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Author/s:
TSARENKO, MARIA
Title:
A solvable lattice model: the square-triangle-rhombus random tiling
Date:
2013
Citation:
Tsarenko, M. (2013). A solvable lattice model: the square-triangle-rhombus random tiling.
Masters Research thesis, Department of Mathematics and Statistics, Faculty of Science, The
University of Melbourne.
Persistent Link:
http://hdl.handle.net/11343/37981
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A solvable lattice model: the square-triangle-rhombus random tiling
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a solvable lattice model:
the square-triangle-rhombus
random tiling
Maria Tsarenko
Submitted in total fulfilment of the requirements of
the degree of Master of Philosophy
March 2013
Department of Mathematics and Statistics
The University of Melbourne
Abstract
The aim of this thesis is the study of a new model in statistical mechanics, which
belongs to a class of exactly solvable lattice models. This model is defined on the square
lattice with ten possible configurations at each vertex. It belongs to a hierarchy of integrable systems, constructed by taking a singular limit of certain extensions of models
related to the sl(n) algebra. By the integrability of a system, we mean that the R-matrix
associated to a lattice model solves the Yang–Baxter equation. The limit indicated above
is singular in the sense that the R-matrix granting integrability of the original model does
not satisfy the Yang–Baxter equation of the new model after taking this limit. However, for the model of interest (with n = 3) another matrix R? is found, establishing the
integrability of the model.
The first few systems in this new ‘singular’ hierarchy had already been identified with
some known random tiling models. The study of the latter is independently driven by the
desire to model the atomic structure of physical substances termed quasicrystals (or quasiperiodic alloys). The 10-vertex model has now also been identified with a new random
tiling, called the square-triangle-rhombus tiling model. This new tiling model is the focus
of this thesis.
The two-dimensional lattice models referred to as ‘exactly solvable’ are closely related
to one-dimensional quantum integrable systems. Over the past few decades, the (coordinate and algebraic) Bethe Ansatz method emerged as a powerful tool in the study of
quantum integrable systems. In its modern variant, it is based on the Yang–Baxter algebra and it is the quantum analog of the inverse scattering method. For our problem,
the algebraic Bethe Ansatz is used to obtain the solution for the square-triangle-rhombus
tiling model.
Although the Yang–Baxter equation for the tiling model is established for all values
of parameters, the solvability of the model is, however, restricted by our ability to solve
the Bethe equations. We were able to do so in closed form in a two-parameter subspace,
restricted by the region of solutions of the Bethe equations.
We compute the exact expression for the bulk entropy of the new square-trianglerhombus random tiling in the thermodynamic limit, within the solvable region. In this
limit, the phases of maximal symmetry of the tiling are identified and the value of the
maximum entropy is found to lie on this line. The square-triangle-rhombus tiling exhibits
properties that are qualitatively different from its specialisation – the square-triangle tiling.
The introduction of rhombi stabilises the system, making the homogeneous tiling entropically favourable over the phase separated tiling, for which the entropy would be a linear
combination of the entropies of the quasi-periodic and purely crystalline phases.
i
Declaration
This is to certify that
(i) the thesis comprises only my original work towards the MPhil,
(ii) due acknowledgement has been made in the text to all other material used,
(iii) the thesis is less than 50 000 words in length.
Maria Tsarenko
ii
Acknowledgments
First and foremost, I wish to express my deepest gratitude to my supervisor Jan de Gier for
his guidance and unceasing enthusiasm over the course of this project. I am thankful for many
discussions that shaped my knowledge of solvable lattice models and continue to inspire my interest
in the area. I am grateful for the time he gave generously, including his tireless reading and editing
of my drafts. His persistent help, his support and encouragement is what made it all possible.
I would like to thank the Department of Mathematics and Statistics for providing me with the
funding that enabled me to complete this thesis. Also, I would like to thank the many members
of staff who have taught me over the years and who made me feel at home here.
A big thank you to all my friends and fellow students, who I have been fortunate to work
alongside over the years and who have made the journey an enjoyable and a memorable one. I
especially want to thank Ana Dow for being my rock from before the beginning; Nick Davis, Leigh
Humphries, Joanna Hutchinson, Anthony Mays and Shanil Ramanayake for their energy and
encouragement; Nick Beaton, Anita Ponsaing and Gus Schrader for mathematical and personal
inspiration.
I am indebted to my friend and colleague Michael Wheeler for introducing me to the algebraic
Bethe Ansatz and for explaining the subtleties of the field that significantly aided my progress. A
special thank you to Paul Zinn-Justin for many valuable comments and suggestions.
Finally, I wish to thank my parents for their perpetual support.
iii
Contents
1 Introduction
1
1.1
Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Solvable Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.1
Transfer-matrix method . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.2
The Bethe Ansatz method . . . . . . . . . . . . . . . . . . . . . . . .
7
Integrable Random Tiling Models . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.1
Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3.2
Quasi-periodic tilings
9
1.3.3
Random tilings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3
1.4
. . . . . . . . . . . . . . . . . . . . . . . . . .
Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Algebraic Bethe Ansatz for the Six-vertex Model
2.1
2.2
14
Integrable six-vertex model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1
L - matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2
Transfer matrix T (u)
2.1.3
Monodromy matrix Tα (u) . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.4
R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5
RT T relation and the Yang–Baxter equation . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . 18
Algebraic Bethe Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1
Pseudo-vacuum state: |Ωi . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2
Commutation relations for the operators A, B and D . . . . . . . . . 27
iv
2.2.3
ABA Solution: Eigenvectors of T (u) and Bethe equations . . . . . . 29
3 Algebraic Bethe Ansatz for the Fifteen-vertex Model
33
3.1
Integrable fifteen-vertex model . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2
First step in the method of nested Bethe Ansatz . . . . . . . . . . . . . . . 34
3.3
3.2.1
The first set of commutation relations . . . . . . . . . . . . . . . . . 36
3.2.2
Proposed form for the first level eigenvector |Ψ1,...,N i . . . . . . . . . 40
3.2.3
Actions of A(1) and D(1) on |Ψ1,...,N i . . . . . . . . . . . . . . . . . . 40
3.2.4
First level expressions for eigenvalues Λ(1) (u) and Bethe equations . 41
(1)
(1)
Second and final step in the method of NBA
. . . . . . . . . . . . . . . . . 42
(2)
3.3.1
Actions of A(2) and D(2) on |Ψα1 ...αM ,1...N i . . . . . . . . . . . . . . . 42
3.3.2
Second level expressions for the sub-eigenvalues Λ(2) (u) and Bethe
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.3
Final form for the eigenvalues Λ(1) (u) and the Bethe equations . . . 45
3.A Proof of (3.14) and (3.15) in Proposition 2 . . . . . . . . . . . . . . . . . . . 47
3.B Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 From Perk–Schultz type Models to Integrable Random Tilings
56
4.1
A generalised Perk–Schultz type model . . . . . . . . . . . . . . . . . . . . . 58
4.2
A family of singular integrable models . . . . . . . . . . . . . . . . . . . . . 60
4.3
4.2.1
Parameter rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2
Diagonalisation of the new model . . . . . . . . . . . . . . . . . . . . 61
Random tiling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1
Rhombus random tilings: eµ = 0 . . . . . . . . . . . . . . . . . . . . 63
4.3.2
Square-triangle random tilings: u = 0 . . . . . . . . . . . . . . . . . 66
5 Square–Triangle–Rhombus Tiling Model
70
5.1
The square-triangle-rhombus tiling . . . . . . . . . . . . . . . . . . . . . . . 70
5.2
Linear and bilinear constraints in tile densities . . . . . . . . . . . . . . . . 72
5.3
The eigenvalue and the Bethe Ansatz equations . . . . . . . . . . . . . . . . 76
v
5.3.1
Cauchy–Stieltjes transform . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2
Analytic continuation and monodromy . . . . . . . . . . . . . . . . . 80
5.4
Solution of the monodromy problem . . . . . . . . . . . . . . . . . . . . . . 82
5.5
Expression for the largest eigenvalue of the square-triangle random tiling:
u = 0 and arbitrary µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.6
Entropy of the square-triangle random tiling: u = 0 . . . . . . . . . . . . . . 91
5.7
Lattice dependence and area conversions . . . . . . . . . . . . . . . . . . . . 93
5.8
Entropy of the square-triangle-rhombus tiling for arbitrary weights of the
rhombus r and square s+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.8.1
The line of maximum symmetry and entropy for arbitrary u > 0 and µ 99
5.8.2
Boundary points and lines on the solvable surface σ ? . . . . . . . . . 102
5.A Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Conclusion and Outlook
108
vi
Chapter 1
Introduction
1.1
Statistical Mechanics
The aim of Statistical Mechanics is to explain and predict macroscopic properties of matter given the knowledge of microscopic behaviour of individual particles. It provides a
framework for the calculation of a wide variety of observable properties of a system (such
as density, pressure, entropy, free energy and magnetisation), using the information about
local interactions between atoms or molecules to obtain global, thermodynamic quantities
of the system.
A typical system under consideration consists of N ∼ 1023 particles. If we were given
initial states (say position and velocity) of individual particles and asked to find their
positions some time later, we would struggle to give an exact answer even for a three
particle system (the ‘three-body problem’ has not been completely solved). Numerical
computations are also out of reach. Given a futuristic computer with 104 Gb of memory,
we would still be 13 orders of magnitude short of being able to specify even one point in
phase space.
It is a relief, then, to know that in this field one is less concerned with trajectories of
individual particles and more interested in the average properties of a system. For instance,
one may wish to express the density of some substance as a function of temperature and
pressure, or to know whether a medium undergoes a phase transition, and if so, what is
the value and nature of the critical point at which it occurs.
Phase transitions are the more striking consequences of interactions among particles
and the study of their behaviour is of particular interest. Perhaps the simplest examples
are the boiling point of water (eg. 100◦ C at 1 atm) and spontaneous magnetisation of a bar
of iron at zero magnetic field. Near a critical point Tc (of a continuous phase transition)
1
the correlations between particles become long-ranged and the strength of the divergence
of some global state function f (t), as the system approaches Tc , is characterised by the
value of a critical exponent α:
f (t) ∝ tα , as t → 0,
with t =
T − Tc
.
Tc
The critical behaviour of a system near a continuous phase transition is classified by
universality classes. The critical exponents of each class do not depend on the details of
the interactions but rather on some general features such as the spatial dimension of the
system, its symmetry, and the range of forces.
Let us begin with the definitions of some fundamental quantities in the field of statistical mechanics that are of relevance in the current context. Let σ denote a state or
a configuration of the system. Then the energy of state σ is given by the Hamiltonian
H(σ). For a system in thermodynamic equilibrium at a temperature T , the probability P
of observing a state σ decreases, as the energy of that state increases
P (σ) =
1 − H(σ)
e kT ,
Z
where k is the Boltzmann constant, T is the temperature and e−
H(σ)
kT
is called the Boltz-
mann weight of a configuration σ. The normalisation Z is called the partition function
and is given by the weighted sum of all states of a system:
Z=
X
e−
H(σ)
kT
.
(1.1)
σ
Many statistical properties of the system can be derived from Z. The free energy F of
the system is given by
F = −kT ln Z
(1.2)
and is a measure of the amount of work a closed system (at constant temperature and
volume) can perform.
Let N be the size of the system (eg. the number of lattice points) and let us denote
the partition function of a finite system by ZN . As the size of the system gets large, ie.
N → ∞, the free energy is proportional to the size of the system, and we can define the
free energy per site or the bulk free energy f by the following limit, ie.
1
ln ZN .
N →∞ N
f = −kT lim
(1.3)
Note that Z is always analytic, hence phase transitions can only be associated with infinitely many particles, that is, in the thermodynamic limit and have their origins in
singularities associated with f .
2
The observable energy of the system, that is, the expected value of energy hHi, called
the internal energy is given by
P
U = hHi =
σ
H(σ)e−H(σ)/kT
.
Z
The free energy F can be thought of as Legendre transform of the internal energy. Using
the definitions above, we can rewrite
F = U − T S,
(1.4)
where S = ∂F/∂T is the entropy. In general, the Legendre transform can be used to
express a function in terms of its derivative. We will use it later, to shift from a function
in certain independent variables, to a new function with a dependence on new variables.
In the instance when all states have the same energy E, which must therefore be equal
to the average energy U , the partition function Z is simply given by
Z = ΩeE/kT ,
where Ω is the total number of configurations. Then, from (1.2) and (1.4) we can write
the expression for the entropy S as follows:
F = U − TS
−kT ln Z = E − T S
−kT ln Ω − kT ln(e−E/kT ) = E − T S
=⇒ S = k ln Ω.
(1.5)
Intuitively, entropy is the amount of freedom a system has to adopt a state. That is, for
a macro-state of high entropy, a system has a larger number of ‘choices’ of micro-states to
be in.
Outlined above are the main ingredients of what makes up a statistical mechanical
model. Naturally, these models are simplifications or approximations of real physical
systems. Ideally, the models are designed to be “as simple as possible, but no simpler”1 ,
so as to capture the essential characteristics of real systems, and at the same time to
(hopefully) be mathematically tractable.
To solve these models and find closed form expressions for the quantities of interest is
very difficult in general, even for those which are seemingly simply formulated. We say
that a lattice model is solvable if it is possible to find an exact expression for the partition
1
The quote is usually attributed to Albert Einstein via indirect references. For example, see How a
Difficult Composer Gets That Way by Roger Sessions, New York, 1950.
3
function (or equivalently, entropy or free energy). It is the possibility of obtaining exact
non-perturbative results that makes these models interesting to study from a mathematics
point of view.
Finally, universality predicts that a solution of a model near criticality gives precise
results for a real system under consideration, provided that the model and the physical
system possess the same dimensionality and symmetry.
1.2
Solvable Lattice Models
In theoretical physics, lattice models have proven to be fruitful for modelling many kinds
of physical phenomena and make understanding of interacting systems through exact
solutions more attainable.
A lattice model is defined by an ensemble of particles or spins placed on the edges of
a lattice (usually square, hexagonal or triangular). All allowed states of spins (eg. ±1
for the Ising model) as well as the Hamiltonian describing the interactions between them
need to be specified. Amongst the famous lattice models are the Ising model, the Potts
model, the six- and eight-vertex models and the O(n) loop model.
There are various classes of lattice models. One-dimensional models are simpler and do
not exhibit phase transition for any ‘physical’ interactions between spins. Two-dimensional
models are useful for a broader variety of real systems and are very rich mathematically. It
is also believed that two-dimensional systems can behave quite like the three-dimensional
ones (sometimes by the incorporation the third dimension as an internal degree of freedom).
There is a subset of lattice models that are exactly solvable and the computation of
their partition function Z is accessible. Our interest lies with a subclass of these – the
two-dimensional models on a square lattice.
1.2.1
Transfer-matrix method
A powerful method for tackling the computation of Z for solvable models is the transfermatrix method.
First formulated for one-dimensional systems by Montroll [39], and simultaneously by
Lassettre and Howe [34] and Kramers and Wannier [33] in 1941, a transfer matrix is an
operational technique for describing the propagation of physical elements (spins) along
a chain or a lattice. The fundamental observation is that the partition function can be
4
expressed as the trace of some power of the so-called transfer matrix T . Thus the problem
of calculating the partition function can be reformulated as an eigenvalue problem for T .
Let us demonstrate a simple use of a transfer matrix on the example of a onedimensional ‘lattice’ model, that is, a model of spins on a chain – the Ising model.
Solution of the one-dimensional Ising model
To introduce the concept of the transfer matrix and to give it some context, we look at
the one-dimensional Ising model. It is the oldest solvable model and also the simplest
and the most well-known. Introduced by Wilhelm Lenz as mathematical description of a
ferromagnet, it was solved in one-dimension by his student Ernst Ising in 1924.
In one dimension the Ising model is defined as follows. At each site j of the chain of
length N , there is a spin that can take one of two values, say sj ∈ {−1, 1}. The interactions
between spins is described by the Hamiltonian H
−βH =
N
X
Ksj sj+1 +
N
X
j=1
Bsj ,
j=1
where β = 1/kT , (k – the Boltzmann constant, T – temperature), K is the coupling
constant and B is the magnetic field. We consider the model with periodic boundary
conditions: s1 = sN +1 . The partition function is then equal to


N
N
X
X
X
ZN =
exp 
Ksj sj+1 +
Bsj  ,
s1 ,...,sN
j=1
j=1
N
X
N
1X
or equivalently rewritten as

X
ZN =
s1 ,...,sN
exp 
Ksj sj+1 +
j=1
2

B(sj + sj+1 ) .
j=1
The partition function can be easily calculated by introducing the transfer matrix
1
0
0
Ts,s0 = exp Kss + B(s + s )
2
"
# "
#
T11 T12
eK+B e−K
=
=
.
T21 T22
e−K eK−B
Now the partition function can be rewritten in terms of T as
ZN =
X
Ts1 ,sN TsN ,sN −1 . . . Ts2 ,s1 = Tr T N ,
s1 ,...,sN
5
and can be calculated in a closed form in terms of the eigenvalues of T
N
ZN = ΛN
+ + Λ− ,
Λ± = eK cosh B ±
p
e2K sinh2 B + e−2K .
Although in this example the expression for the partition function is exact, in general one
can only hope to find analytic solutions in the thermodynamic limit, as N → ∞, where
ZN is given by the largest eigenvalue of T :
Z ∼ ΛN
max as N → ∞.
The bulk free energy f is then given by
1
ln Z
N →∞ N
h
f = − kT lim
i
= −kT ln eK cosh B + (e2K sinh2 B + e−2K )1/2 .
As can be seen, f is an analytic function of temperature and magnetic field, thus the
one-dimensional Ising model has no phase transition for any positive T .
The transfer-matrix method can be extended to two-dimensions. In 1967, Lieb [36]
used this method in his pioneering work to find the exact solution for the entropy of
the two-dimensional ‘square ice’ model, now more commonly referred to as the six-vertex
model.
The transfer matrix T encodes all possible spin configurations of one row of a lattice.
The full M × N lattice is built by ‘stacking’ the rows on top of each other or equivalently,
multiplying M copies of T in a consistent way. In the limit as M → ∞, the partition
function Z is determined by the largest eigenvalue of T
Z ∼ ΛM
max , as M → ∞
and in the thermodynamic limit (as N → ∞ also) we are interested in computing the bulk
free energy f :
f =−
1
ln Z
= − lim
ln Λmax .
N →∞ N
M,N →∞ M N
lim
A lattice model is solvable if it is possible to calculate the exact expression for the function
f above.
We are thus faced with a diagonalisation problem for T . For two-dimensional models,
T is too large to be diagonalised directly, even with the aid of modern computers. Consider
for example one row of a square M × N lattice where each of the vertical edges can be
either empty or occupied. Then one row has 2N possible states. The partition function
is given by Z = TrT M , where T is 2N × 2N transfer matrix. Thus Λmax is the largest
of the 2N eigenvalues of T , which are incalculable for large values of N with standard
diagonalisation methods.
6
1.2.2
The Bethe Ansatz method
In 1931 Bethe [6] managed to diagonalise large matrices in the process of solving a spin-1/2
Heisenberg model. His method gave rise to a whole new approach to the study of solvable
lattice models and enabled one to efficiently find the spectrum of quantum Hamiltonians
in models such as the Heisenberg spin chain, the one-dimensional Bose gas with point
interaction, as well as for the classical transfer matrices, such as the six-vertex model.
Bethe Ansatz was the name given to a method for the exact calculation of eigenvalues and
eigenvectors of a certain class of solvable lattice models.
The essence of Bethe’s original method (coordinate Bethe Ansatz ) lies in ‘guessing’ the
form of the eigenvectors |Ψi of the quantum Hamiltonian and imposing certain constraints
that ensure that |Ψi are indeed the eigenvectors of H. These constraints are equivalent
to a system of non-linear simultaneous equations and are usually referred to as the Bethe
equations. The 2N dimensional eigenvalue problem is thus reduced to a problem of solving
N non-linear simultaneous equations.
Since then, the Bethe Ansatz method has been generalised and restated in an algebraic
framework, incorporating the transfer-matrix methodology. In 1979 Faddeev, Sklyanin,
and Takhtadzhyan [52, 18] formulated the quantum analogue of the inverse scattering
method, and the algebraic version of the Bethe Ansatz is a part of this method. It
is this reformulated version of the method, called the algebraic Bethe Ansatz, that is
applied to solve the lattice model studied in this thesis. The principal requirement for the
diagonalisation of T is that the transfer matrices (one for each row of the lattice) commute
for all values of associated row parameters.
Today the term algebraic Bethe Ansatz does not just mean the method of constructing
the eigenvectors of the Hamiltonians, but stands for any manipulations in the algebra
of matrix elements of the monodromy matrix. (The creation of the quantum inverse
scattering method stimulated a new understanding of the relationship between quantum
and classical integrable systems and led to the creation of the theory of quantum groups
[8]).
The central goal in the theory of solvable lattice models is to find models whose transfer
matrices belong to a family of commuting operators. The existence of infinitely many
conserved quantities is a general consequence of integrability. There is no single accepted
definition of what makes a model integrable. We shall see that a sufficient condition in
practice is that the Yang–Baxter equation holds. This equation is a consistency condition
for the so-called factorised scattering, meaning that an integrable model does not have any
true three-particle interactions. Any such interaction can be decomposed into consecutive
7
interactions between pairs of particles and, importantly, the order in which the interactions
take place is inconsequential. Checking this latter condition is the most important test to
determine whether a model can be integrable or not.
1.3
Integrable Random Tiling Models
The lattice model at the focus of this thesis is, in fact, equivalent to a random tiling
model. A random tiling model is a filling of a region of the plane with elementary tiles
without gaps or overlaps. Each tile comes from a fixed finite set of prototiles of specified
orientations.
In mathematical physics, the motivation for the study of random tilings came with the
discovery of quasicrystals. In this section we give some background and provide a physical
motivation for the study of random tilings as models for quasicrystals.
1.3.1
Quasicrystals
In the early 80s, Shechtman and collaborators [51] made a remarkable discovery. They
observed that a solid they had made in their laboratory by rapid solidification, produced
discrete diffraction patterns, like a periodic crystal. To their surprise, however, the diffraction pattern possessed icosahedral symmetry, impossible for any periodic structure.
Rotational symmetries compatible with translational symmetry in two and three dimensions are known to be exclusively 2-, 3-, 4- and 6-fold. As crystals are invariant under
both, discrete translations and rotations, the study of their atomic structure was therefore
confined to models based on periodic lattices. With no exceptions, each (recorded) crystal
found up until then conformed to one of 32 ways of combining these symmetry axes.
However, the aperiodic ‘crystal’ (AlPdMn alloy) discovered by Shechtman exhibited
‘forbidden’ icosahedral rotational symmetry with its six intersecting 5-fold axes. In two
dimensions, one can observe a 5-fold rotational symmetry (see Figure 1.1). Shechtman’s
discovery (very controversial at the time) had sparked much interdisciplinary scientific activity in physics, materials science, mathematics and crystallography2 . Soon after, many
other alloys with ‘forbidden’ rotational symmetries were discovered and are now collectively termed quasicryastals 3 .
Examples of symmetries other than pentagonal include octagonal symmetry found in
2
Dan Shechtman was awarded the Nobel Prize in chemistry in 2011 “for the discovery of quasicrystals”
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/shechtman.html
3
The term “quasi-periodic crystal” or “quasicrystal” was first used by Levine and Steinhardt [35].
8
Figure 1.1: Left: The atomic model of the aluminium-palladium-manganese quasicrystal
surface (Wikipedia). Right: Mosaic on the wall of Darb-i Imam Shrine in Isfahan, Iran,
circ. 1450 [37]
VNiSi, CrNiSi [58] and MnSiAl [5] alloys, decagonal symmetry in AlCuM [23] , AlCoNi
[4] and dodecagonal symmetry in NiCr [25] and VNi(Si) [9]. (For some reviews, see
[54, 26, 16]).
All quasicrystals mentioned so far were created in laboratories. The first quasicrystals
were unstable, so when heated, they formed regular crystals. However since 1987, many
stable quasicrystals have been created, opening the door to potential applications. Also,
the discovery of naturally occurring khatyrkite and cupalite [7] in 2009 was an additional
reassurance of quasicrystals’ ability to form and remain stable under geological conditions.
Quasicrystalline alloys possess properties quite unlike usual metals, and as such, have
a potentially vast range of applications. Some of these properties are stability at high temperatures, ability to withstand pressure in all directions equally, hardness and resistance
to corrosion and to wear [16]. With the increasing understanding of their structure and
properties, there is an ever-increasing potential for their applications in nanotechnology
and materials science.
The first step towards mathematical understanding of the atomic structure of quasicrystals comes from looking at quasi-periodic tilings.
1.3.2
Quasi-periodic tilings
The knowledge of quasi-periodic tilings of the plane dates back to long before the discovery
of quasicrystals. A certain 10-fold quasi-periodic tilings is decorating the entrance of the
Darb-i Imam Shrine in Iran (See Figure 1.1) and other quasi-periodic tilings appear in
islamic architecture from as early as the 12th century [37]. In the West, these tilings lived
9
in the realm of recreational mathematics, and it was not until the 70s, with the famous
construction of an aperiodic tiling by Penrose [43, 21], that they were investigated more
rigorously.
Figure 1.2: Left: A quasi-periodic Penrose tiling with 5-fold rotational symmetry (http://
www.alienscientist.com/penrose.html). Right: A quasi-periodic square-triangle tiling
with 12-fold symmetry [1].
Penrose’s tiling (in Figure 1.2) consists of a set of two tiles, tiling the plane only
aperiodically and exhibiting 5-fold rotational symmetry [35]. Thus the Penrose tiling
became the first mathematical model for the atomic structure of a quasicrystal. By now,
many quasi-periodic tilings are known, such as the quasi-periodic square-triangle tiling
in Figure 1.2, and very complicated quasi-periodic tilings can be easily generated 4 . For
some reviews on the subject, see [21, 22].
In general, quasi-periodic tilings are geometric objects without translational symmetry,
but which can have any of the infinite set of point group symmetries. Mathematically
quasi-periodicity can be more easily understood through higher dimensional periodicity.
Given a hypercube, a quasi-periodic tiling is the set of all lattice points within a slice of
that hypercube, bounded by two parallel hyper-planes of an irrational gradient. A physical
example of a two-dimensional quasi-periodic tiling is the surface obtained by cutting a
cubic lattice with an irrational plane. In the case of the Penrose tiling, one needs to go up
to five dimensions before a zig-zag path along the edges of the tiling becomes a Cartesian
path and each corner of the hypercube is specified with a five-coordinate vector.
As a simple example illustrating the idea of quasi-periodicity with higher dimensional
4
Quasi-periodic lozenge tiling generator on P.Zinn-Justin’s webpage http://www.lpthe.jussieu.fr/
~pzinn/tilings/
10
periodicity, consider the function f (x):
f (x) = cos(x) + cos(bx),
b ∈ R.
If b is rational, then f is periodic and the two cosine functions, or rather the two delta
functions of their Fourier transforms can be indexed by a single vector of the reciprocal
lattice. If b is irrational, then f is quasi-periodic and the two incommensurate lengths
require two index vectors. The function
f (x, y) = cos(x) + cos(y)
is always periodic in two dimensions. And the quasi-periodic one-dimensional f (x) can be
recovered from f (x, y) by setting y = bx, with irrational b.
Interesting and beautiful as they may be, quasi-periodic tilings have some serious flaws
as models of quasicrystals. One critique of the perfect quasi-periodicity is the stringent local matching rules imposed on the tiles that are required to achieve global quasi-periodicity.
These rules are infeasible to be realised under physical conditions and are equivalent to requiring unrealistic long-range interactions between the atoms. Ideally, one would hope for
a thermodynamically stable model of a quasicrystal. An important early observation made
by Elser [17] is that sharp diffraction peaks and rotational symmetry can be preserved by
introducing a general kind of disorder, which serves to (hopefully) stabilise a quasicrystal.
This led to an idea of random tiling models as entropic models for quasicrystals.
1.3.3
Random tilings
We define a random tiling model as an ensemble of edge-to-edge tilings of a plane with
tiles taken from a fixed set of prototiles, joined without gaps or overlaps, see the right hand
side of Figure 1.3. Another assumption is that the number of allowed configurations in a
given region should grow exponentially with the area of that region. This means that for
some fixed set of prototiles there is a large ensemble of possible tilings, including periodic,
quasi-periodic and disordered. It is believed [24, 17] that the dominant ‘macrostate’ – a
typical state of greatest entropy – will be the one of maximum symmetry (compatible with
the tiles and the Boltzmann weights), possibly exhibiting ‘forbidden’ rotational symmetry
and long-range quasi-periodic translation order. The guiding principle that the state
of maximal entropy is also the one of maximal symmetry follows from group-theoretic
arguments [48, 10].
In the language of statistical mechanics, allowing for local random rearrangements of
tiles leaves the total energy U unchanged, but gives rise to a finite entropy contribution
that will lower the free energy F of the system, potentially stabilising the quasicrystal.
11
Figure 1.3: Left: A quasi-periodic rhombus tiling constructed by slicing a 9-dimensional
hypercube with an irrational plane Right: A random tiling of rhombi constructed by
slicing a 9-dimensional hypercube with some surface (images courtesy of P. Zinn-Justin)
The macroscopic state is quasi-periodic only in a statistical sense. Models of random
tilings also allow for impurities and defects observed in real quasicrystals. We are thus led
to study ensembles of random tilings as more realistic, entropically stabilised models for
quasicrystals.
Over the last two decades a vast body of work has been dedicated to the study of
various random tiling models. Many results were found numerically to high precision, and
many exact results are now known as well. For a sample of some of the earlier literature
for the latter case, see [24, 28, 29, 60, 40, 13, 14, 12, 15, 42, 48].
Our interest lies in the exact calculations of the thermodynamic properties of random
tilings. Random tilings of rectangles and triangles exhibiting 8-, 10-, and 12-fold symmetric
phases have been extensively studied by de Gier and Nienhuis in [12, 14, 13]. They
succeeded in computing analytic expressions for the entropy in all of these models. In
the case of the square-triangle random tiling model, de Gier and Nienhuis made a crucial
connection with an integrable lattice model, implementing the method of the algebraic
Bethe Ansatz mentioned in the previous section to give exact results.
The main attention of this thesis is focused on the study of the so-called squaretriangle-rhombus tiling model – an extension of the model in [13, 28], with the aim of
calculating an exact expression for the entropy and description of the phase diagram.
The results presented in this thesis build on the previous work of de Gier and Nienhuis
[13], specifically their observation that this random tiling is equivalent to a lattice model,
and we use the method of algebraic nested Bethe Ansatz to diagonalise the transfer matrix.
We also use an extension of Kalugin’s method [28] for the computation of a closed form
12
expression of the bulk entropy of the square-triangle-rhombus tiling model.
1.4
Outline
The layout of this is thesis is as follows. In chapter 2 we introduce the concept of a
lattice model and explain precisely what is meant by integrability. Next, we demonstrate
c2 )
the method of the algebraic Bethe Ansatz on the example of a six-vertex (i.e., Uq (sl
symmetry) lattice model to obtain expressions for the eigenvalues of T . In chapter 3
the reader is introduced to the method of nested Bethe Ansatz (a generalisation of the
method used in chapter 2) for the case of the integrable 15-vertex lattice model with
c3 ) symmetry.
Uq (sl
In chapter 4, we introduce the square-triangle-rhombus tiling and formulate the main
problem. The connection is made between the tiling model and an integrable lattice model
related to the singular limit of the one discussed in chapter 3. In chapter 5 we calculate
an expression for the entropy of the tiling model in the thermodynamic limit, find the line
of maximum entropy and symmetry, discuss the physical meaning of the limiting curves
and the phase diagram. Finally, in the conclusion, we synthesise our findings and discuss
possible extensions and directions for future research.
13
Chapter 2
Algebraic Bethe Ansatz for the
Six-vertex Model
Introduction
The purpose of this chapter is to provide a brief introduction to the algebraic Bethe
Ansatz (ABA) in a sufficiently general setting. The ABA method is the quantum version
of the Inverse Scattering Method pioneered by the Leningrad school in the 70’s by Faddeev
and collaborators. It uses the Yang–Baxter algebra satisfied by matrix elements of the
monodromy matrix to generate eigevectors by applying certain operators to a reference
state, called the pseudo-vacuum. The Bethe equations emerge as consistency conditions for
these states to be eigenvectors of the transfer matrix. For some reviews, see [18, 32, 53, 20].
An important idea behind solving integrable models is the extension of the physical
Hilbert space H with an auxiliary space Vα . The introduction of Vα serves to decouple the
interaction between the physical spaces, so that they do not interact amongst themselves,
but only with the auxiliary space. This simplifies the problem and allows for exact solution.
What is meant by ‘integrability’ of a model is the existence of a family of infinitely
many commuting operators, the Hamiltonian H of the system being one of them. The
generating function for these operators is the transfer matrix T (u), acting in H, an object
that plays a key role in this method.
In the first section of this chapter we introduce a local operator Lαn (u) (which encodes
Boltzmann weights of a lattice vertex), followed by the monodromy matrix Tα (u) which
describes the interaction of the auxiliary spin with the physical chain of spins via a product
of L matrices. Next we define the transfer matrix T (u) which denotes the weight of one
row of the lattice and, importantly, is expressed as the trace over auxiliary space α of
14
Tα (u), i.e., T (u) is a ‘projection’ of T on the physical space. We then introduce the Rmatrix acting in Vα ⊗ Vβ , the existence of which allows for commuting transfer matrices,
or equivalently for diagonalisation of the whole family T (u) and hence the Hamiltonian
H. In the second section, we describe the algebraic Bethe Ansatz for calculating the
eigenvectors and eigenvalues of the transfer matrix. We show that if the R-matrix satisfies
the Yang–Baxter equation, it then satisfies the so called intertwining relation which yields
the commutation relations on the operators in Tα (u). Finally, the Bethe equations are
derived as the necessary conditions for the proposed Ansatz indeed to be a solution of the
eigenvalue equation.
Note
Solutions of the Yang–Baxter equation arise naturally in the study of quasi-triangular
Hopf algebras. In this chapter and the next we discuss models related to the quantum
cn ) in tensor products of fundamental representations. Uq (sl
cn ) is a
affine algebras Uq (sl
Hopf algebra that is a q-deformation of the universal enveloping algebra of the affine Lie
cn . Since the actual Hopf algebraic structure of the models is not used anywhere
algebra sl
in the thesis, we shall omit its definition and refer the reader to the literature on the
subject of quantum groups, for example, see [8], [30] and [38].
2.1
Integrable six-vertex model
In this section we describe the construction of a discrete integrable system possessing
c2 ) symmetry. We define a model on a square M × N lattice (see Figure 2.1), where
Uq (sl
each edge can take one of two values, 1 or 2. Each vertex configuration has an associated
energy, and therefore a Boltzmann weight. We demonstrate the ABA approach on the
case of a well-known system – the six-vertex model [2, 55]. The model obeys a restriction
on the vertex configurations called the ice rule, which states that any vertex configuration
where the number of 1s and 2s on the left and top edges does not equal the number of
1s and 2s on the bottom and right edges, has a Boltzmann weight 0. Therefore these
vertex configurations do not appear on the lattice, so of the possible eight configurations,
it leaves six that are shown on the right hand side of Figure 2.1.
Each lattice site represents a two-dimensional vector space Hn ' C2 , and the full row
denoting the complete Hilbert space of lattice states is taken to be the tensor product of
N
2
those at each site: H = N
n=1 Hn . Names Vα , Vβ ' C are given to auxiliary spaces.
A diagrammatic intersection of lines is simply a convenient notation for a tensor prod15
uct of corresponding vector spaces associated with the lines.
1
α1
2
1
..
.
1
1
1
2
αM
1
2
2
1
2
1
1
2
1
b
2
a
N
1
2
1
2
···
2
2
1
2
c
Figure 2.1: The six-vertex model is defined on a M × N square lattice, where each vertex can be in one of six configurations shown on the right hand side. The horizontal
(physical) sites of the lattice are labelled by {1, . . . N } and auxiliary spaces are labelled
by {α1 , . . . , αM }.
2.1.1
L - matrix
Consider a six-vertex vertex model on a square lattice with periodic boundary conditions
in both directions. The Boltzmann weights associated with each vertex are denoted by a
matrix:
X
Lαn (u) =
0 0
0
0
Lαn (u)ii jj Eii ⊗ Ejj .
(2.1)
i,i0 j,j 0
0
where Eii is an elementary matrix with 1 in ith row and i0 th column and zeros elsewhere.
Each entry of L(u) represents the Boltzmann weight of a particular vertex configuration.
c2 ) symmetry, each edge of the lattice can be in one of two states,
For models with Uq (sl
thus L is a 4 × 4 matrix with its entries given by:
j0
i0 j 0
Lαn (u)i
j
:=
α i0
i
j
n

11
Ln (u)11


Ln (u)11
12
∈ 

Ln (u)11
21


Ln (u)11
22

Ln (u)12
11
Ln (u)21
11
Ln (u)12
12

22 
Ln (u)21
L
(u)
n
12
12 


21
22
Ln (u)21 Ln (u)21 


22
Ln (u)21
L
(u)
n
22
22
Ln (u)12
21
Ln (u)12
22
Ln (u)22
11 
α
The arrows represent the direction of the flow of time, in other words, the order in which
the operators act. So with our convention we ‘read’ the diagrams left to right and top to
bottom.
16
In diagrammatic form we shall often omit the indices denoting components, but with
an understanding that an unlabelled edge stands for all allowed states of that edge:


Lαn (u) :=
1



= 


1
1 2
22
1






2


= 
Ln (u)11
Ln (u)21
Ln (u)12
Ln (u)22


 .
α
α
The Lαn (u)-matrix is an important operator involving the local quantum (physical)
space Hn and the auxiliary space Vα , that is, Lαn (u) : Vα ⊗Hn → Vα ⊗Hn . So Lαn (u) acts
in Vα with its entries being operators on Hn . These entries are functions of a parameter u
called the rapidity. In the case of the six-vertex model, the ice rule and invariance under
1 ↔ 2 reversal implies that the Boltzmann weights are as follows:
22
L(u)11
11 = L(u)22 = a(u),
21
L(u)12
12 = L(u)21 = b(u),
12
L(u)21
12 = L(u)21 = c(u),
and when represented diagrammatically L has the following block structure:

1


0
0
0
1
1



1
2
1



1
1
2
1
0
0



2
2
= 
Lαn (u) =


2
1


0
0

1
2
2
2



1
1
2



2

0
0
0
2


0
 a(u)

 0
b(u)

=
 0
c(u)


0
0
17
0
0
c(u)
0
b(u)
0
0
a(u)








α
























2 

α
All Lαn (u) are equivalent, but with their entries acting in different spaces Hn , where
n ∈ {1, . . . , N } represents which site of the row the L-operator acts on
N −n times
n−1 times
}|
{
z }| {
z
Lαn (u) =I ⊗ · · · ⊗ I⊗ Lα (u)⊗ I ⊗ · · · ⊗ I .
2.1.2
Transfer matrix T (u)
The transfer matrix T plays a key role in ABA. It is an operator that propagates a given
configuration from one row of the lattice to the next. T encodes a Boltzmann weight of
configuration on N sites with fixed states of vertical edges. Fixing sets j = {j1 , j2 , . . . , jN }
0 }, T is defined as follows:
and j0 = {j10 , j20 , . . . , jN
0
T (u)jj ≡
X
···
X
i1
0
iN −1 jN
−1
jN −1
i j0
LN (u)i1NjNN LN −1 (u)iN
i j0
i j0
· · · L2 (u)i23 j22 L1 (u)i12 j11 .
iN
Diagrammatically this is
j10
i1
j20
i2
j1
0
0
jN
−1 jN
···
i3
j2
iN
i1
jN −1 jN
Every entry of T corresponds to a particular choice of spins on the vertical edges. In the
case of the six-vertex model, there are two possible values of spins, so T has dimension
2N × 2N . Note that the transfer matrix acts only on the physical space H.
If we know the transfer matrix for our model, then we can find the partition function
by taking the trace of M products of the same T
Z=
X
···
j1
Z = Tr T
X
Tjj12 Tjj23 . . . TjjMM−1 TjjM1
jM
M
,
where the products correspond to linking the rows and taking the final trace then closes
the top and bottom lines with periodic boundary conditions.
The strategy at this point is not to diagonalise the transfer matrix directly, but instead
to look for conditions, under which two transfer matrices with different values of spectral
parameter u commute, that is, to diagonalise a whole family T (u) simultaneously for every
u. Although this seems like an even harder problem than the one we started with, it turns
out that this approach brings out the underlying algebraic structure (due to integrability)
18
which helps to find the eigenvectors that are shared by all T (u). We can then compute the
eigenvalues for each vector for all values of parameter u. In the end, that is our main goal.
In particular, we are interested in the largest eigenvalue Λmax as, in the thermodynamic
limit as M → ∞, it gives the partition function Z,
#
"
M M
Λ
Λ
3
2
+
+ ... .
Z = ΛM
max 1 +
Λmax
Λmax
2.1.3
Monodromy matrix Tα (u)
We introduce an additional object, the monodromy matrix T , which is the operator that
propagates one open horizontal row to the next, but without imposing periodic boundary
conditions. Thus T possesses an additional degree of freedom compared to T , corresponding to the states at both ends of the horizontal line. Hence T is a 2N +1 × 2N +1 matrix
defined by the usual matrix product of Ls in Vα
N
Tα (u) = LαN (u) · · · Lα1 (u) ≡
←−
Y
Lαn (u)
n=1
and diagrammatically as:
j10
Tα (u)
i1
=
0
0
jN
−1 jN
j20
i2
j1
···
i3
iN
j2
iN +1
jN −1 jN
(Note, that while the diagrams are read left to right, the corresponding operators in the
equations are ordered from right to left.)
c2 ) model and two-dimensional representation of the auxiliary
In the case of a Uq (sl
space, the monodromy matrix is represented as:


Tα (u)
=

1






1

...
...
1
2
2
2
...
...

1






2

α

=
A(u)
B(u)
C(u)
D(u)
19

 .
α
Tracing over the auxiliary space amounts to requiring the in and out state to coincide,
thus reproducing the transfer matrix
Trα Tα (u) = Trα (LαN (u) · · · Lα1 (u))
= A(u) + D(u)
= T (u).
The operators A, B, C and D are 2N × 2N matrices acting in the full quantum space H.
They are complicated in general, but we will not need their exact form. As we shall see,
the ABA construction only uses the commutation relations that they obey. Their role is
to encode the effect of the interaction of the auxiliary spin with the physical spins. We
can think of B and C as spin creation/annihilation operators respectively. Taking the
trace amounts to closing the system and requiring that the auxiliary spin emerges from
the interaction in the same state as it entered.
We return to our task of finding a condition under which two transfer matrices T ≡
T (u) and T 0 ≡ T (v) commute
0
(T T 0 )jj =
X
0
00
Tjj Tjj00
j00
=
N
X Y
X
0
in+1 i∗n+1 jn
)jn ,
i∗n
(Sin
i1 ,...,iN i∗1 ,...,i∗N n=1
where S is the double-row L-matrix, defined as follows:
0 ∗0
0
(Sii ii∗ )jj ≡
X
∗0 0
0 00
Lii jj L0 ii∗ j 00j .
(2.2)
j 00
Hence we can write
0
j0
j0
j0
(T T 0 )jj = Tr Sj11 Sj22 . . . SjNN
(2.3)
and similarly
0
j0
j0
j0
(T 0 T )jj = Tr S 0 j11 S 0 j22 . . . S 0 jN
N
and ask for when do these two expressions commute, i.e. when
T T 0 = T 0T .
20
(2.4)
2.1.4
R-matrix
T (u) and T (v) will certainly commute if there exists a non-singular matrix R such that
j0
0
Sjj = R S 0 j R−1 .
(2.5)
0
This is easily observed by substituting the right hand side of (2.5) into (2.3) for every Sjj :
j0
j0
j0
0
R−1 ),
(T T 0 )jj = Tr(R S 0 j11 S 0 j22 . . . S 0 jN
N
and due to the cyclic property of trace, we obtain (2.4) as claimed:
j0
j0
j0
= Tr(S 0 j11 S 0 j22 . . . S 0 jN
R−1 R)
N
0
= (T 0 T )jj .
Condition (2.5) is called the local intertwining relation (or RLL relation) and can be
rewritten as
Rαβ (u, v)Lαn (u)Lβn (v) = Lβn (v)Lαn (u)Rαβ (u, v)
(2.6)
and more explicitly, in component notation
X
Rii
00 j 00
j
0
0
(u, v)Lii00 (u)Ljj 00 (v) =
i00 j 00
X
00
00
0 0
Ljj (v)Lii (u)Rii00jj 00 (u, v).
(2.7)
i00 j 00
The subscripts α and β denote two different vector spaces Vα and Vβ that the Ln act on
at the nth lattice site. The relation can be seen more clearly in its diagrammatic form:
v
j0
j 00
u
i0
i00
i
v
j0
i00
u
i0
j 00
i
=
j
n
j
n
The R-matrix is said to intertwine the Lαn and Lβn , as it only acts on the auxiliary
spaces associated with both of these, and as an identity on the physical (vertical) space.
Rαβ (u, v) depends on two rapidities, u and v, and is an element of End(Vα ⊗ Vβ ), with
V α ' Cn .
c2 ) models, n = 2 and R is a 4 × 4 matrix. In its component notation, it is
For Uq (sl
easy to see that the intertwining relation gives rise to 24 commutation relations involving
elements of Lα (as given in Section (2.1.1), where each equation corresponds to a fixed
set of external indices {i, i0 , j, j 0 }. The internal indices {i00 , j 00 } are being summed over on
both sides, contributing 2 × 22 terms to each equation.
21
2.1.5
RT T relation and the Yang–Baxter equation
The monodromy matrix Tα satisfies the global intertwining relation:
Rαβ (u, v)Tα (u)Tβ (v) = Tβ (v)Tα (u)Rαβ (u, v)
(2.8)
which is a generalisation of the RLL relation, with R intertwining two auxiliary spins over
all sites in the lattice.
v
v
=
u
u
1
N
1
N
This result can be derived as a consequence of its local version given in (2.6). To see this,
we write

Rαβ (u, v)Tα (u)Tβ (v) = Rαβ (u, v) 
N
←−
Y

Lαn (u) 
n=1
N
←−
Y

Lβn (v)
n=1
and since Ls acting on different spaces commute, we write
N
= Rαβ (u, v)
←
−
Y
(Lαn (u)Lβn (v))
n=1
and now applying the RLL equation N times successively, proves (2.8):
N
=
←
−
Y
(Lβn (v)Lαn (u)) Rαβ (u, v)
n=1
N

=
←−
Y

Lβn (v) 
n=1
N
←−
Y

Lαn (u) Rαβ (u, v)
n=1
= Tβ (v)Tα (u)Rαβ (u, v).
Taking the trace over α of (2.8) and recalling that Trα Tα (u) = T (u) we find
[T (u), T (v)] = 0.
Not any R(u, v) can be an R-matrix for some integrable model. The sufficient condition
for it to be the case is the Yang–Baxter equation (YBE). It is an equation in R-matrices
alone and is the fundamental equation defining an integrable model. We derive it here,
with the assumption that matrix R(u, v) satisfies RT T relation (2.8).
22
Consider a tensor product of the three monodromy matrices Tα (u)Tβ (v)Tγ (w) acting
on Vα ⊗ Vβ ⊗ Vγ . Also, Rαβ (u, v) in understood to be (Rαβ (u, v) ⊗ I), that is, acting
non-trivially in Vα ⊗ Vβ , and as an identity in Vγ , hence we can write Rαβ Tγ = Tγ Rαβ .
Then, in the light of (2.8), Tα (u)Tβ (v)Tγ (w) can be reduced to Tγ (w)Tβ (v)Tα (u) in two
ways:
Tα (u)Tβ (v)Tγ (w) =
−1
−1
−1
= Rβγ
(v, w)Rαγ
(u, w)Rαβ
(u, v)Tγ (w)Tβ (v)Tα (u)Rαβ (u, v)Rαγ (u, w)Rβγ (v, w),
but equivalently, we can write
−1
−1
−1
= Rαβ
(u, v)Rαγ
(u, w)Rβγ
(v, w)Tγ (w)Tβ (v)Tα (u)Rβγ (v, w)Rαγ (u, w)Rαβ (u, v).
Clearly, the right hand sides of the equalities above are not equal for arbitrary matrices
R.
A sufficient condition for them to be so is
Rαβ (u, v)Rαγ (u, w)Rβγ (v, w) = Rβγ (v, w)Rαγ (u, w)Rαβ (u, v)
(2.9)
or more explicitly, in component notation:
Rii
00 j 00
j
0 0
0 00
(u, v)Rii00kk (u, w)Rjj 00kk00 (v, w) = Rjj
00 k 00
k
(v, w)Rii
00 k 0
k00
0 0
(u, w)Rii00jj 00 (u, v),
(2.10)
where the summation over repeated indices {i00 , j 00 , k 00 } is implied. Diagrammatically, it
can be represented as
k0
v
k0
j 00
j0
i
k 00
u
j0
i00
i
k 00
=
i00
i0
v
j
u
k
w
i0
j
j 00
k
w
This is called the Yang–Baxter equation. By leaving the state of external edges {i, j, k, i0 , j 0 , k 0 }
unspecified, we can write (2.9) as a single equation, representing 26 equations (in the case of
c2 ) in the entries of the R-matrix, with {i, j, k, i0 , j 0 , k 0 } fixed, and {i00 , j 00 , k 00 } summed
Uq (sl
over). So the YBE is a very strong restriction on a, b and c as scalar functionss in the
rapidities u, v and w.
For example, consider an external configuration fixed by {i, j, k, i0 , j 0 , k 0 } = {2, 1, 2, 2, 2, 1}.
Summation over internal configurations yields the non-linear equation
12
21
21
21
22
21
21
21
22
R21
(u, v)R12
(u, w)R21
(v, w) + R21
(u, v)R22
(u, w)R12
(v, w) = R12
(v, w)R21
(u, w)R22
(u, v)
23
giving the following identity in the weight functions:
c(u, v)c(u, w)b(v, w) + b(u, v)a(u, w)c(v, w) = c(v, w)b(u, w)a(u, v).
(2.11)
Integrable models having the same R-matrices, but different L-operators (corresponding to different physical spaces) are closely related and can be derived from different
d
representations of the quantum affine algebra Uq (sl
2 ). The YBE is an analogue of the
Jacobi identity, and the R-matrix is analogous to the set of structure constants.
There are many solutions to the YBE. In principle, one could take a general n2 × n2
R matrix and look for solutions. It turns out that the first non-trivial solution occurs for
n2 = 4, and YBE is equivalent to a system of 64 equations in 16 unknowns. Classifying
all the solutions to the YBE is not easy. Usually, at this stage an Ansatz for the R-matrix
is made to see whether it solves the YBE and under which conditions.
As a side remark, we mention a fundamental example of an L-matrix. Given that R is
a solution of the YBE, it is natural to look for a candidate for the L-matrix of the same
form as R. It is due to the fact that equations (2.6) and (2.9) are really just different forms
of the Yang–Baxter equation and the ice rule applies to both. The simplest choice of an Lmatrix satisfying RLL relation is L(u) = R(u − u0 ), where u0 is some fixed inhomogeneity
parameter.
2.2
Algebraic Bethe Ansatz
The algebraic Bethe Ansatz method supposes that we have solved the YBE for a given
model and know the R-matrix. Once the solution is found, it specifies the L-operator
and thus the monodromy and the transfer matrix. Hence we have a family of commuting
transfer matrices T (u), that share the eigenvectors with the Hamiltonian H. In the
next step we proceed to diagonalise these simultaneously. The goal is to construct the
eigenfunctions of the Hamiltonian via the creation and annihilation operators, (B(u) and
C(u)) which act on some reference state.
Although the Bethe Ansatz approach described in this section is not universal, it can
been modified and extended as has been done for the XY Z spin-chain in a famous paper
of Takhtadzhan and Faddeev [56]. There is also a generalisation of algebraic Bethe Ansatz
d
method for models with Uq (sl
n ) symmetry for all n ≥ 3, called the Nested Bethe Ansatz;
d
the Uq (sl
3 ), case is the subject of the next chapter.
The Bethe Ansatz described in this section works for the 6-vertex model (or XXZ/XXX
spin chains). Thus, for the remainder of the chapter, the R-matrix under consideration is
24
the six-vertex model R-matrix obeying the ice rule, given by:

0
0
0
 a(u, v)


0
b(u, v)
c(u, v)
0

Rαβ (u, v) = 

0
c(u, v)
b(u, v)
0


0
0
0
a(u, v)









αβ
d
For the XXZ spin-chain (Uq (sl
2 ) R-matrix in spin 1/2 representation), the parametrisation of weights for the R-matrix is:
a(u, v) = sinh(u − v + φ),
b(u, v) = sinh(u − v),
c(u, v) = sinh(φ),
(2.12)
where φ is a constant. The Heisenberg chain XXX (or Y(sl2 ) spin 1/2) weights are given
by
a(u, v) = u − v + φ,
b(u, v) = u − v,
c(u, v) = φ.
(2.13)
The RXXX above is one of the simplest solutions of the Yang–Baxter equation.
The reader is referred to [8] for a review of Quantum Groups and more particularly,
to Section 5 of Chapter 12 for the construction of rational and trigonometric solutions
of the quantum Y–B equation from the finite-dimensional representations of Yangians
cn ) respectively. (Yangians may be viewed as
Y(sln ) and Quantum Affine Algebras Uq (sl
‘degenerate’ versions of the quantum affine algebras).
2.2.1
Pseudo-vacuum state: |Ωi
The diagonalisation of T (u) relies on the RT T relation, some simple properties of the
L-operators and, importantly, on the existence of a highest weight vector with respect to
c2 ) representations in the configuration space H = QN N Hn .
the tensor product of Uq (sl
n=1
We call |Ωi the pseudo-vacuum and require it to be an eigenstate of the transfer matrix
T (u) as well as to be annihilated by the operators C(u):
C(u) |Ωi = 0.
To find this state, we observe that in each Hn there exists a local state vector |1in such
that the action of Lαn on it is upper-triangular in the auxiliary space. This is easily
observed diagrammatically, and is shown below for the L-matrix of the six-vertex model:
25

Lαn (u) |1in


1








1


Ln (u)21 |1in 
 =

2
Ln (u)2 |1in
 Ln (u)11 |1in
= 

Ln (u)12 |1in
1
α
1
1
2
1
1
2
2



1








2
α

1
 Ln (u)1 |1in
=
0
where
Ln (u)12 |1in

Ln (u)21 |1in

 ,
Ln (u)22 |1in
α
= 0, since this block’s vertex configurations contradict the ice rule (see
Figure 2.1). Explicitly:
!
0 c(u)
Ln (u)12 |1in =
0
0
!
1
= 0.
0
n
n
Next we also observe that this local vacuum state is the eigenvector of Ln (u)11 and Ln (u)22
!
!
1
a(u)
0
Ln (u)11 |1in =
= a(u) |1in
0
b(u)
0
n
n
and
Ln (u)22 |1in =
b(u)
0
0
a(u)
!
1
!
0
n
= b(u) |1in .
n
We summarise the action of the L-operator on the local vacuum state:


Ln (u)21 |1in
 a(u) |1in
L(u)αn |1in = 

0
b(u) |1in
where we will not need to find the exact form of the
Ln (u)21 |1in



α
block.
We expect there to be a vector in H on which the monodromy matrix T has the same
action as L has on |1i, making it upper triangular, i.e., to have C(u) |Ωi = 0. Define
therefore a global vacuum state:
|Ωi = |1i1 ⊗ · · · ⊗ |1iN
Indeed, we find that the action of T (u) on |Ωi is upper triangular, and |Ωi is the eigenvector
of the operators A(u) and D(u). This is a direct consequence of the upper-triangular action
of the L operators:

T (u) |Ωi
=
N
←−
Y


Lαn (u) |1i1 ⊗ · · · ⊗ |1iN
n=1
26


=


LN (u)11 |1iN
LN (u)21 |1iN
LN (u)12 |1iN
LN (u)22 |1iN


 a(u) |1iN

=


0



 × ··· × 


α

LN (u)21 |1iN
b(u) |1iN

L1 (u)11 |1i1
L1 (u)21 |1i1
L1 (u)12 |1i1
L1 (u)22 |1i1




α




 a(u) |1i1


 × ··· × 




0
L1 (u)21 |1i1
b(u) |1i1
α



 ,


α
where ‘×’ stands for the usual matrix product in auxiliary space α and reverse tensor
product in physical Hilbert spaces Hn


 N
 a (u) |Ωi

=


0
∗
bN (u) |Ωi






α
Hence |Ωi is an eigenvector of T (u) with the eigenvalue aN (u) + bN (u):
T (u) |Ωi = Trα T (u) |Ωi = [A(u) + D(u)] |Ωi
= (aN (u) + bN (u)) |Ωi .
Observe that B(u) and C(u) play the role of global creation and annihilation operators:
B(u) |Ωi =
6 0,
2.2.2
C(u) |Ωi = 0.
Commutation relations for the operators A, B and D
The global intertwining relation RT T (2.8) puts severe restrictions on the operators A, B
and D:
Rαβ (u, v)Tα (u)Tβ (v) = Tβ (v)Tα (u)Rαβ (u, v)
It generates 16 equations, which are given explicitly by the following matrix multiplication:

a(u, v)
0
0
0


0
b(u, v) c(u, v)
0



0
c(u, v) b(u, v)
0

0
0
0
a(u, v)

A(u)A(v) A(u)B(v) B(u)A(v) B(u)B(v)

  A(u)C(v) A(u)D(v) B(u)C(v) B(u)D(v)


  C(u)A(v) C(u)B(v) D(u)A(v) D(u)B(v)

C(u)C(v) C(u)D(v) D(u)C(v) D(u)D(v)
27








A(v)A(u) B(v)A(u)

 C(v)A(u)

=
 A(v)C(u)

C(v)C(u)
A(v)B(u)
B(v)B(u)

a(u, v)


D(v)B(u) 



A(v)D(u) B(v)D(u) 

D(v)A(u)
C(v)B(u)
B(v)C(u)
D(v)C(u)
C(v)D(u)
D(v)D(u)
0
0
0
0
b(u, v) c(u, v)
0
0
c(u, v) b(u, v)
0
0
0
0
a(u, v)




.


Of all these, three equations will be of particular use to us, being the defining generalised commutation relations for the operators A, B and D:
B(u)B(v) = B(v)B(u),
a(u, v)B(u)A(v) = c(u, v)B(v)A(u) + b(u, v)A(v)B(u)
c(u, v)B(u)D(v) + b(u, v)D(u)B(v) = a(u, v)B(v)D(u)
which we rewrite below in a more convenient form (exchanging u ↔ v in the second
relation) and include their diagrammatic representations:
B(u)B(v) = B(v)B(u)
v
2
u
2
1
···
A(u)B(v) =
v
u
2
1
=
1
u
2
v
2
a(v,u)
b(v,u) ×
=
1
− c(v,u)
b(v,u) ×
D(u)B(v) =
v
u
2
2
···
1
u
1
v
2
v
1
u
2
(2.15)
1
···
a(u,v)
b(u,v) ×
=
2
− c(u,v)
b(u,v) ×
28
u
2
v
2
v
2
u
2
1
1
···
a(u, v)
c(u, v)
B(v)D(u) −
B(u)D(v)
b(u, v)
b(u, v)
1
···
1
a(v, u)
c(v, u)
B(v)A(u) −
B(u)A(v)
b(v, u)
b(v, u)
1
···
(2.14)
1
(2.16)
2
···
1
2
···
1
2.2.3
ABA Solution: Eigenvectors of T (u) and Bethe equations
We want to know the eigenfunctions of T (u) and begin with the construction of its eigenvectors. So we look for vectors |Ψi ∈ H, such that:
T (u) |Ψi = [A(u) + D(u)] |Ψi = Λ(u) |Ψi
(2.17)
To achieve this, we make an educated guess for the form of the eigenvectors, or an Ansatz
(name borrowed from the German language) in the form of the product of B(vm ) acting
on the pseudo-vacuum and derive conditions under which (2.17) is satisfied.
Proposition 1. Given that R is a solution of the YBE in (2.9), whose weights b and c
obey the symmetry and antisymmetry properties respectively:
c(u, v) = c(v, u)
and
b(u, v) = −b(v, u),
(2.18)
then the global state vectors
|Ψi =
M
Y
B(vm ) |Ωi
m=1
are the eigenvectors of the equations (2.17), where the eigenvalues Λ(u) are given by
Λ(u) = Λ(u; v1 , . . . , vM ) = aN (u)
M
M
Y
Y
a(u, vm )
a(vm , u)
+ bN (u)
,
b(vm , u)
b(u, vm )
(2.19)
m=1
m=1
provided that the following constraints on the rapidities {vi , . . . vM } are satisfied:
N
a (vi )
M
Y
M N
a(vm , vi ) + (−1) b (vi )
m=1
m6=i
M
Y
a(vi , vm ) = 0.
(2.20)
m=1
m6=i
The constraints in (2.20) above are a system of coupled non-linear equations called the
Bethe equations.
Proof. Acting on the vector |Ψi with the transfer matrix T (u), we obtain
T (u) |Ψi = [A(u) + D(u)] |Ψi = A(u)
M
Y
B(vm ) |Ωi + D(u)
m=1
M
Y
B(vm ) |Ωi
(2.21)
m=1
where due to the first commutation relation (2.14) the product of Bs is unordered. The
first step is to calculate the two terms on the right hand side of (2.21). We begin with the
first term and make a proposition PM , to be proved by induction for general M ≥ 1:
M
M Y
Y
a(vm , u)
B(vm ) |Ωi
(2.22)
A(u)
B(vm ) |Ωi =aN (u)
b(vm , u)
m=1
m=1
M
M X
Y
c(vi , u)
a(vm , vi )
N
−
a (vi )
B(u)
B(vm ) |Ωi .
b(vi , u)
b(vm , vi )
m=1
i=1
m6=i
29
Recalling that A(u) |Ωi = aN (u) |Ωi and the second commutation relation (2.15) with
v → v1 , establishes that P1 is true:
c(v1 , u)
a(v1 , u)
B(v1 )A(u) −
B(u)A(v1 )
b(v1 , u)
b(v1 , u)
a(v1 , u)
c(v1 , u)
=⇒ A(u)B(v1 ) |Ωi = aN (u)
B(v1 ) |Ωi − aN (v1 )
B(u) |Ωi .
b(v1 , u)
b(v1 , u)
Q
Now consider A(u) kj=1 B(vj ) |Ωi for 1 ≤ k ≤ M . Applying the same commutation
A(u)B(v1 ) =
relation once, gives:
A(u)
k
Y
B(vj ) |Ωi =
k−1
k−1
Y
Y
c(vk , u)
a(vk , u)
B(vk )A(u)
B(u)A(vk )
B(vm ) |Ωi −
B(vm ) |Ωi
b(vk , u)
b(vk , u)
m=1
j=1
m=1
(2.23)
Suppose Pk−1 is true for some integer k, 1 ≤ k ≤ M . Then substituting Pk−1 into the
right hand side of the equation above, obtain:
A(u)
k
Y
j=1
k−1
Y a(vm , u)
a(vk , u)
N
B(vk )a (u)
B(vm ) |Ωi
B(vj ) |Ωi =
b(vk , u)
b(vm , u)
m=1
a(vk , u)
−
B(vk )
b(vk , u)
k−1
X
i=1
k−1
Y a(vm , vi )
c(vi , u)
B(u)
B(vm ) |Ωi
a (vi )
b(vi , u)
b(vm , vi )
m=1
N
m6=i
k−1
Y
a(vm , vk )
B(vm ) |Ωi
b(vm , vk )
m=1
k−1
k−1
X
Y a(vm , vi )
c(vk , u)
c(vi , vk )
N
+
B(u)
a (vi )
B(vk )
B(vm ) |Ωi .
b(vk , u)
b(vi , vk )
b(vm , vi )
m=1
c(vk , u)
B(u)aN (vk )
−
b(vk , u)
i=1
m6=i
Simplifying the first and the third terms, and collecting the second and the fourth, gives:
A(u)
k
Y
j=1
k Y
a(vm , u)
B(vm ) |Ωi
B(vj ) |Ωi =a (u)
b(vm , u)
N
(2.24)
m=1
k−1
Y a(vm , vk )
c(vk , u)
− a (vk )
B(u)
B(vm ) |Ωi
b(vk , u)
b(vm , vk )
m=1
k−1
k
k−1
X
Y a(vm , vi ) Y
a(vk , u) c(vi , u) c(vk , u) c(vi , vk )
B(vm ) |Ωi .
−
aN (vi )
−
B(u)
b(vk , u) b(vi , u) b(vk , u) b(vi , vk )
b(vm , vi ) m=1
m=1
N
i=1
m6=i
m6=i
(2.25)
Recall the scalar equation in the weight functions (2.11) coming from the Y-B equation,
and substituting u → vk , v → vi and w → u, we find that
a(vk , u)c(vi , u) c(vk , u)c(vi , vk )
a(vk , vi )c(vi , u)
−
=
.
b(vk , u)b(vi , u) b(vk , u)b(vi , vk )
b(vk , vi )b(vi , u)
30
(2.26)
where we have used the properties of b and c weights given in (2.18). Hence we can simplify
the last term in (2.22) by bringing
A(u)
k
Y
j=1
a(vk ,vi )
b(vk ,vi )
inside the product, yielding
k Y
a(vm , u)
B(vm ) |Ωi
B(vj ) |Ωi =a (u)
b(vm , u)
N
m=1
k−1
Y a(vm , vk )
c(vk , u)
− a (vk )
B(u)
B(vm ) |Ωi
b(vk , u)
b(vm , vk )
m=1
k−1
k X
Y
c(vi , u)
a(vm , vi )
N
−
B(u)
B(vm ) |Ωi .
a (vi )
b(vi , u)
b(vm , vi )
m=1
N
i=1
m6=i
Noticing that the second term in the expression can be taken into the sum as i = k term,
proves Pk :
A(u)
k
Y
j=1
k Y
a(vm , u)
B(vm ) |Ωi
B(vj ) |Ωi =a (u)
b(vm , u)
N
m=1
−
k
X
i=1
k Y
a(vm , vi )
c(vi , u)
B(u)
B(vm ) |Ωi .
a (vi )
b(vi , u)
b(vm , vi )
m=1
N
m6=i
Therefore, by induction it is true that PM is true for every integer M ≥ 1.
By analogous arguments, relying on the third commutation relation (2.16) we are also
able to show that:
D(u)
M
Y
j=1
M Y
a(u, vm )
B(vm ) |Ωi
B(vj ) |Ωi =b (u)
b(u, vm )
N
(2.27)
m=1
−
M
X
i=1
M Y
c(u, vi )
a(vi , vm )
b (vi )
B(u)
B(vm ) |Ωi .
b(u, vi )
b(vi , vm )
m=1
N
m6=i
Adding equations (2.22 and 2.27), we observe that the sum of the two leading terms is
precisely the eigenvalue Λ(u) as stated in (2.19). Moreover, |Ψi is an eigenvector of T (u) if
and only if the sub-leading terms cancel via satisfying the following system of M coupled
equations:
M
M
c(u, vi ) Y a(vi , vm )
c(vi , u) Y a(vm , vi )
N
+ b (vi )
= 0,
a (vi )
b(vi , u) m=1 b(vm , vi )
b(u, vi ) m=1 b(vi , vm )
N
m6=i
for 1 ≤ i ≤ M (2.28)
m6=i
Again, using the symmetry and antisymmetry of c and b, we can cancel the common
factors in the expressions above and obtain the Bethe Equations as given in (2.20):
aN (vi )
M
Y
a(vm , vi ) + (−1)M bN (vi )
m=1
m6=i
M
Y
m=1
m6=i
31
a(vi , vm ) = 0.
(2.29)
Thus, the ABA produces eigenstates of the transfer matrix by action of a product
of operators B(vi ) on a reference eigenstate (pseudo-vacuum), provided their arguments
satisfy certain equations (Bethe equations). In general it is expected that considering all
solutions of the Bethe equations leads to all eigenstates of the transfer matrix, so that the
diagonalisation problem is completely solved [3].
32
Chapter 3
Algebraic Bethe Ansatz for the
Fifteen-vertex Model
Introduction
A fundamental step in solving integrable models is finding the eigenvectors and eigenvalues
of a transfer matrix. In the previous chapter this was done for the six-vertex model using
the algebraic Bethe Ansatz (ABA). The six-vertex R-matrix is a solution of the Yangc2 ) symmetry. Here we will consider higher rank
Baxter equation with underlying Uq (sl
solutions.
The nested Bethe Ansatz (NBA) is a method for diagonalisation of the transfer matrices that expresses higher rank solutions in terms of lower ones by successive application
of ABA. The key ingredient of the technique is the existence of a highest weight vector,
which we previously called the pseudo-vacuum. (Knowing a pseudo-vacuum for the model
is equivalent to knowing a highest weight vector for the representation of the algebra which
underlies the model.)
The NBA was introduced by Kulish and Reshetikhin [46], who diagonalised the transfer
matrices for systems of interacting gl(n) spins with arbitrary highest weight. They reduced
the problem of diagonalisation of the gl(n)-invariant transfer matrix to the same problem
for the matrix with gl(n − 1) symmetry. The reader is also referred to a paper by Schultz
cn )-invariant R-matrix.
[50], for an explicit derivation of results for models with Uq (sl
In this chapter we present an overview of the NBA technique for integrable systems
c3 ). In the spirit of NBA our objective
associated with representations of the algebra Uq (sl
c3 ) problem to one with a Uq (sl
c2 )-invariant R matrix. The
is the reduction of an Uq (sl
solution to the latter is known and was the subject of the previous chapter.
33
We give an explicit derivation of the eigenvectors and eigenvalues, and of the Bethe
equations for this class of models. Our exposition closely follows that of Wheeler [59],
Section 3, which presents a clear and concise summary of the method.
Throughout this chapter we will be referring back to some concepts defined or derived
in the previous chapter and as such, rely on the reader’s familiarity with it.
3.1
Integrable fifteen-vertex model
c2 ) models, equip the nth lattice site
Consider a M × N lattice and, analogously to Uq (sl
with a Hilbert space Hn ' C3 . The complete space of lattice states is the tensor product
N
of those at each site: H = N
n=1 Hn . As before, the auxiliary spaces are labelled by Vα ,
Vβ ' C3 . Vertex weights are assigned with the local L-matrix:
X
0 0
0
0
L(u) =
L(u)ii jj Eii ⊗ Ejj ,
i,i0 j,j 0
0
where Eii is an elementary matrix with 1 in ith row and i0 th column and zeros elsewhere.
c3 ) local L-operator can be viewed as a 9×9 matrix. Diagrammatically its entries
The Uq (sl
can be represented by a vertex, where a horizontal line is associated with an auxiliary
vector space and a vertical line with a quantum space:
j0
0 0
Lαn (u)ii jj =
α i0

i
=
j
n
Ln (u)11 Ln (u)21 Ln (u)31


 Ln (u)12 Ln (u)22 Ln (u)32

Ln (u)13 Ln (u)23 Ln (u)33



 .

α
Lαn (u) acts in Vα ⊗ Hn and the indices now take the values i, i0 , j, j 0 ∈ {1, 2, 3}.
The monodromy matrix Tα (u) acts in Vα with its entries acting in H. It is defined by
the following ordered product of L-matrices:


1 T (u)2 T (u)3
N
T
(u)
1
1
1
←
−
Y


1
2
3 
Tα (u) =
Lαn (u) = 
 T (u)2 T (u)2 T (u)2  .
n=1
T (u)13 T (u)23 T (u)33
α
3.2
First step in the method of nested Bethe Ansatz
c3 )
The procedure of the nested Bethe Ansatz method is based on the reduction of the Uq (sl
c2 ). It comprises of two steps and the two ‘levels’ of
eigenvector problem to one for Uq (sl
34
the operators, which are characterised by the dimension of the spaces in which they act.
To distinguish between analogous operators at different levels, the superscripts (1) and
(2) are used. We begin with the decomposition of the monodromy matrix as an operator
(a)
on (Vα
(b)
(a)
(b)
' C and Vα ' C2 are the auxiliary spaces.
!
(1)
A(1) (u) Bα (u)
,
(1)
(1)
Cα (u) Dα (u)
⊕ Vα ) ⊗ H = Vα ⊗ H, where Vα
Tα(1) (u) =
where we defined
A(1) (u) = T (u)11 ,
,
Bα(1) (u) = T (u)21 T (u)31
α
!
T (u)12
,
Cα(1) (u) =
T (u)13
α
!
2
3
T
(u)
T
(u)
2
2
Dα(1) (u) =
,
T (u)23 T (u)33
A(1) (u) : Vα(a) ⊗ H → Vα(a) ⊗ H,
Bα(1) (u) : Vα(b) ⊗ H → Vα(a) ⊗ H,
Cα(1) (u) : Vα(a) ⊗ H → Vα(b) ⊗ H,
Dα(1) (u) : Vα(b) ⊗ H → Vα(b) ⊗ H.
α
The transfer matrix is defined as before, by a trace over auxiliary space of the monodromy matrix:
T (u) = Trα Tα(1) (u) =
3
X
T (u)ii .
i=1
Then the partition function on a lattice of size M × N can be written as
Z = TrH T (u)M .
Again, we seek to diagonalise the family T (u) simultaneously for all u. We look for
conditions under which the transfer matrices commute. It will be the case if there exists
an R-matrix, such that the local operators satisfy the intertwining relation:
(1)
(1)
Rαβ (u, v)(Lα (u) ⊗ L(v)β ) = (Lβ (v) ⊗ Lα (u))Rαβ (u, v)
and moreover, if the R-matrix satisfies the Yang-Baxter equation
(1)
(1)
(1)
(1)
(1)
(1)
Rαβ (u, v)Rαγ
(u, w)Rβγ (v, w) = Rβγ (v, w)Rαγ
(u, w)Rαβ (u, v).
(3.1)
A simple solution to the Y–B equation is an R-matrix whose weights obey a generalised
version of the ice rule, which states that the number of 1s, 2s and 3s on the left and bottom
edges of a given vertex must equal the number of 1s, 2s and 3s on the top and right edges
ii , Rij and Rji . In
of the same vertex. That is, the only non-zero entries are of the form Rii
ij
ij
particular, we let
11
22
33
R11
= R22
= R33
= a(u, v),
21
31
32
R12
=R13
= R23
= c+ (u, v),
12
13
23
R21
= R31
= R32
= c− (u, v),
12
13
21
23
31
32
R12
= R13
= R21
=R23
= R31
= R32
= b(u, v).
35
Therefore R(1) is given by

a(u, v) 0
0

 0 b(u, v)
0


 0
0
b(u, v)


 0 c (u, v) 0
−


(1)
Rαβ (u, v) = 
0
0
 0

 0
0
0


 0
0 c− (u, v)


 0
0
0

0
0
0
0
0
0
0
0
0

c+ (u, v)
0
0
0
0
0
0
0
0
c+ (u, v)
0
0
b(u, v)
0
0
0
0
0
0
a(u, v)
0
0
0
0
0
0
b(u, v)
0
c+ (u, v)
0
0
0
0
b(u, v)
0
0
0
0
c− (u, v)
0
b(u, v)
0





















0
0
0
0
0
a(u, v)
(a)
(b)
(a)
αβ
(b)
and it is an element of End(Vα ⊗Vβ ), and Vα ⊗Vβ = (Vα ⊕Vα )⊗(Vβ ⊕Vβ ). Solutions
c3 ) type are known and are given by
for the R-matrices of Y(sl3 ) and Uq (sl
a(u, v) = sinh(u − v + φ),
b(u, v) = sinh(u − v),
c± (u, v) = e±(u−v) sinh(φ)
c3 ), and by
in the case of Uq (sl
a(u, v) = u − v + φ,
b(u, v) = u − v,
c(u, v) = φ
in the case of Y(sl3 ). (As in chapter 2, the reader is referred to [8] for a review of
the construction of rational and trigonometric solutions of the quantum Y–B equation
equation from the finite-dimensional representations of Yangians Y(sl3 ) and Quantum
c3 ) respectively.)
Affine algebras Uq (sl
[ problem to an Uq (sl(2))
[ one, we will use
In the process of reduction of the Uq (sl(3))
(2)
the Rαβ (u)-matrix, familiar to us from the previous chapter.


0
0
a(u,
v)
0




 0
b(u, v)
c+ (u, v)
0 


(2)
Rαβ (u, v) = 

 0
c− (u, v)
b(u, v)
0 




0
0
0
a(u, v)
αβ
Here, the auxiliary spaces are Vα , Vβ ' C2 .
3.2.1
The first set of commutation relations
We take the local L-operator to be given by
(1)
Lαn (u) = Rαn
(u, 0).
36
(3.2)
Then the monodromy matrix is given by the following product of R-matrices
Tα(1) (u) = LαN (u) . . . Lα1 (u)
(1)
(1)
= RαN (u, 0) . . . Rα1 (u, 0).
The R(1) -matrix being a solution of the Y–B equation, ensures that the monodromy matrix
satisfies the global intertwining relation:
(1)
(1)
(1)
(1)
Rαβ (u, v)(Tα(1) (u) ⊗ Tβ (v)) = (Tβ (v) ⊗ Tα(1) (u))Rαβ (u, v).
(3.3)
From the RT T relation (3.3) we can extract the commutation relations between the operator entries of the monodromy matrix. The relations are easily proved in a diagrammatic
form, using definitions from the previous chapter. The first commutation relation we need
is the one between the B (1) operators:
(1)
(1)
(2)
a(u, v)Bα(1) (u)Bβ (v) = Bβ (v)Bα(1) (u)Rαβ (u, v).
(3.4)
For a diagrammatic proof of the commutation relation (3.4), the external edges on the
right are fixed to be 1s, and on the left the indices are free, k, l ∈ {2, 3}. The unmarked
edges stand for a summation over all possible internal configurations.
v k ∈ {2, 3}
1
v k ∈ {2, 3}
1
u l ∈ {2, 3}
1
=
1
u l ∈ {2, 3}
1
1
N
The two 1s on the right of the left hand side of the relation above imply that the possible
weights associated to this vertex are in the first row of the first block row of R(1) . The only
possibility is R(u, v)11
11 = a(u, v). On the left of the right hand side of the commutation
relation, the values of the unmarked edges of the R-matrix vertex are summed over all
k, l ∈ {2, 3}, that is, the tensor product of two monodromy matrices is multiplied by all
the weights in R(1) (u, v)kk0 ll0 , with k, l ∈ {2, 3}. Here, we make an important observation
37
N
that this is equivalent to a multiplication by the R(2) -matrix as defined in (3.2):

0
0
0
0
0
0
a(u, v) 0
0


 0 b(u, v)
0
c+ (u, v) 0
0
0
0
0


 0
0
0
0
c+ (u, v)
0
0
0
b(u, v)


 0 c− (u, v) 0
b(u, v)
0
0
0
0
0


(1)
Rαβ (u, v) =  0
0
0
0
a(u, v)
0
0
0
0


 0
0
0
0
0
b(u, v)
0
c+ (u, v)
0


 0
0 c− (u, v)
0
0
0
b(u)
0
0


0
0
0
0
c− (u, v)
0
b(u, v)
0
 0

0
0
0
0
0
0
0
0
a(u, v)























αβ
So we have a reduced diagram below
v k ∈ {2, 3}
1
1
v k ∈ {2, 3}
1
u l ∈ {2, 3}
1
=
u l ∈ {2, 3}
1
1
1
1
N
N
which is precisely the commutation relation in (3.4). Algebraically, (3.4) is a compact way
of writing the four equations in the operators T12 , T13 , which can be obtained simply by
equating the entries on both sides of (3.3), in particular, the following four:
a(u, v)T (u)21 T (v)21 = a(u, v)T (v)21 T (u)21
a(u, v)T (u)21 T (v)31 = b(u, v)T (v)31 T (u)21 + c− (u, v)T (v)21 T (u)31
(3.5)
a(u, v)T (u)31 T (v)21 = c+ (u, v)T (v)31 T (u)21 + b(u, v)T (v)21 T (u)31
a(u, v)T (u)31 T (v)31 = a(u, v)T (v)31 T (u)31
Each of these has an obvious diagrammatic representation. For example, (3.5) can be
represented as follows:
v
3
u
2
1
1
1
1
···
v
3
2
u
2
3
1
···
=
1
b(u, v)
a(u, v)
v
3
3
u
2
2
+
···
c− (u, v)
38
1
1
The second commutation relation tells us how to exchange the order of A(1) and B (1)
a(u, v)Bα(1) (u)A(1) (v) = c+ (u, v)Bα(1) (v)A(1) (u) + b(u, v)A(1) (v)Bα(1) (u)
We rewrite it in a more convenient form for later use:
A(1) (v)Bα(1) (u) =
a(u, v) (1)
c+ (u, v) (1)
B (u)A(1) (v) −
B (v)A(1) (u)
b(u, v) α
b(u, v) α
(3.6)
Diagrammatically, this second relation can be proved by fixing external edges on the right
to be 1s as before, while on the left to be 1 on the top and l ∈ {2, 3} on the bottom edge:
v
1
1
v
1
1
1
u l ∈ {2, 3}
1
=
1
u l ∈ {2, 3}
1
1
1
N
N
Next, we require a commutation relation describing the exchange of B (1) and D(1) operators:
(1)
(1)
(1)
(2)
b(u, v)Dα(1) (u)Bβ (v) + c− (u, v)Bα(1) (u)Dβ (v) = Bβ (v)Dα(1) (u)Rαβ (u, v)
(3.7)
The relation is easily observed diagrammatically by fixing the external edges on the left
to be l = 1 and k = {2, 3} and summing over all possible internal configurations.
(2)
(2)
For later use, we also introduce the permutation matrix Pαβ , where Pαβ =


1 0 0 0





0 0 1 0 
(2)


Pαβ = 

 0 1 0 0 


0 0 0 1
(2)
1
a(u,u) Rαβ (u, u):
αβ
We use the space permuting property of P , in particular we write
(2)
(1)
Bα(1) (u)Pαβ = Bβ (u)
(3.8)
as well as
(2)
(2)
(1)
Dα(1) (u)Pαβ = Pαβ Dβ (u)
(3.9)
and
(2)
(2)
(2)
(2)
Rαγ
(u, v)Pαβ = Pαβ Rβγ (u, v) .
(3.10)
Finally, we will be needing the unitarity relation:
(2)
(2)
Rαβ (u, v)Rβα (v, u) = a(u, v)a(v, u).
39
(3.11)
(1)
Proposed form for the first level eigenvector |Ψ1,...,N i
3.2.2
(1)
Our goal is to diagonalise the transfer matrix T (u) = Trα Tα (u) =
Ansatz eigenvector will be denoted by
(1)
|Ψ1,...,N i,
P3
i
i=1 T (u)i .
The
and is an element of the complete sets of
states H1 ⊗ · · · ⊗ HN . The corresponding eigenvalue is denoted by Λ(1) (u).
Thus, we seek solutions to the eigenvalue equation
h
i
(1)
(1)
(1)
T (u) |Ψ1,...,N i = A(1) (u) + Trα Dα(1) (u) |Ψ1,...,N i = Λ(1) (u) |Ψ1,...,N i
(3.12)
(1)
The first step in making an Ansatz is to propose a form for the eigenvector |Ψ1,...,N i:
(1)
(2)
|Ψ1,...,N i = Bα(1)
(v1 ) . . . Bα(1)
(vM ) |Ψα1 ...αM ,1...N i
1
M
(3.13)
(2)
where |Ψα1 ...αM ,1...N i ∈ Vα1 ⊗ · · · ⊗ VαM ⊗ H1 ⊗ · · · ⊗ HN to be defined later. Here,
H1 , . . . , HN ' C3 and Vα1 , . . . , VαM ' C2 .
(1)
Actions of A(1) and D(1) on |Ψ1,...,N i
3.2.3
(1)
Proposition 2. Given a solution Rαβ (u, v) of YBE (3.1), the action of A(1) and B (1)
satisfy the following commutation relations:

M
−→ a(v , u)
Y
i
A(1) (u)Bα(1)
B (1) (vi ) A(1) (u)
(v1 ) . . . Bα(1)
(vM ) = 
1
M
b(vi , u) αi
i=1



M
(2)
M
−
→
X
Y
Y
(v
,
v
)
R
a(v
,
v
)
c+ (vi , u) (1)
α
α
j
i


j i
j i
 A(1) (vi ) (3.14)
B (u) 
Bα(1)
(vj ) 
−
j
b(vi , u) αi
b(v
,
v
)
a(v
,
v
)
j
i
j
i
j6=i
i=1
j<i
j=1
and a similar equation for D(1) and B (1) :
M

−→ B (1) (v )
Y
αi
i  (2)
Dα(1) (u)Bα(1)
(v1 ) . . . Bα(1)
(vM ) = 
Tα (u)
1
M
b(u, vi )
i=1



M
(2)
M
(2)
−
→
X c− (u, vi )
Y
Y
(v
,
v
)
R
1
Tα (vi )
αj αi j i


(1)

−
Bα(1)
(u)
B
(v
)

j 
i
b(u, vi )
b(vi , vj ) αj
a(vj , vi )
a(vi , vi )
j6=i
i=1
(3.15)
j<i
j=1
(2)
c2 )-type given by
where Tα (u) is the monodromy matrix of Uq (sl
(2)
(2)
Tα(2) (u) = Dα(1) (u)Rαα
(u, vM ) . . . Rαα
(u, v1 ).
1
M
2, 3
•
...
(2)
Tα (u) = 2, 3
v1
vM
40
...
1
2, 3
N
(2)
Tα (u) is the next level monodromy matrix in the NBA technique, therefore in the next
(2)
step we will be interested in the trace of Tα (u) on Vα . For the proof of Proposition 2,
the reader is referred to the appendix 3.A at the end of this chapter.
3.2.4
First level expressions for eigenvalues Λ(1) (u) and Bethe equations
To make further progress at this point we need to make an assumption that the reference
(2)
(2)
state |Ψα1 ...αM ,1...N i is an eigenvector of both A(1) (u) and Trα Tα (u) satisfying:
(2)
(2)
(3.16)
(2)
(2)
(3.17)
A(1) (u) |Ψα1 ...αM ,1...N i = Γ(1) (u) |Ψα1 ...αM ,1...N i
T (2) (u) = Trα Tα(2) (u) |Ψα1 ...αM ,1...N i = Λ(2) (u) |Ψα1 ...αM ,1...N i
Our goal is to construct solutions to (3.17), thus our subsequent choice of the ref(2)
erence state |Ψα1 ...αM ,1...N i will be such as to validate these assumptions. Having constructed it, we will later verify (3.16). Adding (3.14) and (3.15) of Proposition 2 applied
(2)
to |Ψα1 ...αM ,1...N i, and employing identities (3.16) and (3.17), we obtain
T
(1)
(u) |Ψ1,...,N i
=
(1)
Γ
(u)
M
Y
a(vi , u)
b(vi , u)
i=1
+Λ
(2)
(u)
M
Y
i=1
1
b(u, vi )
!
(1)
|Ψ1,...,N i
(1)
where we have assumed that all terms not proportional to |Ψ1,...,N i cancel. Then the
eigenvalue can be preliminarily expressed as
Λ(1) (u) = Γ(1) (u)
M
Y
a(vi , u)
i=1
b(vi , u)
+ Λ(2) (u)
M
Y
i=1
1
.
b(u, vi )
(3.18)
The unwanted terms in (3.14) and (3.15) cancel if the variables {v1 , . . . , vM } satisfy the
Bethe equations
(1)
Γ
M
M
Y
Y
a(vj , vi )
1
(2)
− Λ (vi )
= 0,
(vi )
b(v
,
v
)
b(v
j i
i , vj )
j6=i
j6=i
j=1
j=1
where we have used
c+ (u,v)
b(u,v)
− (v,u)
= − cb(v,u)
. We rewrite this condition as
M
Y
b(vj , vi )
Γ(1) (vi )
=
(2)
a(vj , vi )b(vi , vj )
Λ (vi )
j6=i
j=1
= (−1)M −1
M
Y
j6=i
j=1
41
1
.
a(vj , vi )
(3.19)
3.3
Second and final step in the method of NBA
The second level monodromy matrix (with explicit dependence on Vα ) is given by


(2)
(2)
A (u; v1 , . . . , vM ) B (u; v1 , . . . , vM )
 .
Tα(2) (u; v1 , . . . , vM ) = 
C (2) (u; v1 , . . . , vM ) D(2) (u; v1 , . . . , vM )
α
The monodromy matrix
T (2)
obeys the intertwining relation as derived in the previous
chapter:
(2)
(2)
(2)
(2)
Rαβ (u, v)(Tα(2) (u) ⊗ Tβ (v)β ) = (Tβ (v) ⊗ Tα(2) (u))Rαβ (u, v).
(3.20)
Recall that from this relation we can extract the following commutations relations between
the A(2) , B (2) and D(2) operators:
B (2) (u)B (2) (v) = B (2) (v)B (2) (u),
(3.21)
A(2) (v)B (2) (u) =
a(u, v) (2)
c+ (u, v) (2)
B (u)A(2) (v) −
B (v)A(2) (u),
b(u, v)
b(u, v)
(3.22)
D(2) (u)B (2) (v) =
c− (u, v) (2)
a(u, v) (2)
B (v)D(2) (u) −
B (u)D(2) (v).
b(u, v)
b(u, v)
(3.23)
Our goal now is to obtain solutions to (3.17), which can be achieved using the ordinary
Bethe Ansatz from the previous chapter. Therefore, let
(2)
|Ψα1 ...αM ,1...N i = B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i
(3.24)
where the B (2) operators act on Vα1 ⊗ · · · ⊗ VαM ⊗ H1 ⊗ · · · ⊗ HN and the pseudo-vacuum
is defined to be

|2iα ⊗ |1i =
M
Y


j=1
1
0


αj
1

N 

Y
 
⊗
 0  .
 
i=1
0
i
3.3.1
(2)
Actions of A(2) and D(2) on |Ψα1 ...αM ,1...N i
Using commutativity of the B (2) operators (3.21) and relation (3.22), we obtain the two
relations derived in Chapter 2 (2.22 and 2.27):
(2)
A(2) (u) |Ψα1 ...αM ,1...N i = A(2) (u)B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i =
#
"L
Y a(ui , u)
B (2) (ui ) A(2) (u) |2iα ⊗ |1i
b(ui , u)
i=1


L
L
X
Y
a(uj , ui ) (2)
c+ (ui , u) (2)


−
B (u) 
B (uj ) A(2) (ui ) |2iα ⊗ |1i (3.25)
b(ui , u)
b(uj , ui )
j6=i
i=1
j=1
42
Similarly, there is the formula for the action of the D-operator
(2)
(2)
D(2) (u) |Ψα1 ...αM ,1...N i = D(2) (u)B (2) (u1 ) . . . B (2) (uL ) |Ψα1 ...αM ,1...N i =
"L
#
Y a(u, ui )
(2)
B (ui ) Dα(2) (u) |2iα ⊗ |1i
b(u, ui )
i=1


L
L
X c− (u, ui )
 Y a(ui , uj ) (2)

−
B (2) (u) 
B (uj ) D(2) (ui ) |2iα ⊗ |1i (3.26)
b(u, ui )
b(u
,
u
)
i j
j6=i
i=1
3.3.2
j=1
Second level expressions for the sub-eigenvalues Λ(2) (u) and Bethe
equations
The pseudo-vacuum |2iα ⊗ |1i is chosen in such a way as to be an eigenvector of the
operators A(2) (u) and D(2) (u):
A(2) (u) |2iα ⊗ |1i = Γ(2) (u) |2iα ⊗ |1i
= b(u)N
M
Y
a(u, vj ) |2iα ⊗ |1i
(3.27)
j=1
D(2) (u) |2iα ⊗ |1i = Λ(3) (u) |2iα ⊗ |1i
= b(u)N
M
Y
b(u, vj ) |2iα ⊗ |1i .
(3.28)
j=1
For a diagrammatic proof, observe that the left hand side of (3.27) corresponds to the
following state
2
α
2
2
2
2
1
1
2, 3
•
...
u 2
1
v1
vM
1
1
...
2
1
N
Recalling all the allowed vertex configurations, it is evident that there is only possible
configuration for the unlabelled edge states, being:
2
α
2
2
u 2
2
2
...
2
2
2
2
2
2
•
2
2
v1
43
1
1
2
2
1
vM
1
1
...
2
1
1
1
1
2
1
2
1
N
The weight of this lattice configuration is the product of all its vertex weights, as in
(3.27).
There is an analogous diagrammatic proof of (3.28):
2
α
2
2
2
2
v1
2
2
3
u 3
2
2
2
...
3
2
2
1
1
2, 3
•
...
u 3
=
α
1
1
2
1
3
•
2
v1
1
...
vM
3
1
N
1
3
2
1
vM
1
3
1
...
3
1
1
1
1
3
1
3
1
N
The expression for the Λ(2) is obtained by adding (3.25) and (3.26) to give
!
L
L
Y
Y
a(u
,
u)
a(u,
u
)
(2)
(2)
i
i
T (2) (u) |Ψ1,...,N,α1 ,...,αM i = Γ(2) (u)
+ Λ(3) (u)
|Ψ1,...,N,α1 ,...,αM i ,
b(ui , u)
b(u, ui )
i=1
i=1
where we have assumed that all the terms not proportional to |Ψ(2) i cancel. This requirement for cancellation of the sub-leading terms in (3.25) and (3.26) will give rise to the
second set of the Bethe equations. Provided they are satisfied, substituting in the expressions from (3.27) and (3.28) above, the explicit expression for the eigenvalue is given
by
Λ(2) (u) = Γ(2) (u)
L
Y
a(ui , u)
i=1

= b(u)N 
b(ui , u)
M
Y
+ Λ(3) (u)
a(u, vj )
j=1
L
Y
a(u, ui )
i=1
L
Y
a(ui , u)
i=1
b(ui , u)
b(u, ui )
+
M
Y
b(u, vj )
j=1
L
Y
a(u, ui )
i=1
b(u, ui )
(2)

.
(3.29)
Finally, |Ψ1,...,N,α1 ,...,αM i is an eigenvector of T (2) (u) if the rapidity variables {u1 , . . . , uL }
in (3.25 and 3.26) satisfy the Bethe equations:
(2)
Γ
L
L
Y
Y
a(uj , ui )
a(ui , uj )
(3)
(ui )
− Λ (ui )
= 0,
b(u
,
u
)
b(ui , uj )
j
i
j6=i
j6=i
j=1
j=1
44
c+ (u,v)
b(u,v)
where we have again used the property of the weights
− (v,u)
= − cb(v,u)
. Alternatively,
we can rewrite the Bethe equations in a multiplicative form
L
Y a(ui , uj )
Γ(2) (ui )
L−1
.
=
(−1)
a(uj , ui )
Λ(3) (ui )
j6=i
(3.30)
j=1
Final form for the eigenvalues Λ(1) (u) and the Bethe equations
3.3.3
Now, it is a good time to validate the assumption in (3.16). We do so by direct calculation
of Γ(1) (u) and then use the result to write an explicit expression for Λ(1) (u).
(2)
Proposition 3. If the global reference state |Ψα1 ...αM ,1...N i is given by the following product of the B-operators acting on pseudo-vacuum |2iα ⊗ |1i
(2)
|Ψα1 ...αM ,1...N i = B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i ,
then Γ(1) (u) is an eigenvalue of A(1) (u), and |Ψ(2) i an eigenvector:
(2)
(2)
A(1) (u) |Ψα1 ...αM ,1...N i = Γ(1) (u) |Ψα1 ...αM ,1...N i
(2)
= a(u)N |Ψα1 ...αM ,1...N i .
For the proof of Proposition 3 the reader is referred to the appendix 3.B at the end of
this chapter.
We can now give an explicit solution for the original eigenvalue problem in (3.12).
From (3.13) and (3.24), the eigenvectors of T (u) are given by
(1)
|Ψ1,...,N i = Bα(1)
(v1 ) . . . Bα(1)
(vM )B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i
1
1
Combining (3.18) and (3.29) and using the result (3.35) of the proposition above, the full
expression for the eigenvalues Λ(1) (u) of T (u) in terms of the R(1) -matrix weights is given
by
Λ
(1)
(u) = a(u)
N
M
L
L
Y
Y
a(u, vj ) Y a(ui , u)
a(u, ui )
N
+ b(u)
+ b(u)
.
b(vi , u)
b(u, vj )
b(ui , u)
b(u, ui )
M
Y
a(vi , u)
i=1
N
j=1
i=1
i=1
Finally, we give the two systems of Bethe equations necessary for the success of the Ansatz.
Writing the ratio of
Λ(2) (vi )
,
Γ(1) (vi )
noting that the second term in (3.29) vanishes at u = vi , as
b(vi , vi ) = 0 and substituting (3.19) into the right hand side, we arrive at the following
system of equations in rapidities {v1 , . . . , vM } and {u1 , . . . , vL }:
L
M
M
Y
Y
a(uj , vi )
b(vi )N Y
M −1
a(vi , vj )
= (−1)
a(vj , vi )
a(vi )N
b(uj , vi )
j6=i
j=1
j=1
j=1
45
and putting it in a usual form, obtain
M
L
Y
Y
a(vj , vi )
a(uj , vi )
M −1
N
= (−1)
a(vi )
.
b(vi )
b(uj , vi )
a(vi , vj )
j6=i
N
j=1
Writing
Γ(2) (ui )
Λ(3) (ui )
j=1
and substituting (3.30) into the right hand side, results in the second set
of Bethe equations in the same sets of rapidities
M
L
Y
Y
a(ui , uj )
a(ui , vj )
= (−1)L−1
.
b(ui , vj )
a(uj , ui )
j6=i
j=1
j=1
In summary, we have found an explicit expression for the eigenvalues of the transfer
matrix via the action of a certain product of B-operators on the pseudo-vacuum, provided
their arguments {ui } and {vi } satisfy a system of two sets of Bethe equations. Hence we
c3 ) symmetry.
have solved the transfer matrix diagonalisation problem of a model with Uq (sl
The picture below is a diagrammatic representation of all the Bethe eigenvectors and
the pseudo-vacuum for the model associated with solutions of the quantum Yang–Baxter
c3 ) algebra. The edge states in grey highlight the part of the
equation based on Uq (sl
diagram that ‘freezes’ (and factors out):
2
2
2
2
2
···
1
1
1
1
···
2
3
2
2
3
3
1
2
2
..
.
2
2
3
2
2
3
u1
1
1
1
1
1
1
2
uL
1
v1
2, 3
2, 3
1
2, 3
..
.
2, 3
2, 3
1
1
1
1
46
N
vM
3.A
Proof of (3.14) and (3.15) in Proposition 2
Proof. Let us begin with the proof of the first statement of the proposition by considering
(1)
(1)
the left hand side: A(1) (u)Bα1 (v1 ) . . . BαM (vM ). We make a proposition PM , which is
the expression in (3.14), to be proved by induction for any integer M ≥ 1. The second
commutation relation (3.6) with u → v1 and v → u, establishes that P1 is true:
c+ (v1 , u) (1)
a(v1 , u) (1)
(1)
(1)
(1)
(1)
B (v1 )A (u) −
B (u)A (v1 ) .
A (u)Bα (v1 ) =
b(v1 , u) α
b(v1 , u) α
Establishing the M = 2 case
In the first step of the proof we establish that P2 is true. This is the first non-trivial
induction step. Proposition 2 for M = 2 states:
a(v2 , u) a(v1 , u) (1)
B (v1 )Bα(1)
(v2 )A(1) (u)
2
b(v2 , u) b(v1 , u) α1
c+ (v1 , u) a(v2 , v1 ) (1)
B (u)Bα(1)
−
(v2 )A(1) (v1 )
2
b(v1 , u) b(v2 , v1 ) α1
c+ (v2 , u)
B (1) (u)Bα(1)
−
(v1 )Rα(2)
(v1 , v2 )A(1) (v2 ) (3.31)
1
1 α2
b(v2 , u)b(v1 , v2 ) α2
(v1 )Bα(1)
(v2 ) =
A(1) (u)Bα(1)
1
2
To show (3.31), observe that with the aid of (3.4) we can permute the order of the B
-operators
(2)
A(1) (u)Bα(1)
(v1 )Bα(1)
(v2 ) = A(1) (u)Bα(1)
(v2 )Bα(1)
(v1 )
1
2
2
1
Rα1 α2 (v1 , v2 )
a(v1 , v2 )
Applying (3.6) to the equation above once, gives:
(1)
A
(2)
Rα1 α2 (v1 , v2 )
(1)
(1)
(u)Bα2 (v2 )Bα1 (v1 )
a(v1 , v2 )
(2)
(2)
|Ψ
a(v2 , u) (1)
Rα α (v1 , v2 )
i=
B (v2 )A(1) (u)Bα(1)
(v1 ) 1 2
1
b(v2 , u) α2
a(v1 , v2 )
(2)
−
c+ (v2 , u) (1)
Rα α (v1 , v2 )
Bα2 (u)A(1) (v2 )Bα(1)
(v1 ) 1 2
1
b(v2 , u)
a(v1 , v2 )
Applying the same commutation relation once again gives:
(2)
a(v2 , u) a(v1 , u) (1)
Rα α (v1 , v2 )
=
B (v2 )Bα(1)
(v1 )A(1) (u) 1 2
1
b(v2 , u) b(v1 , u) α2
a(v1 , v2 )
1
(2)
−
a(v2 , u) c+ (v1 , u) (1)
Rα α (v1 , v2 )
Bα2 (v2 )Bα(1)
(u)A(1) (v1 ) 1 2
1
b(v2 , u) b(v1 , u)
a(v1 , v2 )
−
c+ (v2 , u) a(v1 , v2 ) (1)
Rα α (v1 , v2 )
(v1 )A(1) (v2 ) 1 2
Bα2 (u)Bα(1)
1
b(v2 , u) b(v1 , v2 )
a(v1 , v2 )
2
(2)
3
(2)
c+ (v2 , u) c+ (v1 , v2 ) (1)
Rα α (v1 , v2 )
+
B (u)Bα(1)
(v2 )A(1) (v1 ) 1 2
1
b(v2 , u) b(v1 , v2 ) α2
a(v1 , v2 )
47
4
1 and 3
Simplifying a(v2 , u) a(v1 , u) (1)
B (v1 )Bα(1)
(v2 )A(1) (u)
2
b(v2 , u) b(v1 , u) α1
c+ (v2 , u)
3 =−
B (1) (u)Bα(1)
(v1 )Rα(2)
(v1 , v2 )A(1) (v2 )
1
1 α2
b(v2 , u)b(v1 , v2 ) α2
1 =
2 and
immediately yields the first and last terms in (3.31) respectively. We will combine (1)
4 , but first rewrite these in the desired form bringing Bα1 (u) to the front using (3.4) and
(3.11):
(2)
2 =−
c+ (v1 , u)
Rα α (v1 , v2 ) (1)
Bα(1)
(u)Bα(1)
(v2 )Rα(2)
(v2 , u) 1 2
A (v1 )
1
2
2 α1
b(v1 , u)b(v2 , u)
a(v1 , v2 )
(2)
4 =
c+ (v2 , u) c+ (v1 , v2 ) (1)
Rα1 α2 (v1 , v2 ) (1)
Bα1 (u)Bα(1)
(v2 )Pα(2)
A (v1 )
2
2 α1
b(v2 , u) b(v1 , v2 )
a(v1 , v2 )
An important observation is that the sum of the following two terms simplifies
c+ (v1 , u)
c+ (v2 , u) c+ (v1 , v2 ) (2)
Rα(2)
P
(v2 , u) +
2 α1
b(v1 , u)b(v2 , u)
b(v2 , u) b(v1 , v2 ) α2 α1
c+ (v1 , u)
=−
R(2) (v2 , v1 ),
b(v1 , u)b(v2 , v1 ) α2 α1
−
with the use of the relations in the R matrix weights, a consequence of the Y–B equation:
c+ (vi , u)
a(vM , u)c+ (vi , u) c+ (vM , u)c+ (vi , vM )
−
=
a(vM , vi )
b(vM , u)b(vi , u)
b(vM , u)b(vi , vM )
b(vM , vi )b(vi , u)
c+ (vM , u)c+ (vi , u) c+ (vM , u)c+ (vi , vM )
c+ (vi , u)
−
=
c+ (vM , vi )
b(vM , u)b(vi , u)
b(vM , u)b(vi , vM )
b(vM , vi )b(vi , u)
c+ (vM , u)c− (vi , u) c+ (vM , u)c+ (vi , vM )
c+ (vi , u)
−
=
c− (vM , vi )
b(vM , u)b(vi , u)
b(vM , u)b(vi , vM )
b(vM , vi )b(vi , u)
2 and 4 simplifies to:
Therefore the sum of (2)
−
c+ (v1 , u)
Rα α (v1 , v2 ) (1)
Bα(1)
(u)Bα(1)
(v2 )Rα(2)
(v2 , v1 ) 1 2
A (v1 ).
1
2
2 α1
b(v1 , u)b(v2 , v1 )
a(v1 , v2 )
2 and 4 in its final form
Employing the unitarity relation (3.11) we rewrite the sum of −
c+ (v1 , u)a(v2 , v1 ) (1)
B (u)Bα(1)
(v2 )A(1) (v1 )
2
b(v1 , u)b(v2 , v1 ) α1
recovering the middle term in (3.31) and hence establishing that P2 is true.
Proof by induction for all integers M ≥ 1
To proceed with the proof by induction, consider A(1) (u)
→
−
QM
(1)
j=1 Bαi (vj )
for M ≥ 3 and let
the proposition PM denote the proposition in (3.14). Analogously with the M = 2 case,
48
(1)
we begin by bringing the BαM (vM )-operator to the front by applying the commutation
relation in (3.4) M − 1 times:
M −1
A
(1)
(u)Bα(1)
(v1 ) . . . Bα(1)
(vM )
1
M
(1)
=A
(u)Bα(1)
(vM )
M
−→ Bα(1) (vj )
Y
j
R(2) (vj , vM )
a(vj , vM ) αj αM
j=1
Now applying the second commutation relation (3.6) to the right hand side of the expression above once, gives:
M −1

−
→ Bα(1) (vj )
Y
a(v
,
u)
M
j
B (1) (vM )A(1) (u) 
R(2) (vj , vM )
A(1) (u)
Bα(1)
(vj ) =
j
b(vM , u) αM
a(vj , vM ) αj αM
j=1
j=1
M −1

(1)
−
→
Y
B
(v
)
c+ (vM , u) (1)
αj
j
−
B (u)A(vM ) 
R(2) (vj , vM )
b(vM , u) αM
a(vj , vM ) αj αM
M
−
→
Y
j=1
Assume that proposition PM −1 is true for some M ≥ 2, then substituting PM −1 into the
right hand side of the equation above gives the following four terms:
M
A
(1)
(u)
−
→
Y
Bα(1)
(vi ) =
j
j=1
M −1
 M −1

(2)
−
→
−
→
Y
Y
Rαj αM (vj , vM )
a(vj , u) (1)
a(vM , u) (1)
 A(1) (u)
B (vM ) 
B (vj ) 
b(vM , u) αM
b(vj , u) αj
a(vj , vM )
j=1
j=1


M −1
M
−1
−
→
X c+ (vi , u)
a(vM , u) (1)
 Y a(vj , vi ) (1)

BαM (vM )
Bα(1)
(u) 
Bαj (vj )
−
i
b(vM , u)
b(vi , u)
b(v
,
v
)
j
i
j6=i
i=1
1
j=1

 M −1

−→ Rα(2)α (vj , vi )
−→ Rα(2)α (vj , vM )
Y
Y
j i
j M


 A(1) (vi )
2
a(vj , vi )
a(vj , vM )
j<i
j=1
M −1
 M −1

(2)
−
→
−
→
Y
Y
Rαj αM (vj , vM )
a(vj , vM ) (1)
c+ (vM , u) (1)
 A(1) (vM ) 3
−
BαM (u) 
Bαj (vj ) 
b(vM , u)
b(vj , vM )
a(vj , vM )
j=1
j=1


M −1
M
−1
−→
X c+ (vi , vM )
c+ (vM , u) (1)
 Y a(vj , vi ) (1)

BαM (u)
Bα(1)
(vM ) 
Bαj (vj )
+
i
b(vM , u)
b(vi , vM )
b(v
,
v
)
j i
j6=i
i=1
j=1

 M −1

−→ Rα(2)α (vj , vi )
−→ Rα(2)α (vj , vM )
Y
Y
j i
j M


 A(1) (vi )
a(vj , vi )
a(vj , vM )
j<i
j=1
49
4
1:
Simplifying  k

M −1
M
−1
−
→ a(v , u)
−→ R(2) (v , v )
Y
Y
Y
αi ,αM i M
i
(1)
 Bα(1) (vM )
1 =
A(1) (u)
B
(v
)
i
αi
M
b(vi , u)
a(vi , vM )
i=1
i=1
i=1
M

−
→ a(v , u)
Y
i
=
B (1) (vi ) A(1) (u)
b(vi , u) αi
i=1
3 is immediately recognised as the i = M th
we immediately obtain the first term in PM . 2 and 4.
term of the sum in the second term in PM . The strategy now is to combine (1)
2 with Bαi (u) brought to the front with the help of (3.4)
First, let us rewrite 

M −1
M
−1
−
→
X
c+ (vi , u)

 Y a(vj , vi ) (1)
2 =−
Bα(1)
(u)Bα(1)
Bαj (vj )
(vM )RαM ,αi (vM , u) 
i
M
b(vM , u)b(vi , u)
b(v
,
v
)
j
i
j6=i
i=1
j=1

 M −1

−→ Rα(2)α (vj , vi )
−→ Rα(2)α (vj , vM )
Y
Y
j i
j M


 A(1) (vi ).
a(vj , vi )
a(vj , vM )
j<i
j=1
(1)
4 with Bαi (u) brought to the front by permuting the spaces of αM and
We also rewrite αi , using (3.8)
4 =
M
−1
X
i=1

M −1

−→ a(v , v )
c+ (vM , u) c+ (vi , vM ) (1)
Y

j i
Bαi (u)Bα(1)
(vM )Pα(2)
(vj )
Bα(1)

j
M
M αi
b(vM , u) b(vi , vM )
b(v
,
v
)
j i
j6=i
j=1

 M −1

−→ Rα(2)α (vj , vi )
−→ Rα(2)α (vj , vM )
Y
Y
j i
j M


 A(1) (vi )
a(vj , vi )
a(vj , vM )
j<i
j=1
Analogously with the M = 2 case, the sum of the following two terms
−
c+ (vi , u)
c+ (vM , u) c+ (vi , vM ) (2)
Rα(2)
P
(vM , u) +
M αi
b(vi , u)b(vM , u)
b(vM , u) b(vi , vM ) αM αi
with the help of the following Y–B relations in the weights is equal to
−
c+ (vi , u)
R(2) (vM , vi ).
b(vi , u)b(vM , vi ) αM αi
2 and 4 simplifies to:
Therefore the sum of −
M
−1
X
i=1


−
→
Y
a(vj , vi ) (1)
c+ (vi , u)


Bα(1)
(u)Bα(1)
(vM )Rα(2)
(vM , vi ) 
Bαj (vj )
i
M
M αi
b(vi , u)b(vM , vi )
b(vj , vi )
j6=i
M −1
j=1

 M −1

−
→ Rα(2)α (vj , vi )
−
→ Rα(2)α (vj , vM )
Y
Y
j i
j M


 A(1) (vi )
a(vj , vi )
a(vj , vM )
j<i
j=1
50
=−
M
−1
X
i=1




M
M −1
−
→
−
→
Y
Y
a(vj , vi )  (1)
c+ (vi , u) (1)



Bαi (u) 
Bα(1)
(vj )
 BαM (vM ) 
j
b(vi , u)
b(vj , vi )
j6=i
j6=i
j=1
j=1

 M −1

(2)
→ Rα(2)α (vj , vi )
−
→ Rα(2)α (vj , vM )
Y
Y
RαM αi (vM , vi ) −
j i
j M

 A(1) (vi )
a(vM , vi )
a(vj , vi )
a(vj , vM )
j<i
j=1


M
M
−1
−
→
X
c+ (vi , u) (1)
 Y a(vj , vi ) (1)

=−
Bαi (u) 
Bαj (vj )
b(vi , u)
b(vj , vi )
j6=i
i=1

M −1
j=1


 M −1

(2)
(2)
(2)
(2)
←
−
−
→
−
→
Y
Y
Y
R
(v
,
v
)
R
(v
,
v
)
R
(v
,
v
)
αM αj M j  RαM αi (vM , vi )
αj αi j i
αj αM j M



 A(1) (vi ).


a(v
,
v
)
a(v
,
v
)
a(v
,
v
)
a(v
,
v
)
j i
j M
M j
M i
j6=i
j<i
j=1
j=1
(3.32)
Applying the YBE with rapidities vi , vj , vM
Rα(2)
(vM , vi )Rα(2)
(vj , vi )Rα(2)
(vj , vM ) = Rα(2)
(vj , vM )Rα(2)
(vj , vi )Rα(2)
(vM , vi )
j αi
j αM
j αM
j αi
M αi
M αi
followed by the unitarity relation
(2)
(2)
RαM αj (vM , vj ) Rαj αM (vj , vM )
= 1,
a(vM , vj )
a(vj , vM )
starting with j = 1 and for all j < i simplifies the last line of products in (3.32) to give


−→ Rα(2)α (vj , vi )
Y
j i


a(vj , vi )
j<i
To see this, observe that Rαj αM trivially passes past all Rαj+1 αi , . . . , Rαi−1 αi from right to
left for all 1 ≤ j < i, and also that Rαj αi passes past all RαM α1 , . . . , RαM αi−1 , RαM αi+1 , . . . , RαM αM −1
for all j < i from right to left. Thus, after some calculations we have find



 M −1

M −1 (2)
−→ Rα(2)α (vj , vi )
−→ Rα(2)α (vj , vM )
←
− Rα α (vM , vj ) R(2) (v , v ) Y
Y
Y
 αM αi M i 

j i
j M
M j




a(v
,
v
)
a(v
,
v
)
a(v
,
v
)
a(v
,
v
)
j
i
j
i
j
M
M
M
j6=i
j=1
j<i
j=1


−
→ Rα(2)α (vj , vi )
Y
j i
.
=
a(vj , vi )
j<i
Interpreted diagrammatically, these operations correspond to ‘pulling tight’ and ‘untangling’ of the M th strand from past all the others
51
v1
vi
vM −1 vM
)
→
−
QM −1
j=1
)
···
→
−
Q
v1
vM
R(vj , vM )
=
)
j<i R(vj , vi )
R(v , v )
) M i
←
Q
−M −1
j6=i R(vM , vj )
···
vi
j=1
···
···
2 +
4 simplifies to give the following expression
Hence 


k
M
−1
−
→
−→ Rα(2)α (vj , vi )
X
Y
Y
a(v
,
v
)
c+ (vi , u) (1)


j i
j i
 A(1) (vi ) |Ψ(2) i .
−
B (u) 
Bα(1)
(vj ) 
j
b(vi , u) αi
b(v
,
v
)
a(v
,
v
)
j
i
j
i
j6=i
i=1
j<i
j=1
3 as the i = M th term into the sum in the expression above, proves PM
Incorporating and therefore expression (3.14) of Proposition 2.
Next, we need to prove the second expression (3.15) in Proposition 2. However, as the
proof of the action of D(1) on |Ψ(2) i is largely the same as that for the action of A(1) on
|Ψ(2) i, we shall only give the details in the case M = 2. We apply twice commutation
relation (3.7):
Dα(1) (u)Bα1 (v1 )Bα2 (v2 )
c− (u, v1 ) (1)
1
(1)
(1)
(2)
B (v1 )Dα (u)Rαα1 (u, v1 ) −
B (u) Bα(1)
(v2 )
=
2
b(u, v1 ) α1
b(u, v1 ) α
1
(2)
(2)
=
B (1) (v1 )Bα(1)
(v2 )Dα(1) (u)Rαα
(u, v2 )Rαα
(u, v1 )
2
2
1
b(u, v1 )b(u, v2 ) α1
c− (u, v2 )
(2)
B (1) (v1 )Bα(1) (u)Dα(1)
(v2 )Rαα
−
(u, v1 )
2
1
b(u, v1 )b(u, v2 ) α1
c− (u, v1 )
B (1) (u)Bα(1)
(v2 )Dα(1)
(v1 )Rα(2)
(v1 , v2 )
−
2
1
1 α2
b(u, v1 )b(v1 , v2 ) α
c− (u, v1 )c− (v1 , v2 ) (1)
+
B (u)Bα(1)
(v1 )Dα(1)
(v2 )
1
2
b(u, v1 )b(v1 , v2 ) α
2:
We apply (3.4) to 2 =−
c− (u, v2 )a(u, v1 ) (1)
B (u)Bα(1)
(v1 )Dα(1)
(v2 )
1
2
b(u, v1 )b(u, v2 ) α
1 . The second term at i = 1 of (3.15) is
The first term in (3.15) is precisely −
c( u, v1 )
(2)
B (1) (u)Bα(1)
(v2 )Dα(1) (v1 )Rαα
(v1 , v2 )Pα,α1
2
2
b(u, v1 )b(v1 , v2 ) α
52
1
2
3
4
→
−
Q
j<i R(vj , vi )
3.
By applying (3.10), (3.8) and (3.9) we find The second term at i = 2 of (3.15) is
−
c(u, v2 )
(2)
(v2 , v1 )
B (1) (u)Bα(1)
(v1 )Rα(2)
(v1 , v2 )Dα(1) (v2 )Pαα2 Rαα
1
1
1 α2
b(u, v2 )b(v2 , v1 )a(v1 , v2 ) α2
Using again (3.10), (3.8) (3.9) as well as (3.11) this simplifies to
−
c− (u, v2 )a(v2 , v1 ) (1)
B (u)Bα(1)
(v1 )Dα(1)
(v2 )
1
2
b(u, v2 )b(v2 , v1 ) α
2 +
4 provided
This is equal to −
c− (u, v2 )a(v2 , v1 )
c− (u, v2 )a(u, v1 ) c− (u, v1 )c− (v1 , v2 )
=−
+
b(u, v2 )b(v2 , v1 )
b(u, v1 )b(u, v2 )
b(u, v1 )b(v1 , v2 )
or c− (u, v1 )b(u, v2 )c− (v2 , v1 ) + a(u, v1 )c− (u, v2 )b(v2 , v1 ) = b(u, v1 )c− (u, v2 )a(v2 , v1 ), which
is a consequence of the Y–B equation.
3.B
Proof of Proposition 3
Proof.
(2)
A(1) (v) |Ψα1 ...αM ,1...N i = T (v)11 B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i .
(2)
Let us study at the action of T (v)11 on Bα (u). Recall that
Bα(2) (u) =
T (u)22 T (u)32
α
From the RT T relation, we can extract the following commutation relation:
b(u, v)T (u)22 T (v)11 + c− (u, v)T (u)21 T (v)12 = b(u, v)T (v)11 T (u)22 + c+ (u, v)T (v)21 T (u)12
=⇒ T (v)11 T (u)22 = T (u)22 T (v)11 +
c− (u, v)
c+ (u, v)
T (u)21 T (v)12 −
T (v)21 T (u)12
b(u, v)
b(u, v)
(3.33)
As to the second component of B, it simply commutes with T (v)11
T (v)11 T (u)32 = T (u)32 T (v)11
(3.34)
So it is clear that as A(1) (v) propagates through the product of B (1) (ui )s, it will eventually
reach |1i ⊗ |2iα .
However, (3.33) tell us that every time T (v)11 propagates past T (v)22 , there are two
extra terms being generated in the process. We will now show that all the extra terms
53
are annihilated when acting on the pseudo-vacuum. Firstly, observe that T (u)12 operator
acting on the pseudo-vacuum annihilates it:
T (u)12 |2iα ⊗ |1i = 0,
since T (u)12 is one of the operators in C. Next let us see what happens as the extra terms
act on the remaining product of B(ui )s and, in particular study the action of the T (v)12
on T (v)22 and on T (v)32 . For this we need another two commutation relations:
a(u, v)T (u)22 T (v)12 = c+ (u, v)T (v)22 T (u)12 + b(u, v)T (v)12 T (u)22
and
a(u, v)T (u)32 T (v)12 = c+ (u, v)T (v)32 T (u)12 + b(u, v)T (v)12 T (u)32
and rearranging, gives
T (v)12 T (u)22 =
c+ (u, v)
a(u, v)
T (u)22 T (v)12 −
T (v)22 T (u)12
b(u, v)
b(u, v)
T (v)12 T (u)32 =
a(u, v)
c+ (u, v)
T (u)32 T (v)12 −
T (v)32 T (u)12
b(u, v)
b(u, v)
and
From the two relations above, we see that as T (v)12 acts on B, even though more terms
have been generated, all these have T (v)12 permuted to the end of the product, whose
action on |2iα ⊗ |1i gives the 0 vector, as mentioned above. Hence all the extra terms
disappear. Consequently, we have that
A(1) (v)B (2) (u1 ) . . . B (2) (uL ) |2iα ⊗ |1i = B (2) (u1 ) . . . B (2) (uL )A(1) (v) |2iα ⊗ |1i
Lastly,
A(1) (v) |2iα ⊗ |1i = Γ(1) (v) |2iα ⊗ |1i
= a(v)N |2iα ⊗ |1i
Diagrammatically, this can be represented as follows, with A(1) acting only on the quantum
54
spaces:
1
α
vM
v1
vM
1
N
1
1
1
v1
1
1
1
•
v
1
...
1
α
1
1
•
v
=
1
1
...
1
1
1
1
1
1
1
1
1
1
N
Thus, we have shown that
(2)
(2)
A(1) (u) |Ψα1 ...αM ,1...N i = a(u)N |Ψα1 ...αM ,1...N i .
55
(3.35)
Chapter 4
From Perk–Schultz type Models
to Integrable Random Tilings
Introduction
[ generalisation of the
The purpose of this chapter is to pave the path from the Uq (sl(3))
[ Perk–Schultz model, to the square-triangle-rhombus
XXZ chain, namely the Uq (sl(3))
integrable random tiling model, the study of which is the primary objective of this thesis.
The calculations of the exact results relating to this tiling problem are detailed in the next
chapter. Here we shall cover the preliminary material, with the aim of formulating the
main problem in the context of relevant known models and previous studies.
Most of this chapter is based on some unpublished work of de Gier and Nienhuis. The
structure of the chapter is as follows. In the first section we generalise a known solution
[ algebra to a Perk–Schultz [44] type model by
for the R-matrix based on a Uq (sl(n))
\
way of introducing some additional field variables. Perk–Schultz models are Uq (sl(m|n))
models with non-standard co-products given by Reshetikhin twists. See [41] and [47].
In the second section, we rescale the L-matrix weights, taking a certain singular limit.
The generalisation (with additional field variables) of the diagonalisation method by the
algebraic Bethe Ansatz of the new model is possible and we write down the eigenvalue
and the Bethe equations.
In the third section, we identify the new model with some previously studied tiling
models. In particular, the n = 2 case corresponds to a rhombus random tiling and the
n = 3 case with spectral parameter u = 0 corresponds to the square-triangle random tiling.
The correspondence between rhombus and square-triangle random tilings and 5− and
9−vertex models respectively had been observed previously. In particular, the connection
56
of the 9−vertex model with the square-triangle random tiling model was made in [13],
where the model was also solved using algebraic Bethe Ansatz.
Lastly, we consider the case n = 3 but now for general values of u. In unpublished
work, de Gier and Nienhuis proposed a generalisation of the square-triangle random tiling
model to a square-triangle-rhombus tiling by the introduction of a new rhombus tile. The
spectral parameter u plays the role of a Boltzmann weight associated with the new tile.
They also identified the square-triangle-rhombus random tiling with a 10−vertex model.
Ř matrix
Integrable models considered in this chapter are again given on an M × N square lattice
with periodic boundary conditions. We introduce a new operator, Ř, for assigning vertex
weights which we find convenient later. One can recover the R-matrix as defined previously
by multiplication by a permutation matrix:
R = P Ř
where P = R(0) permutes factors of the tensor products. In other words, the vertex
weights are labelled as follows
j0
0 0
0 0
α i0
Ř(u)ij ji ≡ Rαβ (u)ii jj = u
i
(4.1)
j
β
or equivalently, Ř(u) (in terms of the R-matrix weights) is given by
X
0 0
0
0
Ř(u) =
R(u)ij ji Eii ⊗ Ejj ,
(4.2)
i,i0 j,j 0
0
where Eii is an elementary matrix as before.
Analogously, we define local operators Ľαk (u) = Pαk Lαk (u) where Pαk permutes spaces
α and k. The transfer matrix T is defined as before by a trace over α of the ordered product
of L operators. Then the family T (u) can be diagonalised simultaneously for all u if there
exists an R-matrix, such that the local operators L satisfy the intertwining relation:
Rαβ (u, v)Lα (u)Lβ (v) = Lβ (v)Lα (u)Rαβ (u, v)
and moreover, if the R-matrix satisfies the Yang–Baxter equation
Rαβ (u − v)Rαγ (u)Rβγ (v) = Rβγ (v)Rαγ (u)Rαβ (u − v).
57
We have written R(u, v) = R(v − u) above, as the R-matrices are of this form for all the
systems considered in this chapter.
4.1
A generalised Perk–Schultz type model
In chapters 2 and 3 we diagonalised the transfer matrices associated with the lattice models
obeying the ‘ice-rule’ with two and three degrees of freedom for each bond of the lattice.
In 1981, Perk and Schultz [49, 44] presented a new family of integrable models, the ncomponent generalisation of the ice-type models, with the additional field variables xi . In
[50], Schultz computed the most general solutions of the Yang–Baxter equation and the
eigenvalue under restriction of the ‘ice-rule’, but without the extra field variables. The
family of integrable models presented in this chapter is obtained by taking a singular limit
of the Ľ matrix generalised in a manner similar to that of [44].
There is a well-known solution for a Řsl matrix based on the sl(n) algebra, given by
Řsl (u) = P Rsl (u) = I + uP
=
n X
n
X
Eii ⊗ Ejj + u
i=1 j=1
n X
n
X
Eij ⊗ Eji
i=1 j=1
Observe that Rsl (u) = P Řsl (u) is the Y(sl(n)) matrix (discussed in the two previous
chapters for n = 2 and n = 3). For this model, the weights a, b and c are more special
and are given by the following entries of the Řsl matrix
ii
a(u) = Řsl (u)ii
ii = Rsl (u)ii = u + 1
ij
b(u) = Řsl (u)ij
ji = Rsl (u)ij = u,
i 6= j
jj
c(u) = Řsl (u)ij
ij = Rsl (u)ii = 1,
i 6= j.
Note that the weights c and b satisfy the symmetry and antisymmetry properties respectively, previously used in the derivation of the eigenvalue Λ(u) and the Bethe equations.
58
Here is an example of Řsl (u) for n = 2:

1

 1
1




1



1
0



Řαβ (u) = 




0

1







0

0
0
2
1
2
0
2
1
1
2
1
1
2
2
2
0
0
2
0
1
0
2
2























2 

2

0
0
 a(u)

 0
c(u)

=
 0
b(u)


0
0










 u+1 0

 0
1


 0
u


0
0
0
b(u)
0
c(u)
0
0
a(u)
αβ
=
αβ

0
0
u
0
1
0
0 u+1








.
αβ
Řsl also obeys the generalised ice-rule, meaning that the states entering in the lower
and right-hand bonds must emerge in the upper and left-hand bonds. This property
ensures that the method of nested Bethe Ansatz demonstrated in the previous chapters
(for n = 2, 3) is also applicable here for any n.
De Gier and Nienhuis generalised the model, by extending the solution based on the
Řsl matrix to include field variables (in the Perk–Schultz fashion [44]) in the following
manner:
Ř(u) =
n
X
Eii
⊗
Ejj
+u
n
X
i,j=1
Eii ⊗ Eii
i=1


−


n
X
yij Eij ⊗ Eji +
i,j=1
i<j
n
X
i,j=1
j<i

−1 j
yji
Ei ⊗ Eji 
.
Setting yij = −1 in Ř above, we recover Řsl .
This solution for Ř allows for a more general expression for the Ľ operator satisfying
59
the intertwining the relation. We can also include additional variables xi as follows
n X
n
n
X
X
j
i
Ľ(u) =
xi xj Ei ⊗ Ej + u
x2i Eii ⊗ Eii
i=1 j=1
i=1

−

X
x2j yij Eij ⊗ Eji +
i<j
X
−1 j
x2j yji
Ei ⊗ Eji  .
(4.3)
j<i
Setting xi = 1 for all i, we recover Ř(u). Let us write explicitly the n = 3 Ľ matrix:


2 (u + 1)
0
0
0
0
0
0
x
0
0

 1


2


0
x
x
0
−ux
y
0
0
0
0
0
1
2
12
2






2
0
0
0
−ux
y
0
0
0
0
x
x
13
1
3


3




−1
2
0
−ux1 y12
0
x2 x1
0
0
0
0
0






2


0
0
0
0
x2 (u + 1)
0
0
0
0




2


0
0
0
0
0
x2 x3
0
−ux3 y23
0




−1
2


0
0
−ux1 y13
0
0
0
x3 x1
0
0






−1
2
0
0
0
0
0
−ux
y
0
x
x
0


3 2
2 23


2
0
0
0
0
0
x3 (u + 1)
0
0
0
The transfer matrices for the lattice models given by Ľ above have been diagonalised by
the method of nested Bethe Ansatz in [50], although without the additional field variables
xi .
4.2
A family of singular integrable models
In this section we describe a family of integrable models constructed by de Gier and
Nienhuis by taking a certain limit of the generalised Perk–Schultz type model described
above. The transfer matrices obtained this way are singular, but nevertheless belong to a
family of commuting operators.
4.2.1
Parameter rescaling
In the generalised Perk–Schultz model above one can take a particular limit of the spectral
parameter as well as the additional field variables. Define new rescaled parameters x̄i and
ȳij by the following transformations
√
xi → x̄i , for i ≥ 2
yij →
ȳij
.
and
x̄1
x1 → √ ,
(4.4)
(4.5)
60
The new local operator L̄ is defined by taking the following limit in the rescaled variables
L̄(ū) = lim Ľ(−1 + ū).
→0
In this limit, the only term containing the rescaled spectral parameter ū is in the E11 ⊗ E11
position in the L̄-matrix. And all the terms on and below the diagonal, except for the
those containing x1 are annihilated. We thus have L̄ given by
L̄(ū) = x̄21 ūE11 ⊗ E11 +
n
X
−1 1
x̄1 x̄i (E11 ⊗ Eii + Eii ⊗ E11 ) + x̄21 ȳi1
Ei ⊗ E1i
i=2
+
n
X
x̄2j ȳij Eij ⊗ Eji .
(4.6)
i<j
4.2.2
Diagonalisation of the new model
The eigenvalue of the new model for arbitrary n and generic filed variables can be obtained
by generalisation of the construction in [50] and is given by
Λ(u) =
n
X
Λ(r) (u),
r=1
where
Λ
(r)
Mr−1
M1
r2 (u − u(1) )
Y Rrr (u − u(r−1) )
Y
Rr2
rr
r1 N
i
i
·
·
·
×
(u) = L(u)r1
(1)
(r−1)
r,r−1
r1
)
i=1 Rr,r−1 (u − ui
i=1 Rr1 (u − ui )
Mn−1
Mr
rr (u(r) − u)
Y
Rrr
i
(r)
i=1
r+1,r
(ui − u)
Rr+1,r
···
n−1r (n−1)
Y Rn−1r
− u)
(ui
(n−1)
nr (u
Rnr
i
i=1
. (4.7)
− u)
The Bethe equations can be obtained from an analytic requirement, that Λ(u) has no
poles. Thus the n − 1 sets of simultaneous equations can be formulated as n − 1 conditions
(r)
(r)
on the sets of parameters {ui } stated in terms of the residues of intermediate Λ(r) (ui ):
Resu(r) Λ(r) (u) + Λ(r+1) (u) = 0,
for all i = 1, . . . , Mr ,
n ∈ {1, . . . , n − 1}.
i
More explicitly, for the generalised model in (4.3) the eigenvalue is given by,
Λ(1) (u) = (x21 (1 + u))N
n
Y
(−y1j )−Nj
j=2
Λ(r) (u) = (x2r u)N
r−1
Y
(−yjr )Nj
j=1
M1
(1)
Y
u−u −1
i
(1)
i=1
n
Y
,
u − ui
Mr−1
(−yrj )−Nj
j=r+1
Mr
(r)
Y u − u(r−1) + 1 Y
u − ui − 1
i
(r−1)
u − ui
i=1
(r)
i=1
,
u − ui
(4.8)
Λ(n) (u) = (x2n u)N
n−1
Y
j=1
Mn−1
(−yjn )Nj
Y u − u(n−1) + 1
i
(n−1)
i=1
u − ui
61
,
where 2 ≤ r ≤ n − 1. The transfer matrix parameters Ni are related to the types of Bethe
Ansatz root numbers Mi by
Nj = Mj−1 − Mj ,
where M0 = N, and Mn = 0.
(4.9)
Now for the new model (after rescaling and talking the limit as → 0), the expressions
for the eigenvalue reduce to
(1)
Λ
(ū) =
n
Y
(x̄21 ū)N
−Nj
(−ȳ1j )
Λ
(ū) =
(−x̄2r )N
r−1
Y
Nj
(−ȳjr )
n−1
Y
,
ū − ūi
i=1
j=1
Λ(n) (ū) = (−x̄2n )N
−1
(1)
j=2
(r)
M1
Y
Mr−1
n
Y
(−ȳrj )
−Nj
j=r+1
Mn−1
(−ȳjn )Nj
j=1
Y
1
i=1
ū − ūi
(n−1)
Mr
Y
Y
1
i=1
(r−1)
ūi
i=1
ū −
−1
(r)
,
(4.10)
ū − ūi
.
Hence, we shave obtained a general expression for the eigenvalue for the rescaled model.
We shall now look in more detail at the models arising in the cases of n = 2 and n = 3,
with particular values of the field variables xi and yij .
4.3
Random tiling models
As it happens, the new model in (4.10) corresponds to some well-known physical systems.
In the case of n = 3, if we set x̄i = ȳ12 = ȳ13 = 1, ȳ23 = eµ and M1 = N2 + N3 , M2 = N3 ,
and writing ū = u, the L-matrix becomes

 u 0

 0 1


 0 0



 0 1


L̄α (u) =  0 0


 0 0


 0 0



 0 0

0 0
0
0 0 0
0
0
0
1 0 0
0
0
1
0 0 0
1
0
0
1 0 0
0
0
0
0 0 0
0
0
0
0 0 0
0 eµ
1
0 0 0
1
0
0
0 0 0
0
0
0
0 0 0
0
0

0 

0 


0 



0 


0 


0 


0 



0 

0
α
L̄ constructed in this fashion is a singular limit of an integrable model. We cannot simply
set Ř(u) = L̄(u) that would ensure that the local intertwining relation holds. However,
62
we were able to find a R? matrix that satisfies the intertwining relation
?
?
Rαβ
(v − u)Lα (v)Lβ (u) = Lβ (v)Lα (u)Rαβ
(v − u),
(4.11)
with L(u) = P L̄(u).
This R? (u) is given by














R? (u) = 











1 0 0
0 0 0
0
0 0 0
1 0 0
0
0 0 0
0 0 0
1
0 1 0
u 0 0
0
0 0 0
0 1 0
0
0 0 0
0 0 0
0
0 0 1
0 0 0
u
0 0 0
0 0 1
0
0 0 0
0 0 0
0
0
0 

0 0 


0 0 



0 0 


0 0 


1 0 


0 0 



eµ u 0 

0 1
.
αβ
At µ = 0, the R? as given above is closely related to that found by Zinn-Justin in [61].
For systems of size n = 3, the eigenvalue is the sum of three terms as given by (4.10) that
is
N
Λ(u) = u
M1
Y
i=1
M
M1
Y
1 Y2
1
1
−M2 µ
+e
u − ui
ui − u
u − vj
+ e(M1 −M2 )µ
i=1
M2
Y
j=1
j=1
1
.
vj − u
(4.12)
Let us now briefly look at the two systems which are special cases of the one given by
(4.12). These two systems have been previously discussed in the literature and identified
with random tiling models.
4.3.1
Rhombus random tilings: eµ = 0
The first system corresponds to the case of n = 2, the 5-vertex model which has been
identified with an ensemble of rhombus tilings (see Figure 4.2). This system can be
obtained from (4.12) by setting eµ = 0 which implies N3 = M2 = 0 and the eigenvalue is
thus given by
Λ(u) = uN + (−1)M
M
Y
j=1
63
1
.
u − uj
(4.13)
1
1
2
1
1
1
2
2
2
1
1
1
2
a=u
c=1
c=1
1
1
1
2
b=1
2
2
1
b=1
Figure 4.1: Correspondence between rhombus tiling and 5-vertex weights (as given by
Ľ(u) for n = 2, with xi = yij = 1 )
Rhombus tilings have been extensively studied in the mathematics and physics literature alike. Boxed rhombus tilings are equivalent to plane partitions, non-intersecting
lattice paths, and viewed as projections from three dimensions, model the melting of cubic crystals. Since rhombus tilings are equivalent to non-intersecting lattice paths, a large
toolbox of determinantal methods and random matrix theory is available to analyse them,
see e.g. [19, 27] and references therein.
Figure 4.2: Typical rhombus tiling with underlying triangular (or sheared square) lattice
of length N = 10 and with M = 3 domain walls.
We use the rhombus tiling as an example to illustrate the computation for the bulk
free entropy σ for the system. The analogous computation for the square-traingle-rhombus
tiling is much more complicated, however the primary goal is essentially the same – finding
a closed form expression for σ.
The rhombus tiling is placed on an M × N cylinder, ie. with periodic boundary
conditions. We think of the tiling being built layer by layer, top to bottom. This process
is encoded in the transfer matrix T whose matrix elements give the possible number of
64
ways of adding a layer. The bulk entropy σ (defined in the Introduction of this thesis)
here has a meaning of the growth constant in the limit as the area of the tiled cylinder
goes to infinity. Thus σ is given by
ln Z
ln TrT M
= lim
M,N →∞ M N
M,N →∞
MN
1
= lim
ln Λmax (u).
N →∞ N
σ(u) =
lim
We have already computed the eigenvalue in (4.13) using algebraic Bethe Ansatz. Also,
M −1 for each
uj satisfy the Bethe equations, which in this case simplify to uN
j = (−1)
j ∈ {1, . . . , M }.
In the case that all tiles are equally weighted, i.e., for u = 1, fixing M to be even for
the time being, the Bethe equations can be rewritten as
(iπ+2πij)(M −1)
uN
.
j =e
We need to choose solutions for uj , that are ‘closest together’ ensuring that they correspond
to the largest eigenvalue. Hence we take
iπ
(M − 2j + 1) .
uj = exp
N
The eigenvalue can be rewritten as


M
X
1
1 
Λmax (u) =
ln uN + (−1)M −
ln(u − uj ) .
N
N
j=1
Using the notation m = M/N for the density of domain walls, it follows that in the limit
N → ∞, where the sum can be replaced with an integral, the maximum eigenvalue for
fixed m and with u = 1 is given by
1
lim
ln Λmax = −
N →∞ N
Z
1
=−
2
m
ln(1 − eiπx )dx
−m
Z m
ln 2 − 2 cos(πx) dx .
0
This is maximal for the special value of m when 2 − 2 cos(πm) = 1, i.e. m = 1/3. For
larger values of m the integral will have negative contributions. Hence we find the bulk
entropy or the growth constant to be
1
σ=−
2
Z
1/3
ln 2 − 2 cos(πx) dx ≈ 0.323.
0
65
1
1
1
1 1
2
1
1
1
1
2
1
1 1
2
2
1 2
3
1
2
1
1
2
3
1
1
2
3
1 1
3
1 3
1
1
1
2
3
1
1
2
2
3
3
3
1
1
1
1
2
2
2
1
3
2
3
1
1
1
2 2
2
3
1
3
1
3
3
1
1
3
2
3
3
1
1
3
3
2
Figure 4.3: Correspondence between 9-vertex lattice model and the square-triangle tiling.
4.3.2
Square-triangle random tilings: u = 0
The second specialisation of the integrable system is given by setting u = 0 in (4.12). This
is the 9-vertex model, known to be equivalent to the square-triangle random tiling model.
It is generally regarded as a model for the structure of the 12-fold symmetrical quasicrystal.
It has been first studied by Widom [60] who diagonalised the transfer matrix of the model
using coordinate Bethe Ansatz. This resulted in coupled non-linear equations, which
Widom solved numerically, in particular finding the value for the entropy. Soon after,
Kalugin [28] found exact solutions to these non-linear equations in the thermodynamic
limit, giving analytic expressions for the entropy as a function of macroscopic parameters.
Later, de Gier and Nienhuis [13] solved the model using algebraic Bethe Ansatz. More
recently, Purbhoo [45] and Zinn-Justin [61] related square-triangle tilings to the puzzles
of Knutson–Tao–Woodward [31] which compute Littlewood–Richardson coefficients.
66
s0
s+
s−
t+
t−
Figure 4.4: A typical patch of the square-triangle tiling and labeled prototiles.
In principle, there are five parameters of the model corresponding to the three square
and two triangle densities ns0 , ns+ , ns− and nt+ , nt− (since two types of triangles always
occur in pairs). However, of these densities only three are independent. One degree of
freedom is removed by fixing the area and another via a non-linear constraint, which stems
from imposing periodic boundary conditions. The same constraint is believed to hold
(asymptotically) in the case of the free boundary conditions. Due to these constraints
some care has to be taken in choosing a set of independent parameters. For example,
choosing the three square densities allows for determination of the triangle densities only
up to a permutation. This subtlety does not arise if two square and one triangle density
is chosen.
In our case, we assign a non-trivial Boltzmann weight of eµ only to one type of tile,
specifically to the s+ square tile, that is: assign a chemical potential of µ to and zero
chemical potential to all other tiles. If the largest eigenvalue Λmax has been found, then
one is able to vary the average density ns+ of tile s+ in the model via the Legendre
conjugate of its chemical potential µ:
ns+ (µ) =
∂ 1
log Λmax (µ),
∂µ N
(4.14)
and compute the bulk entropy σ in terms of the tile density as follows
σ(ns+ ) =
1
log Λmax − µ ns+ .
N
(4.15)
As formulated above, the model can be obtained from (4.12) by setting u = 0. In this
case, the eigenvalues of the transfer matrix are given by
Λ = (−1)M2 e−M2 µ
M1
Y
i=1
u−1
i
M2
Y
vj−1 + e(M1 −M2 )µ
j=1
M2
Y
vj−1 ,
j=1
and the corresponding Bethe Ansatz equations are:
0=
uN
i
M1
+ (−)
M2
Y
j=1
0 = 1 + (−)M2
M1
Y
i=1
e−µ
,
ui − vj
e−µ
.
ui − vj
The equations as given above were derived by de Gier and Nienhuis in [13]. The bulk
67
σ?
r
n∗s+ = ns0 = ns− = m2
0.12
0.10
0.08
0.06
0.04
0.02
√
3 ?
4 nt
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4.5: Bulk entropy σ ? as a function of the total triangle area fraction of the original
square-triangle lattice.
entropy σ has been calculated by Kalugin in [28] in terms of a model parameter γ ∈ (0, π/2)
(where all nsi and nti and µ are functions of γ) to be:
√ !
π γ π
π γ 6 3
γ σ(γ) = 2 log
+
log tan
+
tan
+
+ 2 cos
+
cos γ
6
3
4
6
12 6
π γ π γ π
γ +2 cos
−
log tan
−
tan
−
.
6
3
4
6
12 6
(4.16)
See Figure 4.5 for a plot of the bulk entropy σ ∗ (γ) as a function of the total triangle area
fraction of the original square-triangle lattice.
The square-triangle tiling has three distinct symmetric phases. The first one, corresponding to the point of maximum symmetry has the four triangle densities and the three
square densities equal, so the square and triangular tiles occupy half of the area each. The
point of maximum symmetry is also the point of maximum entropy with the tiling exhibiting the ‘forbidden’ 12-fold rotational symmetry. This means that tilings with average
12-fold quasiperiodic symmetry are entropically favoured.
The other two incommensurate phases are a hexagonal phase with 6-fold rotational
symmetry and a square phase with 4-fold rotational symmetry (as can be seen in Figure
4.6). The concavity of the entropy curve means that intermediate phases are unstable.
For example, if there is an excess of triangles, the system phase separates into a linear
combination of a random quasiperiodic state and a purely crystalline state with triangles
only. This tiling model has been extensively studied by de Gier in [11] and we refer an
interested reader to that manuscript for details.
68
Figure 4.6: Typical configurations in the six-fold and four-fold phases of the tiling.
Finally, the lattice model given in (4.12) with non-zero u, is a 10-vertex lattice model
and has been identified with a new type of random tilings, the square-triangle-rhombus
tiling model. It is a generalisation of both rhombus tiling and square-triangle tiling models.
The next and final chapter of this thesis is dedicated to the study of this model.
69
Chapter 5
Square–Triangle–Rhombus Tiling
Model
We introduce a new integrable random tiling model, the square-triangle-rhombus tiling.
This model is an extension of the known square-triangle tiling by the introduction of
a new ‘thin rhombus’ tile in one orientation. As we saw in the previous chapter, this
generalisation preserves integrability and the Boltzmann weight u of the new tile plays
the role of the spectral parameter in the R∗ matrix as given in (4.3). The eigenvalue is a
function of two independent tile weights u and eµ . Here we compute the entropy of the
model on a two-parameter subspace, and study the tiling phase diagram on that subspace.
5.1
The square-triangle-rhombus tiling
The tiling consists of filling a region of the plane with three types of squares s0 , s+ , s− ,
two types of triangles t+ , t− (which occur in pairs), and a new thin rhombus tile r.
r
s0
s+
s−
t+
t−
Figure 5.1: Types of tiles in allowed orientations on the original and deformed triangular
lattices. Triangles that always occur in pairs are grouped accordingly and are labelled by
a single parameter.
70
Figure 5.2: A typical patch of the square-triangle-rhombus tiling (left) is deformed to a
triangular lattice (right), with world lines shown (N1 -dashed and N2 -solid lines).
The ‘thin rhombus’ acquires a weight of u, the s+ type square a weight of eµ . All other
tile weights are set to 1. The edges of all tiles have length 1. This tiling can be deformed
to one on a triangular lattice, meaning that every vertex of the original tiling is mapped
to a vertex of a triangular lattice. See the right hand side of Figure 5.2. In principle,
one can work on any lattice, however the choice of the triangular lattice simplifies many
calculations (it provides a natural choice of scale for the length of one row of tiles; the areas
of all squares are equal, as are the areas of all the triangles). It is quite straightforward to
convert between the original and the deformed lattices, thus the calculations are performed
on the ‘deformed’ triangular lattice and we convert between the two when necessary.
As mentioned earlier, de Gier and Nienhuis identified the square-triangle-rhombus
tiling model with a 10-vertex lattice model (see also the appendix of [61]). The equivalence
between the 10-vertex model and the square-triangle rhombus tiling on the original and
deformed triangular lattices is shown in Figure (5.3).
The vertex model underlying this random tiling obeys the generalised ice-rule, so there
is a conservation of two different types of particles or, equivalently, two kinds of lines
propagating from each row of the tiling to the next (with periodic boundary conditions
in the horizontal direction). This is made evident by decorating the tiles with solid and
dashed lines as shown in Figures 5.1 5.2 and 5.3.
It is important to note here that with this choice of tile decorations, the pseudo-vacuum
state does not correspond to the tiling made up of ‘empty’ tiles. With the pseudo-vacuum
corresponding to horizontal edges being labelled “1”, it is the state filled with dashed lines
only, which is a combination of two states, one comprised entirely of rhombi r and the
other comprised entirely of triangles of type t− (or the 1st and 9th vertices in Figure 5.3).
71
1
1
2
1
1
3
2
1
1
3
2
1
1
3
1
1
1
1
2
3
2
3
1
1
2
1
1
2
1
1
3
2
2
1
3
3
3
1
3
2
Figure 5.3: Correspondence between 10-vertex model, STR tiling, and STR tiling deformed
to a triangular lattice.
5.2
Linear and bilinear constraints in tile densities
In this section we derive the linear and bilinear constraints satisfied by the tile densities.
The underlying lattice model is again of size M × N with periodic boundary conditions.
Thus on the original (and on the deformed triangular) lattice N is the number of tiles per
row (or layer). Let N1 (N2 ) be the number of dashed(solid) lines in any given row. Then
N3 = N −N1 −N2 is the number of empty sites or squares of type s0 . In the corresponding
vertex model (Figure 5.3) Ni is the number of vertical edges labelled i.
72
The parameters Ni of the transfer matrix are related to the Bethe Ansatz root numbers
Mi by Ni = Mi−1 − Mi , as given by (4.9). In the case of n = 3, for the square-trianglerhombus tiling model, we write these relations explicitly:
N1 = N − M1 ,
N2 = M1 − M2 ,
(5.1)
N3 = M2 .
In what follows, we write ni =
Ni
N
and mi =
Mi
N .
Let us emphasise once more that the
propagating lines do not directly correspond to the Bethe Ansatz particles, but this choice
of decoration is useful for our purposes.
Let nsi , nti , nr denote the densities per unit area of the triangular lattice of squares,
triangles and rhombi. The three squares and the rhombus now have area 1 and triangles
have area 1/2.
There are six independent parameters of the model corresponding to the three square,
two triangle and the rhombus densities. The two particle numbers M1 and M2 , are to be
fixed by solving the Bethe equations.
We shall see that amongst these densities, only four are independent. One degree of
freedom is removed by fixing the area, and another via a non-linear constraint, which
stems from imposing periodic boundary conditions.
From Figure 5.3 it is clear that ns0 = n3 = m2 , that is the density of ‘empty’ s0 squares
is equal to the density of type ‘3’ particles propagating in the underlying lattice model,
which is also equal to the density of Bethe root numbers m2 .
We can express the density ni of world lines per row in terms of tile densities. Observe
that the average number n1 of dashed lines in any row (or “time-step”) is equal to the
density of squares of type s− plus 1/2 of the t− type triangles (since the line crosses two
triangles per time-step) plus the density nr of thin rhombi. Analogously, for the density
of solid lines n2 . Thus we have:
1
n1 = ns− + nt− + nr ,
2
1
n2 = ns+ + nt+ ,
2
n3 = ns0 .
(5.2)
The sum of all tile densities is equal to 1, ie n1 + n2 + n3 = 1, giving the area constraint:
1
nr + ns− + ns+ + ns0 + (nt− + nt+ ) = 1.
2
(5.3)
Note that the area constraint depends on the choice of the lattice, since tiles have different
areas on different lattices. The condition (5.3) is true on the deformed triangular lattice.
73
Theorem 1. The tile densities satisfy the following quadratic relation:
nt− nt+ = 4(ns0 ns+ + ns0 ns− + ns+ ns− + ns+ nr ).
(5.4)
Proof. Let p denote the number of layers in which each solid line crosses each dashed line
exactly once. Then the total number of crossings of the dashed and solid lines in a patch
of size pN is equal to N1 N2 . The crossings only take the form of squares s+ and s− . Hence
the total number of crossings in the patch is given by the number of these two types of
squares, i.e.
N1 N2 = pN (ns− + ns+ ),
p=
n1 n2
N.
ns− + ns+
In order for every solid line to cross every dashed line once, the average displacement
of a solid line to the left and a dashed line to the right must add up to N . Following the
lines from bottom to top, a dashed line moves half a step to the right with each time step,
unless it encounters a square of type s− , which pushes it one step further to the right,
or if it encounters a thin rhombus, which pushes it one step to the left (relative to where
it would end up otherwise). There are pN ns− squares of type s− and N1 dashed lines.
Therefore, on average, a dashed line encounters
pN ns−
N1
of s− type squares. Thus, the total
shift of the dashed line to the right is given by:
p pN (ns− − nr )
+
,
2
N1
similarly, the total shift of the solid lines to the left is given by:
p pN ns+
+
.
2
N2
Finally, the average shift of the world lines on the triangular lattice (in a patch of size pN
where all the lines have crossed once) must add up to N :
N n s+
N ns−
N nr
p 1+
+
−
= N.
N1
N2
N1
Dividing by N and substituting (5.3) into the right hand side of the equation above, we
have
n1 n2
ns− + ns+
ns−
ns+
nr
1
1+
+
−
= nr + ns− + ns+ + ns0 + (nt− + nt+ ).
n1
n2
n1
2
Now using the linear relations (5.2) to eliminate ni , we obtain the required quadratic
relation in the tile densities:
nt− nt+ = 4(ns0 ns+ + ns0 ns− + ns+ ns− + ns+ nr ).
74
Six- and four-fold symmetric phases of the square-triangle tiling; nr = 0
In the case of the square-triangle tiling (setting nr = 0) the bilinear equation above reduces
to a constraint on the Bethe Ansatz densities in two important special cases. First is the
case where all the square densities are equal: ns+ = ns− = ns0 = m2 . In this case the
linear relations in (5.2) reduce to:
nt−
,
2
nt
n2 = m2 + + ,
2
n3 = m2 .
n1 = m2 +
Rewriting the line densities ni in terms of mi using (5.1) and substituting these into the
equations above, we express the triangle densities nt− and nt+ in terms of mi alone
nt− = 2(1 − m1 − m2 ),
nt+ = 2(m1 − 2m2 ).
Substituting these into the bilinear relation (5.4), we recover the constraint for the 6-fold
symmetric phase:
1 + m1 (1 + m2 − m1 ) = (1 + m2 )2 .
(5.5)
In this phase one pair of triangles dominates over the other with all square densities being
equal. The squares form domain walls between patches of triangles of type t− or t+ as
shown in Figure 5.4.
The second important case is where the two triangle densities are equal: nt+ = nt− ≡
nt± as well as the two square densities: ns+ = ns− = m2 . In this case we have:
nt±
,
2
nt
+ ±,
2
n1 = m2 +
n2 = ns0
n3 = m2 .
Rewriting these in terms of mi using (5.1)
nt± = 2(1 − m1 − m2 ),
ns0 = 2m1 − 1
and substituting into (5.4), we obtain the non-linear constraint for the 4-fold symmetric
phase:
2m1 m2 = (1 − m1 )2 .
75
Figure 5.4: Typical configurations in the six-fold and four-fold phases of the tiling
In this phase the triangles are equal in numbers and form domain walls between patches
of ns0 type squares. The other two types of squares make up the corners of domain wall
boundaries and equal in numbers.
5.3
The eigenvalue and the Bethe Ansatz equations
In this section, we solve the Bethe equations for m1 and m2 (following Kalugin’s method
[28]) by converting the system of M1 + M2 simultaneous non-linear equations to coupled
integral equations in the limit as the size of the system goes to infinity. For special choices
of m1 and m2 , we can solve the integral equations and express all tile densities as well as
m1 and m2 in terms of a single new parameter γ.
The eigenvalue Λ(u) for the square-triangle-rhombus tiling model was given in (4.12)
by:
N
Λ(u) =u
M1
Y
i=1
M
M1
Y
1
1 Y2
1
−M2 µ
+e
u − ui
ui − u
u − vj
i=1
M2
Y
+ e(M1 −M2 )µ
j=1
1
.
vj − u
j=1
(5.6)
The expression for the eigenvalue is given in terms of two types of Bethe Ansatz roots {ui }
and {vj } and is a function of Boltzmann weights u and eµ of two tiles r and s+ . The two
sets of simultaneous non-linear Bethe equations in M1 of {ui } and M2 of {vj } variables
are:
uN
i
M1
+ (−)
M2
Y
e−µ
= 0,
ui − vj
(5.7)
M1
Y
e−µ
= 0.
ui − vj
(5.8)
j=1
1 + (−)
M2
i=1
76
These Bethe equations depend explicitly on parameter µ. As the size of tiling gets large,
the system of M1 +M2 equations in (5.7 & 5.8) is difficult to solve. However, it can be done
in the thermodynamic limit (see [28]) by converting this system of nonlinear equations to
coupled integral equations. To do this, we first rewrite (5.7 & 5.8) in logarithmic form:
N ln(ui ) +
M2
X
ln(u − vj ) − (M1 − 1)iπ + µM2 = 2πiIi ,
Ii ∈ Z,
(5.9)
ln(ui − v) − (M2 − 1)iπ + µM1 = 2πiI˜j ,
I˜j ∈ Z.
(5.10)
m2 =
M2
,
N
(5.11)
m1 =
M1
.
N
(5.12)
j=1
M1
X
i=1
Next, let us define new functions F1 and F2 by
F1 (u) = ln(u) +
M2
1 X
ln(u − vj ) + m2 µ,
N
j=1
F2 (v) =
1
N
M1
X
ln(ui − v) + m1 µ,
i=1
Then the definitions above together with (5.9 & 5.9) imply that the functions Fi satisfy
the following equations:
Re [F1 (ui )] = Re [F2 (vj )] = 0.
(5.13)
The equations in (5.13) have many solutions and one must choose those that correspond
to the eigenvalue of the transfer matrix. For the largest eigenvalue (or for the ground
state) we assume that successive roots are as close together as possible. This assumption
is made on the basis of numerical solutions for small systems. This implies the following
conditions:
N [F1 (ui+1 ) − F1 (ui )] = 2πi,
(5.14)
N [F2 (vj+1 ) − F2 (vj )] = 2πi.
(5.15)
If we assume N, M1 and M2 to be even, the eigenvalue Λ(u) in (5.6) can be rewritten
in terms of F1 and F2 . We take a logarithm of each term on the right hand side of (5.6)
then exponentiate, rewriting Λ as follows:
Λ(u) = uN eN m1 µ e−N F1 (u) + e−N F1 (u)−N F2 (u) + e−N F2 (u) .
(5.16)
Differentiating F1 (u) in (5.11), we have
F10 (u)
M2
1 X
1
1
.
= −
u N
vj − u
j=1
77
(5.17)
b∗1
0.5
b∗2
b∗2
U
b∗1
V
0.2
0.4
0.6
0.8
-0.4 -0.2
b2
b2
0.2
0.4
0.6
0.8
1.0
-0.5
b1
b1
0.5
V
1.0
-0.5
U
1.0
-1.0
Figure 5.5: Numerical solutions for the Bethe roots {ui } and {vj } lying on the curves U
and V respectively in the complex plane. Left: N = 233, M1 = 73, M2 = 17, and γ = 0.4.
Right: N = 157, M1 = 113, M2 = 11, and γ = −0.8.
We can turn this sum into an integral in the following way. First, in the thermodynamic
limit (as N, M1 , M2 → ∞) as all the roots vj form a finite curve V in the complex plane,
R
2
we define f2 as the density of vj roots, with the normalisation V f2 (v)dv = 2πiM
N . This
implies that
M2
X
j=1
N
1
=
vj − u
2πi
Z
V
f2 (v)
dv.
vj − u
We do the same for the roots ui and their density f1 , with the normalisation
(5.18)
R
U
f1 (u)du =
2πiM1
N .
Next, equation (5.14) in the thermodynamic limit can be rewritten as
F10 (ui ) ∼
1
F1 (ui+1 ) − F1 (ui )
2πi
=
∼f1 (ui ).
ui+1 − ui
N ui+1 − ui
(5.19)
Using the expression above and substituting it into the left hand side of (5.17), and using
(5.18) we arrive at the following integral equation in f1 (u) and f2 (v):
Z
1
f2 (v)
1
f1 (u) = −
dv.
u 2πi V v − u
(5.20)
An analogous expression can be derived by considering (5.15). Hence, in the thermodynamic limit we arrive at the following system of two coupled integral equations in the root
density functions f1 (u) and f2 (v):
Z
1
f2 (v)
1
−
dv,
u 2πi V v − u
Z
1
f1 (u)
f2 (v) = −
du.
2πi U u − v
f1 (u) =
(5.21)
(5.22)
In Figure 5.5 V is a directed curve from b∗1 to b1 while U is directed from b2 to b∗2 , where
b∗i is the complex conjugates of bi .
78
5.3.1
Cauchy–Stieltjes transform
In this subsection we briefly recall the properties of Cauchy–Stieltjes transforms, appearing in the integral equations (5.21 & 5.22).
Definition: Given a function f (u) defined on a finite curve C in the complex plane
C, the Cauchy–Stieltjes transform F (u) is given by:
Z
f (v)
1
dv.
F (u) :=
2πi C u − v
F (u) satisfies the following three properties:
• F (u) is analytic on C\C.
• F (u) =
k
u
+ O( u12 ), where k is a constant.
• F (u) has a jump discontinuity across C, given by:
lim [F (u− ) − F (u+ )] = f (u).
u± →u
We briefly check the properties above:
• If u ∈
/ C, then the function
f (v)
u−v
has no poles in C.
• C is a finite curve in complex plane, hence. In the limit as
u→∞
=⇒
1
1
∼ for all v ∈ C
u−v
u
therefore
1
2πi
Z
C
f (v)
1 1
dv ∼
u−v
u 2πi
Z
k
,
u
C
Z
1
where k =
f (v)dv.
2πi C
f (v)dv =
• Let u ∈ C and b and b∗ be the endpoints of a directed curve C. If u− and u+ are
points just below and just above the point u respectively, then the integral from b
to b∗ is given by:
1
lim [F (u− ) − F (u+ )] =
u± →u
2πi
I
= f (u).
79
f (v)
dv
v−u
b∗1
b∗2
V
U
b2
ΓV
b1
ΓU
Figure 5.6: Diagram of the curve V - on which f1 is non-analytic, U - on which f2 is
non-analytic, and oriented paths ΓV and ΓU crossing these cuts in complex plane.
5.3.2
Analytic continuation and monodromy
We are interested in finding analytic continuations of the functions f1 and f2 across the
cuts V and U respectively. The root density functions f1 and f2 are analytic on all of
C\(V ∪ {0}) and C\U respectively. Analytic continuation of f1 along the path ΓV crossing
V is computed from the third property of section 5.3.1, and is given by:
ΓV : f1 → f1 − f2
f2 → f2 .
Similarly, analytic continuation of f2 along the path ΓU across the cut U , is given by:
ΓU : f1 → f1
f2 → f1 + f2 .
The analytic continuations over the curves U and V for a general linear combination
G(z) = a1 f1 (z) + a2 f2 (z)
can be summarised with the following monodromy matrices:
 

 
 

 
a1
1 0
a1
a1
1 1
a
   , ΓU :   7→ 
  1 .
ΓV :   7→ 
a2
−1 1
a2
a2
0 1
a2
For general curves U and V in the complex plane that do not touch or intersect, ie.
U ∩ V = ∅, moving along closed paths starting and finishing at z ∈
/ U ∪ V , corresponds
−1
to taking appropriate words in the group generated by ΓU , ΓV , Γ−1
U and ΓV . The group
generated by the two monodromy matrices ΓU and ΓV is in fact SL(2, Z) which is infinite,
making it difficult to compute f1 and f2 in general.
However, if the endpoints of the curves U and V coincide, (i.e b1 = b2 ≡ b =⇒ b∗1 = b∗2 )
the situation is much simpler. A path Γ that starts at some point z ∈
/ U ∪V in the complex
80
plane in the ‘outside’ region (in the vicinity of ∞) necessarily crosses both U (‘outside’ →
‘inside’) and V (‘inside’ → ‘outside’). So from the point of view of point z in the ‘outside’
region, there is one combined U − V cut. The analytic continuations of G(z) along the
−1
joined path Γ = Γ−1
U ΓV across the combined cut, are summarised by the monodromy
matrix:

 
 
0 −1
a
a1
  1 .
Γ :   7→ 
a2
a2
1 1
−1
The monodromy group, with the generator Γ = Γ−1
U ΓV , is isomorphic to Z6 . (The
condition that endpoints coincide characterises the solvable sector of the tiling model.
Furthermore, in the case of the square-triangle tiling, it gives the sector exhibiting the
6-fold rotational symmetry, including the 12-fold-symmetric tiling.)
From now on, consider G(z), with a1 , a2 ∈ {−1, 0, 1} for simplicity of computation of
f1 and f2 . Finite monodromy of G(z) means that the problem of solving coupled integrals
is reduced to an algebraic one. The root density functions f1 (z) and f2 (z) can be made
single-valued with an appropriate change of variables. It will be convenient to parameterise
b in terms a new parameter γ:
b(γ) = i|b|e−iγ ,
γ ∈ (−π/2, π/2).
γ is the angle made by the straight line from the origin to b and the imaginary axis
of the complex plane. To make fi s single-valued, first we transform z via a Möbius
transformation, fixing two points: z = b 7→ t = 0 and z = b∗ 7→ t → ∞,
az + c
,
ae − dc 6= 0
dz + e
zb−1 − 1
1
1
2i
7→
,
− ∗ = − cos γ 6= 0,
∗−1
1 − zb
b b
|b|
z 7→
where the restriction of cos γ 6= 0 means the endpoints b, b∗ ∈
/ R. Now the image of the cut
under the Möbius transformation is analogous to that of the function y(z) = z 1/6 , thus
with respect to the new variable t,
1/6
−1
zb − 1
,
t=
1 − zb∗−1
z=b
1 + t6
,
1 + bb∗−1 t6
(5.23)
the functions f1 and f2 are single-valued.
In the original variable z, f1 and f2 lie on different sheets of the Riemann surface of
the same function G(z). One can fix a particular part of G to lie on a particular sheet
of the Riemann surface and here we set G equal to f2 at z = 0 on the sheet with the
point t = i for both γ > 0 and γ < 0. On the other sheets of the Riemann surface, G is
determined by the monodromy generator Γ.
81
5.4
Solution of the monodromy problem
In this section we solve for f1 and f2 or, rather, for their primitives F1 and F2 with the
aim of computing the eigenvalue (5.16) in terms of these.
Based on numerical calculations we assume that for −π/2 < γ < π/2, functions f1
and f2 have no singularities at b or b∗ . To compute F1 and F2 consider the one-forms
f1 (z)dz and f2 (z)dz. The one-form G(z)dz has only simple poles at z = 0 and z → ∞.
Thus G(z)dz is meromorphic and as such is uniquely determined by its residues:
G(z)dz =
12
X
k=1
rk
dt.
t − tk
(5.24)
We compute all the residues of G(z)dz at its poles, giving only a couple of examples here,
the other computations being analogous. Let us find the residues of G(z)dz = f1 (z)dz at
z = 0 and z → ∞:
Res(f1 (z)dz, z = 0)
Z
dz
f2 (v)
dz
−
dv,
=Res
z
2πi V v
z=0
=1.
And the residue of f1 (z)dz as z → ∞ is given by the residue of f ( w1 ) at w = 0, with the
change of variables z =
1
w:
1
1
f1 (z)dz = − 2 f1
dw
w
w
Z
1
1 1
f2 (v)
=−
−
dv dw.
w 2πi w V wv − 1
Hence
Res(f1 (z)dz, z → ∞) =
Z
dw
1 dw
(f2 (v)dv),
= Res −
−
w
2πi w V
w=0
= − (1 + m2 ).
The values of the residues of f2 (z)dz or any linear combination can be found similarly.
The residues of G(z)dz, with a1 , a2 ∈ {−1, 0, 1} are listed in Table 5.1.
82
z
tn
G, γ < 0
G, γ > 0
rn = Res−1 (Gdz)
0
eiπ/6
f1 + f2
f1
1
∞
eiπ/3+iγ/3
f2
f2
−m1
0
i
f2
f2
0
∞
−e−iπ/3+iγ/3
−f1 + f2
−f1 + f2
1 + m2 − m1
0
−e−iπ/6
−f1
−f1 + f2
−1
∞
−eiγ/3
−f1
−f1
1 + m2
0
−eiπ/6
−f1 − f2
−f1
−1
∞
−eiπ/3+iγ/3
−f2
−f2
m1
0
−i
−f2
−f2
0
∞
e−iπ/3+iγ/3
f1 − f2
f1 − f2
−1 − m2 + m1
0
e−iπ/6
f1
f1 − f2
1
∞
eiγ/3
f1
f1
−1 − m2
Table 5.1: Poles and residues of the one-form G(z)dz. The two leftmost columns list the
poles of G(z)dz in the z- and in the t-plane respectively. The third and fourth columns
show the functional form of G(z) near those poles for γ < 0 and γ > 0 respectively. And
the fifth column has the values of the residues of these poles.
Hence, we find
G(z) =
12
X
k=1
=
12
X
k=1
1
=
z
rk dt
t − tk dz
2
ie−iγ e2iγ − t6
rk
×−
t − tk
6|b| (1 + e2iγ ) t5
t + t−1
−5
∗ −1
5
√
− C(t + t ) − C (t + t ) ,
3
(5.25)
where C is given by,
1
C= √
eiγ + e2iγ/3 (im1 − eiπ/6 (1 + m2 )) ,
2 3 cos γ
and C ∗ is the complex conjugate of C.
The finite monodromy holds only when the curves U and V have the same endpoints (b
and b∗ ) in complex plane. This condition has not been imposed as yet and we do so now.
The solutions of Re[G(z)dz] = 0 are given by at most a single curve through any point
(for G(z)dz/dt 6= 0). More curves may meet at some point z0 if G(z0 ) = 0 or G(z0 ) → ∞.
83
For the problem under consideration, the multiple images of the curves U and V meet at
z = b(t = 0) and z = b∗ (t → ∞). Using (5.23), compute
dz
dt ∼ 6C(1 + e−2iγ )dt as t → 0,
dt
dz
dt
G dt ∼ −6C ∗ (1 + e2iγ ) 2 as t → ∞.
dt
t
G
(5.26)
(5.27)
The curves meeting at t = 0 and t = ∞ in the t-plane implies that the one-form (5.24)
should vanish there so we require that G(z)dz/dt = 0 at t = 0 in (5.26). This implies that
C = 0.
Solving C = 0 for real and imaginary parts fixes m1 and m2 to be:
2
π+γ
m∗1 = √ cos
,
3
3
γ
2
m∗2 = √ cos − 1,
3
3
(5.28)
π π
for all γ ∈ − ,
.
2 2
(5.29)
In what follows we omit the superscript and it is understood that m1 and m2 are always
fixed in the solvable sector m∗1 and m∗2 .
As an aside, we mention that in the case of the square-triangle tiling with u = 0, the Bethe
Ansatz particle densities parameterised by γ as given by (5.28 & 5.29) automatically satisfy
the 6-fold symmetry constraint in (5.5) for all γ ∈ (−π/2, π/2). Hence the entropy can be
computed analytically precisely in the 6-fold symmetric phase, that is, when all the square
densities are equal i.e. ns− = ns+ = ns0 = m2 . This also includes the 12-fold symmetry
point where ns− = ns+ = ns0 = m2 as well as nt− = nt+ .
In what follows, we need the following change of variables for u in terms of θ and γ:
u(θ, γ) = |b|
cos 3θ
.
sin(γ − 3θ)
(5.30)
On the interval −π/6 ≤ θ < π/6, this relation can be inverted and we will denote by θ̄
the solution of
θ̄(u) =
1
|b|
arctan tan γ −
,
3
u cos γ
for − π/6 ≤ θ̄ < π/6.
(5.31)
When b = b∗ , the eigenvalue can be calculated from (5.25) by considering the following
integral
eiθ
Z
I(θ, γ) = Re
0
t + t−1 dz
√
dt.
z 3 dt
Since z = b(t = 0) is a root of the Bethe Ansatz equation and
Fi (b) = 0
84
mod 2πi.
(5.32)
b∗
V
b∗
0.5
0.2
0.4
0.6
U
1.0
0.5
u− r
b
U
1.0
0.8
ru+
-0.5
-0.5
-1.0
-1.0
V
ru−
0.2
0.4
0.6
0.8
r u+
1.0
b
Figure 5.7: Numerical solutions for the Bethe roots {ui } and {vj } lying on the curves
U and V respectively in the complex plane. Left: N = 152, M1 = 91, M2 = 23, and
γ = −0.07. Right: N = 164, M1 = 91, M2 = 25, and γ = 0.07.
The functions Fi (u(θ)) are given precisely by I(θ, γ) for appropriate intervals of values of
θ (and γ), as can be seen from Table 5.1 and Figure 5.8. For example, for γ > 0, F1 (u(θ))
is given by I(θ) in the sector of the t-plane that lies between the two images of the curve
V and containing an image of the curve U as well as the points t = eiγ/3 (z → ∞) and
t = eiπ/6 (z = 0).
One can see from Figure 5.8 that inside the sector of the t-plane with
R
G(z)dz = F1 (z),
the image of numerical solutions {ui } (in red) forming the curve U is given precisely by
the analytic solution to the Bethe equation I(θ, γ) = 0 mod 2πi (in blue). Similarly,
R
inside G(z)dz = F2 (z) sector of the t-plane, the image of numerical solutions {vj } (in
blue) forming the curve V coincides with the analytic solution to the Bethe equation
I(θ, γ) = 0 mod 2πi (in blue). Outside these two regions the solutions to I(θ, γ) = 0 may
or may not coincide with the images of the Bethe roots.
Thus I(θ, γ) contains all the information required to find the eigenvalue Λ, and is given
by:
θ
π
θ π
−
tan
+
I(θ, γ) = ln tan
12 2
12 2 γ
−
π
θ
γ
θ
−
tan
−
− (1 + m2 − m1 ) ln tan
6
2
6 2 γ+π θ
γ θ − m1 ln tan
.
−
tan
−
6
2
6 2 85
2
2
V
V
2
2
F2
1
F2
F2
F1 + F2
F1 + F2
1
F2
1
1
F1
F1
�2
�1
�2
�1
1
2
1
2
F1
�1
�2
�2
�1 �2
1
�1
21
F1
U
�1
�1
�2
�2
U
�1
�2
Figure 5.8: Multiple images of the Bethe Ansatz roots ({ui }- red, {vj }- black) in the
t-plane for γ < 0 and γ > 0 on the left and the right hand sides respectively. The solid
(blue) lines are the solutions of I(θ, γ) = 0. Grey and black dots on the unit circle are
the images of u = 0 and u → ∞ respectively. The solid (blue) lines are the solutions of
I(θ, γ) = 0.
We shall see later that the allowed domain for θ and γ are
π
6
≤θ<
π
2
and − π2 < γ <
π
2,
in which I(θ, γ) is real and equal to:
θ
π
θ
π
−
tan
+
I(θ, γ) = ln − tan
12 2
12 2
γ−π θ
γ θ
− (1 + m2 − m1 ) ln tan
−
tan
−
6
2
6 2
γ+π θ
γ θ
− m1 ln tan
−
tan
−
.
6
2
6 2
What remains to be done is fixing the constants µ and |b|. As we will see, these can
be expressed as functions of the single parameter γ. First, note that the comparison of
the constants in F1 and I can be conveniently done in the limit as limu→∞ F1 (u), ie.
limθ→γ/3 I(θ, γ), since at that point F1 (u) is independent of vj s. However, the functions
diverge there, so when taking the appropriate limit, a term is added to cancel the divergent
term.
Recall, that F1 (u) was given by:
M2
1 X
ln(u − vj ) + m2 µ
F1 (u) = ln(u) +
N
j=1
so in the limit as u → ∞,
F1 (u) ∼ (1 + m2 ) ln(u) + m2 µ
86
2
Subtracting only the divergent part of ln(u) from F1 , we evaluate the following limit:
Re lim [F1 (u(θ)) + (1 + m2 ) ln(eiθ − eiγ/3 )]
θ→γ/3
cos 3θ
iθ
iγ/3
=Re lim (1 + m2 ) ln |b|
(e − e ) + m2 µ
sin(γ − 3θ)
θ→γ/3
!
#
"
eiθ − eiγ/3
+ m2 µ
=Re (1 + m2 ) ln |b| cos γ lim
θ→γ/3 sin(γ − 3θ)
−|b| cos γ iγ/3
=Re (1 + m2 ) ln
ie
+ m2 µ
3
|b| cos γ
+ m2 µ.
=(1 + m2 ) ln
3
Taking the corresponding limit of I(θ), we have
Re lim [I(θ, γ) + (1 + m2 ) ln(eiθ − eiγ/3 )]
θ→γ/3
"
#
tan( 12 ( γ3 − θ))
= − (1 + m2 )Re lim ln
θ→γ/3
eiθ − eiγ/3
π
π
γ
γ 1
+ ln tan
−
tan
+
.
− (1 + m2 ) ln √
12 6
12 6
3
Hence, we find that
h
i
Re lim I(θ, γ) + (1 + m2 ) ln(eiθ − eiγ/3 )
θ→γ/3
π
π
√
γ
γ = ln tan
−
tan
+
+ (1 + m2 ) ln 2 3.
12 6
12 6
Similarly, taking the limit of F2 (u), as u → ∞ and equating it to the expression for the
limit of I(θ, γ), as θ →
i(π+γ)
3 ,
we arrive at the following system of two equations in four
parameters:
(1 + m2 ) ln
|b| cos γ
3
π
π
√
γ
γ + m2 µ = ln tan
−
tan
+
+ (1 + m2 ) ln 2 3
12 6
12 6
and
m1 ln
|b| cos γ
3
π
π γ √
γ
+ m1 µ = ln tan
+
tan
+
+ m1 ln 2 3.
12 6
4
6
Substituting the expressions for m1 and m2 in terms of γ from (5.28 & 5.29), we solve the
equations above for µ and |b| as functions of γ:
π
π
π γ π
cos γ3
γ
γ γ −
+
+
+
tan
+
ln
tan
tan
,
µ(γ) = − ln tan
12 12
12 6
4
6
12 6
cos π+γ
3
(5.33)
87
√
π
6 3
γ
1
|b(γ)| =
tan
−
tan
(π + 2γ) ×
cos γ
12 6
12
√
γ
π+γ
π γ π
γ 12 ( 3−2 cos 3 ) sec 3
× tan
+
tan
+
,
4
6
12 6
π π
for all γ ∈ − ,
. (5.34)
2 2
µ
2.0
1.5
1.0
0.5
γ
-1.5
-1.0
0.5
-0.5
1.0
1.5
Figure 5.9: Plot of µ as a function of γ ∈ (−π/2, π/2). The model is solvable, that is, the
entropy can be computed analytically for values of µ > µmin due to restrictions imposed
by the Bethe Ansatz solutions.
|b| 1.10
1.08
1.06
1.04
1.02
γ
-1.5
-1.0
0.5
-0.5
1.0
1.5
Figure 5.10: Plot of |b| as a function of γ ∈ (−π/2, π/2). The model is solvable for the
values of b: 0 < |b| < |b|max due to restrictions imposed by the Bethe Ansatz solutions.
In the case when γ = 0, we recover Kalugin’s result:
√
√ 3
√ |b(0)| = 6 3 2 − 3
.
88
The model is thus solvable on a surface where all tile densities and other quantities can
be expressed in terms of the model parameter γ and spectral parameter u.
Next we seek to compute some quantities of interest for the square-triangle-rhombus
tiling, in particular, the value of the maximum bulk entropy. Let us begin with computation of the bulk entropy in the special case of the square-triangle tiling and re-derive some
previously known results in [60, 28, 13].
5.5
Expression for the largest eigenvalue of the square-triangle
random tiling: u = 0 and arbitrary µ
We compute the largest eigenvalue of the transfer matrix of the square-triangle tiling – a
special case of the square-triangle-rhombus tiling model in the absence of rhombi, ie. with
u = 0. Thus, we will recover some previously known results about this tiling model.
The eigenvalue Λ(0) of the transfer matrix given in (5.16), here reduces to
1
ln Λ(0) = m1 µ + lim (ln u − min Re{F1 (u), F1 (u) + F2 (u)}),
u→0
N
(5.35)
where F2 term has been discarded in the expression above, since in the limit as u → 0,
ln(u) − F2 (u) → −∞, so F2 does not contribute to the largest eigenvalue.
The task is to find for which values of parameter γ, which of F1 , F1 + F2 in (5.35)
contribute to the largest eigenvalue Λmax at u = 0. Observe that u = 0 belongs to different
sectors of I(θ, γ) (as given by Table 5.1) subject to the sign of γ. Therefore, the expression
in (5.35) has to be evaluated in three different regions depending on whether u = 0 lies
on V or is in- or outside of the region enclosed by V and U .
Let us denote the points of intersection of the curves V and U with the real axis by u−
and u+ (see Figure 5.7). Points u− = u(θ− , γ) and u+ = u(θ+ , γ) are given by solutions
of the Bethe equations:
I(θ+ , γ) = I(θ− , γ) = 0,
γ ∈ (−π/2, π/2).
One cannot solve for θ− and θ+ in closed form, but from Table 5.1 we know that these
are confined to the following intervals:
π γ
θ+ ∈ − ,
6 3
and θ− ∈
γ + π γ + 2π
,
3
3
,
This information is enough to prove the following proposition used in the computation of
the largest eigenvalue.
89
0.2
Λθ=π/6 (γ)
F1
0.1
F1 + F2
γ
-1.5
-1.0
0.5
-0.5
1.0
1.5
-0.1
-0.2
Figure 5.11: Contributions to eigenvalue Λ for u = 0 θ →
π
6
. The solid curve corresponds
to taking F1 in (5.35), while the dashed curve corresponds to F1 + F2 .
Proposition 4. Let θ− and θ+ be solutions of the Bethe equations:
I(θ+ , γ) = I(θ− , γ) = 0,
γ ∈ (−π/2, π/2),
where
π γ
θ+ ∈ − ,
6 3
and
θ− ∈
γ + π γ + 2π
,
3
3
.
Then
π π
u+ > 0 for all γ ∈ − ,
2 2
and
π u− < 0 for γ ∈ − , 0 ,
2π .
u− > 0 for γ ∈ 0,
2
See appendix (5.A) for proof of proposition 4.
Next, we find out which of F1 and F1 + F2 contributes to the largest eigenvalue in
(5.35) in the two regions: γ < 0 and γ > 0 (or equivalently for u− < 0 and u− > 0).
To calculate the contribution from F1 in the limit as u → 0, corresponds to taking the
following limits of I(θ, γ) (see Table 5.1 or Figure 5.8) in the expression for the eigenvalue
in (5.35):
m1 µ+ lim ln(u(θ)) − I(θ, γ), for γ < 0, and
θ→−π/6
m1 µ+ lim
ln(u(θ)) − I(θ, γ), for γ > 0.
θ→ π/6
90
and for F1 + F2 we calculate the following:
m1 µ+ lim ln(u(θ)) − I(θ, γ), for γ < 0, and
θ→π/6
m1 µ+ lim [ln(u(θ)) − I(θ, γ)] − lim I(θ, γ), for γ > 0.
θ→π/6
θ→π/2
As can be seen from Figure 5.11, when γ < 0 (u− < 0), the largest eigenvalue is given
by F1 + F2 , while for γ > 0 (u− > 0), it is given F1 in (5.35). Therefore, the eigenvalue
Λmax (0, γ) is a smooth curve over the entire interval, given by
1
Λmax (0, γ) = m1 µ+ lim ln(u(θ)) − I(θ, γ), for all γ ∈ (−π/2, π/2).
N
θ→π/6
5.6
Entropy of the square-triangle random tiling: u = 0
We can now compute the bulk entropy σ of the square-triangle random tiling in the
thermodynamic limit, as the size of the lattice goes to infinity: M, N → ∞. (The order
in which these limits are taken is not important here). Recall, that with u = 0, there is
only one independent tile density ns+ of the s+ type square with its associated Boltzmann
weight eµ . All other tile densities will be expressed in terms of ns+ .
The partition function Z over the ensemble of configurations on a lattice of size A =
M × N is given by
Z=
X
weight(config) = TrT M .
configs
Recall that in the limit as M → ∞
Z ∼ ΛM
max ,
and in the thermodynamic limit (as N → ∞ also) the limits converge and we have that
1
ln Z
= lim
ln Λmax (0).
N →∞ N
M,N →∞ M N
lim
The entropy S of the random tiling as a function of ns+ is given by
S = ln #{configuration where the # of squares s+ = Ns+ } .
The bulk entropy σ is then defined by the following limit:
S(ns+ )
,
A→∞
A
σ(ns+ ) = lim
where A = M × N.
91
(5.36)
The partition function can now be re-expressed in terms of σ, as a sum over configurations
of fixed ns+ as follows:
Z=
X
weight(config) =
=
eµ(Ns+ ) eS(Ns+ )
Ns+
configs
X
X
eA(µns+ +σ(ns+ )) .
ns+
In the limit as A = M × N → ∞
Z ∼ max eA(µns+ +σ(ns+ ))
ns+
A µn∗s+ +σ(n∗s+ )
=⇒ Z = e
(5.37)
,
where n∗s+ (γ) is the value maximising (5.37), given by the solution of
∂σ(ns+ )
.
∂ns+
(5.38)
ln Z
= µn∗s+ + σ(n∗s+ ),
A
(5.39)
µ=−
Therefore
lim
A→∞
and from (5.36) and (5.39) above, σ is given by
1
ln Λmax (0) − µn∗s+
N
= lim (ln |u(θ)| − I(θ)) + µm1 − µn∗s+ .
σ(n∗s+ ) =
(5.40)
(5.41)
θ→π/6
Differentiating (5.40) with respect to µ
∂ns+
∂σ ∂ns+
∂ 1
ln Λmax = ns+ + µ
+
∂µ N
∂µ
∂ns+ ∂µ
(5.42)
and substituting in the point (5.38), we compute n∗s+ parameterised by γ
n∗s+ (γ)
−1
∂ 1
1 ∂ ln Λ(0)
∂µ
=
ln Λ(0) =
∂µ N
N
∂γ
∂γ m1 ,m2
m1 ,m2
γ
2
= √ cos − 1 = m2 .
3
3
(5.43)
The fact that we keep m1 and m2 in (5.43) constant at their values of m∗1 and m∗2 while
varying γ, actually means that we allow for infinitesimal values of C. It is non-trivial, but
proven in [57] that G(z) in (5.25) indeed gives the lowest order corrections away from the
solvable line.
Thus, we have found that n∗s+ = ns0 = m2 . And with the help of linear and bilinear
relations (5.3 & 5.4) we can solve for ns− to establish: n∗s+ = ns0 = ns− = m2 , showing
92
that on the line of exact solution the chemical potential is precisely tuned so that all
square densities are equal. Substituting ns+ into (5.41) we give an explicit expression for
the entropy σ as a function of γ alone:
σ = lim (ln |u(θ)| − I(θ)) + µ(m1 − m2 )
θ→π/6
√ !
π γ π γ π
6 3
γ + 2 cos
=2 ln
+
ln tan
+
tan
+
+
cos γ
6
3
4
6
12 6
π γ π γ π
γ −
−
−
+ 2 cos
ln tan
tan
, for γ ∈ (−π/2, π/2).
6
3
4
6
12 6
(5.44)
The expression in (5.44) is equal to Kalugin’s result in [28].
5.7
Lattice dependence and area conversions
Recall that nsi , nti and nr represent the tile densities on the deformed triangular lattice.
Ultimately, we are interested in the computation of densities and bulk entropy on the
original square-triangle and square-triangle-rhombus lattices. Let us re-express the bulk
entropy σ as a function of the total triangle area fraction of the original tiling.
The linear and bilinear relations in Section 5.2 were derived on the deformed triangular
lattice. The bilinear relation (5.4) is homogeneous and as such is independent of the
choice of the underlying lattice. However, the linear relations and the area constraint
(5.2) and(5.3) are lattice dependent since the relative areas of different prototiles vary
with the deformation of the lattice. Let nx and ax denote the density and the area of tile
x respectively on the deformed triangular lattice, and let n?x and a?x denote the tile density
and tile area on the underformed tiling. We can rewrite (5.3) in a general form on the
deformed and original lattices as:
X
nx ax = 1, and
x
X
n?x a?x = 1,
x
however, the area fractions, in general, vary:
nx ax 6= n?x a?x .
To convert between the lattices, we can simply multiply σ and nx by the area ratio
the two lattices:
Nx
Nx A
A
=
= nx ?
?
?
A
A A
A
S
S
A
A
σ? = ? =
=σ ?
A
A A?
A
n?x =
93
A
A?
of
σ?
r
n∗s+ = ns0 = ns− = m2
0.12
0.10
0.08
0.06
0.04
0.02
√
3 ?
4 nt
0.0
0.2
0.4
0.6
0.8
1.0
Figure 5.12: Bulk entropy σ ? as a function of the total triangle area fraction of the original
square-triangle lattice.
where
A? X
=
nx a?x
A
x
(5.45)
In the absence of rhombi, nr = 0, (5.1) and (5.2) give the following linear relations in
particle (world lines) and tile densities:
1
n1 = 1 − m1 = ns+ + nt+
2
1
n2 = m1 − m2 = ns− + nt−
2
n3 = m2 = ns0
(5.46)
Imposing conditions for the 6-fold symmetric phase (ns− = ns+ = ns0 = m2 ) where the
model is solvable, and recalling the areas of prototiles a?t n?t =
√
3 ?
4 nt
and a?si n?si = n?si we
find the area conversion factor (5.45) for the square-triangle lattice to be given by:
√
A?
3
=
nt + 3m2
A
√4
√
3
γ
nt + 2 3 cos
(5.47)
=
4
3
where nt is the total triangle density: nt = nt+ + nt− . With the help of (5.46), one can
express nt as a function of γ alone:
√
γ
nt = 2(1 − 3m2 ) = 4 2 − 3 cos
.
3
(5.48)
See Figure (5.12) for a plot of the bulk entropy σ ? vs the total triangle area fraction
√
3 ?
4 nt
of the original square-triangle lattice. The maximum occurs precisely at the point
of 12-fold symmetric phase, where the area fractions of triangles and squares are equal.
94
5.8
Entropy of the square-triangle-rhombus tiling for arbitrary weights of the rhombus r and square s+
For general values of µ and u > 0 , the eigenvalue in (5.16) is given by:
1
ln Λ(u, µ) = m1 µ + ln(u) − min Re {F1 (u), F1 (u) + F2 (u), F2 (u)} .
N
After some calculations, we find that the largest eigenvalue (in the integrable sector:
m1 = m∗1 and m2 = m∗2 ) is given by
1
ln Λmax (u, µ) = m1 µ + ln(u) − I (θ, γ) ,
N
π
π
π
π
where ≤ θ <
and − < γ < .
6
2
2
2
(5.49)
The domains for θ and γ above together with (5.30) give
γ > 3θ − π =⇒ −π < γ − 3θ < 0 =⇒ u > 0.
This confirms that the domain corresponds to the physically feasible, positive weight of
rhombus tile r. As before, in the thermodynamic limit the partition function can be
expressed in terms of bulk entropy σ:
ln Z
1
=
ln Λmax (u, µ) = n∗s+ µ + n∗r ln u + σ(n∗s+ , n∗r )
A→∞ A
N
lim
and using the Legendre transform, write the bulk entropy σ for the square-trianglerhombus tiling as a function of two independent tile densities as
σ(n∗s+ , n∗r ) = (m1 − n∗s+ )µ + (1 − n∗r ) ln u − I (θ, γ) ,
π
π
π
π
for ≤ θ < and − < γ < ,
6
2
2
2
(5.50)
where the values of n∗s+ and n∗r that maximise (5.50) are given, respectively, by the solutions of
µ=
∂σ(ns+ , nr )
∂σ(ns+ , nr )
and ln u = −
.
∂ns+
∂nr
(5.51)
We may compute n∗s+ and n∗r by taking appropriate derivatives of the largest eigenvalue.
Note that θ and γ are not independent variables; u and µ are, or, rather u and γ. So we
need to be careful that when we are differentiating with respect to one of them, the other
is kept fixed. The average density of s+ squares can be calculated from (5.25) and is given
by the derivative of with respect to µ(γ):
ns+
∂ 1
=
ln Λ(u, µ) = m1 −
∂µ N
(
95
∂I (u, γ)
∂γ
)
u,m1 ,m2
∂µ
∂γ
−1
m1 ,m2
On a curve of fixed u in the γ, t-plane, du = 0, so
∂u
∂γ dγ
+
∂u
∂t dt
= 0, hence
dt
dγ
= − ∂u/∂γ
∂u/∂t ,
and we can now evaluate the expression for ns+ as follows:
( Z
t(u,γ)
∂ 1 t + t−1
∂u
√
ns+ = m1 − Re
− C(t + t−5 ) − C ∗ (t−1 + t5 )
dt+
∂γ u
∂t
3
0
−1
)
∂µ
1 t + t−1
∂u dt(u, γ)
−5
∗ −1
5
√
+ Re
− C(t + t ) − C (t + t )
.
u
∂t
dγ
∂γ m1 ,m2
3
u,m ,m
1
2
Once evaluated, the expression above can be re-expressed back in terms of θ and γ:
8
π + 2θ
π − 2γ
π + γ − 3θ
ns+ = √ cos
sin
sin
,
4
12
6
3
π
π
π
for ≤ θ < and − π/2 < γ < .
6
2
2
The average density of thin rhombi nr can be calculated from (5.32) and is given by
nr =
Evaluating nr at θ =
∂ 1
∂I
ln Λ(u, µ) = 1 −
u
∂ ln u N
∂u Z t
t + t−1 du
d
√
dt u
= 1 − Re
du 0 u 3 dt
Z u
t + t−1
d
√ du
= 1 − Re
du 0
3
−1
t+t
√
= 1 − Re
3
π π 2
,
.
= 1 − √ cos θ, for θ ∈
6 2
3
π
6
– the value corresponding to sending u → 0, we recover nr = 0 as
expected.
Recall (5.1) and (5.2), and now with nr > 0, we have
1
n1 = 1 − m1 = ns− + nt− + nr
2
1
n2 = m1 − m2 = ns+ + nt+
2
n3 = m2 = ns0 .
There are six tile densities, nr , ns− , ns0 , ns+ , nt− , nt+ , of which, due to linear and quadratic
relations (5.3 & 5.4), only four are independent quantities. We can parametrise all six in
terms of the following four: m1 , m2 , nr and ns+ .
The densities n1 and n2 are fixed via the Bethe Ansatz root numbers m1 and m2 , while
96
nr and ns+ can be varied via their Legendre conjugate variables µ and ln u. Thus, we find
nt+ = 2(n2 − ns+ ) = 2(m1 − m2 − ns+ )
nt = (nt− + nt+ ) = 2(1 − m2 − ns− − ns+ − nr )
=
2(m21 + ns+ − 2m1 (m2 + ns+ ) + m2 (1 − nr + ns+ ))
m1
ns0 = 1 − n1 − n2 = m2
ns− =
ns+ (m1 − m2 − 1) + (m2 − m1 )(nr + m1 − 1)
.
m1
The areas of a rhombus and a triangle on the original lattice are
1
2
√
and
3
4
respectively.
Hence, area conversion factor is given by
√
1
3
A?
=
nt + m2 + ns− + ns+ + nr
A
4 h
2
γ
√
γ
γ √
γ i
= cos
3 − 2 2 cos
2 3 − 3 cos −
− θ − sin
.
6
6
6
6
Lastly, the expression for the bulk entropy σ ? =
A
A? σ(θ, γ)
as a function of rhombi and
triangle area fractions has been computed and although it is not pretty, we state below
for completeness.
π
θ
π
θ
A 1
−3
ln
−
tan
−
tan
+
+
A? 3
12 2
12 2
√
cos(3θ)
+2 3 cos θ ln(|b|) + µ + ln
+
sin(γ − 3θ)
γ γ+π θ
γ θ
ln tan
−
tan
−
+
+ cos
3
6
2
6 2
π+γ
π + γ − 3θ
2π + γ − 3θ
π + γ − 3θ
+ cos
ln tan
tan
− µ sin
,
3
6
6
3
(5.52)
σ ? (θ, γ) =
where
π
π
π
π
≤ θ < , − < γ < and µ and |b| are as given by (5.33 & 5.34)
6
2
2
2
See Figure 5.13 for the graph of σ ? (θ, γ).
Let us now briefly study the symmetries of the original square-triangle-rhombus tiling.
Recall that in the absence of rhombi, the integrable line was precisely the one where all
square densities are equal. However, the introduction of a ‘thin’ rhombus r breaks the
symmetry between s+ on one hand, and s0 and s− on the other. This can be observed by
looking at the images of tiles under the reflection in either diagonal of the rhombus tile r:
r ↔ r,
s+ ↔ s+ ,
s0 ↔ s− ,
97
t+ ↔ t− .
(5.53)
σ?
1 ?
2 nr
√
3 ?
4 nt
Figure 5.13: Bulk entropy σ ? as a function of the total triangle area fraction
rhombi area fraction
1 ?
2 nr
√
3 ?
4 nt
and
of the original square-triangle-rhombus lattice.
Furthermore, if one breaks the symmetry between the triangles t− and t+ , then it also
breaks the symmetry between between s− and s0 squares, since t+ tiles fit on the same
side of the rhombus as s0 tiles do (and t− fit on the same side as s− ). Thus, thinking
naively, if there is an abundance of t+ tiles, then there is less room for s0 tiles.
The next important observation is that the tiles which are invariant under this symmetry (being r and s+ ) are precisely the ones the Boltzmann weights (u and eµ ) have
been assigned to.
One might ‘predict’ that a typical state assumed by the system free to arrange itself into
a configuration (for some fixed u and µ), will be one obeying the symmetry in (5.53). The
symmetry allows for a greater number of tiling configurations and the state of maximum
entropy indeed occurs at some point possessing this symmetry.
98
n?rmax ≈ 0.359
σ?
n?r = 0.5
0.4
n?r = 0.1
0.3
n?r = 0.8
0.2
n?r
0.1
n?r = 0.01
=0
n?r = 0.95
√
3 ?
4 nt
0.0
0.0
0.2
0.4
0.6
0.8
Figure 5.14: Plot of σ ? as a function of the total triangle area fraction
1.0
√
3 ?
4 nt
at various
fixed values of rhombus densities n?r . n?rmax is the value of rhombus density yielding the
maximum entropy of the system at some value of n?t .
5.8.1
The line of maximum symmetry and entropy for arbitrary u > 0
and µ
On the solvable surface parameterised by θ and γ, there is a line: θ =
2
3γ
+
π
6
which
preserves the symmetries between t− and t+ as well as s− and s0 tiles:
ns− = ns0 ,
nt− = nt+ .
This line of symmetry is the line of maximal entropy regardless of the choice of the
underlying lattice, however the value of maximum entropy on this line and its location are
lattice dependent. The results for the densities and the tile area fractions are presented
here on the original square-triangle-rhombus lattice, which is indicated as before, with ?
in the superscript. So we have that
n?s− = n?s0 ,
n?t− = n?t+ .
It is possible to compute exactly all the tile densities on this line. The densities and the
line itself can be written in terms of any one parameter and (where possible) we choose γ
99
θ
θ=
1.5
γ
3
+
π
3
( π2 , π2 )
B
c
a
θ = 23 γ +
(γmax , θmax )
1.0
d
C
(− π2 , π6 )
b
0.5
π
6
A
( π2 , π6 )
D
γ
-1.5
-1.0
0.5
-0.5
1.0
1.5
Figure 5.15: Domain of σ in terms of γ and θ, showing the boundary lines a, b, c of the
domain, the line of symmetry d and the point (γmax , θmax ) at which the entropy σ ? of the
system (on the solvable surface) is maximal.
for convenience:
π γ A
4
γ
√ sin
+
sin
,
A?
6
3
3
3
4
π + 2γ
π + 4γ
A
A
A
?
?
√
2−
cos
+ 4 cos
,
nt− = nt+ = ? nt− = ? nt+ =
A
A
A?
6
6
3
A
A
4
π+γ
2
π + 4γ
√ cos
n?s+ = ? ns+ =
− √ cos
,
?
A
A
3
6
3
3
A
A
A
2
γ
A
?
?
√ cos − 1 ,
ns0 = ns− = ? ns0 = ? ns− = ? m2 =
(5.54)
A
A
A
A?
3
3
where γ is now restricted to lie in γ ∈ 0, π2 . The area conversion factor on this line is
n?r
A
= ? nr =
A
given by
γ √
γ √
γ A?
= 2 cos2
2 3 − 3 cos
+
3 − 2 sin
.
A
6
3
3
(5.55)
The entropy on this line parameterised by γ is given by:
?
σmax
line (γ)
A
4µ
π+γ
γ
π + 2γ
= ? − √ cos
− ln tan tan
A
3
3
6
3
2
π + 4γ
+ √ cos
(ln(|b|) + µ + ln[2 sin γ])
6
3
π − 2γ
π + 2γ
tan
+m1 µ + ln tan
12
12
π γ π
π + 2γ
+(1 − m1 + m2 ) ln tan
+
tan
.
, where γ ∈ 0,
4
6
12
2
100
A
A?
in the expression above is given by (5.55) and µ and |b| are as in (5.33 & 5.34).
The point (θmax , γmax ) which gives the maximum value of the entropy of the tiling
?
model lies on this line, that is σmax
= σ ? ( 32 γmax +
π
6 , γmax ).
Unfortunately, it is not
?
possible to compute the exact value of γmax . The values of γmax and σmax
are solutions of
transcendental equations and are approximately equal to:
γmax = 0.68849(5),
σmax = σ ? (γmax ) = 0.43865(4).
?
Note that σmax
= σ ? (γmax ) 6= σmax 6= σ(γmax ).
The approximate values of all the tile area fractions of the original tiling at γmax are
as follows:
1 ?
n (γmax ) = 0.23279(5)
2 r
n?s+ (γmax ) = 0.03724(7)
n?s0 (γmax ) = n?s− (γmax ) = m?2 (γmax ) = 0.16123(4)
√
√
3 ?
3 ?
nt− (γmax ) =
n (γmax ) = 0.20374(5)
4
4 t+
?
σmax
line
0.4
0.3
0.2
0.1
1 ?
2 nr
0.2
0.4
0.6
0.8
1.0
? (γ) vs. rhombi area fraction
Figure 5.16: Plot of σmax
1 ?
2 nr
Looking at Figure 5.16 we observe that on the line of maximum symmetry the bulk
entropy σ ? is a convex function of rhombi tile density (and also of the total triangle
density). The convexity of the entropy signifies total stability of the phases of the system
possessing this symmetry. Note, that this was not the case for the square-triangle tiling,
where the maximum entropy was a concave function of the total triangle density (see
101
Figure 5.11). In that case, the entropy of the homogeneous tiling was less than the sum of
the entropies of the pure crystalline and quasi-periodic phases. It was only the geometry
of the system that prohibited it from separating into a combination of two distinct phases.
Therefore, we conclude that introducing rhombi into the system, stabilises it for all values
of parameters on the line of maximum symmetry.
5.8.2
Boundary points and lines on the solvable surface σ ?
The limiting lines and points on the boundary of the solvable region are labelled a, b, c
and A, B, C respectively and the images of these on the solvable surface σ ? are shown in
Figure 5.17. The lines are parametrised by γ, with the exception line c, which is given in
terms of θ.
Line a: θ →
γ
3
+ π3 ;
σa? (γ) =
n?s+ = n?s− = n?t− = 0
lim
σ ? (θ, γ) = 0,
θ→γ/3+π/3
and the tile area fractions in terms of γ are:
1 ?
A
nr = ?
2
A
"
√
#
π+γ
1
3
−
cos
2
3
3
A
m2
A? "
#
√
√
A
γ
γ
3 1
3
= ?
− cos −
sin
A
2
2
3
2
3
n?s0 =
√
3 ?
n
4 t+
π π
.
where γ ∈ − ,
2 2
This curve corresponds to the tiling being made up of one type of square s0 , one type of
triangle t+ and rhombi r.
Line b: θ → π6 ;
n?r = 0
σb? (γ) = lim σ ? (θ, γ) =
θ→π/6
"
√ !
π γ h
π γ π
6 3
γ i
A
= ? 2 ln
+ 2 cos
+
ln tan
+
tan
+
+
A
cos γ
6
3
4
6
12 6
π γ h
π γ π
π π
γ ii
+2 cos
−
−
−
ln tan
tan
, γ∈ − ,
.
6
3
4
6
12 6
2 2
The expression σb? (γ) =
A
A? σ(γ)
above is, of course, precisely the expression for the entropy
of the square-triangle tiling with σ as given in (5.44). The area conversion factor is on
102
σ?
d
D
B
b
1 ?
2 nr
a, c
√
3 ?
4 nt
A, C
Figure 5.17: Bulk entropy σ ? as a function of the total triangle area fraction
rhombi area fraction
1 ?
2 nr
√
3 ?
4 nt
and
with labelled limiting lines and points
this line is
A? √
γ
= 2 3 − 3 1 + cos
.
A
3
The area fractions of various tiles on the square-triangle lattice in terms of γ are as follows:
A
n?s0 = n?s+ = n?s− = ? m2
A
"
#
√
√
√
3
3 ?
A
γ
3
γ
n = ?
3 − cos −
sin
4 t+
A
2
3
2
3
" √
#
√
2 3 cos γ3 − 3 2 sin γ3 + 1
3 ?
A
√
n = ?
.
4 t−
A
2 cos γ3 − 3 sin γ3
Line c: γ → π2 ;
n?s+ = n?s0 = n?t+ = 0
The curve c is parameterised by θ (it cannot be parameterised by γ being perpendicular
to the direction of γ).
σc? (γ) = limπ σ ? (θ, γ) = 0.
γ→ 2
103
The area conversion factor on this curve in terms of θ ∈
π π
6, 2
is given by
√ A?
1 √
3 − 1 + cos θ + 2 − 3 sin θ
=
A
2
And the tile are fractions are:
A 1 cos θ
1 ?
n = ?
− √
2 r
A 2
3
A
cos θ
?
ns− = ? sin θ + √ − 1
A
3
√
√
A 1
3 ?
n = ?
cos θ − 3(sin θ − 1) ,
4 t−
A 2
θ∈
π π ,
.
6 2
The line c is similar to a in the sense that it also corresponding to the tiling being made
up of only one type of square, one pair of triangles and rhombi, but now with s− and t−
instead of s0 and t+ .
Point A: (θ, γ) → ( π6 , π2 )
n?t− = 1
n?r = n?s0 = n?s+ = n?s− = n?t+ = 0
This point corresponds to the tiling made up entirely of t− type triangles.
Point B: (θ, γ) → ( π2 , π2 )
n?r = 1
n?s0 = n?s+ = n?s− = n?t− = n?t+ = 0
This point corresponds to the tiling made up entirely of rhombi r.
Point C: (θ, γ) → ( π6 , − π2 )
n?t+ = 1
n?r = n?s0 = n?s+ = n?s− = n?t− = 0
This point corresponds to the tiling made up entirely of t+ type triangles.
104
Figure 5.18: A phase of the square-triangle-rhombus random tiling dominated by t− triangles.
Point D: (θ, γ) → ( π6 , 0)
n?r = 0
n?s0 = n?s+ = n?s− =
n?t− = n?t+ =
1
6
1
4
This point corresponds to the 12-fold symmetric phase of the square-triangle tiling. At
this point the area fractions of triangles and squares are equal.
5.A
Proof of Proposition 4
Proof. Recall that u can be parameterised in terms of θ and γ and was given in (5.30) by
u(θ, γ) = |b|
cos 3θ
.
sin(γ − 3θ)
The allowed interval for θ+ gives
−
π
γ
π
< θ+ < <
=⇒ cos(3θ+ ) > 0, and
6
3
6
γ−π
π
γ
< − < θ+ <
=⇒ 0 < γ − 3θ+ < π =⇒ sin(γ − 3θ+ ) > 0
3
6
3
105
hence we conclude
π π
for all γ ∈ − ,
.
2 2
u+ > 0
Next, for θ− we have
γ+π
γ + 2π
< θ− <
=⇒ sin(γ − 3θ− ) > 0,
3
3
for all
γ ∈ (−π/2, π/2).
Also, from the intervals, we have that
γ+π
π
γ + 2π
5π
< <θ− <
<
3
2
3
6
for
γ ∈ (−π/2, π/2),
giving
π γ + 2π
θ− ∈
, and
,
2
3
γ+π π
θ− ∈
.
,
3
2
cos(3θ− ) > 0
for
cos(3θ− ) < 0
for
Hence we conclude that
π γ + 2π
θ− ∈
,
=⇒ u− > 0, and
2
3
γ+π π
θ− ∈
,
=⇒ u− < 0.
3
2
(5.56)
(5.57)
Finally, we need to prove that
π γ + 2π
,
=⇒ u− > 0, and
2
3
γ+π π
γ < 0 =⇒ θ− ∈
,
=⇒ u− < 0.
3
2
γ > 0 =⇒ θ− ∈
(5.58)
(5.59)
To this end, we show the following:
• I(θ, γ) is monotonically decreasing in θ in the whole interval θ ∈
γ∈
γ+π γ+2π
3 , 3
for all
(− π2 , π2 ),
• I(θ, γ) has a root in the interval θ− ∈
π γ+2π
2, 3
for all γ ∈ − π2 , π2 ,
• the series expansion of I(θ, γ) in γ at θ = π2 , the monotonicity of I and the existence
of θ− imply the expression in (5.58 & 5.59).
106
Differentiating (5.32) with respect to θ, we obtain
" Z iθ
#
e
t + t−1 dz
d
∂I(θ, γ)
√
= Re
dt
∂θ
dθ 0
z 3 dt
2
i(t + 1) dz
√
= Re
z 3 dt
"
#
√
2 3i(1 + e2iγ )t5
= Im
(eiγ − t3 )(eiγ + t3 )(t4 − t2 + 1)
√
2 3i(1 + e2iγ )t5
= iγ
.
(e − t3 )(eiγ + t3 )(t4 − t2 + 1)
∂I(θ,γ)
∂θ
vanishes only at t ≡ eiθ = 0 which is outside the domains of parame
γ+2π
ters γ and θ. Since the derivative is negative at a point inside the interval θ ∈ γ+π
,
,
3
3
π
π
for example ∂I | π (θ = ei(( 6 +π)/3+ 12 ) ) ≈ −1.43), it is negative everywhere in the interval.
The derivative
∂θ γ= 6
I(θ, γ) is a continuous function in θ in the same interval, hence we conclude that I(θ, γ)
is monotonically decreasing in the whole interval.
Next, we have that I(θ, γ) is positive at a point inside the interval, eg: I(θ =
π
6)
17π
36 , γ
=
≈ 0.25, and negative at the boundary: limθ→ γ+2π I(θ, γ) = −∞. This implies that
3
γ+2π
I(θ, γ) must have one root inside the interval, that is: θ− ∈ ( γ+π
3 , 3 ).
Finally, at θ = π2 , I(θ, γ) has the following series expansion in γ around γ = 0:
I
2 √
√ 3 − ln(2 + 3) γ + O[γ]3 ,
,γ =
2
3
π
where the coefficient in front of γ is positive:
√ 2 √
3 − ln(2 + 3) ≈ 0.28.
3
Hence, given a small > 0,
I
Since I
π
2,
π π
, > 0, and I
, − < 0.
2
2
> 0 and given that I(θ, γ) is monotonically decreasing in θ in the whole
interval, it means that I(θ, γ) takes only positive values everywhere to the left of π/2 in
that interval, hence we conclude that θ− > π/2, and (5.58) gives that u− > 0. Analogous
arguments for γ < 0 give I π2 , − < 0 and imply that u− < 0. This completes the
proof.
107
Chapter 6
Conclusion and Outlook
The square-triangle-rhombus tiling model has been solved exactly in the thermodynamic
limit on a two parameter subspace of the four parameter space. The two independent
variables defining this subspace are the tile weights eµ and u assigned to one type of
square and the rhombus tile respectively.
An integrable extension of the square-triangle tiling to the square-triangle-rhombus
tiling is possible with the inclusion of the spectral parameter, which can be interpreted
as the weight us of the new tile. This guarantees that the new model, like the original,
is integrable for all values of its parameters. Although the physical quantities of the new
model are now functions of four independent variables, the solvable sector restricted by
the solutions of the Bethe equations is the same for both models. In this sector the Bethe
Ansatz particle densities are parameterised by a single parameter γ. Staying in this sector
imposes restrictions on the computable range of the quantities for the system, including
tile densities, with a consequence that they are expressed in terms of parameterisations of
µ(γ) and u only.
We were successful in obtaining a two-dimensional exactly solvable surface, computing
the entropy and the densities as functions of u and µ. It was found that introducing
rhombi into the system breaks the symmetry between the three types of squares exhibited
by the square-triangle tiling. This is due to the fact that the rhombus tile is restricted to
one fixed orientation, unlike the squares and the triangles. We have derived an expression
for the entropy on a curve characterised by the symmetries between two types of squares
and two pairs of triangles. The maximum entropy for the system was found to lie at a
point on this curve for particular values of u and µ.
An important observation is that the injection of rhombi into the system stabilises
it, making the homogeneous tilings entropically favourable over phase-separated tilings.
108
For the tilings dominated by triangles (in the thermodynamic limit), the entropy function
becomes convex immediately upon the introduction of rhombi. There exists a (small) value
of rhombi density, above which, the entropy function is convex for all triangle or square
densities, signifying the overall stability of the system. The equation of the critical line
(as a function of rhombi density) on which this transition occurs has not been computed
and the problem remains outstanding.
Another observation is that although the tiling has maximum entropy on the line of
maximum symmetry, this symmetry is no longer quasiperiodic, but is simply two-fold.
This is due to the fact that the rhombus tile posses only a two-fold symmetry and appears
in only one orientation. A natural desire then, is to extend the model to include rhombi
in more than one orientation, however such models are no longer integrable.
We believe that the approach of generalising tiling models by introducing a spectral
parameter and associating it with a new tile may be applicable to other random tiling
models, like the 8- and 10-fold quasi-periodic tiling of rectangles and triangles studied in
[11], [12], and [14]. It would be interesting to identify the new tiles and investigate the
entropy, phase diagrams and the underlying symmetries of these models.
With regards to the model studied in this thesis, ideally we would like to expand the
solvable sector by finding general solutions to the Bethe equations for arbitrary values
of the Bethe Ansatz particle densities. This problem has recently been understood to
be related to the Painlevé VI equation for certain values of its parameters. Solving this
equation is an interesting open problem.
In the past few years, an interesting connection has been made between random tilings
and combinatorics. The square-triangle tiling has been related in [45] to so-called puzzles
[31], whose enumeration reproduces Littlewood–Richardson coefficients, see Figure 6.1.
See also [61] for the connection between tilings and Schur functions.
The extension of the square-triangle tiling to the square-triangle-rhombus tiling by
0 0
0
1 1
1
1
0
1 1 1 0 0 0
1
0
1 1 1 0 1 1
1
1
1
0
1
0
0 0 1 1 1 1
1
1
1
0
0
1 1 1 1
1 1 1 1 0 0 0
1
1
1
0
0
1
0
0
0 0 0
0 0 1 1
1
1
1
1
0
0
0
Figure 6.1: A patch of the square-triangle random tiling in a finite domain on the left is
related to a Knutson–Tao–Woodward puzzle on the right.
109
inclusion of spectral parameters naturally produces a generalisation of the Littlewood–
Richardson coefficients in the equivariant case where they live in the ring of polynomials
of the inhomogeneities. This method has been used to recover some known formulae as
well as to discover a new one [61] for double Schur functions. One could hope to be able
to extend this method to other integrable random tilings mentioned earlier, with the goal
of understanding the ring structure of the underlying polynomials.
110
Bibliography
[1] M. Baake, R. Klitzing, and M. Schlottmann, Fractally shaped acceptance domains of
quasiperiodic square-triangle tilings with dedecagonal symmetry, Physica A: Statistical
Mechanics and its Applications 191 (1992), no. 1–4, 554 – 558.
[2] R. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.
[3]
, Completeness of the bethe ansatz for the six and eight-vertex models, Journal
of Statistical Physics 108 (2002), 1–48.
[4] C. Beeli and W. Steurer, Proceedings of the workshop on quasicrystal structure analysis, Oldenbourg Wissenschaftsverlag, 2000.
[5] S. Ben-Abraham and F. Gähler, Covering cluster description of octagonal mnsial
quasicrystals, Phys. Rev. B 60 (1999), 860–864.
[6] H. Bethe, Zur theorie der metalle, Zeitschrift fr Physik A Hadrons and Nuclei 71
(1931), 205–226, 10.1007/BF01341708.
[7] L. Bindi, P. Steinhardt, N. Yao, and P. Lu, Natural quasicrystals, Science 324 (2009),
1306–1309.
[8] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press,
Cambridge, 1995, Corrected reprint of the 1994 original.
[9] H. Chen, D. Li, and K. Kuo, New type of two-dimensional quasicrystal with twelvefold
rotational symmetry, Phys. Rev. Lett. 60 (1988), 1645–1648.
[10] R. Christoph, An alternative view on quasicrystalline random tilings, Journal of
Physics A: Mathematical and General 32 (1999), no. 50, 8823.
[11] J. de Gier, Random tilings and solvable lattice models, 1998.
[12] J. de Gier and B. Nienhuis, The exact solution of an octagonal rectangle-triangle random tiling, Journal of Statistical Physics 87 (1997), 415–437, 10.1007/BF02181494.
111
[13]
, Integrability of the square-triangle random tiling model , Phys. Rev. E (3) 55
(1997), no. 4, 3926–3933.
[14]
, Bethe Ansatz solution of a decagonal rectangle-triangle random tiling, J.
Phys. A 31 (1998), no. 9, 2141–2154.
[15] N. Destainville, M. Widom, R. Mosseri, and F. Bailly, Random tilings of high symmetry: I. mean-field theory, Journal of Statistical Physics 120 (2005), 799–835,
10.1007/s10955-005-6989-y.
[16] J. Dubois, Useful quasicrystals, World Scientific, 2005.
[17] V. Elser, Comment on “Quasicrystals: A new class of ordered structures”, Phys. Rev.
Lett. 54 (1985), 1730–1730.
[18] L. Faddeev, How the algebraic Bethe Ansatz works for integrable models, Symétries
quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, pp. 149–219.
[19] P. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010.
[20] F. Franchini, Notes on Bethe Ansatz techniques, http://people.sissa.it/
~ffranchi/BAnotes.pdf.
[21] M. Gardner, Penrose tiles to trapdoor ciphers, Freeman, 1989.
[22] B. Grunbaum and G. Shephard, Tilings and patterns, Dover Books on Mathematics
Series, Dover Publications, 2012.
[23] L. He, Y. Wu, and K. Kuo, Decagonal quasicrystals with different periodicities along
the tenfold axis in rapidly solidified Al65 Cu20 M15 ; (M=Mn, Fe, Co or Ni), Journal
of Materials Science Letters 7 (1988), 1284–1286, 10.1007/BF00719959.
[24] C. Henley, Random tilings with quasicrystal order: transfer-matrix approach, Journal
of Physics A: Mathematical and General 21 (1988), no. 7, 1649.
[25] T. Ishimasa, H. Nissen, and Y. Fukano, New ordered state between crystalline and
amorphous in ni-cr particles, Phys. Rev. Lett. 55 (1985), 511–513.
[26] C. Janot, Quasicrystals: A primer , Monographs on the Physics and Chemistry of
Materials, Oxford University Press, 1997.
[27] K. Johansson, Non-intersecting paths, random tilings and random matrices, Probability Theory and Related Fields 123 (2002), 225–280, 10.1007/s004400100187.
112
[28] P. Kalugin, The square-triangle random-tiling model in the thermodynamic limit, J.
Phys. A 27 (1994), no. 11, 3599–3614.
[29]
, Low-lying excitations in the square - triangle random tiling model , Journal
of Physics A: Mathematical and General 30 (1997), no. 20, 7077.
[30] C. Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, SpringerVerlag, New York, 1995.
[31] A. Knutson, T. Tao, and C. Woodward, The honeycomb model of GLn (C) tensor
products. II. Puzzles determine facets of the Littlewood-Richardson cone, J. Amer.
Math. Soc. 17 (2004), no. 1, 19–48 (electronic).
[32] V. Korepin, N. Bogoliubov, and A. Izergin, Quantum inverse scattering method and
correlation functions, Cambridge Monographs on Mathematical Physics, Cambridge
University Press, Cambridge, 1993.
[33] H. Kramers and G. Wannier, Statistics of the two-dimensional ferromagnet. I, Phys.
Rev. (2) 60 (1941), 252–262.
[34] E. Lassettre and J. Howe, Thermodynamic properties of binary solid solutions on
the basis of the nearest neighbor approximation, The Journal of Chemical Physics 9
(1941), no. 10, 747–754.
[35] D. Levine and P. Steinhardt, Quasicrystals: A new class of ordered structures, Phys.
Rev. Lett. 53 (1984), 2477–2480.
[36] E. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967), 162–172.
[37] P. Lu and P. Steinhardt, Decagonal and quasi-crystalline tilings in medieval islamic
architecture, Science 315 (2007), no. 5815, 1106–1110.
[38] S. Majid, A quantum groups primer , London Mathematical Society Lecture Note
Series, Cambridge University Press, 2002.
[39] E. Montroll, Statistical mechanics of nearest neighbor systems, The Journal of Chemical Physics 9 (1941), no. 9, 706–721.
[40] B. Nienhuis, Exact solution of random tiling models, Physics Reports 301 (1998),
no. 13, 271 – 292.
[41] M. Okado and H. Yamane, R-matrices with gauge parameters and multi-parameter
quantized enveloping algebras, Special functions (Okayama, 1990), ICM-90 Satell.
Conf. Proc., Springer, Tokyo, 1991, pp. 289–293.
113
[42] M. Oxborrow and C. Henley, Random square-triangle tilings: A model for twelvefoldsymmetric quasicrystals, Phys. Rev. B 48 (1993), 6966–6998.
[43] R. Penrose, Pentaplexity a class of non-periodic tilings of the plane, The Mathematical Intelligencer 2 (1979), 32–37, 10.1007/BF03024384.
[44] J. Perk and C. Schultz, New families of commuting transfer matrices in q-state vertex
models, Phys. Lett. A 84 (1981), no. 8, 407–410.
[45] K. Purbhoo, Puzzles, tableaux, and mosaics, J. Algebraic Combin. 28 (2008), no. 4,
461–480.
[46] N. Reshetikhin, Calculation of the norm of Bethe vectors in models with SU(3) symmetry, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 150 (1986),
no. Voprosy Kvant. Teor. Polya i Statist. Fiz. 6, 196–213, 223.
, Multiparameter quantum groups and twisted quasitriangular Hopf algebras,
[47]
Lett. Math. Phys. 20 (1990), 331–335.
[48] C. Richard, M. Höffe, J. Hermisson, and M. Baake, Random tilings: Concepts and
examples, J. PHYS. A: MATH. GEN 31 (1998), 3–1.
[49] C. Schultz, Solvable q-state models in lattice statistics and quantum field theory, Phys.
Rev. Lett. 46 (1981), 629–632.
, Eigenvectors of the multicomponent generalization of the six-vertex model ,
[50]
Phys. A 122 (1983), no. 1-2, 71–88.
[51] D. Shechtman, I. Blech, D. Gratias, and J. Cahn, Metallic phase with long-range
orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984), 1951–
1953.
[52] E. Sklyanin, L. Takhtadzhyan, and L. Faddeev, Quantum inverse problem method. I ,
Theoretical and Mathematical Physics 40 (1979), 688–706, 10.1007/BF01018718.
[53] N. Slavnov, The algebraic Bethe Ansatz and quantum integrable systems, Uspekhi
Mat. Nauk 62 (2007), no. 4(376), 91–132.
[54] J. Suck, M. Schreiber, and P. Häussler, Quasicrystals: An introduction to structure,
physical properties, and applications, Springer Series in Materials Science, Springer,
2002.
[55] B. Sutherland, Exact solution of a two-dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett. 19 (1967), 103–104.
114
[56] L. Takhtadzhan and L. Faddeev, The quantum method of the inverse problem and the
heisenberg XY Z model , Russian Mathematical Surveys 34 (1979), no. 5, 11.
[57] A. Verberkmoes and B. Nienhuis, Bethe Ansatz solution of triangular trimers on the
triangular lattice, Phys. Rev. E 63 (2001), 066122.
[58] N. Wang, H. Chen, and K. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59 (1987), 1010–1013.
[59] M. Wheeler, Scalar products in generalized models with SU(3)-symmetry, (2012),
arXiv:1204.2089.
[60] M. Widom, Bethe Ansatz solution of the square-triangle random tiling model , Phys.
Rev. Lett. 70 (1993), 2094–2097.
[61] P. Zinn-Justin, Littlewood-Richardson coefficients and integrable tilings, Electron. J.
Combin. 16 (2009), no. 1, Research Paper 12, 33.
115