Dilute loop models and the BMW algebra

(1)

Dilute loop models and the BMW algebra

Jeroen Dekker

July 1, 2016

Bachelor Thesis Mathematics, Physics and Astronomy Supervisors: Prof. dr. Bernard Nienhuis, Prof. dr. Jasper Stokman

Korteweg-de Vries Institute for Mathematics Faculty of Science

(2)

Abstract

In this thesis the link between the BMW algebra and the Izergin-Korepin 19-vertex model is studied. An explicit representation of the BMW algebra on (C3)⊗n and its Baxterization are presented. The R-matrix for a critical 19-vertex model is explicitly constructed. Under certain constraints this model is shown to be equivalent to a dilute loop model without self-intersections. It is then demonstrated that under these constraints the general 19-vertex model reduces to the Izergin-Korepin model. Moreover, the R-matrix of this model under those constraints is equal to the Baxterized representation of the BMW algebra. This demonstrates the relation between a dilute loop model and the BMW algebra. Furthermore, the definition of the BMW category and a generalization of skein modules are introduced. These give rise to a natural action of the BMW algebra on a generalization of the vector space of perfect matchings.

Title: Dilute loop models and the BMW algebra Author: Jeroen Dekker, eikenrode02@gmail.com, 10617345

Supervisors: Prof. dr. Bernard Nienhuis, Prof. dr. Jasper Stokman Second graders: Dr. Raf Bocklandt, Prof. dr. Kareljan Schoutens Date: July 1, 2016

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math

(3)

1 Introduction

Symmetry plays a major role in modern-day physics. Many exactly solvable statistical models exhibit some form of symmetry. This, however, is usually not apparent at first sight. Our goal is to relate mathematical structures with different models in statistical physics by means of their symmetry. This can be used as an approach to resolve and also construct quantum integrable systems. First we will explain what is meant with a symmetry.

In general, a symmetry may be described as an invariance of the system under certain operations. We can interpret this in different ways. On the one hand we have topological symmetry. Here the topological picture is left invariant under different transformations. For instance a triangle is invariant under rotations of 2π₃ or 4π₃ as well as under three different reflections. On the other hand we have a more algebraic notion of symmetry. This is expressed in the form of equations, e.g. two operators A and B that commute satisfy the simple equation AB = BA.

The set of transformations that leave the system invariant form an algebra. We therefore want to use the language of algebras to identify the different symmetries in statistical models. Moreover, this language allows us to switch between these two interpretations of symmetry.

We will focus on statistical models that are solvable using the transfer matrix method of calculating the partition function. To determine if a model is solvable the Yang-Baxter equa-tion is of great importance: whenever a given system is consistent with this equaequa-tion it is integrable. It owes its name to the work of Yang (1969) [21] and Baxter (1972) [15]. The equation is based upon the Bethe Ansatz, which is a method of finding exact solutions for certain statistical models. It is in the Yang-Baxter equation that the symmetry of such a statistical model is manifest.

There are two complementary goals of this text: Firstly we want to find representations of the BMW algebra using its algebraic and topological definition. Secondly we want to explore the Izergin-Korepin spin-1 model and its corresponding loop model, and relate their symmetry to some algebraic structure. Surprisingly, it turns out that sometimes these two goals are interlinked. Representations can give rise to statistical models and conversely the algebraic structure of these models can be given by representations of algebras.

We illustrate this deep connection between the mathematics and physics by considering the Temperley-Lieb algebra and certain statistical models. It is not of vital importance to com-pletely comprehend the following result, this is solely a motivation for this thesis. However, we strongly encourage the interested reader to examine [27] for the necessary derivations.

(6)

• The Hamiltonian of the one-dimensional Heisenberg spin-1/2 chain of n particles may be described in terms of the transfer matrix of the 6-vertex model [8, 15]. The state space of this quantum spin-system is given by (C2)⊗n, each spin in the chain is represented by a vector in C2. The transfer matrix acts on this space and gives the statistical probability of transitioning from one state to another. It is possible to write the transfer matrix as product of so-called R-matrices which only act non-trivially on 2 neighboring tensor components. These R-matrices satisfy the Yang-Baxter equation. Furthermore they provide the defining relations of the Temperley-Lieb algebra, that is they form a representation of this algebra!

The Temperley-Lieb algebra consists of diagrams with two rows of n points and non-intersecting lines connecting these points. The multiplication is given by concatenation of diagrams and each closed loop may be resolved by a factor δ. It can be defined by its generators ei = at strands i and i + 1, modulo the relations e2i = δei, eiej = ejei

for |i − j| > 1 and eiei+1ei = ei. A typical element is shown in figure 1.1.

Figure 1.1: A typical element of the Temperley-Lieb algebra.

By means of Baxterisation one could have also followed above method in reverse, by starting from a representation of the Temperley-Lieb algebra one is able to construct an R-matrix that describes the spin-1/2 chain.

• We know that the 6-vertex model is equivalent to a completely packed loop model [16, 27]. This loop model is a tiling of a square diagonal lattice with two different loop tiles, see figures 1.2 and 1.3. Notice that these tiles are exactly the unit and

Figure 1.2: The two loop tiles that make up the completely packed loop model.

the generator ei of the Temperley-Lieb algebra. The rows in a loop configuration (c.f.

figure 1.3) can therefore be considered as elements of this algebra. Let us now define a non-intersecting perfect matching as a row of 2k points on the x-axis of the plane and k non-intersecting lines in the upper half-plane pairing these points. We see that the Temperley-Lieb algebra on 2k points has a natural action on the vector space of these

(7)

Figure 1.3: A configuration in the completely packed loop model.

matchings, given by composition of diagrams. The rows of the loop model we saw as elements of this algebra, so we conclude that this model gives rise to a representation on these matchings.

The matchings can be transformed in states of the spin-1/2 chain. The representation of the Temperley-Lieb algebra carries down under this map, which yields a representation of the Temperley-Lieb algebra on the level of the spin-1/2 chain, i.e. on (C2₎⊗n_.

• We now have two representations of the Temperley-Lieb algebra on (C2₎⊗n_{, one algebraic}

given by the R-matrix and one topological given by the action on the matchings. The miraculous result is that these representations are equivalent and that their intertwiner is given by the transformation that maps the loop model on the 6-vertex model! This connection between both models is deep in the sense that these are based upon the same algebraic structure, the Temperley-Lieb algebra and the corresponding solution to the Yang-Baxter equation, but in different representations.

Inspired by the above results we will try to follow the same reasoning and apply it to the more general Izergin-Korepin spin-1 model and the BMW algebra. We expect to find a similar link between these two. It is non-trivial that this should be the case, we base this expectation mainly on two articles. The first article by Kulish, Manojlovi´c and Nagy [18] shows that the R-matrix of the Izergin-Korepin model is a representation of the BMW algebra. The second article, written by Nienhuis [13], treats an equivalence between this model and a dilute loop model.

All components we encounter in this process of understanding the symmetry are also in-teresting in itself. Chromatic polynomials naturally appear in the study of the equivalences between vertex and loop models [16]. Loops in a loop model have many interpretations, they can for example be seen as polymer rings or self avoiding random walks. Loop models are also of great interest in percolation theory. This theory tries to describe the connected components of random graphs and also finds applications in the study of porous materials. Moreover, many well-known statistical models can be mapped onto a loop model. A simple

(8)

example is the Ising model in two dimensions, where the loops describe the domain walls separating regions of up and down spins [19]. There have also been found loop models which have critical points described by conformal field theory [20].

The study of ‘braid-like’ algebras such as the BMW algebra and Temperley-Lieb algebra originate from knot theory and have many applications in this area of mathematics. The algebras are used to construct knot invariants like Kauffman’s Dubrovnik invariant [4]. See also [1, 2, 3, 9] and references therein.

The outline of the rest of this thesis is as follows. In chapter 2 we will start by providing a intuitive introduction into the theory of braid groups and the BMW algebra. We then give the topological and algebraic definition of the BMW algebra and show that both definitions agree. Here also a representation of the BMW algebra and its Baxterization are treated. This representation turns out to be connected to the Izergin-Korepin model. Chapter 3 deals with the theory of vertex and loop models. We start by shortly recapping the theory of statistical physics and critical phenomena. Then we will define the 19-vertex model and calculate the corresponding R-matrix. Thereafter loop models are introduced and in the final section we establish an equivalence between the 19-vertex model and a dilute loop model. Chapter 4 is mainly concerned with the quantum inverse scattering method and the Yang-Baxter equation. We will only touch the surface of this theory. We hope to give a general idea why the Yang-Baxter equation is a sufficient relation for integrability of a system. We refer to other literature for the precise results. In chapter 5 we will summarize the established links between the different models. Furthermore we will introduce a generalization of the Temperley-Lieb category and skein modules. This will serve as a ‘topological’ representation of the BMW algebra, it gives us a natural generalization of perfect matching on which the BMW algebra may act.

The first chapters of this thesis form mainly an introductory review of the already estab-lished theory behind the BMW algebra and statistical physics. In section 3.3 and 3.5 we expand on the theory and give explicit methods of calculating the equivalences between the treated models. The fifth chapter summarizes the found link between the BMW algebra and dilute loop models. It also introduces the definition of the BMW category.

I would like to thank Jasper Stokman and Bernard Nienhuis for supervising this project, for their valuable help and their enlightening feedback on the texts I wrote. I am also truly grateful to them for introducing me to this wonderful area of mathematical physics.

Jeroen Dekker, Amsterdam, July 1, 2016.

(9)

2 BMW algebra

In this chapter we will introduce the reader to the Birmann-Murakami-Wenzl (BMW) algebra. In sections 2.2.1 and 2.2.2, we give the definitions of the BMW algebra and its topological counterpart, Kauffman’s Tangle algebra. Thereafter in section 2.3 we will give a specific rep-resentation of the BMW algebra which is related to a spin-1 chain. Here we will also give the Baxterization of a generator of the BMW algebra and then conclude that the representation can also be Baxterized. This representation then satisfies the braid relation with spectral parameter which is also known by physicists as the Yang-Baxter equation.

The main result is the fact that the algebraically defined BMW algebra and the topological Kauffman’s Tangle algebra are isomorphic. Besides giving a purely algebraic representation we wonder if the topological interpretation also gives us a ‘natural’ representation of the BMW algebra. This will be elaborated upon in chapter 5 where we introduce the BMW category. But first we will introduce the braid group.

2.1 Intuitive introduction to the braid group

The symmetric group Sndescribes all the permutations of a set of n distinguishable elements,

usually {1, . . . , n}. Elements of Sn are often denoted by either the two line notation or the

cycle notation, for example 1 2 3 4 5

4 2 1 5 3 and (1 4 5 3) in S5, respectively. A more insightful

representation may be to simply draw a diagram with two rows of n dots and connecting the k-th dot in the upper row with its image under the permutation in the lower row. The per-mutation (1 4 5 3) in S5 for example can be represented by the diagram shown in figure 2.1.

Now multiplication simply has become the concatenation of such diagrams. Given two

per-1 2 3 4 5

1 2 3 4 5

Figure 2.1: The permutation (1 4 5 3) in S5

mutations we find their product by drawing their diagrams above each other and then joining the lower row of the upper diagram via straight lines with the upper row of the lower diagram.

(10)

We know that Sn is generated by the 2-cycles σi ··= (i i + 1) for i = 1, . . . n − 1; any

permutation can be made by consecutively swapping two neighboring elements. Because Sn

is generated by these σi we have an algebraic way of defining Sn, namely the group generated

by σ1, . . . σn modulo the following relations:

(1) σ2 i = 1.

(2) σiσi+1σi= σi+1σiσi+1 and σiσj = σjσi for |i − j| > 1 (braid relations).

Both the topological and algebraic description allow for generalization to the so-called braid group Bn. A real-life braid consists of a number of strands, e.g. made of hair or rope, woven

together such that we have as many strands coming out of the braid as we started with. As the name `braid group´ suggests this group can be interpreted as all the possible braids on n strands.

A braid of order n consists of two separate, parallel planes in each of which lay n points on a line. The n points in the first plane each are connected to a different other point in the second plane via a smooth curve. These curves are allowed to be twisted around each other finitely many times without intersecting, but we demand that the braid can be drawn such that all strands only descend (otherwise this could form knots). By a projection to a perpendicular plane we can draw these braids in diagrams similar to the ones we drew for the symmetric group, keeping in mind that there is a difference between over- and under-crossings. In figure 2.2 an example of a braid with four strands is shown.

Figure 2.2: A braid consisting of four strands.

Clearly any braid can be represented by infinitely many different braid diagrams. By continuous movement of the strings (no cutting) we can get a new diagram which depicts essentially the same braid as we started with. These movements can do two things, they either move two strands over another or they move a strand over or under a crossing of two other strands. We call these the Reidemeister moves of type II and III respectively. The two Reidemeister moves are depicted in figure 2.3.

It is important to distinguish between different braid diagrams of the same braid and diagrams of fundamentally different braids. Therefore we will only consider equivalence classes

(11)

(R2)

(R3)

Figure 2.3: The Reidemeister moves that define an equivalence between braid diagrams.

of braids where two braids are equivalent if they can be transformed into each other by a sequence of these Reidemeister moves. Similar to the symmetric group, we can now define a multiplication by simple concatenation of diagrams, connecting the lower ends of the first braid to the upper ends of the second, as illustrated in figure 2.4.

=

Figure 2.4: The product of two braids is defined as the concatenation of their diagrams.

Every braid has an inverse as well, which we create by horizontally mirroring the diagram. The set of all braids on n strands forms a group. It was first considered by Emil Artin in 1925 to study link diagrams, which he thought could be helpful in knot theory for finding knot invariants [1]. In this article Artin proved that, much like the symmetric group, all braids can be generated by elements σi (where i = 1, . . . , n − 1) that twist the i-th and (i + 1)-th

strand, see figure 2.5.

(12)

Furthermore, he showed that the braid group is exactly described by the following algebraic definition.

Definition 2.1. The braid group Bn is the group generated by elements σ1, . . . , σn−1 with

the braid relations: σiσi+1σi= σi+1σiσi+1 and σiσj = σjσi for |i − j| > 1.

We have sketched the braid relations in figure 2.6. The first of the two braid relations states that one can reorder the crossings without changing the braid itself. This natural relation will play a critical role in this thesis. It is one of the defining relations in different “braid-like” algebra’s, but, as we will see shortly, it also shows up when determining if a physical model is solvable.

=

(a) σiσi+1σi= σi+1σiσi+1

=

(b) σiσj = σjσi for |i − j| > 1 Figure 2.6: The braid relations.

Notice that the symmetric group is a quotient of the braid group, namely Sn = Bn/N ,

where N is the smallest normal subgroup containing σ2_i for i = 1, . . . , n − 1. Graphically this is projecting all the crossings on a plane, i.e. forgetting whether a crossing was an over- or under-crossing. We see that the generators σi∈ Bn exactly correspond to the transpositions

that generate the symmetric group.

So far we have seen two equivalent ways of describing the symmetric and braid group. On the one hand we described these groups purely algebraic in terms of generators and equations. On the other hand we have seen a topological description. The latter is more natural in the sense that it is very intuitive to work with as it gives us a topological picture. The algebraic description, however, is more rigorous because it is independent of how this picture is drawn. In the following section we will find a similar equivalence for the algebraic and topological interpretation of the BMW algebra.

(13)

2.2 The Birman-Murakami-Wenzl algebra

We will now introduce the Birman-Murakami-Wenzl algebra BM Wn. It was first constructed

independently by Birman and Wenzl (1989) and Murakami (1986) [2, 3]. It is similar to the braid group, but it also has certain relations which allow us to break up crossings. We start off by giving the algebraic definition and derive some interesting relations for it. Then we look at Kauffman’s tangle algebra M Tn, constructed by Morton and Traczyk [5]. This is the

topological interpretation of the BMW algebra and we will give the isomorphism between these two algebra’s. Before we do so, we begin by recalling the definition of an algebra. Definition 2.2. Let R be a commutative ring. An associative R-algebra with unit 1 is an R-module A endowed with a R-bilinear multiplication map

A × A → A, (x, y) 7→ x · y

such that 1 · x = x · 1 = x for all x ∈ A and satisfying associativity, that is: x · (y · z) = (x · y) · z ∀x, y, z ∈ A

Henceforth we will call an associative algebra over a ring R with a unit element simply an R-algebra. We will give some examples which we will use throughout this text.

Example 2.3. (i) The group algebra C[G] of any group G is an associative C-algebra. It consists of all formal linear combinations of finitely many group elements with coefficients in C_.

(ii) The set of all endomorphisms EndC(V ) of a complex vector space V form a C-algebra

with multiplication given by the composition of endomorphisms.

(iii) The free algebra as defined below is an associative R-algebra as well.

Definition 2.4. Let R be a commutative ring. The free algebra over R with generators g1, . . . , gn is the free R-module with a basis consisting of words with letters g1, . . . , gn,

in-cluding the empty word (the unit of this algebra). The multiplication is defined by simple concatenation of words, for example for two basis elements we have:

(gi1· · · gik) · (gj1· · · gjℓ) = gi1· · · gik · gj1· · · gjℓ.

This algebra is often denoted by Rhg1, . . . gni.

2.2.1 The BMW algebra

Let ν, z and δ be parameters in C satisfying ν−1 _{− ν = z(δ − 1). Here z and δ are not}

necessarily invertible.

Definition 2.5. The Birman-Murakami-Wenzl algebra BM Wn(ν, z, δ) is the quotient of the

free algebra over C with generators σ±1₁ , . . . , σ_n±1 and e1, . . . , en modulo the ideal generated

(14)

(B1) (Kauffman skein relation) σi− σi−1= z(1 − ei).

(B2) (Idempotent relation) e2_i = δei.

(B3) (Braid relations) σiσi+1σi= σi+1σiσi+1 and σiσj = σjσi if |i − j| > 1.

(B4) (Tangle relations) eiei±1ei = ei and σiσi±1ei = ei±1ei.

(B5) (Delooping relations) σiei = eiσi = νei and eiσi±1ei = ν−1ei.

The parameter δ is called the loop fugacity and the parameter ν the twist.

If we assume z to be invertible then we can define δ = (ν − ν−1_{)/z + 1 and we write}

BM Wn(ν, z) instead of BM Wn(ν, z, δ). Also the idempotent relation then can be derived

from the other relations. This follows from multiplying the skein relation with ei and then

applying the delooping relation. Henceforth we shall assume z to be invertible.

From the Kauffman skein relation (B1) it follows now that eiej = ejei for |i − j| > 1.

Another useful relation that results from definition 2.5 is the cubic relation for σi.

Proposition 2.6. Let q ∈ C\{0} be the complex parameter determined by the equation z = q − q−1_{. Then the following cubic equation holds for σ}

i:

(σi− q)(σi+ q−1)(σi− ν) = 0 (2.1)

Proof. Multiplying the Kauffman skein relation (B1) with σi and reordering the equation we

find:

zeiσi= −σ2i + zσi+ 1

Then using the delooping relation (B5) we get: ei = −z−1ν−1

σ_i2− zσi− 1

(2.2) From the delooping relation (B5) we also know that ei(σi− ν) = 0. If we substitute equation

(2.2) in this delooping relation we end up with:

(σ_i2− zσi− 1) (σi− ν) = 0

By writing z = q − q−1 and factorizing we find equation (2.1).

In literature also BM Wn(q, ν) and Wn(q, ν) are often used as alternative notation for

BM Wn(ν, z = q − q−1).

We recognize different braid-like algebra’s in the BMW algebra. For example there exists an algebra homomorphism from the Temperley-Lieb algebra T Ln(δ) (see [7]) that sends the

elements hei | i = 1, . . . , n − 1i in T Ln to ei in BM Wn. Furthermore, BM Wn modulo the

ideal generated by these ei is isomorphic to the Hecke algebra (of type A) [9, 11]. We also

see an homomorphism of the group algebra of the braid group to the BMW algebra, given by the map C[Bn] → BM Wn: σi 7→ σi. It is clear that the braid relations are conserved under

this map. This map is not injective; C[Bn] has no finite basis, but as shown by Morton and

Williamson [4, 6], BM Wn does have a finite basis. Its dimension is 2n!! = (2n)!/(2nn!), i.e.

(15)

2.2.2 Kauffman´s tangle algebra

The definition 2.5 of the BMW algebra is rather abstract and the relations given therein are not that insightful. Therefore we will try once more to give a topological interpretation to it, now by looking at tangle diagrams.

Definition 2.7. An (n, k)-tangle diagram is a rectangular piece of knot diagram in the plane consisting of strands and closed curves. It has n strands entering at the top and k strands ending at the bottom.

An example of a (n, k)-tangle diagram is given in figure 2.7. Such a tangle diagram is a two-dimensional picture with some extra information at each crossing about the nature of this crossing, i.e. if it is an over or under-crossing. It is important to note that there exist no (n, k)-tangle diagrams whenever n + k is odd. As we did before with the braid diagrams, we need to distinguish between ‘topologically’ equal and ‘topologically’ different tangle diagrams.

Figure 2.7: A (4, 2)-tangle diagram.

Definition 2.8. Two tangle diagrams are called regular isotopic if they are related by a sequence of Reidemeister moves of type (R2) and (R3) (figure 2.3). We write U_kn for the set of (n, k)-tangles up to regular isotopy.

These Reidemeister moves encode some extra three-dimensional structure in the tangle diagrams which would exist naturally in three-dimensional tangles. We have a unit element in U_nn, namely the diagram with n straight lines, denoted by 1 or simply 1. Similar to the braid group the set of (n, n)-tangles Un

n admits an associative multiplication by concatenation

of diagrams, making it into a monoid. It is not too difficult to see that Bn is the complete

group of units in Un

n under this multiplication.

An important concept used in knot theory is that of the closure of a tangle diagram T ∈ U_nn. This is a link diagram defined as the (0, 0)-tangle diagram ˆT that we get by joining the upper ends of the n strands with the lower ends by disjoint arcs lying outside the braid diagram without further crossings.

(16)

Now we construct Kauffman’s tangle algebra M Tnfrom Unnby factoring out some relations.

Let ν±1_{, z and δ be complex parameters such that ν}−1_{− ν = z(δ − 1) as before.}

Definition 2.9. Kauffman’s tangle algebra M Tn is the C-module constructed from C[Unn] by

factoring out the following three relations: The Kauffman skein relation:

− = z −

!

(2.3) The delooping relations:

= ν and = ν−1 (2.4)

And the idempotent relation:

T ⊔ O = δT, (2.5)

where T ⊔ O adds a disjoint circle O to any tangle T with O having no crossings with T or itself.

These relations apply to the tangle diagrams at a local level. That means, for instance, whenever we have an over-crossing in a tangle diagram we can, using relation (2.3), locally substitute this for a linear combination of an under-crossing, two straight lines and a cup and cap.

It is not hard to see that relations (2.3)-(2.5) carry down under composition of diagrams, i.e. these relations commute with the composition of tangle diagrams. Therefore M Tn is a

well-defined algebra with a multiplication given by a linear extension of the composition of tangle diagrams. This multiplication is associative and unital. The proof of this is fairly straightforward and will not be given here.

Proposition 2.10. The composition of tangle diagrams induces a C-bilinear multiplication on M Tn, making M Tn an associative algebra over C. The diagram consisting of only straight

lines is the unit of this algebra.

2.2.3 Isomorphism between M T_n and BM W_n

Having defined the algebra’s BM Wn and M Tn, we now work to wards the crucial theorem

that gives us an isomorphism between these two. The observant reader may have noticed that the defining relations of both algebra’s already seem very similar. We will use the following suggestive notation.

Definition 2.11. Write Si and Ei respectively for the tangle diagrams in Unn as pictured in

figure 2.8. These tangle diagrams Si and Ei are also elements of the algebra M Tn and we

denote them with the same letters.

The elements Si have obvious inverses S_i−1. These are given by swapping the over-crossing

(17)

i i + 1 1 ... ... n ... ... Si = i i + 1 1 ... ... n ... ... Ei =

Figure 2.8: The tangles Si and Ei in Unn.

It follows from relation (2.3) from the definition of Kauffman’s tangle algebra that in M Tn

we have

Si− S_i−1 = z(1 − Ei). (2.6)

Moreover, from (2.4) we deduce that

SiEi = EiSi = ν−1Ei,

S_i−1Ei = EiSi−1 = νEi,

(2.7) and relation (2.5) shows that

E_i2= δEi. (2.8)

Graphically, these relation are even more apparent, as is illustrated in figure 2.9.

−

= z

−

!

= δ

=

= ν

−1

Figure 2.9: The relations (2.6)-(2.8) that are satisfied by the elements Si and Ei in Kauffman’s tangle algebra M Tn.

So these elements Si and Ei of the algebra M Tn satisfy the Kauffman skein, idempotent

and delooping relations, similar to the elements σi and ei in BM Wn. Furthermore, the braid

relations were already shown to be satisfied by Si in figure 2.6 (here the elements σi∈ Bncan

be replaced with Si ∈ M Tn). In figure 2.10 is shown that Si and Ei also satisfy the tangle

relations.

The BMW algebra and Kauffman’s tangle algebra seem almost indistinguishable, they agree on all their defining relations. This similarity is formalized in the following two theorems. First we may define a map from BM Wn to M Tn such that all relations are respected.

(18)

=

Figure 2.10: The tangle relations for Kauffman’s tangle algebra.

Theorem 2.12. There exists a homomorphism ϕ : BM Wn → M Tn defined by ϕ(σi) = Si

and ϕ(ei) = Ei.

Proof. Figures 2.6, 2.9 and 2.10 show that the relations (B1)-(B5) in BM Wn are respected

by the elements Si and Ei in M Tn. Thus the map ϕ is a well-defined homomorphism.

An even stronger statement holds. The Birmann-Murakami-Wenzl algebra and Kauffman’s tangle algebra are isomorphic.

Theorem 2.13. The homomorphism ϕ : BM Wn → M Tn as defined in theorem 2.12 is an

isomorphism of algebra’s over C.

The proof is rather lengthy, so here we will only give the idea. We strongly recommend reading [4] for the whole proof.

First the surjectivity of ϕ is proven. For this we define a n-connector as being a pairing of 2n points. These n-connectors form a generalization of the permutations of n points, see also figure 2.1. For each n-connector c we can make different (n, n)-tangle diagrams Tc such that

these diagrams correspond to the connector c, i.e. Tc connects the same points as c.

Now, given a tangle T , choose a sequence of base-points, consisting firstly of a point on one end of each strand in the tangle, and then of one point on each closed loop. We call a tangle totally descending (for this choice of base-points) if on traversing all the strands, starting from the base-points of each component in order, each crossing is first met as an overcrossing. It can be shown that M Tn is spanned by the (finite) set of totally descending

tangles Tc without closed loops, where c ranges over all n-connectors. Furthermore each of

these totally descending Tc can be written as monomial in Ei and Si. This then proves that

M Tn is generated by Ei and Si and it also proves the surjectivity of the map φ.

The proof of injectivity is a bit more involved. Let us consider the ideal M Tn(r) consisting

of all tangle diagrams with no more than r ‘through’ strands (strands that connect from top to bottom). It is not hard to see that M Tn(r) as an ideal is generated by the element

E1E3· · · E2k−1, where k is given by r = n − 2k. Now we define BM Wn(r), unsurprisingly, as

(19)

Injectivity of ϕ : BM Wn(r) → M Tn(r) for all n and r now follows from double induction

on both n and r. For fixed n the injectivity can be proven for r = 0 or r = 1 (depending on the parity of n) by the injectivity of ϕ on BM Wn−1 = BM Wn−1(n−1) (using induction on

n). Now the induction on r is performed. For this it is necessary to construct a subspace Vn(r) ⊂ BM Wn(r) that complements BM Wn(r−2). This space Vn(r) can be shown to have the

following properties:

(1) Vn(r)+ BM Wn(r−2) is a two-sided ideal in BM Wn(r),

(2) ϕ|Vn(r)→ M Tn is injective,

(3) e1e3· · · e2k−1∈ Vn(r).

These three properties then establish the induction step: using (1) and (3) we are able to show that Vn(r)+ BM Wn(r−2) = BM Wn(r), then using (2) and the induction hypothesis we

find that ϕ : BM Wn(r) → M Tn(r) is injective. Property (1) can be proven with some extra

lemmas and (3) is immediate from the construction. For property (2) a spanning set for Vn(r)

can be constructed whose image under ϕ is an independent set of totally descending tangles. This last theorem is the crux of this chapter. It explicitly shows that we can use our topological intuition to make statements about an algebra that first only had an algebraic definition. Our hope is that this allows us to find representations of the BMW algebra which naturally follow from the topological picture. Hereafter we will call both algebra’s the BMW algebra and we will use the description that is the most natural or easiest to work with depending on the situation.

2.3 Representations of BM W

_n

In this section we will consider a specific representation of BM Wn. We will see later that this

representation has a direct connection to a spin-1 model, but for now we will forget about this application. We begin with a short generalization of the theory of representations of groups to algebra’s.

Definition 2.14. Let A be a C-algebra. A representation of A is a pair (π, V ), where V is a vector space and is called the representation space and π : A → End(V ) is an algebra homomorphism and is called the representation map.

Notice that any C-algebra A in particular is also a ring. Therefore we can define a repre-sentation to be a left A-module V where the bilinear multiplication is defined as:

A × V → V : (a, v) 7→ a · v ··= π(a)v

Example 2.15. Let G be a discrete group. We have seen that C[G] is an C-algebra as well as a vector space. Then we define the regular representation of C[G] simply as multiplication

(20)

in C[G], that is: π : C[G] × C[G] → C[G] :   X g∈G λgg, X h∈G µhh  7→ X g,h∈G λgµh(gh)

Let us now look at a representation of BM Wn(ν, z). We specify the parameters to be

z = q − q−1 and ν = q−2 and then write q = e2η for a complex number η. This is a special case for which we only have one free parameter η.

We now consider C3 _{with the standard (Euclidean) ordered basis {e}

1, e2, e3}. For the

representation space we take the vector space (C3)⊗n. We define the representation map as follows:

π : BM Wn(ν, z) → End (C3)⊗n : σi7→ ˇRi(η), ei 7→ Ei(η), (2.9)

where both ˇRi(η) and Ei(η) only act non-trivially in the i-th and (i + 1)-th tensor component

of (C3)⊗n as the matrices ˇR(η) and E, respectively. These matrices, acting on (C3)⊗2 with the ordered basis given by {e1⊗ e1, e1⊗ e2, ..., e3⊗ e3}, are given by 9 × 9 matrices. For σi

we have: ˇ R(η) =                  e2η 0 1 0 e−2η 1 ω 1 e−ηω 0 1 e−2η e−ηω (1 − e−2η)ω 1 ω e2η                  (2.10)

Here ω = e2η_{− e}−2η _{and all vacant entries are 0. For e}

i we have: E(η) =                  0 0 e2η −eη 1 0 −eη ₁ _−e−η 0 1 −e−η e−2η 0 0                  (2.11)

(21)

So the representation of the BMW algebra is given by: π(σi) = ˇRi(η) = 1 ⊗ · · · ⊗ 1 ⊗

i-th and (i+1)-th component

ˇ

R(η) ⊗1 ⊗ · · · ⊗ 1 π(ei) = Ei(η) = 1 ⊗ · · · ⊗ 1 ⊗ E(η) ⊗1 ⊗ · · · ⊗ 1

We checked using MATLAB that these tensors ˇRi = ˇRi(η) and Ei = Ei(η) satisfy all the

BMW relations (B1)-(B5) from definition 2.5 (see also [18]): ˇ

Ri− ˇR−1i = (q − q−1) (1 − Ei) , (2.12)

E_i2= δEi, (2.13)

with δ = q + 1 + q−1, and:

ˇ

RiRˇi+1Rˇi= ˇRi+1RˇiRˇi+1, (2.14)

EiEi±1Ei= Ei, (2.15) ˇ RiRˇi±1Ei= Ei±1Ei, (2.16) ˇ RiEi = EiRˇi = 1 q2Ei, (2.17) EiRˇi±1Ei = q2Ei. (2.18)

Furthermore, it is trivial that ˇRi and ˇRj commute for |i − j| > 1. By linear extension this

then proves that π is indeed a representation of BM Wn(q, ν = q−2). As we will see at the end

of chapter 3, ˇR(η) is the Izergin-Korepin R-matrix. In the context of statistical models the braid relation 2.14 is nothing else than the constant Yang-Baxter equation in the so-called braid group form.

We shall now construct elements of the BMW algebra that depend on a spectral parameter λ and satisfy a parameter-dependent version of the braid relation. Mathematically these elements could be seen as a result of the natural structure that appears in a larger algebra (the affine BMW algebra) of which the BMW algebra is a subalgebra. It could give more insight in the structure of the algebra. The precise reason why, however, falls well beyond the scope of this thesis.

The procedure of adding a spectral parameter is called the Yang-Baxterization or more commonly the Baxterization. For the BMW algebra there has not yet been found a construc-tive method to find such a parameter dependent element. But [10] has found (presumably by brute-force) the following Baxterized elements (for general ν and q):

σ_i(±)(u) = 1 q − q−1 u−1σi− uσ−1_i + ν ± q ±1 uν ± u−1_q±1 ei (2.19)

(22)

These are elements of BM W (ν, q − q−1) with spectral parameter u ∈ C. We can retrieve the σi and σ−1_i by taking the proper limit:

σi = lim u→0(q − q −1_)uσ(±) i (u), σ−1_i = lim u→∞ −(q − q −1_)u−1_σ(±) i (u)

Notice furthermore that σ(+)(u) and σ(−)(u) are related by the transformation q ↔ −q−1, which corresponds to the algebra isomorphism BM Wn(q, ν) ∼= BM Wn(−q−1, ν) [11].

The Baxterized elements satisfy a remarkable version of the braid relation:

σ_i(±)(u)σ(±)_i+1(uv)σ(±)_i (v) = σ_i+1(±)(v)σ_i(±)(uv)σ_i+1(±)(u) (2.20) The existence of these elements implies that we also have a Baxterized version of ˇR(η) that satisfies a similar braid relation. In order to have the relations for the spectral parameters in additive form (instead of multiplicative) let us write u = eλ−µ _{and v = e}µ_{. Then we define}

ˇ

Ri(λ, η) = π

σ(+)_i (eλ/2). The braid relation (2.20) then implies that the following relation holds:

ˇ

Ri(λ − µ, η) ˇRi+1(λ, η) ˇRi(µ, η) = ˇRi+1(µ, η) ˇRi(λ, η) ˇRi+1(λ − µ, η) (2.21)

Many physicists know this as the (spectral parameter-dependent) Yang-Baxter equation. As shown by [18] this is the R-matrix with spectral parameter of the Izergin-Korepin model. This fact is truly extraordinary. Apparently the representation of the Birman-Murakami-Wenzl al-gebra provides a realization of the hidden symmetry embodied by this statistical model. To explain this connection is one of the main goals of this thesis.

The matrix of this Baxterized representation ˇRi(λ, η) = π

σ_i(+)(eλ/2₎_{is given by ˇ}_R

i(λ, η) =

P Ri(λ, η), where P is the permutation operator given by P v ⊗ w = w ⊗ v and R(λ, η) is given

by (see [18]): R(λ, η) =                  a1 a2 b1 a3 b2 b3 c1 a2 c2 a4 b2 a2 b1 c3 c2 a3 c1 a2 a1                  (2.22)

The entries are given by the following formulas:

(23)

a2 = sinh(λ − 3η) + sinh(3η), b3 = −2e−λ+2ηsinh(η) sinh(2η) − e−ηsinh(4η)

a3= sinh(λ − η) + sinh(η), c1= sinh(2η)(eλ−3η+ e3η)

a4 = sinh(λ − 3η) + sinh(3η) − sinh(5η) + sinh(η), c2= e−2ηsinh(2η)(1 − eλ)

b1 = − sinh(2η)(e−λ+3η+ e−3η), c3= 2eλ−2ηsinh(η) sinh(2η) − eηsinh(4η)

It is important to remember from this section that, given above representation of the BMW algebra, we can construct a Baxterized version of ˇR that depends on a spectral parameter and that satisfies the Yang-Baxter equation (2.21). In the following chapters we will show how this same matrix also arises from statistical physics.

Up to here we have only given one purely algebraic representation of the BMW algebra. Now we can follow two different paths to find other representations. Firstly, we can use the topological interpretation of this algebra to construct a ‘natural’ representation on a yet undefined vector space of matchings on n points. Here natural means that this representation is given by concatenation of such a matching and tangle diagrams in the BMW algebra. In particular we wonder if such a representation is equivalent to the algebraic one.

Secondly, as argued in the introduction, the equivalence between the Izergin-Korepin model and a dilute loop model suggests that there exists an intertwining between the representation given by ˇR and a yet unknown action of the BMW algebra on the dilute loops. Similar to the case of the Temperley-Lieb algebra and a completely packed loop model, this intertwiner could be given by the transformation map of the Izergin-Korepin model into this loop model. By applying this transformation to above representation (2.9) one should be able to construct this second representation. This is treated in more detail in chapter 5.

(24)

3 Vertex and loop models

In this chapter we will discuss the theory of vertex and loop models in statistical physics.These are versatile models that can be represented by paths on a lattice. It is this fact that shows a link with the topological picture of the BMW algebra.

We start off with a preliminary on the theory of statistical models an critical points. After a recap of the partition function we define the correlation length of a variable in a model and with this we shortly discuss criticality. Then in section 3.2 we will introduce the 19-vertex model. Our definition has an apparent generalization to system with more states. In particular we look at certain weights such that the 19-vertex model is critical. In the next section we introduce the transfer- and R-matrix. These give a somewhat easier method of finding the partition function. As we will see in chapter 4 this also yields an easy way to verify if a model is solvable.

Then in section 3.4 we introduce loop models. In particular we are interested in dilute loop models where not the whole lattice is covered by loops. In the last section we will show an equivalence between a certain 19-vertex model from section 3.2, the Izergin-Korepin model, and a dilute loop model. On the latter we hope to find a action of the BMW algebra.

3.1 Preliminaries on critical phenomena in statistical physics

It can be shown using statistical physics that the probability of finding a system in a certain state is proportional to e−βE(x). Here E(x) is the energy associated with the state x of that system and β = _k1

BT with T the temperature and kB Boltzmann’s constant. We call these

exponentials e−βE(x) the Boltzmann weights W(x). The probability of finding a system in the state x is then given by

P_{(x) =} 1 Ze

−βE(x)_, _(3.1)

where Z is the sum over all Boltzmann weights. This Z is what we call the partition function of our system:

Z =X

x

e−βE(x) (3.2)

Here the sum is carried out over all possible states x of the system. We will assume the temperature to be fixed, in this case this partition function is referred to as the canonical en-semble. The partition function is used to give information about the statistical properties of a system in thermodynamic equilibrium. Certain thermodynamical variables can be calculated from the partition function, such as the free energy, entropy and heat capacity.

(25)

We can define a similar formalism for quantum statistical models. If we have a system with an Hamiltonian H we associate the eigenvalues of eigenstates with the energy of that state. More generally, if H is an observable, e.g. an operator in the Hilbert space of all states, then we write

Z = Tr [exp(−βH)], (3.3) this is independent of the basis and equivalent to (3.2).

We are particularly interested in how the partition function behaves at critical points or in the thermodynamical limit. One of our goals in the following sections is to find a method to calculate the partition function for a lattice model in the limit of its size going to infinity.

We are interested to measure the order in a system. This can be measured by looking at the correlation between two random variables s₁ and s₂ at different locations and different times in our model. The general form of the correlation of these variables at positions R and R + r is given by:

hs₁(R) · s₂(R + r)i − hs₁(R)ihs₂(R + r)i

Here hXi denotes the expected value of a random variable X. Let us consider the correlations in a system of spin particles. We define the correlation function of the spin operators s as:

Γ(r) = hs0· sri − hs0ihsri (3.4)

where sr then denotes the spin of a particle at position r. In ferromagnetic materials we

know that at low temperatures the spins like to align with their neighbors. The ground state is where all spins are pointing up. At low temperatures we therefore find islands of particles of which the spin is aligned down in a sea of spin up particles. At high temperatures there is almost no order in the system and we find that even for small r that s0 and sr are

uncorrelated.

We can summarize the correlation exhibited by this spin system as a combination of a exponential and a power-law dependence on the distance r:

Γ(r) ∝ |r|−(d−2+η)e−|r|/ξ(T ) (3.5) Here d is the dimension of the system, η is a critical exponent and ξ(T ) is the correlation length as function of the temperature. The correlation length is, as the name suggests, the typical distance over which two variables are correlated for a given temperature. For T far away from the critical point we have that ξ(T ) < ∞ and the correlation Γ(r) therefore decays exponentially for larger r.

Statistical models as well as experiments have shown that near the critical temperature TC of

a system the correlation shows a power-law behavior, i.e. for T → TC we must have ξ(T ) → ∞

so that the exponential term goes to 1 [14]. For T close to the critical temperature we see that the correlation length itself exhibits a power-law behavior too, that is ξ ∝ |T − TC|−ν where

ν is another critical exponent. Systems at a temperature close to their critical temperature are simply called critical [12]. For example, a system becomes critical when it nears a phase transition. It is important to note that these critical exponents η and ν are not related to the parameters used in chapter 2.

(26)

3.2 Ice-type vertex models

In this section we define an ice-type vertex model for spin-1 particles on a square lattice. This is based on the square ice model by Elliott Lieb [8]. He used it to exactly calculate the free energy and entropy of a two-dimensional crystal of water molecules (also known as ice) at zero temperature.

Let S be a square lattice of N × M vertices, with M rows and N columns, and toroidal boundary conditions. We have 2N M edges (lines that connect the vertices) in this lattice: M rows of N vertical edges and N columns of M horizontal edges. The idea is that we think of the edges as being the classical particles. We then define a state on the lattice S as an assignment of state variables to the edges. For classical spin-1 particles we want to assign a spin (±1 or 0) to each edge.

Instead, but completely equivalent, we can also define a state on this lattice of spin-1 particles as an assignment of an orientation to the edges. This is done by placing three types of arrows on them: up and right, down and left or the ‘empty arrow’ (see figure 3.1). These then correspond to the spin of the particle at that edge.

Figure 3.1: An example of an assignment of three different state variables ±1 and 0 to a part of a square lattice. The corresponding arrows are also shown.

The dynamics of this model are characterized by the interactions amongst the variables on the edges. These interactions take place on the vertices, hence the name vertex model. We require locality for these interactions; the energy and therefore also the Boltzmann-weight associated to a vertex only depends on the state of the edges meeting in that vertex. This gives us the following definition of a spin-1 ice-type vertex model:

(a) Let S be a square lattice. We place arrows on the edges of the lattice such that around each vertex we have as many arrow pointing toward as away from the vertex. (This is called the ice condition.) We allow for three types of arrows to be place: ±1 and 0, that is up/right, down/left and the empty arrow, respectively. In figure 3.2 we see

(27)

the 19 different configurations allowed for each vertex of the lattice. We will call a configuration of arrows on the whole lattice an ice configuration or plainly a state of our model.

(b) To each vertex of the lattice we now associate a weight ωiwhich corresponds to the i-th

configuration in figure 3.2 as allowed by condition (a). The weight associated with an ice configuration is given by the product over all the weights of the different vertices.

Figure 3.2: All the possible configurations of the vertices in the spin-1 vertex model.

The term ice condition in (a) originates from the original ice model by Lieb [8]. In ice the crystalline structure arises from the forming of hydrogen bonds between two different molecules of water. For (2-dimensional) square ice we put oxygen atoms on the vertices and hydrogen atoms on the edges of a square lattice. Each oxygen atom will be surrounded by four hydrogen atoms. The wave function of an electron on an edge is centered closer to one of the two oxygen atoms. By the Coulomb interaction we must have exactly 2 electrons close to each oxygen atom. Therefore this can be pictured as a square lattice with arrows drawn on the bonds such that we have exactly two arrows pointing in and two pointing out of each vertex. This model is also known as a spin-1₂ or 6-vertex model.

The definition of the spin-1 vertex model we just gave is a generalization thereof. The ice condition can also be interpreted as the conservation of the total spin quantum number (this will be explained in section 3.3).

(28)

We will use the following weights: ω1= v + w − u(1 − 2 cos 2ϕ) ω2 = ω4= eiϕ/2+iχ ue−2iϕ+ v ω3 = ω5= e−iϕ/2−iχ ue2iϕ+ v ω6 = ω8= eiϕ/2−iχ ue−2iϕ+ w ω7 = ω9= e−iϕ/2+iχ ue2iϕ+ w ω10= ω11= ω12= ω13= −u ω14= ω15= v ω16= ω17= w

ω18= e2iχ 2u cos ϕ + veiφ+ we−iϕ

ω19= e−2iχ 2u cos ϕ + ve−iφ+ weiϕ

(3.6)

Here u, v, w, ϕ and χ are complex parameters with ϕ determined by e−iϕ _{+ e}iϕ _{= q}1/2 _for

some natural number q. Henceforth, we will call the ice-type vertex model with these weight the 19-vertex model.

These weights seem rather arbitrary at first sight. They were derived by Nienhuis [13], he rigorously proved an equivalence between a q-state Potts model with weights x, u, v, w and a complicated spin-1₂ vertex model. Our spin-1 model is a ‘sub-model’ of the latter. In the Potts model the weight x was taken to be x = −1 so that the weights at the vertices with four connections decoupled to above (relative) nice weights. As proposed by Nienhuis, a consequence of the equivalence is that (3.6) is a critical manifold within the larger class of 19-vertex models, i.e. our 19-vertex model is critical. It has a connection to a dilute loop model as well, which we will show in section 3.4.

Let A ∈ A be an ice configuration, where A denotes the set of all possible ice configurations of S. By part (b) of the definition the Boltzmann weight WA for this configuration is given

by the product of the weights (3.6) over all vertices of the lattice. The partition function then is given by:

Z =XWA=

X Y

ωvn (3.7)

Here the sum is carried out over all states A ∈ A and the product is performed over all vertices n in the lattice. We let vn denote the vertex configuration of vertex n in the lattice,

so vn = 1, . . . , 19. We have already given an example of an ice configuration in this model

in figure 3.1. One can check easily that the weight of the middle row in this configuration is W = ω2ω12ω16ω19, assuming the cut-off edges carry no arrow.

(29)

3.3 Transfer matrix

In the thermodynamic limit of the lattice size going to infinity the calculation of the partition function becomes a rather challenging problem. To simplify things we will now introduce the concept of a transfer matrix. Baxter applied this method to the 6- and 8-vertex model [15, 26], we will here generalize this method and apply it to the 19-vertex model.

The partition function (3.7) can be rearranged to

Z = X vertical states Y rows     X horizontal states Y vn∈row ωvn     , (3.8)

where we first take the sum over the horizontal variables, which only involves the Boltzmann weights in a single row, and then perform the sum over the vertical variables. Suppose that the vectors λ and λ′ are two consecutive rows of vertical states, where the entries λnand λ′n

of these vectors denote the states of the n-th vertical edge in such a row: λ1 λ2 λ3 λN −1 λN

λ= · · ·

Similarly we can write the vector µ for a horizontal state. Introduce the 3N × 3N transfer matrix t with matrix elements

t_λ,λ′ =

X

µ

ωv1ωv2· · · ωvN, (3.9)

where the sum is carried out over all possible states µ of the row of horizontal edges between λ and λ′ so that it is in accordance with the ice condition. The ωvn are the weights (3.6)

corresponding to the vertices vn in this row µ. When there are no states µ possible between

λand λ′ satisfying the ice condition, the sum (3.9) is empty and yields zero.

Notice that the transfer matrix (3.9) is the factor in between the brackets in (3.8). The partition function can therefore be written as:

Z =X λ1 · · ·X λM tλ1,λ2tλ2,λ3· · · tλM,λ1 = TrtM = N X i=1 ΛM_i (3.10)

The problem of finding the partition function has now been translated in that of the diago-nalization of a matrix. Here Λi denotes an eigenvalue of t. Let’s look at the transfer matrix

in more detail. In a picture the expression (3.9) is given by figure 3.3.

If we see the vertical dimension as time, the transfer matrix gives the probability of going from state λ to λ′ in a single time step. So t plays the role of a discrete evolution operator

(30)

λ′₁ λ′₂ λ′₃ λ′ N −1 λ′N λ1 λ2 λ3 λN −1 λN t_λ,λ′ = X µ · · · · · · µ1 µ2 µ3 µN −1 µN µ1

Figure 3.3: The transfer matrix for the 19 vertex model.

acting in the Hilbert space H(M ) that is spanned by the vertical states λ. The trace in (3.10) is taken over this space, so this is just the Hamiltonian formulation of the model. We may now regard our original two-dimensional classical model as a one-dimensional quantum system.

Because our system is determined by local configurations we can expand the transfer matrix as follows: tλ,λ′= X µ1 · · ·X µN N Y j=1 R(λj, λ′j | µj, µj+1), (3.11) where R is given by R(λ, λ′ | µ, µ′) = ( ωi 0 (3.12)

where the value is determined by whether or not the configuration (λ, λ′ | µ, µ′) is an allowed arrow configuration of type i (c.f. figure 3.4 and 3.2):

v λ′

λ (λ, λ′ | µ, µ′) = _µ

µ′

Figure 3.4: R acting on this state (λ, λ′_{| µ, µ}′_{) gives ω}

v if this state is allowed and 0 if it is not.

We choose an ordered basis for the horizontal states µ, µ′ _{using the standard basis {e}

1, e2, e3} of C3: e1= v+ = +1 = → e2= v0 = 0 = e3= v− = −1 = ← (3.13)

If we regard µ and µ′ in R(λ, λ′ | µ, µ′) as indices, we can define R(λ, λ′) as a matrix (for fixed λ and λ′_{) in the following way:}

R(λ, λ′)vµ··=

X

µ′_∈{+,0,−}

(31)

The transfer operator (3.11) can then be written as:

t_λ,λ′ = Tr R(λ₁, λ′₁)R(λ₂, λ′₂) · · · R(λ_N, λ′_N)

(3.15) For each fixed λ and λ′ we have seen that R(λ, λ′) has a matrix representation. If we now regard λ and λ′ _{as indices too, we can interpret R as a tensor of matrices. Let us choose,}

using the standard basis {e1, e2, e3} of C3, a basis for these vertical states λ, λ′:

e1 = v+ = +1 = ↑

e2 = v0 = 0 = |

e3 = v− = −1 = ↓

(3.16)

Using definition (3.14) we now define R as: Rvλ⊗ vµ··=

X

λ′_,µ′_∈{+,0,−}

R(λ, λ′ | µ, µ′)vλ′⊗ v_µ′ (3.17)

Now R is an operator acting on C3⊗ C3. If we choose {e1⊗ e1, e1⊗ e2, ..., e3⊗ e3} as basis

for C3⊗C3 we can write R as a 9 by 9 matrix in the standard representation of C3⊗C3as C9. We will now derive this matrix explicitly for the 19-vertex model. Assume that a vertical state λ, λ′ is given. We then need to find all possible horizontal states µ, µ′ that make a valid vertex configuration to find the explicit matrix representation of R(λ, λ′_{). Then by}

considering all possible states λ, λ′ we will be able to write down R.

For example consider the vertical state given by λ = 0, λ′ = +1. The only two horizontal state compatible with this are µ = +1 and µ′ _{= 0 or µ = 0 and µ}′ _{= −1. The corresponding}

weights are ω2 and ω9, respectively. Using definition (3.14 ) we have that the R matrix for

this vertical state is given by:

R(λ = 0, λ′ = +1) =    0 ω2 0 0 0 ω9 0 0 0   . (3.18)

Similarly one can calculate this matrix for all other vertical states λ, λ′. By using either these matrices or just definition (3.17 ) we find:

R =                  ω14 ω12 ω5 ω17 ω6 ω19 ω2 ω1 ω9 ω1 ω8 ω11 ω3 ω18 ω7 ω16 ω4 ω13 ω15                  (3.19)

(32)

We see that the just calculated R(λ = 0, λ′ = +1) corresponds to the (2, 1) 3 × 3-block in this tensor product of matrices.

This R-matrix is extremely important in the area of integrable models. It will turn up in the so-called Yang-Baxter equation which gives a sufficient condition on this R-matrix for the model to be solvable. In chapter 5 we will verify that it is a solution to the Yang-Baxter equation, thus proving that our model is integrable. Furthermore, we show at the end of this chapter for what choice of parameters u, v and w above R-matrix gives the matrix ˇR(λ, η) from equation (2.10), the latter of which was shown to be a representation of the BMW al-gebra.

3.4 Loop models

In this section we will introduce loop models. The statistical mechanics of these models have been studied extensively for the last decades. They are very versatile because many other statistical problems can be mapped to a loop model. For example, the two-dimensional Ising model can be solved using a loop model to represent terms of its partition function [19]. We will first discuss the loop models an sich, whereafter in the next section we will show an equivalence to the 19-vertex model. For a more general overview of loop models and renormal-ization one should look at the lecture notes by Nienhuis [12] and at this article by Fendley [20]. Let us consider a diagonal square lattice. We create loops by placing ‘plaquettes’ on this lattice. These plaquettes can be thought of as puzzle pieces or tiles with lines on them. We should place these puzzle pieces such that the lines all fit together and create loops; we do not want any loose ends except at the boundary. Every tile carries 0, 1 or 2 lines. The lines on the tiles connect the middles of two edges with each other and two lines can either avoid each other or cross over one another. All the allowed plaquettes are depicted in figure 3.5. It is important to note that the orientation of these tiles matters. A generalization that we will not treat here is given by allowing branching in the loops or loops of multiple ‘colors’ [20].

A tiling of the whole lattice is called a loop configuration. Once again we want to associate weights to each configuration in order to study the statistics of the model. Similar to the weights of a vertex configuration we can consider the local configurations from figure 3.5 and associate them with weights ρ1, . . . , ρ10 respectively. But now we can also introduce a

parameter associated with the non-local properties of the loops, the loop-fugacity or in this text simply the fugacity. This is the weight associated with a closed loop, i.e. it is the ‘cost’ of creating a new loop. The fugacity is often denoted by n or τ . Remembering the interpretation of the Boltzmann weights as the probability of certain configurations to occur, we see that in the limit of the fugacity going to zero, the state of the system will be given by one single long loop. By increasing the fugacity it will become more advantageous to have more separate loops.

By tweaking the weights ρi we can create many fundamentally different loop models. Let

(33)

Figure 3.5: All possible plaquettes or tiles of local loop configurations of the loop model.

that all weights ρi are equal to 1 or 0. The fugacity is some parameter n. The Boltzmann

weight of a loop configuration consisting of ℓ loops is thus nℓ.

It is clear that by putting ρ1 = ρ8 = ρ9 = ρ10 = 0 we get that each face of the lattice is

exactly visited once by a loop. This is called a fully packed loop model (FPL model). An example of a fully packed loop configuration is given in figure 3.6 (a). The fully packed O(n) loop model has all other weights equal to 1. An easy way to generalize this model is by taking all these non-zero weights ρi different. The weight of a loop configuration is then given by

the product over all local configuration and then multiplied with nℓ_.

Let us set all but ρ8 and ρ9 to zero. We find then that the loops of any loop configuration

will cover all edges of the lattice without crossing. This is what is called the completely packed loop model (CPL model), see e.g. figure 3.6 (b). In the completely packed O(n) model we have ρ8= ρ9 = 1, but more generally one could consider these weights unequal. As was said

in the introduction, the completely packed loop model is equivalent to the 6-vertex model and its rows can be seen as elements of the Temperley-Lieb algebra.

We have been using the name “O(n) model” without explaining it properly. It originates from a n-dimensional spin model. A loop model is derived from a mapping between the partition function of a lattice of n-component spins and polygons in this lattice. The measure of this spin system is invariant under the global O(n) symmetry. Here O(n) denotes the group of real-valued n × n orthogonal matrices with determinant ±1, where n is non-negative integer. Once we abandon its original interpretation as a spin model, we can consider this loop model with negative or non-integer values of n. To differentiate between the spin model and the generalization of the corresponding loop model, often τ is used as the fugacity. But because of historical reasons n is still used as common as τ [12]. We will follow the notation used by the majority of our literature and keep using n.

(34)

(a) (b)

Figure 3.6: Typical loop configurations of the (a) fully packed and (b) completely packed loop models.

Both the fully and completely packed models had at least a single line of a loop on each tile. When we let ρ1 6= 0 we get a so-called dilute loop model in which some faces of the

lattice are not reached by any loop, see figure 3.7. Dilute loop models are often used in low temperature expansions. Its loops can be considered as domain walls. For example, in a low temperature system of spins we typically see clusters of spin down particles separated by a sea of spin up. A dilute loop then describes the boundary separating both spins.

Figure 3.7: A loop configurations of the dilute loop model.

Dilute loop models can also be created from the 19-vertex model, which is one of the main results we will treat in this chapter. This result is similar to the equivalence between the 6-vertex model and a completely packed loop model as explained in the introduction.

(35)

3.5 Transformation of the 19-vertex model to a dilute loop

model

The 19-vertex model (3.6) can be transformed to a loop model under certain conditions. The method is similar to the one provided by Baxter, Kelland and Wu in [16]. As Nienhuis showed in [13], one of the resulting loop models is equivalent to the Izergin-Korepin model [24]. This is rather remarkable. Initially he did not require the 19-vertex model to be integrable and this will not be required either when constructing the loop models. However, from the Izergin-Korepin model we know that its R-matrix is a solutions to the Yang-Baxter equation and thus that this model is integrable. Normally, when constructing statistical models, the Yang-Baxter equation puts a lot of constraints on the weights. However, the original, critical weights from the Potts model of which the 19-vertex model was derived seem to satisfy this equation naturally. We will only show the equivalence to the Izergin-Korepin model, but the other models can be derived with analog methods [13].

Let S be the square lattice corresponding to the 19-vertex model (3.6), again with toroidal boundary conditions, but now rotated over π/4 anti-clockwise. The plan is to replace all the vertex configurations with the tiles from figure 3.5 on the dual lattice. This cannot be done directly because the dilute loop model has a global parameter, whereas the 19-vertex model is completely local. An oriented loop model lies in between. This is a loop model where all loops are given an orientation. A summation over all possible orientations of the loops then yields the dilute loop model.

When coming from the vertex model there are two cases to be considered, one where the fugacity is fixed and one where it is a consequence of the local factors associated with a turn of a loop we find under the transformation. In the former case, as shown by Nienhuis, the weights are invariant under inversion of the arrows and the fugacity is 2. Our focus is on the latter, where the fugacity is free. In this case we must demand ρ10= 0. Otherwise, if we were

to allow self-intersections, the fugacity would depend on the topology of the crossing.

Suppose we are given a state A in the vertex model, i.e. we have an assignment of arrows to some of the edges of the square diagonal lattice S. The idea is that we can create oriented dilute loops on the dual lattice S′ _{that are given by the edges of S that carry an arrow. For}

the vertex configuration 2 from figure 3.2 (rotated over π/4 anti-clockwise) we could do the following:

7→

In general, however, this decomposition of the lattice is far from unique. For example, as we will see shortly, the vertex configurations 18 and 19 can be decomposed as two different oriented loop configurations. Nonetheless, because of the ice condition in our vertex model, above process of mapping vertex configurations on oriented loop configurations will result in well-defined oriented loops with a consistent orientation.

(36)

We demand that all weights are invariant under a rotation over π. Now we introduce the local factors of the turns of which we assumed that the fugacity would be a consequence. Notice that, under the rotational invariance, there are two different turns in the oriented loop model between which we need to differentiate. By going from one diagonal to a perpendicular one we can either cross over a (imaginary) horizontal or vertical line on a face of the lattice. To these crossings we associate weights q₁±1/2and q₂±1/2respectively, where each right turn is associated with the positive power and each left turn with the negative power, as shown here:

q1/2₁ = = q₁−1/2 = =

q1/2₂ = = q₂−1/2 = =

Now we demand that the weights of the oriented loop model can be written as the product of the underlying loop tiles and the orientation of the lines thereon. For example for the vertex configuration ω2 we can define the map from the vertex to the oriented loop model as

follows:

7→ = q₁−1/2

ω2 7→ q−1/21 ρ2

(3.20)

Whenever we have multiple possible oriented loops that are compatible with one vertex configuration, we just send this vertex configuration to the linear combination of all these possibilities, e.g. for ω18 we have:

7→ + = q₁−1 + q2

ω187→ q1−1ρ8 + q2ρ9

(3.21)

(37)

under above transformation. This results in the following set of relations on the weights: ω1= ρ1 ω2 = ω4= q−1/21 ρ2= q−1/21 ρ3 ω3 = ω5= q1/21 ρ2= q1/21 ρ3 ω6 = ω8= q−1/2₂ ρ4= q−1/2₂ ρ5 ω7 = ω9= q1/22 ρ4= q1/22 ρ5 ω10= ω11= ω12= ω13= ρ6 = ρ7 ω14= ω15= ρ8 ω16= ω17= ρ9 ω18= q−11 ρ8+ q2ρ9 ω19= q1ρ8+ q−12 ρ9 (3.22)

We now explain these relations: The first relation states that the empty vertex configuration is mapped to the empty loop. The second to the fifth relation follow from equation (3.21). Now it follows that for consistency we should define the qi’s as:

q1··= ω3 ω2 = ω5 ω4 q2··= ω7 ω6 = ω9 ω8 (3.23)

When the oriented loops are straight lines we have no factors for the turns of the loop, so ρ6 = ω10 = ω11 and ρ7 = ω12 = ω13. The vertex configurations corresponding to ω14, ..., ω17

are mapped to an oriented loop configuration with one turn to the left and one to the right. This implies that the two factors of the turns are each other’s inverses and thus ρ8 = ω14= ω15

and ρ9= ω16= ω17. The final two relations in this list follow from equation 3.21.

Loops that meet on the same face of the dual lattice S′ (configurations 8 and 9 of figure 3.5) should be independent, i.e. the orientation of one of the line segments in such a tile should not be affected by the orientation of the other line segment. This is required by the summation over all orientations we need to perform to get to the loop model. The independence of the loop orientations can be pictured as:

× = × (3.24)

A similar relation should hold when the tiles are rotated by π/2. When we enforce this independence we must have that, following the conversion shown in equation (3.21), ω18 and

its decomposition in oriented loop tiles is consistent with the previous found equations. In formulas ω18= q2ρ9+ q−11 ρ8 and similar ω19= q−12 ρ9+ q1ρ8. These equation should then be

Dilute loop models and the BMW algebra