EXPLORING TOPOLOGICAL ORDER

IN MATTER

By

Juan Diego Jaramillo Salazar

Supervisor: Prof. Giuseppe Mussardo

A Dissertation

Presented to the Faculty

of The Abdus Salam International Center

for Theoretical Physics

Condensed Matter Physics

January 2010

ii

© Copyright by Juan Diego Jaramillo Salazar, 2009.

All Rights Reserved

Acknowledgements

I want to acknowledge the Abdus Salam International Centre for Theoretical

Physics (ICTP) and its Diploma program for giving me the opportunity to im-

prove my scientific skills, also to Prof. Scandolo director of the condensed matter

programme at the ICTP and my supervisor Prof. Mussardo from the Scuola In-

ternazionale Superiore di Studi Avanzati (SISSA) for his encourage and guide to

approach this promising branch of theoretical physics which is topological order in

matter. Finally a special acknowledge to my family for their unconditional support.

iv

To my family

v

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

2 The statistics of anyons 4

3 Anyons as Quantum Field Theories 8

3.1 Chern-Simons Gauge Fields . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Conformal Field Theories . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 The quantum group Uq(sl(2)) . . . . . . . . . . . . . . . . . . . . . 13

3.4 q-Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . 15

4 A knot-theoretic approach to anyons 19

4.1 The Jones polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Temperley-Lieb Algebra . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Jones-Wenzl Projector . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Recoupling Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 Braiding of anyons . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Fibonacci Anyons 35

vi

6 Conclusion 42

vii

Chapter 1

Introduction

Landau’s theory of symmetry breaking is the underlying description behind a wide

variety of systems: metals, semiconductors, magnets, superconductors, superflu-

ids, etc. Nevertheless, in the last decades condensed matter physics has seen the

rise of a whole new class of phases. Discoveries such as the fractional quantum

Hall (FQH) effect and high-Tc superconductivity cannot be described in terms

of symmetry breaking; there are many ground states, but they all have the same

symmetry! A derivation of FQH Pfaffian state in terms of a topological quantum

field theory (TQFT): the Chern-Simons theory, has been carried out [4]. TQFT

has been around for a while, they are special cases of quantum field theories, where

the action is invariant under continuous deformations of the space. Such relation

between FQH and TQFT suggest that FQH ground states are indeed topological

states of matter. The FQH quasi-particles live on 2 + 1 space-time dimensions

and carry fractional charge and flux [15]. They give rise to a degenerate ground

state. The reason is the nontrivial action of particle exchange, which corresponds

to braiding in 2 + 1 dimensions. The braiding group has multiple representations

1

and they generalize the concept of fermionic and bosonic statistics. While at 3D

space the only possible particles are bosons and fermions, in 2D a richer type

of statistical particles arise corresponding to a continuum of representations for

the braiding group. For example, the abelian representation U(1) assigns to each

type of anyons a multiplicative phase eiθ after exchanging anyons of the same type

i.e. ψ(r2, r1) = eiθψ(r1, r2). Representations of greater dimensions may lead to

non-abelian anyons. Its been experimentally proved that the FQH effect exhibits

abelian anyons at the filling fraction: ν = 1/3, and its been predicted to exhibit

non-abelian anyons at other filling fractions [20]. But anyons may not be exclusive

of the FQH effect, they are also candidates to explain high-Tc superconductivity,

superfluids and the A-phase of 3H films [23, 29, 16]. It is of great interest to

design new experimental settings leading to topological order, its benefits may be

foresight when looking at the multiple technologies based on symmetry-breaking

phases: magnetic memories, liquid crystal displays, low-Tc superconductivity, etc.

A special motivation for topological order comes after the work of Kitaev [12] which

pointed the benefits for the realization of a fault tolerant quantum computer ex-

ploiting the geometric invariance of topological states. Quantum memories and

gates are naturally robust when encoded in topological degrees of freedom since

they remain -up to an exponential decay- invariant under local perturbations. But

the simplest representation which allows for universal quantum computation has 60

dimensions [19]. Besides initialization, processing and measurement of topological

qubits, the first challenge is to realize a physical system with a suitable repre-

sentation in the latter sense. In these respect, there are many challenges in the

categorization of hamiltonian systems with suitable properties such as scalability,

stability of their topological ground states, addressability of the anyons, etc. The

2

partial answer to many of the big questions in this subject has reveal new insight

into the nature of particles and quasi-particles and encourage the continuing effort

to reveal the mysteries behind exotic statistics. In the following chapters I shall

give a very brief introduction to the theory of anyons, trying to give a unified

description encompassing fields from topology, algebra and (of course) physics.

3

Chapter 2

The statistics of anyons

Fermi-Dirac and Bose-Einstein statistics refer to how exchange between identical

particles transforms the total wave-function. Noting that “exchange” is a physical

process, it can be interpreted as derived from a topological action: the location of

particles can be continuously deformed while their statistical factor will not vary

unless a complete exchange has been performed. The fact that this operation is

topological implies that their fixed points in the classical configuration space will

correspond to singularities. Mathematically speaking, if X ≡ Rd is the classical

configuration space of a single particle, and XN the space for N particles, then

the quantum configuration space MdN for N particles in d-space is

MdN =

XN −∆

SN, (2.0.1)

where ∆ are points where at least two particles share the same coordinates and

SN is the permutation group of N objects. To understand how different statistics

emerge from this picture we can take the case of two particles, N = 2. There will

4

be a singularity at (r1, r2) ∈ X2 : r1 = r2. The physical process of exchanging

two particles is associated to a Feynman’s path integral, in particular a Wilson

loop, connecting both initial and final events. In the relative space any exchange

corresponds to a closed path of one particle around the origin. The singularity at

r1 = r2, corresponds to the origin in the relative space. Any “exchange” can either

enclose the singularity or not. If the dimension is d > 2, is easy to check that

there are only two closed paths up to continuous deformations, they correspond

to bosons and fermions depending whether the statistical factor is (+1) or (−1)

after exchange. When d = 1 (interacting) bosons are equivalent to (free) fermions.

But at d = 2 the topological classes associated to closed paths are infinite; paths

differing in the number of loops around the origin belong to a different topological

class. This leads to an infinite set of possible statistical particles. In mathematical

trivia, we have just noted that,

π1(Md>2N ) = SN , (2.0.2)

π1(Md=2N ) = BN ,

where π1(MdN) denotes the fundamental group of the space Md

N while SN and BN

are the symmetric and braid groups on N objects, respectively. Recall that BN

is an infinite group generated by N − 1 elements {σ1, ..., σN−1} satisfying Artin’s

relations:

σiσi+1σi = σi+1σiσi+1, 1 ≤ i ≤ N − 2, (2.0.3)

σiσj = σjσi, |i− j| ≥ 2. (2.0.4)

5

Both SN and BN are in general non-commutative, but in contrast with BN , if

σi ∈ SN then σ2i = 1. This can be interpreted in terms of tangles as depicted in

Fig.2.1. Further explanation on the role of tangles and its relation to the braid

Figure 2.1: In contrast with the symmetric group, elements in the braid group, ingeneral, are s.t. σ2

i 6= 1

group will be covered on chapter 4.

A 2D Magnetic Monopole

In order for the path integral to detect the singularity, we have to introduce a force

field to couple to. Since the singularity depends on the position of the two identical

particles, it is reasonable to associate this field to each particle. But this field is

special, it only contributes to distinguish topological states in the latter sense. An

interpretation of it in terms of a known field can be done on the plane; imagine a

magnetic field crossing perpendicularly the plane along an infinitesimal area δA,

then according to the Aharonov-Bhom effect any charged particle (e.g. electron)

enclosing the area δA will gain a multiplicative phase. This singularity is no more

than the 2D version of Dirac’s magnetic monopole, the latter corresponding to

a magnetic flux “intersecting” the 3D space. Furthermore, since the action is

topological, by using geometric deformations we can split the total flux into two

halves and relocate each half flux along the charged particles creating a charge-flux

composite particle as depicted in Fig.2.2. This equivalence can be interpreted as

the two-dimensional generalization of the Jordan-Wigner transformation, which

6

in one dimension transmutes fermions into bosons [18]. The exchange depicted in

Figure 2.2: Anyon statistics in 2D as the topological action of composite charge-flux particles. Two equivalent settings: (a) flux φ located at singularity r1 = r2;(b) flux φ/2 located in r1 and r2.

Fig.2.2 yields a multiplicative phase made out of the contribution of each particle’s

path:

exp[i(1

2qΦ +

1

2qΦ )] = eiqΦ = e−iπJ (2.0.5)

where q is the charge, Φ is the magnetic flux and J is the coupling strength between

charge and flux. Notice that when J = 0 or 1 we have bosons and fermions,

respectively. For 0 < J < 1 we have anyons, an infinite set of statistical particles.

As we will see in the following chapters it is possible to realize non-conmutative

statistical “factors” by considering multiple ground states. In general anyon theory

is build on any representation of the braid group Bn. The former example is an

abelian realization of the braid group and they are therefore called abelian anyons.

7

Chapter 3

Anyons as Quantum Field

Theories

In chapter 2 a physical, but still heursitic description of anyons was given. In this

chapter I’ll review the connection between anyons and topological field theories by

means of a special case: the realization of abelian anyons through Chern-Simons

fields. This will be done in the 2nd-quantized formalism in order to expose the

connections between anyons and q-deformed lie algebras [17].

3.1 Chern-Simons Gauge Fields

Considered a non-relativistic matter field ψ(~x, t) of mass m and charge e, min-

imally coupled to an abelian gauge field Aα(~x, t) with a Chern-Simons kinetic

term. The action of the system is given by

S =

∫d3x [iψ†D0ψ +

1

2mψ†(D2

1 +D22)ψ +

κ

2εαβγAα∂βAγ], (3.1.1)

8

where Dα = ∂α+ieAα is the covariant derivative. From now on ~ = c = 1. Varying

S with respect to Aα, we obtain

εαβγ∂βAγ =2e

κjα, (3.1.2)

where the 2nd-quantized current operator jα is given by

j0 = ψ†ψ ≡ ρ , ji =i

2m(ψ†Diψ − (Diψ)†ψ). (3.1.3)

and it satisfies the continuity equation

∂0ρ+ ~∇ ·~j = 0. (3.1.4)

Notice that Eq.(3.1.2) at α = 0 fix a Chern-Simons “magnetic” field strength to

every charged particle:

∂1A2 − ∂2A1 = − eκρ. (3.1.5)

Imposing the condition ∂iAi = 0, we can solve Eq.(3.1.2). This can be done using

the Green Function [17] to obtain,

Ai(x) = −ν∫d2y

∂

∂xiΘ(~x− ~y)ρ(y), (3.1.6)

where ν = e/2πκ and Θ(~x− ~y) is the angle under which ~x is seen from ~y, namely

Θ(~x− ~y) = tan−1

x2 − y2

x1 − y1

. (3.1.7)

9

We would like to take the partial derivative outside the integral. Since Θ(~x − ~y)

is a multifunction in the spatial coordinates, we need to fix a branch-cut and a

reference axis to remove the ambiguities; this is allowed, since the charge density

ρ is a sum of δ-functions. We therefore get the expression,

Ai(x) =∂

∂xiΛ(x), (3.1.8)

where

Λ(x) = ν

∫d2y Θ(~x− ~y)(ρ(y)− ρ0). (3.1.9)

Similarly we can find,

A0(x) = ν

∫d2y Θ(~x− ~y)∂0(ρ(y)− ρ0) = −∂0Λ(x), (3.1.10)

in such a way that we can express the former Chern-Simons potential as a pure

gauge: Aα = −∂αΛ(x). This means that we can obtain a free action by means of

the gauge transformation

Aα −→ A′α = Aα + ∂αΛ = 0, ψ(x) −→ ψ′(x) = e−ieΛ(x)ψ(x). (3.1.11)

Recalling the fact that we started with a bosonic field i.e.

[ψ(~x), ψ†(~x)] = δ(~x− ~y), [ψ(~x), ψ(~x)] = 0, (3.1.12)

and that since ρ = ψ†ψ we have

[ψ(~x),Λ(~y)] = −νΘ(~y − ~x)ψ(~x); (3.1.13)

10

we can deduce the following equal-time permutation formula,

Figure 3.1: Θ(~y − ~x)−Θ(~x− ~y) = π.

ψ′(~x)ψ′(~y) = e−ie

2πκ[Θ(~x−~y)−Θ(~y−~x)] ψ′(~y)ψ′(~x). (3.1.14)

If we further notice, as depicted in Fig.3.1, that Θ(~x − ~y) − Θ(~y − ~x) = π, the

former relation reduces to

ψ′(~x)ψ′(~y) = q ψ′(~y)ψ′(~x), q = eiνπ (3.1.15)

providing an algebraic equation for abelian anyons with statistics1 ν.

3.2 Conformal Field Theories

We now turn our attention to Conformal Field Theories (CFT) in our search for

theories exhibiting non-abelian representations of the braid group. It is well known

that Chern-Simons theories in (2+1) dimensions exhibit the same braiding and

1Notice that we are dealing with identical particles and reversing their order along the refer-ence axis should yield an equivalent statistics i.e. q−1 ≡ q.

11

fusion properties of two-dimensional CFT [31, 21], in addition we will gain a body

of formalism based on quantum groups which allows us to treat anyon models in

a purely algebraic way. In a two-dimensional CFT, we have correlation functions

of conformally invariant fields φ(z, z) taking values on the complex coordinate

z = x+ iy. These are real functions and can be decomposed as

〈n∏a=1

φia(za, za)〉 =∑p

|Fp;i1...in(z1, ..., zn)|2, (3.2.1)

where the terms Fp are meromorphic functions also called conformal blocks. They

define a vector bundle on (z1, ..., zn) transforming under the general action of the

braid group after reordering of particles in the correlator i.e.

Fp;i1...in(zi1 , ..., zir , ..., zis , ..., zin) =∑q

Bpq[i1, ..., in]Fq;i1...in(z1, ..., zn), (3.2.2)

where r < s and {ia; a = 1, ..., n} denotes the quantum numbers. Integers p and

q labels the elements of the basis associated to the corresponding orderings of the

quantum numbers. Notice the general notation: only if ir = is, r 6= s the exchange

is between identical particles. The ordering between particles is given by the norm

on C. The operator product expansion (OPE) exploits the conformal symmetry of

the fields φi to express the product of two such fields in the limit of z1 approaching

z2 as

φi(z1)φ(z2) =∑k

ak,dij (z1 − z2)hk,d−hi−hjφk,d, (3.2.3)

where index d labels all possible “descendants” of φi(z1) and φ(z2). The structure

constants ak,dij are constrained by demanding associativity and braiding consistency.

This constraints are important since our quantum numbers are associated to charge

12

of particles and associativity follows from charge conservation. Notice that to every

field φi there is an associated representation of the braid group, this motivates the

formulation of fusion rules in the same spirit of decomposition of irreps for angular

momentum composition. Nevertheless as we shall see, the fusion rules induced by

OPE belong to a more general class than the one associated to the ordinary tensor

product. The fusion rules are written as

φi × φj =∑k

Nkijφk, (3.2.4)

where the number Nkij counts the number of times that the field φk or its descen-

dants appear in the OPE of φi and φj.

3.3 The quantum group Uq(sl(2))

To understand the contrast between the ordinary tensor product and the fusion

rules we can take the example of the quantum group Uq(sl(2)). Recall that a

quantum group is a “q-deformation” of a universal enveloping algebra. In the case

of Uq(sl(2)) its generators {1, H, L+, L−} satisfy the relations,

[H,L±] = ±2L±, (3.3.1)[L+, L−

]=

qH/2 − q−H/2

q1/2 − q−1/2, (3.3.2)

where q may take any non-zero complex value. Notice that at q = 1 we recover

the universal enveloping algebra U(sl(2)) and that the transformation q → q−1

preserves the algebra. The new coproduct or fusion associated to the quasi-Hopf

13

algebra (see Appendix) for Uq(sl(2)) is given by

∆(H) = 1⊗H +H ⊗ 1 (3.3.3)

∆(L±) = L± ⊗ qH/4 + q−H/4 ⊗ L± (3.3.4)

∆ = id⊗ id (3.3.5)

notice that, except for q = 1 the coproduct is not cocommutative. Nevertheless

we will still demand coassociativity. As for the counit and antipode we have the

following values:

ε(1) = 1, ε(H) = 0, ε(L±) = 0, (3.3.6)

S(1) = 1, S(H) = −H, S(L±) = −q∓1/4L±. (3.3.7)

The representation theory when q is not a root of unity is similar to U(sl(2)); for

every total “angular momentum” j ∈ 12Z+ there is an irreducible highest weight

representation of dimension d = 2j + 1. We will denote this representation by

πΛ, where Λ = d − 1 = 2j is the highest weight. The modules V Λ of these

representations have orthonormal basis |j,m〉, with m = −j,−j + 1, ..., j and the

generating elements H, L+, L− act on this basis as follows

H|j,m〉 = 2m|j,m〉 (3.3.8)

L±|j,m〉 =√bj ∓mcqbj ±m+ 1cq |j,m± 1〉, (3.3.9)

14

where the q-number bmcq is defined as

bmcq =qm/2 − q−m/2

q1/2 − q−1/2. (3.3.10)

We are interested in unitary representations, therefore we need to introduce the

structure of hermitian conjugation in the algebra, this is done by means of a star-

structure. The star-structure that leads to Uq(su(2)) (the analogous of SU(2) for

U(sl(2)) is given by

∗(L±) = L∓, ∗(H) = H. (3.3.11)

A unitary representations is s.t. π(∗(x)) = π(x)†. In the star-structure of Eq.(3.3.11),

they corresponds to L± having real matrix elements. From Eq.(3.3.9) we deduce

that this happens only for q real and positive or for q = eiθ with θ ∈ R : |θ| ≤ 2πk+2

,

where k = (2j − 1) and j ∈ 12Z.

3.4 q-Clebsch-Gordan coefficients

When q is not a root of unity, the tensor product representation πΛ ⊗ πΛ′has the

same decomposition into irreps as for q = 1, i.e.

πΛ ⊗ πΛ′=

Λ+Λ′⊕Λ′′=|Λ−Λ′|

πΛ′′, (3.4.1)

where Λ′′ increases in steps of 2. Using Eqs.(3.3.9) we can systematically recon-

struct the q-deformed version of the Clebsch-Gordan coefficients. In the decom-

position of the tensor product (π2j1 ⊗ π2j2), the basis expansion for the irrep π2j

is denoted as,

15

|j,m〉 =∑m1,m2

j1 j2 j

m1 m2 m

q

|j1,m1〉|j2,m2〉. (3.4.2)

To find the associated q-6j-symbols we notice that the functors2 (∆ ⊗ id)∆ and

(id⊗∆)∆ lead to different categories; they transform morphisms into (1⊗ 2)⊗ 3

and 1 ⊗ (2 ⊗ 3), respectively (notice the simplified notation: s ≡ π2js). Their

modular spaces correspond to two different orthogonal bases in V j1 ⊗ V j2 ⊗ V j3

and can be expressed with the recursive use of the Clebsch-Gordan coefficients as:

ej12jm (j1j2|j3) =∑

m1,m2,m3

j12 j3 j

m12 m3 m

q

j1 j2 j12

m1 m2 m12

q

ej1m1⊗ ej2m2

⊗ ej3m3,

ej23jm (j1|j2j3) =∑

m1,m2,m3

j1 j23 j

m1 m23 m

q

j2 j3 j23

m2 m3 m23

q

ej1m1⊗ ej2m2

⊗ ej3m3.

But we can always find a transformation between these basis. Formally, such a

transformation between functors is called a natural transformation. An explicit cal-

culation of the q-6j-symbols will be performed for the so called Fibonacci anyons3

in chapter 5 using diagramatic language. As an advance I show the diagrams of

the q-Clebsch-Gordan coefficients,

−→

j1 j2 j

m1 m2 m

q

(3.4.3)

2These functors act on the category of the quasi-Hopf algebra (morphisms) and its modularrepresentations (objects).

3For the Fibonacci anyons the q-6j-symbols are slightly modified because q is a root of unity.

16

and the q-6j-symbols,

=∑j23

j1 j2 j23

j3 j j12

q

(3.4.4)

We remit the reader to Refs.[11, 30, 7] where a collection of results on recoupling

for Uq(sl(2)) can be found. In the remaining part I will focus on the case when q

is a root of unity. An interesting fact about this case is that it can be shown to

correspond to the operator algebra of the Wess-Zumino model at level k, where

q = ei2π/(k+2) is the deformation parameter [28]. But representations such as the

one discussed above are not always irreducible under ordinary tensor product. The

reason is because when q = ei2π/(k+2) and k is such that j ≥ k+22

, the elements

(L±)k+2 are map to zero in all representations defined by Eq.(3.3.9). To see this,

it is sufficient to notice that the former values of q yields,

bk + 2cq = 0. (3.4.5)

from zero to (k + 2)/2 − j. As a consequence we have an invariant submodule

generated by vectors between (L+)k+2|l〉 and (L−)k+2|h〉, where l and h are the

lowest and highest weight vectors in the representation as illustrated in Fig.3.2.

Alongside we have an orthogonal complement which is not invariant and therefore

not irreducible representation is possible, neither can we decompose it in a direct

sum since no inner product can be defined such that π2j is unitary with respect

17

Figure 3.2: An schematic illustration of indecomposable representations.

to it. This are non-physical representations and the way we get rid of them is

by redefining the tensor product: ⊗ → ⊗ (abusing of language I will call the new

product fusion just as the coproduct ∆ of the Hopf algebra). But we cannot simply

rule out indecomposable modules because there would be now way to recover

associativity i.e. a natural isomorphism like the one depicted in Eq.(3.4.4). Notice

for example that for odd k: (π1⊗πk)⊗πk+1 = 2πk+1 and π1⊗(πk⊗πk+1) = {0}.

Clearly there is no isomorphism to identify both functors. But we can “cut from

the bottom” by also projecting out modules of type πk+1. This additional criteria

assures associativity in the latter sense and defines the truncated product,

πΛ⊗πΛ′=

min{Λ+Λ′,2k−Λ−Λ′}⊕Λ′′=|Λ−Λ′|

πΛ′′, (3.4.6)

which happens to match the fusion rules of the Wess-Zumino model at level k.

18

Chapter 4

A knot-theoretic approach to

anyons

Whenever we exchange anyons we are braiding threads in 2 + 1 space-time dimen-

sions. The world-lines followed by anyons can only differ topologically i.e. in its

homotopy class. It follows that in the case we are able to measure such path-states,

they should be topological invariants. A remarkable result in this respect is the

relation between Chern-Simons theory and the Jones polynomial [31]. The Jones

polynomial is an invariant of knots and links and I will introduce the key elements

for its construction. I will follow Kauffman’s approach based on the bracket poly-

nomial [10]. With these tools I will reformulate the q-6j-symbols introduced in

Eq.(3.4.4). The first step is to obtain a representation of Bn into the Temperley-

Lieb associative algebra AlgTL via the bracket polynomial. Then the Jones-Wenzl

projectors are defined in AlgTL and the recoupling theorem is stated at roots of

unity. The q-6j-symbols follows naturally from the theorem.

19

4.1 The Jones polynomial

All knots can be projected into the plane such that crossings divide locally into

four regions. Each region can have one of two labels: A or A−1. Labeling de-

pends whether you see an under or over-crossing when facing the crossing from

the respective region. The convention is illustrated in the top diagram of Fig.4.1.

To each crossing there are associated two smoothings as shown in the bottom di-

agrams in Fig.4.1. If the smoothing connects two A-regions then we say is an

A-smoothing, likewise for A−1-smoothings. A state S of the diagram K is a choice

Figure 4.1: Top: Four regions with two labels: A and A−1. Bottom: two types ofsmoothing associated to crossings.

of smoothing for each crossing on K. An example is given by the trefoil diagram

K in Fig.4.2. For any state S in a diagram K we can denote the number of disjoint

jordan curves as ||S|| and let 〈K|S〉 be the product of the state labels. In the tre-

foil example ||S|| = 2 and 〈K|S〉 = A3. Now we can define the bracket polynomial

as 〈K〉 ∈ Z[A, A−1] :

20

〈K〉 =∑s

〈K|S〉 d||S|| (4.1.1)

The bracket polynomial is an invariant of regular isotopy of link diagrams. If K

is an oriented link diagram with w(K) the writhe1 of K, then

fk(A) = (−A3)−w(K) 〈K〉/〈0〉 (4.1.2)

is an invariant of ambient isotopy of link diagrams [8]. Some properties of the

bracket polynomial are:

(i) 〈 〉 = A⟨∪∩

⟩+ A−1〈||〉, where the small diagrams stand for parts of larger

ones that differ only as indicated by them.

(ii) 〈0 t K〉 = d 〈K〉, where 0t denotes the disjoint union of the diagram K

with a Jordan curve in the plane, and d = −A−2 − A2.

(iii) If VK(t) is the original Jones polynomial, then

VK(t) = fK(t−1/4)

1The writhe is a Z2-label for crossings on oriented links.

21

Figure 4.2: Diagram K of the trefoil knot and a choice of smoothing or state S.

4.2 Temperley-Lieb Algebra

The relation between knots and braids was already recognized by J. Alexander

who proved that any oriented link is isotopic to the closure of some braid [1].

This is illustrated in Fig.4.3 where the braid is represented in terms of a tangle: a

planar diagram consisting of n ordered points (objects) at the bottom (input) and

top (output) acting as boundaries to curves in 3D space embedded isotopically

in the plane, where crossings corresponds to labeled vertices. The use of braids

Figure 4.3: Braid closure.

to describe knots and links is appealing since they are clearly defined by Artin’s

relations (Eq. (2.0.4)). Because the representation of knots and links from closed

22

Figure 4.4: Tangle-theoretic interpretation of the generators of the Temperley-Liebalgebra (AlgTL).

braids is highly non-unique, it took several decades before a complete knot invari-

ant could be discovered2. This was done by V. Jones using the Temperley-Lieb

associative algebra (AlgTL) an algebra originally found in the study of the Potts

and ice-type models [6]. In Fig.4.4 is shown a tangle-theoretic interpretation of the

generators of AlgTL at level n = 4. Pointwise product is given by vertical stacking

and tensor product is given by horizontal stacking. Combining the property (i)

of the bracket polynomial and the tangle interpretation of AlgTL it is natural to

define the following representation ρ : Bn → Tn of the braid group into AlgTL:

ρ(σi) = A Ui + A−11n,

ρ(σ−1i ) = A−1Ui + A 1n. (4.2.1)

2Alexander provided the first attempt, but his polynomial is not invariant under all Reide-meister moves. Reidemeister moves are all possible “deformations” of a knot diagram.

23

where σi is a clockwise permutation between the i-th and (i+ 1)-th input threads

and A ∈ C and is related with the deformation parameter q as q = A2. Similarly,

σ−1i corresponds to counterclockwise permutation. The bracket polynomial of a

knot induces a “trace” function on AlgTL via braid closure:

tr(α) ≡ 〈α〉, (4.2.2)

where α ∈ AlgTL. For example tr(∪∩) = 〈 〉 = d and tr(||) = 〈 〉 = d2.

Together with a star-structure on AlgTL given by horizontal reflection, the former

trace becomes a positive semi-definite bilinear form at d = 2cos( πk+2

) and we denote

it by 〈·|·〉d [5]. This trace is often called the Markov’s trace.

4.3 Jones-Wenzl Projector

When d is “special” i.e. when d = 2cos( πk+2

), we can define for each level of

AlgTL an element fn−1 ∈ AlgTLn : 〈J(fn−1)|AlgTLn〉d = 0, where J is the

ideal closure. By the ideal closure in this context is meant, the smallest subset of

AlgTLn containing fn−1 s.t. is closed under formal (complex) linear combinations,

top/bottom and left/right stacking. These elements are the Jones-Wenzl projectors

and are illustrated as fn−1 = for every level n in AlgTL. They share the

following properties:

1. f 2i = fi.

2. fiUj = Ujfi = 0 for j ≥ i.

24

3. 〈fn−1〉 = ∆n(−A2), where ∆n(x) =

xn+1 − x−n−1

x− x−1

is the n-th Chebyschev

polynomial.

Such elements can be defined inductively by the following recursion formula:

, (4.3.1)

where µ1 = 1/d and µk+1 = (d−µk)−1. The Jones-Wenzl projectors for n = 1, 2, 3

objects are shown below:

f0 =

[ ∣∣∣∣ ],f1 = =

[ ∣∣∣∣∣∣∣∣ ]+1

d

[∪∩

],

f2 = =

[ ∣∣∣∣∣∣∣∣∣∣∣∣ ]− d

d2 − 1

[ ∣∣∣∣∪∩ +∪∩

∣∣∣∣ ]+1

d2 − 1

[+

].

4.4 Recoupling Theory

As pointed out in section 3.4, coassociativity of fusion is stated up to an isomor-

phism between functors. Such isomorphism corresponds to the q-deformed version

of the 6j-symbols and we provided a diagrammatic representation in Eq.(3.4.4).

25

In the present section I will use the AlgTL and the Jones-Wenzl projectors to pro-

Figure 4.5: 3-vertex are the building blocks of diagrams in the q-6j-symbol trans-formation. They are made from three “compatible” Jones-Wenzl projectors.

vide an interpretation of these diagrams. The specific values of the q-6j-symbols

will follow naturally from diagrammatic constraints. The building blocks of the

diagrams are the 3-vertex. They are made out of three projectors as depicted in

Fig.4.5. Not all combinations of three projectors a, b and c are allowed. Recall

that the label of the projectors correspond to the level of the AlgTL where it

belongs to3. The 3-vertex should obey the constraints:

i = (a+ b− c)/2,

j = (a+ c− b)/2, (4.4.1)

k = (b+ c− a)/2,

where a, b, c, i, j, k ∈ Z+. Physically, these constraints can be associated to

conservation of charge, recalling that threads are world-lines of anyons which are

carriers of fractional charge. The constraints can be rewritten as,

(i) a+ b+ c ≡ 0 (mod 2)

3The angular momentum j associated to a given projector a is given by: j = a/2, the reasonis because a single thread in the tangle representation contributes with angular momentum 1/2.

26

(ii) a+ b− c, b+ c− a, c+ a− b are each ≥ 0.

In the case when q is a root of unity and s.t. q = eiπ/r where r ∈ Z+, we have the

additional constraint:4

(iii) a+ b+ c ≤ 2r − 4.

A triple {a, b, c} which obey the former constraints is said to be a q-admissible

triple and is symbolized by {a, b, c} ∈ ADMq.

From the former construction of 3-vertex blocks and based on the recoupling the-

orem we get a natural derivation of the q-6j-symbols. I will state it and remit the

reader to Ref.[10] for its prove:

Recoupling Theorem Let a, b, c, d and j be non-negative fixed integers, such

that {a, b, j} and {c, d, j} are q-admissible triples. Then there exist unique real

numbers αi. 0 ≤ i ≤ r − 2, such that

=∑i

{b,c,i}∈ADMq

{a,d,i}∈ADMq

αi . (4.4.2)

4It is designed to address the complications discussed in the last paragraph of section 3.4.

27

We can solve the former equation expanding into the Temperley-Lieb algebra by

means of the recursion relation depicted in Eq.(4.3.1). An alternative way is to use

Markov’s trace. I will show the latter method for the case of Fibonacci anyons in

chapter 5. Before going into Fibonacci anyons some important comments on the

q-6j-symbol should be said. As it was pointed out this is a natural transformation.

It can be shown that diagrams on the left and right hand side of Eq.(4.4.2), con-

stitute two different basis. They generate the vector space of diagrams where four

projectors (a, b, c, d) are connected by any possible flat tangle T as depicted in

Fig.4.6. From this perspective it is clear that the natural transformation given by

Figure 4.6: Horizontal and vertical diagrams form two different bases for the vectorspace constituted by four projectors connected by any possible flat tangle T .

the q-6j-symbols is a change of basis. A criteria that assures that this transforma-

tion is consistent with coassociativity as discussed in section 3.4, is to check for its

ergodicity on the set of functors generated by all possible orders of pairwise fusions

on n anyons. The prove of this is reduced by the MacLane coherence theorem to

an array of four anyons. This is the so called pentagon equation. Let F denote the

natural transformation induced by the q-6j-symbols. In the simplified notation:

28

s ≡ π2js , the former constraint implies that the path

(1⊗(2⊗3))⊗4F−→ 1⊗((2⊗3)⊗4)

F−→ 1⊗(2⊗(3⊗4))F−→ (1⊗2)⊗(3⊗4),

(4.4.3)

is equivalent to the alternative path,

(1⊗(2⊗3))⊗4F−→ ((1⊗2)⊗3)⊗4

F−→ (1⊗2)⊗(3⊗4). (4.4.4)

It is not difficult to prove that the definition of F in terms of tangles obey the

former relation, nevertheless we remit the reader to Ref.[10] for details. The former

relation can be stated in a diagrammatic fashion as depicted in Fig.4.7. Notice

29

Figure 4.7: Diagrammatic representation of the pentagon equation.

that the choice of a category is the algebraic equivalent of a choice of coordinates

in a geometric setting.

4.5 Braiding of anyons

In section 3.3 it was noticed that comultiplication on the algebra i.e. fusion of

anyons was no longer cocommutative. This suggest the definition of an operation

for commuting anyons which is known as braiding, R : A ⊗ A → A ⊗ A, where

A is the quasi-hopf algebra. In section 3.3, we had A ≡ Uq(sl(2)) and R takes

the form of an isomorphism between (1 ⊗ 2) and (2 ⊗ 1). This resembles the ac-

tion of the natural transformation F , but their meaning is very different; while

30

braiding is a physical process, the natural transformation is just a mathematical

aim to express the space of anyons; given an array of anyons fusing into, lets say

the vacuum, there is no way we can inquire the system as to know what was the

specific sequence of decays leading to the former array5. That is, all compatible

fusion sequences represented with every functor in the pentagon equation is phys-

ically indistinguishable and therefore the claim that F -moves are not a physical

processes, but a choice of “coordinate system”. At this stage, where the structure

of quantum groups has been reformulated in terms of consistency equations such

as the pentagon equation, it seems natural to define braiding in the same terms.

Indeed, the Yang-Baxter equation

σ1σ2σ1 = σ2σ1σ2, σi ∈ R (4.5.1)

present on Artin’s characterization of the braid group is derived from demanding

consistency of braiding on three objects tensor product as depicted on Fig.4.8.

In this figure, the spaces U , V and W are the representations of the quasi-hopf

algebra A, which after adding this new transformation is to be called a quasi-

triangular quasi-hopf algebra. There is one more consistency equation, the so called

hexagon equation involving both: the natural transformations F and braiding R.

But before we state it, a few more comments should be said about the pairwise

action of R. The first observation is that R reduces to the symmetric group Sn,

with the additional constraint σ2i = id. That would be the case for statistics in

3 + 1 space-time dimensions, but as mentioned in chapter 2, we are interested in

2 + 1 space-time, where its general braid group structure allows for a richer set

5Decay, understood as the inverse process of fusion of two particles into one.

31

Figure 4.8: Diagrammatic representation of the Yang-Baxter equation.

of statistics. In particular it allows for unitary representations which are dense

in SU(2), providing a way to realize quantum computation. I show the action of

braiding on a 3-vertex, since it will be useful for later calculations:

= (−1)(b+c−a)/2A[b(b+2)+c(c+2)−a(a+2)]/2 (4.5.2)

the prove of it can be found in Ref.[10]. The last constraint is the hexagon equation

which assures consistency between braiding R and the natural transformation F .

Whenever we have more than two anyons, the specific action of braiding depends

on the choice of the category or its associated functor6. In particular, for every

braiding of adjacent anyons, there is a category (actually many) which is modular

to the action of R i.e. it becomes a diagonal matrix as in Eq.(4.5.2). In such a case

6Although F acts on functors, by fixing the input category, every functor can be associatedto a given output category.

32

the pairwise action of R will be denoted as Rbc with diagonal elements Rabc. The

usual procedure for the implementation of braiding is to use F -moves to approach

the suitable category were braiding looks simple (R → R), apply R and then use

F -moves to come back to the original category. The Fig.4.9 shows this procedure

when the original category is at one F -move from the suitable category. The final

action is denoted by B to emphasize the fact that the new operation has non-

diagonal matrix elements and therefore it constitutes a non-abelian representation

of the braid group in this case of B3. The hexagon equation is depicted in Fig.4.10.

Figure 4.9: The action of braiding depends on the choice of category used todescribe the array of anyons.

It is defined on an array of three anyons. There are two equivalent paths to braid

anyon 1 around the fusion of anyons 2 and 3. In the modular representations the

equivalence takes the form:

R13F213R12 = F231R1bF123 (4.5.3)

33

where each of these terms is a matrix acting on the space generated by all possible

fusion channels. The anyons involved in each fusion are specified in the subscripts

in their respective order. Considering the fact that R-matrices are diagonal, the

Figure 4.10: The hexagon equation assures the consistency between braiding andrecoupling.

former equation reduces to:

Rc13(F213)caR

a12 =

∑b

(F231)cbR41b(F123)ba. (4.5.4)

34

Chapter 5

Fibonacci Anyons

Now we will use the former tools to build a representation of the braid group Bn

which is dense in SU(m) and therefore suitable for universal quantum computa-

tion. I have followed the approach of Kauffman and Lomonaco in Ref.[9] which

provides a particularly appealing derivation based on recoupling theory through

the Temperley-Lieb algebra. First, lets notice that any recoupling theory based

Figure 5.1: The two non-trivial 3-vertex of the Fibonacci anyons.

35

on AlgTL is fixed after specifying a value of d (or q, or A) and the order n of

the highest Jones-Wenzl projector fn−1, present in the theory. Fibonacci Anyons

happen to be the simplest non-trivial example of a recoupling theory and is fixed

by the values: d = 1+√

52

(“golden number”) and n = 2. I will start with the as-

sumption of n = 2 and later show that the value of d follows, at least for Fibonacci

anyons, after demanding symmetry of the recoupling theory. The two non-trivial

3-vertex are depicted in Fig.5.1 and their notation is simplified with a continuous

line for f1 projector and dashed line for the trivial projector i.e. the vacuum.

There is actually a third 3-vertex made of two 2-projectors and one 4-projector

(a = b = 2, c = 4), but as I will show later, it is ruled out by fixing the value of

d at the “golden number”, where q becomes a root of unity with r = 5. We say

that Fibonacci anyons have two fusion channels. The corresponding recoupling

equations are:

= w + x ,

(5.0.1)

= y + z .

Now we use Markov’s trace together with the star-structure discussed in the last

paragraph of section 4.2 to perform a projection between diagrams. There are

three “indecomposable” net evaluations we can find. All other evaluations can be

reduced to products of the former net values. The first net is the Theta-net. Let

36

≡ H0 and ≡ V2 then,

tr(H0V2) = tr( ) = 〈 〉 ≡ Θ, (5.0.2)

where ∗(V2) = V2. The resulting diagram is called the Theta-net and its bracket

evaluation is denoted by Θ. Then we have the so called Tetrahedron-net ; it is

found in the closure of H2V2, where ≡ H2. We then have,

tr(H2V2) = 〈 〉 ≡ T. (5.0.3)

Finally there is a net associated to the closure of a 2-projector:

tr(H0V0) = 〈 〉 = ∆, (5.0.4)

where V0 is the 90 degrees rotation of diagram H0. The symbol ∆ is not to be

confused with the coproduct in the former context. Using recursion formula from

37

Eq.(4.3.1) we find the values:

∆ = d2 − 1, (5.0.5)

Θ = (d− d−1)2 − ∆

d, (5.0.6)

T = (d− d−1)2(d2 − 2)− 2Θ

2. (5.0.7)

We can solve Eqs.(5.0.1) by performing the following projections: 〈∗V0〉 and 〈∗V2〉,

where ∗ are the elements at each side of the corresponding equation. This leads

to the net equations:

= w −→ ∆ = w ∆2 ⇒ w =1

∆, (5.0.8)

= x −→ Θ = xΘ2

∆⇒ x =

∆

Θ, (5.0.9)

= y −→ Θ = y ∆2 ⇒ y =Θ

∆2, (5.0.10)

= z −→ T = zΘ2

∆⇒ z =

T∆

Θ2. (5.0.11)

Now that we have our natural transformation F , we look for the diagonal form of

the braiding operator R. Since we have only two non-trivial 3-vertex, the linear

operator R acts on a two dimensional space depending on the fusion channel.

38

Using Eq.(4.5.2) we obtain the following values:

= −A−4 , = A8 , (5.0.12)

That is

R =

−A−4 0

0 A8

. (5.0.13)

We now derive the bracket value d of a loop by demanding F : F 2 = 1. This

constraint seems natural since there is no reason for the natural transformation to

be “oriented” in consideration that is not a physical operation. Since,

F ≡

w x

y z

=

1/∆ ∆/Θ

Θ/∆2 T∆/Θ2

. (5.0.14)

Then,

F 2 ≡ 1

(5.0.15)

=⇒ (F 2)11 =1

∆+

1

∆2= 1 ⇐⇒ ∆ =

1±√

5

2.

Further notice that for this values of ∆, we have ∆2 = d2 from Eq.(5.0.5). With

the former constraint, the only way for F to be unitary is that F = F †. This

implies: (i) F : F = F T and (ii) F † = F T . For the first constraint we exploit the

freedom for renormalizing the 3-vertex. Mainly we add a factor α to every 3-vertex

where two 2-projectors fuse into one 2-projector. This changes the recoupling in

39

such a way that Θ→ α2Θ. The corresponding value of F takes the form:

F =

1/∆ ∆/α2Θ

α2Θ/∆2 T∆/Θ2

. (5.0.16)

In particular, if α2 = ∆3/2/Θ then F becomes symmetric:

F −→

1/∆ 1/√

∆

1/√

∆ −1/∆

. (5.0.17)

The second constraint i.e. F † = F T is fulfilled by setting ∆ = d = (1 +√

5)/2

or q = ei6π/5. We can now check whether the 3-vertex with projectors a = b = 2

and c = 4 satisfies the third constraint for a 3-vertex mainly: a + b + c ≤ 2r − 4.

First we should find r ∈ Z+ : qr = −1. The first candidate which is 5/6 is not an

integer, in this case we proceed1 to find the smallest integer power of q yielding

either (+1) or (−1) which for q = ei6π/5 corresponds to r = 5. Replacing this

value in the 3-vertex constraint discards a = b = 2 and c = 4 as an admissible

triple. As we mentioned in the introduction of the hexagon equation, if we have

three anyons we can obtain a representation of the braid group B3 by means of the

conjugation FRF i.e. {id, R, FRF} is a unitary representation of B3. Moreover,

as its been noted, this representation is dense in SU(2) allowing the realization of

one qubit gates with arbitrary precision [22, 9]. If we keep adding anyons we will

end up with a dense “topological” representation of SU(m) and therefore with a

universal quantum computer.

1This procedure follows after noticing that r is meant to be the smallest integer such that theChebischev polynomial ∆r−1(x) = 0, as explained in chapter 5 of [10].

40

41

Chapter 6

Conclusion

I have reviewed the relation between anyons and topological field theories by means

of a special case: the realization of abelian anyons through a Chern-Simons kinetic

term. I used 2nd quantization in order to expose the relation between anyons and

quantum groups. I also showed the approach based on knot theory to the recou-

pling theory of the Wess-Zumino models and applied these tools to derive Fibonacci

anyons; a model which is known to exhibit, at least in principle, fault-tolerant uni-

versal quantum computation. It should be said that the relation between the

former theories and specific hamiltonian systems (such as Kitaev’s honeycomb

lattice) was not presented here, neither the important relation between quantum

groups and quantum complete integrability as advanced by the work of Faddeev,

Sklyanin and Takhtajan using the quantum inverse scattering method [13, 27, 14].

These are important fields of research for statistical and condensed matter physics.

The brief introduction given in this dissertation should be general enough as to be

useful in these latter subjects as well.

42

NOTE

Some of the additional references which provided me with an introduction to

many of the topics discussed above -many of the figures were adapted from their

publications- are [25, 26, 2, 3, 24].

APPENDIX

Definition A Hopf algebra is an associative algebra A with multiplication µ and

unit 1 that has an extra structure ε, S and ∆ called counit, antipode and coprod-

uct. The coproduct or comultiplication ∆ is an algebra map from A to A ⊗ A

with the following property called coassociativity:

(∆⊗ id)∆ = (id⊗∆)∆, (.0.1)

where id is the identity map A. The counit ε of A is an algebra map from A to

C, or equivalently, a one-dimensional representation of A, satisfiying

(ε⊗ id)∆ = (id⊗ ε)∆. (.0.2)

The antipode S of A is a linear map from A to A satisfying

µ(S ⊗ id)∆(a) = µ(id⊗ S)∆(a) = ε(a)1, (.0.3)

for each a ∈ A. Whenever the coassociativity condition is weakened up to a non-

trivial isomorphism the algebra is called a quasi-Hopf algebra. Together with braid-

ing they formed a quasi-triangular Hopf algebra or weak quasi-quantum group.

43

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