by Equation (B.5) automatically implies the same condition with both x1 and y1 interchanged and x2 and y2 interchanged. Proof. Equation (B.5) is invariant under the interchange: x1 ↔ y1 ,
x 2 ↔ y2 ,
a 1 ↔ a2
and
b1 ↔ b2 .
It is possible to show that if the FZ R-matrix to square to the identity then x˜ and y˜ must satisfy one of four constraint equations. Proposition B.3.3. If the FZ R-matrix satisfies Equation (B.4) then x˜ and y˜ satisfy at least one of the following conditions 1.
x1 y1
= 1 or
x1 y1
→ 1,
116
Appendix B. The Fateev–Zamolodchikov (FZ) Model
2.
x2 y2
3.
x1 x2 y1 y2
4.
x1 y1
= 1 or
x2 y2
→ 1, x1 x2 y1 y2
= 1 or
x2 y2
→ ∞ and
→ 1, x1 y1
→ 0 or
→ 0 and
x2 y2
→ ∞.
Proof. Let the FZ R-matrix satisfy Equation (B.4) but none of the conditions 1-3 above. We show that
x1 y1
x2 y2
→ ∞ and
→ 0 or
x1 y1
→ 0 and
x2 y2
→ ∞ which is sufficient to prove the
proposition. If we consider the special case of Equation (B.5) where a1 = a2 = 0 and b1 = b2 = 1 we find
x1 − y 1 λ2 y1 − x1
x2 − y 2 y1 − x1 y2 − x2 = 2 . λ2 y2 − x2 λ x1 − y 1 λ 2 x2 − y 2
Rearranging the above constraint and factoring out a factor of (1 − λ4 ) (as is possible as N ≥ 3 and subsequently λ4 6= 1) we obtain
− 1 xy22 − 1 xy11 xy22 − 1 = 0. λ2 − xy22 λ2 xy11 − 1 λ2 xy22 − 1 λ2 − xy11 x1 y1
Examining this equation leads to three possibilities: 1. the numerator is zero, 2. the numerator goes to zero faster than the denominator or 3. the denominator goes to infinity faster than the numerator. The first two of these options contradict the original assumptions at the beginning of the proof. Thus we need only now consider the third option, which implies the denominator goes to infinity and subsequently that If
x1 y1
→ ∞ and
x2 y2
x1 y1
→ ∞ or
x2 y2
→ ∞.
→ ∞ then it is apparent that
x2 x1 x2 − 1 y2 − 1 −1 y1 y2 1 → 4 6= 0, λ λ2 − xy11 λ2 − xy22 λ2 xy11 − 1 λ2 xy22 − 1 which is contradictory.
x1 y1
B.3 The connection between the general n-state models If
x1 y1
→ ∞ and
x2 y2
117
6→ 0 and is finite then the constraint becomes x2 x2 −1 y2 y2 = 0. λ2 xy22 − 1 −λ2 λ2 − xy22
As the numerator can neither be zero or limit to zero it follows that the denominator x2 y2
must go to infinity i.e.
→ ∞ which contradicts above results. By the symmetry of
the problem this leads to that the case that either x2 y2
x1 y1
→ ∞ and
x2 y2
→ 0 or
x1 y1
→ 0 and
→ ∞. This completes the proposition.
Proposition B.3.4. If the FZ R-matrix satisfies (B.4) then the first two conditions listed in Proposition B.3.3 are equivalent. Proof. By Proposition B.3.2, it is sufficient to prove that condition 1 implies condition 2. We first assume that
x2 y2
6= 1 and
x2 y2
6→ 1 while
x1 y1
= 1 or
x1 y1
→ 1. With this assumption
Equation (B.5) becomes −1 −1 δba12 W (y2 x−1 2 |b2 − a1 )W (y2 x1 |a2 − a1 )W (y1 x2 |b1 − b2 )
= δba21 W (x2 y1−1 |a2 − a1 )W (x2 y2−1 |b1 − a2 )W (x1 y2−1 |b1 − b2 ). We consider the two cases a1 = b2 = b1 = 0 and a2 = 1 and a2 = b1 = a1 = 0 and b2 = 1, which give
y 2 − x2 λ2 x2 − y2
λx2 − y1 =0 λy1 − x2
x2 − y2 λ2 y2 − x2
x2 − y2 6 0. = λ2 y2 − x2
and
respectively. By our assumptions we have that y2 − x2 6= 0 and λ 2 x2 − y 2
λy1 − x2 = 0, λx2 − y1
This implies
λx2 − y1 λy1 − x2 = = 0, λy1 − x2 λx2 − y1
which is a contradiction. Proposition B.3.5. The only FZ R-matrix satisfying conditions 1 or 2 of Proposition B.3.3 is equal to the permutation operator. Proof. By the previous proposition conditions 1 and 2 of Proposition B.3.3 are equivalent. Using this and the identity lim W (x|l) = δl0 ,
x→1
for 0 ≤ l ≤ N − 1, the proposition is straightforward to prove.
118
Appendix B. The Fateev–Zamolodchikov (FZ) Model
Proposition B.3.6. If the FZ R-matrix squares to the identity and satisfies condition 4 of Proposition B.3.3 then condition 3 of Proposition B.3.3 is satisfied. Proof. We first assume that
x1 y1
→ ∞ and
x2 y2
→ 0. Setting a1 = b2 = b1 = 0 and a2 = 1 in
equation (B.5) we find "
λ−
x1 y2
λ xy21 − 1
#
" =
λ−
y1 x2
#
λ xy12 − 1
.
Each side can be viewed as M¨obius transformation, which are injective functions. As such it implies the limit of
x1 y2
and
y1 x2
must be the same or equivalently
x1 x2 y1 y2
→ 1.
The proposition follows by the symmetry of the problem. Proposition B.3.7. If an FZ R-matrix satisfies Equation (B.4) then it is equivalent (up to parameter scaling) to an FZ R-matrix which satisfies x˜ = x˜−T and y˜ = y˜−T . Proof. We first determine that when 1 x˜ = y˜ = , 1 the properties x˜ = x˜−T and y˜ = y˜−T are satisfied, and furthermore the FZ R-matrix corresponding to this is equal to the permutation operator. From Proposition B.3.3 it is known that for the FZ R-matrix to square to the identity there are only four possible conditions on its parameter. By Proposition B.3.4 the first of these conditions are equivalent and furthermore by Proposition B.3.5 we have that these condition imply the R-matrix is the permutation operator. In Proposition B.3.6 it was shown that the 4th condition of Proposition B.3.3 implies the 3rd condition, which we now consider. For generic x˜ and y˜ satisfying
x1 x2 y1 y2
= 1 we have the
general entry of the FZ R-matrix being −1 Rab11ba22 (˜ x, y˜) = W (x1 y1−1 |b2 − a1 )W (x2 y1−1 |a2 − a1 )W (y1 x−1 1 |b1 − a2 )W (y1 x2 |b1 − b2 ).
Rescaling x˜ and y˜ by α ∈ C \ {0}, i.e. x01 = αx1 ,
x02 = αx2 ,
y10 = αy1
leaves the R-matrix unchanged, i.e. R(˜ x, y˜) = R(˜ x0 , y˜0 ).
and
y20 = αy2 ,
B.3 The connection between the general n-state models
119
With this rescaling we have the relation αx1 αx2 x01 x02 x1 x 2 = = = 1. 0 0 y1 y2 αy1 αy2 y1 y2 We choose α = x01
r =
x1 , x2
√
x1 x2 x02
−1
, so
r =
x2 , x1
√ −1 y10 = y1 ( x1 x2 )
We immediately see that x02 = x10−1 and by x1 x2 y1 y2
x01 x02 y10 y20
and
√ −1 y20 = y2 ( x1 x2 ) .
= 1 we also have y20 = y10−1 . When we have
→ 1 the result follows in a similar way.
Proposition B.3.8. An FZ R-matrix which has parameters satisfying x˜ = x˜−T and y˜ = y˜−T has the eigenvalues (including multiplicities) of the permutation operator. Proof. As the R-matrix satisfying the conditions stated in the proposition squares to the identity its eigenvalues are independent of x˜ and y˜. It must therefore have the same eigenvalues as the FZ R-matrix associated with x˜ = y˜ = 11 , which is known to be the permutation operator. Corollary B.3.9. If the FZ R-matrix squares to the identity then it has the eigenvalues (including multiplicities) of the permutation operator. Proof. By Proposition B.3.7 the said R-matrix equivalent (up to parameter scaling) to an FZ R-matrix satisfying x˜ = x˜−T and y˜ = y˜−T . By Proposition B.3.8 this must have the eigenvalues (including multiplicities) of the permutation operator. We have computed exactly which FZ R-matrices will square to the identity. For the FZ R-matrix we first note that R(˜ x, y˜)R(˜ x, y˜) = I ⊗ I is equivalent to R(˜ x, y˜) = R(˜ x−T , y˜−T ). Moreover, we were able to show that if an FZ R-matrix squared to the identity then it is equivalent (up to parameter scaling) to an FZ R-matrix which had parameters satisfying the condition x˜ = x˜−T
and
y˜ = y˜−T .
It is clear that if the parameters of an FZ R-matrix do satisfy the above condition then the R-matrix must also square to the identity. Lastly we showed that eigenvalue spectrum for an FZ R-matrix which squares to the identity is the same as that of the permutation
120
Appendix B. The Fateev–Zamolodchikov (FZ) Model
operator. We recall that the dihedral model depends on the parity of the number of states. If the number of states is odd then the R-matrix squares to the identity and obeys regularity, implying that it has the same eigenvalues and multiplicities as the permutation operator. However, when the number of states is even we obtain two R-matrices R+ (z) and R− (z), defined in their alternate forms in Equations (4.18) and (4.19) respectively. Both R+ (z) and R− (z) square to the identity but neither satisfy regularity. Furthermore, although they both have eigenvalues of 1 and −1 their multiplicities are different and also distinct from that of the permutation operator. This implies that for an even number of states no relationship can exist between the FZ R-matrix and the dihedral R-matrix which preserves eigenvalues of the R-matrix. A limit of the FZ R-matrix If R(˜ x, y˜) is the Fateev–Zamolodchikov R-matrix then we can obtain x1 y1 R(z) = lim R , −1 , −1 x1 ,y1 →∞ x1 y1 where z =
x1 . y1
In the case of N = 3 by previous sections this is equivalent to the R-matrix
from D(D3 ). We will determine limz→0 R(z) in order to compare it to the representations of the canonical elements of the double of the dihedral groups. Firstly we have R(0) = lim
lim
z→0 x1 ,y1 →∞ N X
= lim
z→0
=
R
x1 y1 −1 , x1 y1−1
W (z|b2 − a1 )W (0|a2 − a1 )W (z −1 |b1 − a2 )W (∞|b1 − b2 )eb1 ,a1 ⊗ eb2 ,a2
a1 ,a2 ,b1 ,b2 =1
N X
W (0|b2 − a1 )W (0|a2 − a1 )W (∞|b1 − a2 )W (∞|b1 − b2 )eb1 ,a1 ⊗ eb2 ,a2 ,
a1 ,a2 ,b1 ,b2 =1
where W (0|l) = W (∞|l) = (−1)l λ−l
2
and
with l ∈ Z. This leads to R(0) =
N X a1, a2 ,b1 ,b2 =1
2
W (∞|l) = W (0|l) = (−1)l λl ,
λ2(b2 −a2 )(a2 −a1 −b1 +b2 ) eb1 ,a1 ⊗ eb2 ,a2 .
B.3 The connection between the general n-state models
121
We now consider the operators N 1 X 2j(1−j) S=√ λ ei+j,i N i,j=1
and
N 1 X 2j(j−1) S −1 = √ λ ei,i+j . N i,j=1
Setting U = S ⊗ S, we have U R(0)U
−1
=
N X
ei+j,i−j ⊗ ei,i ,
i,j=1
= (π ⊗ π)R, for odd N where π is the representation πn+ from D(Dn ). We are also able to explicitly calculate in the cases of N = 3, 5, 7, 9 that U R(z)U −1 gives precisely the R-matrices associated with the doubles of dihedral groups (while setting λ = −w−1 ). It should be noted that the R-matrix obtained is associated with the 2( n−1 , n+1 ) 2 2
dimension irrep π2
(1,1)
2-dimensional irrep π2
.
, although this R-matrix is equivalent to that obtained from the
Appendix C
Calculations
This appendix is devoted to detailed calculations.
C.1
Calculations for Chapter 2
Calculation C.1.1. We need to calculate the projection operators. We perform a minor calculation first. For D(Dn ) where n > 2 is odd we calculate pα = (πn ⊗ πn )∆(Eα ) d[α] P ∗ −1 ∗ = (πn ⊗ πn ) |D g,h∈Dn χα (h g )∆(gh ) n| P ∗ −1 −1 ∗ ∗ = d[α] g,h,k∈Dn χα (h g )πn (g(m k) ) ⊗ πn (gk ) 2n P Pn−1 ∗ −1 2i −1 ∗ = d[α] g,h∈Dn i=0 χα (h g )πn (g((σ τ ) h) ) ⊗ πn (g)ei+1,i+1 2n P Pn 2j ∗ −1 = d[α] g∈Dn i,j=1 χα ((σ ) g )πn (g)ei−j+1,i−j+1 ⊗ πn (g)ei+1,i+1 2n P Pn 2j ∗ −1 = d[α] g∈Dn i,j=1 χα ((σ ) g )πn (g)ei−j,i−j ⊗ πn (g)ei,i , 2n where πn = πn+ or πn = πn− .
Calculation C.1.2. For D(Dn ) where n > 2 is odd we calculate the projection operators using πn = πn+ . For α = (0, 0) associated with the representation π1+ we find pα =
1 2n
P
=
1 2n
Pn
=
1 n
g∈Dn
Pn
i=1
i,j=1 [ei+j,i
Pn
i,j=1 ei+j,i
(0,2b)
For α = (0, b) associated with π2
πn (g)ei,i ⊗ πn (g)ei,i ,
⊗ ei+j,i + e2+j−i,i ⊗ e2+j−i,i ],
⊗ ei+j,i . when 1 ≤ 2b ≤ 123
n−1 2
(0,n−2b)
and with π2
when
124
Appendix C. Calculations
1 ≤ n − 2b ≤
n−1 2
we find pα =
1 n
P
Pn
=
1 n
Pn
χα (σ −j )πn (σ j )ei,i ⊗ πn (σ j )ei,i ,
=
1 n
Pn
g∈Dn i,j=1
i=1
i,j=1 (w
(2a,2b)
For α = (a, b) associated with π2 pα =
1 n
P
g∈Dn
Pn
i=1
2b
χα (g −1 )πn (g)ei,i ⊗ πn (g)ei,i ,
+ w−2b )ei+j,i ⊗ ei+j,i .
with 1 ≤ 2a ≤
n−1 2
we calculate
χα (e1,1 g −1 )πn (g)ei−a,i−a ⊗ πn (g)ei,i
+χα (e∗2,2 g −1 )πn (g)ei+a,i+a ⊗ πn (g)ei,i , P = n1 ni,j=1 χα (e1,1 σ −j )πn (σ j )ei−a,i−a ⊗ πn (σ j )ei,i +χα (e∗2,2 σ −j )πn (σ j )ei+a,i+a ⊗ πn (σ j )ei,i , P = n1 ni,j=1 w−2bj ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i . (n−2a,n−2b)
Lastly we have for α = (0, b) associated with π2 pα =
1 n
P
g∈Dn
Pn
i=1
when 1 ≤ n − 2a ≤
n−1 2
χα (e2,2 g −1 )πn (g)ei−a,i−a ⊗ πn (g)ei,i
+χα (e∗1,1 g −1 )πn (g)ei+a,i+a ⊗ πn (g)ei,i , P = n1 ni,j=1 χα (e2,2 σ −j )πn (σ j )ei−a,i−a ⊗ πn (σ j )ei,i +χα (e∗1,1 σ −j )πn (σ j )ei+a,i+a ⊗ πn (σ j )ei,i , P = n1 ni,j=1 w−2bj ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i .
Calculation C.1.3. For D(D2n ) where n > 2 we calculate pα = (πn ⊗ πn )∆(Eα ) P ∗ −1 ∗ −1 ∗ = d[α] g,h,k∈D2n χα (h g )πn (g(kh) ) ⊗ πn (g(k ) ) 4n P Pn−1 ∗ −1 2i −1 ∗ = d[α] g,h∈D2n i=0 χα (h g )πn (g((σ τ ) h) ) ⊗ πn (g)ei+1,i+1 4n P Pn−1 2j ∗ −1 = d[α] g∈D2n i,j=0 χα ((σ ) g )πn (g)ei−j+1,i−j+1 ⊗ πn (g)ei+1,i+1 4n P Pn−1 2j ∗ −1 = d[α] g∈D2n i,j=0 χα ((σ ) g )πn (g)ei−j,i−j ⊗ πn (g)ei,i , 4n
that
125
C.1 Calculations for Chapter 2 (a,b)
where πn = πn,τ , a, b ∈ {0, 1}. For D(D2n ) where n > 2 we calculate pα = (πn ⊗ πn )∆(Eα ) P ∗ −1 ∗ −1 ∗ = d[α] g,h,k∈D2n χα (h g )πn (g(kh) ) ⊗ πn (g(k ) ) 4n Pn−1 P ∗ −1 2i+1 −1 ∗ = d[α] τ ) h) ) ⊗ πn (g)ei+1,i+1 i=0 χα (h g )πn (g((σ g,h∈D 4n 2n P P n−1 2j ∗ −1 = d[α] i,j=0 χα ((σ ) g )πn (g)ei−j+1,i−j+1 ⊗ πn (g)ei+1,i+1 g∈D2n 4n P Pn−1 2j ∗ −1 = d[α] g∈D2n i,j=0 χα ((σ ) g )πn (g)ei−j,i−j ⊗ πn (g)ei,i , 4n (0,0)
where πn = πn,στ , a, b ∈ {0, 1}.
Calculation C.1.4. For D(D2n ) where n is odd we recall the form of the projection operator, n−1 d[α] X X p = χα ((σ 2j )∗ g −1 )πn (g)ei−j,i−j ⊗ πn (g)ei,i . 4n g∈D i,j=0 α
2n
(0,b)
For α = (0, 0) and πn = πn,σ , b ∈ {0, 1}, we have Pn−1
χα (g −1 )πn (g)ei,i ⊗ πn (g)ei,i
Pn−1
πn (g)ei,i ⊗ πn (g)ei,i
pα =
1 4n
P
=
1 4n
P
=
1 4n
P2n Pn−1
=
1 n
g∈D2n g∈D2n
i=0 i=0
j=1
Pn−1
i,j=0
ei+j,i ⊗ ei+j,i + e2−i+j,i ⊗ e2−i+j,i
i=0
πn (g)ej,i ⊗ πn (g)ej,i .
(0,b)
For α = (0, 0) and πn = πn,τ σ , b ∈ {0, 1}, we have Pn−1
χα (g −1 )πn (g)ei,i ⊗ πn (g)ei,i
Pn−1
πn (g)ei,i ⊗ πn (g)ei,i
pα =
1 4n
P
=
1 4n
P
=
1 4n
P2n Pn−1
=
1 n
g∈D2n g∈D2n
i=0 i=0
j=1
Pn−1
i,j=0
ei+j,i ⊗ ei+j,i + e1−i+j,i ⊗ e1−i+j,i
i=0
πn (g)ej,i ⊗ πn (g)ej,i .
(0,b)
(0,b)
We now treat πn = πn,σ and πn = πn,τ σ , b ∈ {0, 1}, simultaneously. For α = (0, b), 1≤b≤
n−1 , 2
we have Pn−1
pα =
1 2n
P
=
1 2n
P2n Pn−1
=
1 n
g∈D2n j=1
Pn−1
i,j=0 (w
i=0
i=0
2bj
χα (g −1 )πn (g)ei,i ⊗ πn (g)ei,i
(w2bj + w−2bj )ei+j,i ⊗ ei+j,i
+ w−2bj )πn (g)ei+j,i ⊗ πn (g)ei+j,i .
126
Appendix C. Calculations
and 1 ≤ b ≤ n − 1, we have For α = (a, b), 1 ≤ a ≤ n−1 2 P Pn−1 1 2a ∗ −1 pα = 2n g∈D2n i,j=0 χα ((σ ) g )πn (g)ei−a,i−a ⊗ πn (g)ei,i +χα ((σ −2a )∗ g −1 )πn (g)ei+a,i+a ⊗ πn (g)ei,i P2n Pn−1 −2bj 1 = 2n ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i j=1 i=0 w P −2bj ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i . = n1 n−1 i,j=0 w
Calculation C.1.5. For D(D2n ) where n is even we recall the form of the projection operator, n−1 d[α] X X p = χα ((σ 2j )∗ g −1 )πn (g)ei−j,i−j ⊗ πn (g)ei,i . 4n g∈D i,j=0 α
2n
(0,c)
For α = (a n2 , b n2 ) a, b ∈ {0, 1} and πn = πn,σ c ∈ {0, 1} we have P Pn−1 1 −1 n n pα = 4n g∈D2n i=0 χα (g )πn (g)ei−a 2 ,i−a 2 ⊗ πn (g)ei,i P2n Pn−1 1 −j j j n n n n = 4n j=1 i=0 χα (σ )[πn (σ ) ⊗ πn (σ )][ei−a 2 ,i−a 2 ⊗ ei,i + e2−i−a 2 ,i−a 2 ⊗ e2−i,i ] P2n Pn−1 1 j n n n n = 4n i=0 (−1) [ei+j−a 2 ,i−a 2 ⊗ ei+j,i + e2−i+j−a 2 ,i−a 2 ⊗ e2−i+j,i ] j=1 P2n Pn−1 1 j n n = 2n i=0 (−1) ei+j−a 2 ,i−a 2 ⊗ ei+j,i j=1 P j n n = n1 n−1 i,j=0 (−1) ei+j+a 2 ,i+a 2 ⊗ ei+j,i . (0,c)
For α = (a n2 , b n2 ), a, b ∈ {0, 1}, and πn = πn,στ , c ∈ {0, 1}, we have P Pn−1 1 −1 n n pα = 4n g∈D2n i=0 χα ((g )πn (g)ei−a 2 ,i−a 2 ⊗ πn (g)ei,i P2n Pn−1 1 −j j j n n n n = 4n j=1 i=0 χα ((σ )[πn (σ ) ⊗ πn (σ )][ei−a 2 ,i−a 2 ⊗ ei,i − e1−i−a 2 ,i−a 2 ⊗ e1−i,i ] P2n Pn−1 1 j n n n n = 4n j=1 i=0 (−1) [ei+j−a 2 ,i−a 2 ⊗ ei+j,i − e1−i+j−a 2 ,i−a 2 ⊗ e1−i+j,i ] P2n Pn−1 1 j n n = 2n j=1 i=0 (−1) ei+j−a 2 ,i−a 2 ⊗ ei+j,i P Pn−1 j n n = n1 2n j=1 i=0 (−1) ei+j+a 2 ,i+a 2 ⊗ ei+j,i . (0,b)
(0,b)
We now treat the cases πn = πn,σ and πn = πn,τ σ , b ∈ {0, 1}, simultaneously. For α = (a n2 , b), with a ∈ {0, 1} and 1 ≤ b ≤ n2 − 1, we have P Pn−1 1 −1 n n pα = 2n g∈D2n i,j=0 χα (g )πn (g)ei−a 2 ,i−a 2 ⊗ πn (g)ei,i P2n Pn−1 1 −j n n = 2n j=1 i=0 χα (σ )ei−a 2 +j,i−a 2 ⊗ ei+j,i P2n Pn−1 2bj 1 = 2n + w−2bj )ei−a n2 +j,i−a n2 ⊗ ei+j,i j=1 i=0 (w P 2bj = n1 n−1 + w−2bj )ei−a n2 +j,i−a n2 ⊗ ei+j,i . i,j=0 (w
127
C.2 Calculations for Chapter 3 For α = (a, b) with 1 ≤ a ≤ pα =
1 2n
P
n 2
g∈D2n
− 1 and 0 ≤ b ≤ n − 1 we have Pn−1 i=0
χα ((σ 2a )∗ g −1 )πn (g)ei−a,i−a ⊗ πn (g)ei,i
+χα ((σ −2a )∗ g −1 )πn (g)ei+a,i+a ⊗ πn (g)ei,i P2n Pn−1 −2bj 1 = 2n ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i j=1 i=0 w P −2bj = n1 n−1 ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a ⊗ ei+j,i . i,j=0 w
C.2
Calculations for Chapter 3
Calculation C.2.1. We recall that d −1 −1 0 R (z ) = R21 (1). dz 12 z=1 This allows us two give the two expressions, N X d ˜ 0 La (z) = PaN ..Rai (1)..Pa1 dz z=1 i=1
N X d ˜ −1 −1 0 La (z ) = Pa1 ..Ria (1)..PaN . dz z=1 i=1
and
These lead to d ˜ − −1 −1 ˜ a (z ) La (z)Ka (1)L c dz z=1 d ˜ d ˜ −1 (1) + L ˜ a (1)K − (1) L ˜ −1 (z −1 ) = La (z) Ka− (1)L a a dz dz a z=1 z=1 0 0 = cRaN (1)Pa(N −1) ..Pa1 Pa1 ..PaN + cPaN ..Pa1 Pa1 ..Pa(N −1) RN a (1) +c
N −1 X
0 0 [PaN ..Rai (1)..Pa1 Pa1 ..PaN + PaN ..Pa1 Pa1 ..Ria (1)..PaN ]
i=1
=
0 cRaN (1)PaN
+
0 cPaN RN a (1)
+c
N −1 X
0 0 R(i+1)i (1)P(i+1)i + P(i+1)i Ri(i+1) (1)
i=1
= 2HaN +
N −1 X
2Hi(i+1) .
i=1
Calculation C.2.2. For K(z) =
X
hi,j (z)ei,j .
i,j
and ˇ 12 (0) = R
X i,j
ei,i−j ⊗ ei+j,i
128
Appendix C. Calculations
we want to determine the constraints imposed by requiring ˇ 12 (0)K2 (0)R ˇ 12 (0) = R ˇ 12 (0)K2 (0)R ˇ 12 (0)K2 (y). K2 (y)R Firstly ˇ 12 (0)K2 (0)R ˇ 12 (0) = Pn−1 R a,b,i,j,k,l=0 ha,b (0)ei,i−j ek,k−l ⊗ ei+j,i ea,b ek+l,k Pn−1 = a,b,i,j,l=0 ha,b (0)ei,i−j−l ⊗ ei+j,i ea,b ei−j+l,i−j Pn−1 = b,i,j,l=0 hi,b (0)ei,i−j−l ⊗ ei+j,b ei−j+l,i−j Pn−1 = i,j,l=0 hi,i−j+l (0)ei,i−j−l ⊗ ei+j,i−j Pn−1 = i,j,l=0 hi,2i−2j−l (0)ei,l ⊗ ei+j,i−j Pn−1 = i,j,l=0 hi,2i+2j−l (0)ei,l ⊗ ei−j,i+j Pn−1 = i,j,l=0 hi,2j−l (0)ei,l ⊗ e2i−j,j Pn−1 ¯2i−j ei,l ⊗ ek,j . = i,j,l,k=0 hi,2j−l (0)δk This leads us to the calculation ¯2i−j ei,l ⊗ ek,j ea,b ˇ 12 (0)K2 (0)R ˇ 12 (0)K2 (y) = Pn−1 R a,b,i,j,l,k=0 hi,2j−l (0)ha,b (y)δk Pn−1 ¯2i−j ei,l ⊗ ek,b = b,i,j,l,k=0 hi,2j−l (0)hj,b (y)δk Pn−1 ¯2i−j ei,l ⊗ ea,b = a,b,i,j,l=0 hi,2j−l (0)hj,b (y)δa Pn−1 ¯2i−j ei,l ⊗ ea,b = a,b,i,j,l=0 hi,4i−2a−l (0)h2i−a,b (y)δa Pn−1 = a,b,i,l=0 hi,4i−2a−l (0)h2i−a,b (y)ei,l ⊗ ea,b . We also make the calculation ¯2i−j ei,l ⊗ ea,b ek,j ˇ 12 (0)K2 (0)R ˇ 12 (0) = Pn−1 K2 (y)R a,b,i,j,l,k=0 hi,2j−l (0)ha,b (y)δk Pn−1 ¯2i−j ei,l ⊗ ea,j = a,i,j,l,k=0 hi,2j−l (0)ha,k (y)δk Pn−1 ¯2i−b ei,l ⊗ ea,b = a,b,i,l,k=0 hi,2b−l (0)ha,k (y)δk Pn−1 ¯2i−b ei,l ⊗ ea,b = a,b,i,l,k=0 hi,2b−l (0)ha,2i−b (y)δk Pn−1 = a,b,i,l=0 hi,2b−l (0)ha,2i−b (y)ei,l ⊗ ea,b . This implies that hi,4i−2a−l (0)h2i−a,b (y) = hi,2b−l (0)ha,2i−b (y), for 1 ≤ a, b, i, l ≤ n, that is, hi,4i−2j−l (0)h2i−j,k (y) = hi,2k−l (0)hj,2i−k (y), for 1 ≤ i, j, k, l ≤ n.
129
C.3 Calculations for Chapter 4
C.3
Calculations for Chapter 4
Calculation C.3.1. For D(Dn ) where n > 2 is odd we apply the representations and find Pn−1 2i ∗ 2i 2i ∗ 2i ± (π2 ⊗ πn )R = i=0 [π2 (σ ) ⊗ πn ((σ ) ) + π2 (σ τ ) ⊗ πn ((σ τ ) )] Pn−1 2i = i=0 [π2 (σ τ ) ⊗ ei+1,i+1 ] Pn−1 2(i−1)k = e1,2 + w−2(i−1)k e2,1 ) ⊗ ei,i ], i=0 [(w and (π2 ⊗ πn )(R−1 )T =
Pn−1 i=0
[π2 ((σ i ))∗ ) ⊗ πn (σ −i ) + π2 ((σ i τ )∗ ) ⊗ πn (σ i τ )]
= [e1,1 ⊗ πn (σ −l ) + e2,2 ⊗ πn (σ l )] Pn−1 = i=0 [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ], (l,k)
where π2 = π2
and πn = πn± .
Calculation C.3.2. Given r(z) = a(z)(e1,1 ⊗ e1,1 + e2,2 ⊗ e2,2 ) + b(z)(e1,1 ⊗ e2,2 + e2,2 ⊗ e1,1 ) +c(z)(e1,2 ⊗ e2,1 + e2,1 ⊗ e1,2 ), where a(z), b(z) and c(z) are some functions. If we consider the basis transformation which yields e1,1 → e1,1 ,
e2,2 → e2,2 ,
e1,2 → wφk e1,2 ,
and
e2,1 → w−φk e2,1 ,
then we can find that e1,1 ⊗ e1,1 → e1,1 ⊗ e1,1 ,
e2,2 ⊗ e2,2 → e2,2 ⊗ e2,2 ,
e1,1 ⊗ e2,2 → e1,1 ⊗ e2,2 ,
e2,2 ⊗ e1,1 → e1,1 ⊗ e2,2 ,
e1,2 ⊗ e2,1 → e1,2 ⊗ e2,1
and e2,1 ⊗ e1,2 → e2,1 ⊗ e2,1 .
This implies that r(z) is invariant under such a transformation.
Calculation C.3.3. Consider the basis transformation which yields e1,1 → e1,1 ,
e2,2 → e2,2 ,
e1,2 → wφk e1,2 ,
and
e2,1 → w−φk e2,1 ,
and the L-operator L(z) =
n−1 X i=0
(w2ik e1,2 + w−2ik e2,1 ) ⊗ ei,i + h(z) [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ] .
130
Appendix C. Calculations
Under the basis transformation we have
L(z) →
n−1 X
(w(2i+φ)k e1,2 + w−(2i+φ)k e2,1 ) ⊗ ei,i + h(z) [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ] .
i=0
Calculation C.3.4. We show a relation here. Recall
A(x, y) =
X
w2(j−i)k ei,i ⊗ ej,j + xyei−l,i ⊗ ej−l,j
i,j
and
(a,b)
p˜
=
n X
[w2jb ei+a+j,i+a ⊗ ei+j,i + w−2bj ei−a+j,i−a ⊗ ei+j,i ].
i,j=1
We first calculate
p˜(a,b) A(x, y) =
=
Pn−1
2jb ei+a+j,i+a ⊗ ei+j,i + w−2bj ei−a+j,i−a i,j,s,t=0 [w [w2(t−s)k es,s ⊗ et,t + xyes−l,s ⊗ et−l,t ]
Pn−1
2(jb−ak) ei+a+j,i+a ⊗ ei+j,i i,j=0 [w +w2(−bj+ak) ei−a+j,i−a ⊗ ei+j,i
+xyw2(jb) ei+a+j−l,i+a ⊗ ei+j−l,i +xyw2(−bj) ei−a+j−l,i−a ⊗ ei+j−l,i ].
⊗ ei+j,i ]
131
C.3 Calculations for Chapter 4 This leads to p˜(a,b) A(x, y)˜ p(c,d) =
Pn−1
2(jb−ak) ei+a+j,i+a ⊗ ei+j,i i,j=0 [w +w2(−bj+ak) ei−a+j,i−a ⊗ ei+j,i
+xyw2(jb) ei+a+j−l,i+a ⊗ ei+j−l,i +xyw2(−bj) ei−a+j−l,i−a ⊗ ei+j−l,i ]˜ p(c,d) =
2(jb−ak+td) ¯0 δa−c es+t+a+j,s+c ⊗ es+t+j,s j,s,t=0 [w 2(−bj+ak+td) ¯0 +w δa+c es+t−a+j,s+c ⊗ es+t+j,s
Pn−1
0 es+t+a+j−l,s+c ⊗ es+t+j−l,s +xyw2(jb+td) δ¯a−c +xyw2(−bj+td) δ¯0 es+t−a+j−l,s+c ⊗ es+t+j−l,s
a+c 2(jb−ak−td) ¯0 δa+c es+t+a+j,s−c ⊗ es+t+j,s +w 2(−bj+ak−td) ¯0 +w δa−c es+t−a+j,s−c ⊗ es+t+j,s
0 es+t+a+j−l,s−c ⊗ es+t+j−l,s +xyw2(jb−td) δ¯a+c +xyw2(−bj−td) δ¯0 es+t−a+j−l,s−c ⊗ es+t+j−l,s ] a−c
=
2(−ak+td) ¯0 ¯0 δb−d δa−c es+t+a,s+c ⊗ es+t,s j,s,t=0 [w 0 ¯0 δa+c es+t−a,s+c ⊗ es+t,s +w2(ak+td) δ¯b+d 2(td) ¯0 ¯0 δb−d δa−c es+t+a−l,s+c ⊗ es+t−l,s +xyw
Pn−1
0 ¯0 δa+c es+t−a−l,s+c ⊗ es+t−l,s +xyw2(td) δ¯b+d +w2(−ak−td) δ¯0 δ¯0 es+t+a,s−c ⊗ es+t,s
b+d a+c 2(ak−td) ¯0 ¯0 +w δb−d δa−c es+t−a,s−c ⊗ es+t,s 2(−td) ¯0 ¯0 +xyw δb+d δa+c es+t+a−l,s−c ⊗ es+t−l,s 0 ¯0 +xyw2(−td) δ¯b−d δa−c es+t−a−l,s−c ⊗ es+t−l,s ].
Through observation of the possible (a, b) and (c, d) we see that p˜(a,b) A(x, y)˜ p(c,d) 6= 0, only if (a, b) = (c, d).
Calculation C.3.5. Here we derive the descendant relation with the operator B(x, y). We recall the operators B(x, y) =
X
xw2ik ei,i ⊗ ej−l,j + yw2jk ei+l,i ⊗ ej,j ,
i,j
and α
p˜ =
n X
[w2jb ei+a+j,i+a ⊗ ei+j,i + w−2bj ei−a+j,i−a ⊗ ei+j,i ].
i,j=1
132
Appendix C. Calculations
We first calculate p˜(a,b) B(x, y) =
=
Pn−1
−2bj ei−a+j,i−a ⊗ ei+j,i + w2bj ei+a+j,i+a i,j,s,t=0 [w [xw2sk es,s ⊗ et−l,t + yw2tk es+l,s ⊗ et,t ]
⊗ ei+j,i ]
Pn−1
2((t−l−a)k−bj) et−l−a+j,t−l−a ⊗ et−l+j,t j,t=0 [xw +xw2((t−l+a)k+bj) et−l+a+j,t−l+a ⊗ et−l+j,t
+yw2(tk−bj) et−a+j,t−a−l ⊗ et+j,t +yw2(tk+bj) et+a+j,t+a−l ⊗ et+j,t ]. This leads to p˜(a,b) B(x, y)˜ p(c,d) =
Pn−1
2((t−l−a)k−bj) et−l−a+j,t−l−a ⊗ j,t=0 [xw xw2((t−l+a)k+bj) et−l+a+j,t−l+a ⊗ et−l+j,t
et−l+j,t +
+yw2(tk−bj) et−a+j,t−a−l ⊗ et+j,t +yw2(tk+bj) et+a+j,t+a−l ⊗ et+j,t ]˜ p(c,d) =
2((t−l−a)k−bj−dv) ¯c δa+l et−l−a+j,t−c−v ⊗ et−l+j,t−v j,t,v=0 [xw 2((t−l+a)k+bj−dv) ¯l−a +xw δc et−l+a+j,t−c−v ⊗ et−l+j,t−v
Pn−1
+yw2(tk−bj−dv) δ¯ca+l et−a+j,t−c−v ⊗ et+j,t−v +yw2(tk+bj−dv) δ¯l−a et+a+j,t−c−v ⊗ et+j,t−v
c 2((t−l−a)k−bj+dv) ¯l+a δ−c et−l−a+j,t+c−v +xw +xw2((t−l+a)k+bj+dv) δ¯ca−l et−l+a+j,t+c−v
⊗ et−l+j,t−v ⊗ et−l+j,t−v
a+l +yw2(tk−bj+dv) δ¯−c et−a+j,t+c−v ⊗ et+j,t−v 2(tk+bj+dv) ¯a−l δ et+a+j,t+c−v ⊗ et+j,t−v ] +yw c
= n
Pn−1
+w +w
d ¯c [xw2((−l−a)k−bl) + y]δ¯k+b δa+l ej−a,v−c ⊗ ej,v [xw2((−l+a)k+bl) + y]δ¯d δ¯l−a ej+a,v−c ⊗ ej,v
j,v=0 {w 2(dv+bj)
2(dv−bj)
−(2dv+bj)
[xw
2((−l−a)k−bl)
+
+w−(2dv−bj) [xw2((−l+a)k+bl) +
k−b c −d ¯l+a y]δ¯k+b δ−c ej−a,v+c −d ¯a−l y]δ¯k−b δc ej+a,v+c
⊗ ej,v ⊗ ej,v }.
We now substitute the above result into the following formula, pα [fα (xy −1 )B(x, y) − fβ (xy −1 )B(y, x)]pβ = 0, and we get the following four constraints, d ¯c 0 = δ¯k+b δa+l f(a,b) (z)(zw2((−l−a)k−bl) + 1) − f(c,d) (z)(w2((−l−a)k−bl) + z) , d ¯l−a 0 = δ¯k−b δc f(a,b) (z)(zw2((−l+a)k+bl) + 1) − f(c,d) (z)(w2((−l+a)k+bl) + z) , −d ¯l+a 0 = δ¯k+b δ−c f(a,b) (z)(zw2((−l−a)k−bl) + 1) − f(c,d) (z)(w2((−l−a)k−bl) + z) , −d ¯a−l 0 = δ¯k−b δc f(a,b) (z)(zw2((−l+a)k+bl) + 1) − f(c,d) (z)(w2((−l+a)k+bl) + z) ,
133
C.3 Calculations for Chapter 4 where (a, b), (c, d) ∈ S.
Calculation C.3.6. This is a more general form of the calculation above, determining an if and only if form. We wish to investigate when the functions h1 (z) =
n Y
z+w
2l((2j−1)k+b)
and
h2 (z) =
j=1
n Y
1 + zw2l((2j−1)k+b)
j=1
are equal for all b ∈ Z. We start under the conditions that gcd(l, n) = 1 and w2 a primitive nth root of unity, n ≥ 2. Here we need to consider two cases, namely whether n is odd or even. If we require h1 (z) = h2 (z), ∀b ∈ Z then we see that for every j there must exist a j 0 which satisfies the following relation −((2j − 1)k + b) ≡ ((2j 0 − 1)k + b) (mod n) 2((j 0 + j − 1)k − b) ≡ 0 (mod n). Thus we have two possibilities ( 0
(j + j − 1)k − b ≡
0 (mod n), 0 (mod
n ), 2
n is odd, n is even.
By considering b ≡ −1 (mod n) we can quickly deduce that for h1 (z) = h2 (z), ∀b ∈ Z we must have gcd(k, n) = 1 gcd(k,
n ) 2
=1
when n is odd, when n is even.
We now know necessary constraints upon k. We now show that these conditions are sufficient. If we enforce our necessary conditions then we are able to to determine that there exists an integer p with the property that 2pk ≡ 2 (mod n). Using this element we can determine h2 (z) =
Qn
z + w−2l((2j−1)k+b)
=
Qn
=
Qn
z + w2l((−2j+1)k−b) z + w2l((2j+1)k−b)
=
Qn
=
Qn
=
Qn
j=1 j=1
j=1 j=1 j=1 j=1
= h1 (z),
z + w2l((2(j+bp−1)+1)k−b) z + w2l((2j−1)k+2bpk−b) z + w2l((2j−1)k+b) ,
134
Appendix C. Calculations
as required. Thus the conditions gcd(k, n) = 1
when n is odd,
gcd(k, n2 ) = 1
when n is even,
are necessary and sufficient.
Calculation C.3.7. We desire to show that f(a,b) (z) = f(n−a,−b) (z) we must recall f(a,b) (z) =
−1 al Y
j=1
2l((2j−1)k+b−akl−1 )
z+w 1 + zw2l((2j−1)k+b−akl−1 )
! f(0,b−akl−1 ) (z).
By previous constraints it holds automatically when a = 0, therefore wlog we consider 1≤a≤n−1 Qn−al−1 z+w2l((2j−1)k−b+akl−1 ))
f(n−a,−b) (z) =
f(0,−b+akl−1 ) (z) −1 −1 Qn z+w2l((2j−1−2al )k−b+akl )) f(0,−b+akl−1 ) (z) −1 −1 −1 j=1+al 1+zw2l((2j−1−2al )k−b+akl ) −1 Qn z+w2l((2j−1)k−b−akl )) f(0,−b+akl−1 ) (z) −1 ) −1 2l((2j−1)k−b−akl j=1+al 1+zw Qal−1 1+zw2l((2j−1)k−b−akl−1 )) f(0,−b+akl−1 ) (z) −1 j=1 z+w2l((2j−1)k−b−akl ) Qal−1 z+w2l(−(2j−1)k+b+akl−1 )) f(0,b−akl−1 ) (z) −1 j=1 1+zw2l(−(2j−1)k+b+akl ) Qal−1 z+w2l((2j−1)k+b−akl−1 )) f(0,b−akl−1 ) (z), −1 j=1 1+zw2l((2j−1)k+b−akl ) j=1
= = = = =
−1 )
1+zw2l((2j−1)k−b+akl
as required. Note. We already have that f(0,b) (z) = f(0,−b) (z).
Calculation C.3.8. Given ˇ R(z) =
n−1 X
g(a,j) (z)ei+a+j,i+a ⊗ ei+j,i ,
a,i,j=0
we calculate ˇ 12 (x)R ˇ 23 (y) = R
n−1 X a,c,j,u,v=0
g(a,j) (x)g(c,v) (y)eu+c+v+a+j,u+c+v+a ⊗ eu+c+v+j,u+c ⊗ eu+v,u .
135
C.3 Calculations for Chapter 4 Defining ˇ 12 (x)R ˇ 23 (xy)R ˇ 12 (y), ΩL = R
ˇ 23 (y)R ˇ 12 (xy)R ˇ 23 (x) ΩR = R
and Ω = ΩL − ΩR , we find ΩL =
Pn−1
a,b,c,j,t,u=0
g(a,j) (x)g(b,c−a) (xy)g(c,t) (y)
eu+j,u−t ⊗ eu−a+j,u−t−c ⊗ eu−a−b,u−c−b
=
Pn−1
a,b,c,j,t,u=0
g(a,j−b) (x)g(b,c−a) (xy)g(c,t+b) (y)
eu+j,u−t ⊗ eu−a+j,u−t−c ⊗ eu−a,u−c and ΩR =
Pn−1
a,b,c,t,u,v=0
g(a,t) (y)g(b,a−c) (xy)g(c,v) (x)
×eu+b+a,u+c+b ⊗ eu+a+t,u+c−v ⊗ eu+t,u−v
=
Pn−1
a,b,c,t,u,v=0
g(a,t+b) (y)g(b,a−c) (xy)g(c,v−b) (x)
×eu+a,u+c ⊗ eu+a+t,u+c−v ⊗ eu+t,u−v . We now adopt the notation that [A]jtv isu is the coefficient of the basis element ei,j ⊗ es,t ⊗ eu,v in the operator A. Using this notation we find [ΩL ]jtv isu
=
n−1 X
j+s+v g(i−s,s−u−k) (x)g(k,j−t−i+s) (xy)g(j−t,v−t+k) (y)δ¯i+t+u
k=0
and [ΩR ]jtv isu
=
n−1 X
j+s+v g(s−u,s−i+k) (y)g(k,i−j) (xy)g(t−v,j−t−k) (x)δ¯i+t+u .
k=0
If we set a = i − s, b = j − t, c = t − v and d = s − u then we find that [Ω]jtv isu =
n−1 X i+t+u g(a,d−k) (x)g(k,b−a) (xy)g(b,k−c) (y) − g(d,k−a) (y)g(k,d−c) (xy)g(c,b−k) (x) δ¯j+s+v . k=0
This implies that n−1 X g(a,d−k) (x)g(k,b−a) (xy)g(b,k−c) (y) − g(d,k−a) (y)g(k,d−c) (xy)g(c,b−k) (x) = 0 k=0
ˇ for 0 ≤ a, b, c, d ≤ n − 1 if and only if R(z) satisfies Equation (2.2). We choose to write it equivalently as n−1 X g(a,k−d) (x)g(k,a−b) (xy)g(b,c−k) (y) − g(c,k−b) (x)g(k,c−d) (xy)g(d,a−k) (y) = 0. k=0
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Appendix C. Calculations
Calculation C.3.9. Here we present an alternate form for the local Hamiltonian: i Pa Pn−1 Pn−1 h 2(2p−1−a)j j ei+a−j,i+a ⊗ ei−j,i w (−1) H ∝ p=1 i=0 a,j=1 h n−1 Pn−1 Pn−1 P 2 Pa j 2(2p−1−a)j ∝ ei+a−j,i+a ⊗ ei−j,i a=1 i=0 j=1 (−1) p=1 w i Pn−a 2(2p−1+a)j + p=1 w ei−a−j,i−a ⊗ ei−j,i hP Pn−1 Pn−1 P n−1 a 2(2p−1−a)j j 2 ei+a−j,i+a ⊗ ei−j,i ∝ a=1 p=1 w i=0 j=1 (−1) i Pn − p=n−a+1 w2(2p−1+a)j ei−a−j,i−a ⊗ ei−j,i hP P n−1 Pn−1 Pn−1 a j 2(2p−1−a)j 2 ∝ ei+a−j,i+a ⊗ ei−j,i a=1 i=0 j=1 (−1) p=1 w i Pn − p=n−a+1 w2(2p−1+a)j ei−a−j,i−a ⊗ ei−j,i hP P n−1 Pn−1 Pn−1 a 2(2p−1−a)j j 2 ei+a−j,i+a ⊗ ei−j,i ∝ a=1 p=1 w i=0 j=1 (−1) i Pa − p=1 w−2(2p−1−a)j ei−a−j,i−a ⊗ ei−j,i hP P n−1 Pn−1 Pn−1 a j 2(2p−1−a)j 2 ei+a−j,i+a ⊗ ei−j,i ∝ a=1 j=1 (−1) p=1 w i=0 i Pa + p=1 w2(2p−1−a)j ei−a+j,i−a ⊗ ei+j,i P n−1 Pn−1 Pn−1 Pa j 2(2p−1−a)j 2 ∝ [ei+a−j,i+a ⊗ ei−j,i + ei−a+j,i−a ⊗ ei+j,i ] . a=1 i=0 j=1 (−1) p=1 w We now consider the action of the element of Dn as consider a subgroup of Sn n−1
σ = [1, 2, 3, ..., n]
and
τ=
2 Y
[k, n − k].
k=1
We have ei+a−j,i+a ⊗ ei−j,i = eσi (a−j),σi (a) ⊗ eσi (n−j),σi (n) , while ei−a+j,i−a ⊗ ei+j,i = eσi τ (a−j),σi τ (a) ⊗ eσi τ (n+j),σi τ (n) . Therefore H∝
n−1 X j=1
n−1
(−1)j
a 2 X X a=1 p=1
" w2(2p−1−a)j
n−1 X
# eγ(a−j),γ(a) ⊗ eγ(n−j),γ(n) .
γ∈Dn
Calculation C.3.10. For D(D2n ) where n ≥ 2 we apply the representations and find (π2 ⊗ πn )R =
Pn−1 i=0
[π2 (σ 2i ) ⊗ πn ((σ 2i )∗ ) + π2 (σ 2i τ ) ⊗ πn ((σ 2i τ )∗ )]
P 2i+1 + n−1 ) ⊗ πn ((σ 2i+1 )∗ ) + π2 (σ 2i+1 τ ) ⊗ πn ((σ 2i+1 τ )∗ )] i=0 [π2 (σ Pn−1 2i = i=0 [π2 (σ τ ) ⊗ ei+1,i+1 ] Pn−1 2(i−1)k = e1,2 + w−2(i−1)k e2,1 ) ⊗ ei,i ] i=0 [(w
C.3 Calculations for Chapter 4 and (π2 ⊗ πn )(R−1 )T =
P2n−1 i=0
[π2 ((σ i ))∗ ) ⊗ πn (σ −i ) + π2 ((σ i τ )∗ ) ⊗ πn (σ i τ )]
= [e1,1 ⊗ πn (σ −l ) + e2,2 ⊗ πn (σ l )] Pn−1 = i=0 [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ], (l,k)
where π2 = π2
(0,b)
and πn = πn,τ .
Calculation C.3.11. For D(D2n ) where n ≥ 2 we apply the representations and find (π2 ⊗ πn )R =
Pn−1 i=0
[π2 (σ 2i ) ⊗ πn ((σ 2i )∗ ) + π2 (σ 2i τ ) ⊗ πn ((σ 2i τ )∗ )]
P 2i+1 + n−1 ) ⊗ πn ((σ 2i+1 )∗ ) + π2 (σ 2i+1 τ ) ⊗ πn ((σ 2i+1 τ )∗ )] i=0 [π2 (σ Pn−1 2i+1 τ ) ⊗ ei+1,i+1 ] = i=0 [π2 (σ Pn−1 2i+1 τ )(w(2i−1)k e1,1 + w−(2i−1)k e2,2 ) ⊗ ei,i ] = i=0 [π2 (σ and (π2 ⊗ πn )(R−1 )T =
P2n−1 i=0
[π2 ((σ i ))∗ ) ⊗ πn (σ −i ) + π2 ((σ i τ )∗ ) ⊗ πn (σ i τ )]
= [e1,1 ⊗ πn (σ −l ) + e2,2 ⊗ πn (σ l )] Pn−1 = i=0 [e1,1 ⊗ ei−l,i + e2,2 ⊗ ei+l,i ], (l,k)
where π2 = π2
(0,b)
and πn = πn,στ .
137
