Exploring Topological Order In Matter – Ictp Dissertation

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 improve my scientific skills, also to Prof. Scandolo director of the condensed matter programme at the ICTP and my supervisor Prof. Mussardo from the Scuola Internazionale 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.

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To my family

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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

3.3

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

3.4

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

. . . . . . . . . . . . . . . . . . . . . . . 11

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, superfluids, 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 ) = eiθ ψ(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 3 H 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 exploiting 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 representation 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 X N the space for N particles, then the quantum configuration space MNd for N particles in d-space is MNd

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 ) ∈ X 2 : 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 (MNd>2 ) = SN ,

(2.0.2)

π1 (MNd=2 ) = BN , where π1 (MNd ) denotes the fundamental group of the space MNd 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:

σi σi+1 σi = σi+1 σi σi+1 , σi σj = σj σi ,

1 ≤ i ≤ N − 2,

|i − j| ≥ 2.

5

(2.0.3) (2.0.4)

Both SN and BN are in general non-commutative, but in contrast with BN , if σi ∈ SN then σi2 = 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, in general, are s.t. σi2 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 chargeflux 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 1 qΦ + qΦ )] = eiqΦ = e−iπJ 2 2

(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, minimally coupled to an abelian gauge field Aα (~x, t) with a Chern-Simons kinetic term. The action of the system is given by Z S=

d3 x [iψ † D0 ψ +

1 † 2 κ ψ (D1 + D22 )ψ + αβγ Aα ∂β Aγ ], 2m 2 8

(3.1.1)

where Dα = ∂α +ieAα is the covariant derivative. From now on ~ = c = 1. Varying S with respect to Aα , we obtain αβγ ∂β Aγ =

2e α j , κ

(3.1.2)

where the 2nd-quantized current operator j α is given by

j 0 = ψ†ψ ≡ ρ ,

i (ψ † Di ψ − (Di ψ)† ψ). 2m

ji =

(3.1.3)

and it satisfies the continuity equation

~ · ~j = 0. ∂0 ρ + ∇

(3.1.4)

Notice that Eq.(3.1.2) at α = 0 fix a Chern-Simons “magnetic” field strength to every charged particle: e ∂1 A2 − ∂2 A1 = − ρ. κ

(3.1.5)

Imposing the condition ∂i Ai = 0, we can solve Eq.(3.1.2). This can be done using the Green Function [17] to obtain,

i

A (x) = −ν

Z

d2 y

∂ Θ(~x − ~y )ρ(y), ∂xi

(3.1.6)

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



 x2 − y 2  Θ(~x − ~y ) = tan−1  . x1 − y 1

9

(3.1.7)

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) =

∂ Λ(x), ∂xi

(3.1.8)

where Z Λ(x) = ν

d2 y Θ(~x − ~y )(ρ(y) − ρ0 ).

(3.1.9)

Similarly we can find, Z A0 (x) = ν

d2 y Θ(~x − ~y )∂0 (ρ(y) − ρ0 ) = −∂0 Λ( x),

(3.1.10)

in such a way that we can express the former Chern-Simons potential as a pure gauge: Aα = −∂α Λ(x). This means that we can obtain a free action by means of the gauge transformation

Aα −→ A0α = Aα + ∂α Λ = 0,

ψ(x) −→ ψ 0 (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); 10

(3.1.13)

we can deduce the following equal-time permutation formula,

Figure 3.1: Θ(~y − ~x) − Θ(~x − ~y ) = π.

e

ψ 0 (~x)ψ 0 (~y ) = e−i 2πκ [Θ(~x−~y)−Θ(~y−~x)] ψ 0 (~y )ψ 0 (~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 0 (~x)ψ 0 (~y ) = q ψ 0 (~y )ψ 0 (~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 1

Notice that we are dealing with identical particles and reversing their order along the reference 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

h

n Y

φia (za , z¯a )i =

a=1

X

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

(3.2.1)

p

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 ) =

X

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

(3.2.2)

q

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 ) =

X

hk,d −hi −hj ak,d φk,d , ij (z1 − z2 )

(3.2.3)

k

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

X

Nijk φk ,

(3.2.4)

k

where the number Nijk counts the number of times that the field φk or its descendants 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± ,  + − q H/2 − q −H/2 , L ,L = q 1/2 − q −1/2

(3.3.1) (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 ∆(L± ) = L± ⊗ q H/4 + q −H/4 ⊗ L± ∆ = id ⊗ id

(3.3.3) (3.3.4) (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:

(L± ) = 0,

(1) = 1,

(H) = 0,

S(1) = 1,

S(H) = −H,

S(L± ) = −q ∓1/4 L± .

(3.3.6) (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 ∈ 12 Z+ there is an irreducible highest weight representation of dimension d = 2j + 1. We will denote this representation by π Λ , where Λ = d − 1 = 2j is the highest weight. The modules V Λ of these representations have orthonormal basis |j, mi, with m = −j, −j + 1, ..., j and the generating elements H, L+ , L− act on this basis as follows

H|j, mi = 2m|j, mi q L± |j, mi = bj ∓ mcq bj ± m + 1cq |j, m ± 1i,

14

(3.3.8) (3.3.9)

where the q-number bmcq is defined as

bmcq =

q m/2 − q −m/2 . q 1/2 − 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 starstructure. 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| ≤

2π , k+2

where k = (2j − 1) and j ∈ 21 Z.

3.4

q-Clebsch-Gordan coefficients 0

When q is not a root of unity, the tensor product representation π Λ ⊗ π Λ has the same decomposition into irreps as for q = 1, i.e.

πΛ ⊗ π

Λ0

0 Λ+Λ M

=

00

πΛ ,

(3.4.1)

Λ00 =|Λ−Λ0 |

where Λ00 increases in steps of 2. Using Eqs.(3.3.9) we can systematically reconstruct the q-deformed version of the Clebsch-Gordan coefficients. In the decomposition of the tensor product (π 2j1 ⊗ π 2j2 ), the basis expansion for the irrep π 2j is denoted as, 15

 |j, mi =



X  j1 j2 j    |j1 , m1 i|j2 , m2 i. m1 m2 m m1 ,m2

(3.4.2)

q

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 ≡ π 2js ). 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:  ejm12 j (j1 j2 |j3 ) =

X m1 ,m2 ,m3

ejm23 j (j1 |j2 j3 ) =

X m1 ,m2 ,m3

 

 j12 j3 j    m12 m3 m q    j1 j23 j    m1 m23 m

q



 j1 j2 j12  j1 j j   em1 ⊗ em2 2 ⊗ em3 3 , m1 m2 m12 q    j2 j3 j23  j1 j j  em1 ⊗ em2 2 ⊗ em3 3 .  m2 m3 m23 q

But we can always find a transformation between these basis. Formally, such a transformation between functors is called a natural transformation. An explicit calculation of the q-6j-symbols will be performed for the so called Fibonacci anyons 3 in chapter 5 using diagramatic language. As an advance I show the diagrams of the q-Clebsch-Gordan coefficients,

 −→

2



 j1 j2 j    m1 m2 m

(3.4.3) q

These functors act on the category of the quasi-Hopf algebra (morphisms) and its modular representations (objects). 3 For the Fibonacci anyons the q-6j-symbols are slightly modified because q is a root of unity.

16

and the q-6j-symbols,

=

X j23

    j1 j2 j23     j j j   3

12

(3.4.4)

q

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+2 , 2

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 |li and (L− )k+2 |hi, 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 π 2j 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 ˆ (abusing of language I will call the new by redefining the tensor product: ⊗ → ⊗ 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 ˆ k )⊗π ˆ k+1 = 2π k+1 and π 1 ⊗(π ˆ k ⊗π ˆ k+1 ) = {0}. for example that for odd k: (π 1 ⊗π Clearly there is no isomorphism to identify both functors. But we can “cut from the bottom” by also projecting out modules of type π k+1 . This additional criteria assures associativity in the latter sense and defines the truncated product, min{Λ+Λ0 ,2k−Λ−Λ0 }

ˆ π Λ ⊗π

Λ0

M

=

00

πΛ ,

(3.4.6)

Λ00 =|Λ−Λ0 |

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 dimensions. 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 polynomial [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 TemperleyLieb associative algebra AlgT L via the bracket polynomial. Then the Jones-Wenzl projectors are defined in AlgT L 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 depends 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 diagrams 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 of smoothing 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 hK|Si be the product of the state labels. In the trefoil example ||S|| = 2 and hK|Si = A3 . Now we can define the bracket polynomial as hKi ∈ Z[A, A−1 ] :

20

hKi =

X

hK|Si d||S||

(4.1.1)

s

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) hKi/h0i

(4.1.2)

is an invariant of ambient isotopy of link diagrams [8]. Some properties of the bracket polynomial are:

(i) h

i = A

∪ ∩

+ A−1 h||i, where the small diagrams stand for parts of larger

ones that differ only as indicated by them.

(ii) h0 t Ki = d hKi, 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 )

1

The 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-Lieb algebra (AlgT L). braids is highly non-unique, it took several decades before a complete knot invariant could be discovered2 . This was done by V. Jones using the Temperley-Lieb associative algebra (AlgT L) 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 AlgT L 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 AlgT L it is natural to define the following representation ρ : Bn → Tn of the braid group into AlgT L: ρ(σi ) = A Ui + A−1 1n , ρ(σi−1 ) = A−1 Ui + A 1n .

(4.2.1)

2 Alexander provided the first attempt, but his polynomial is not invariant under all Reidemeister 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, σi−1 corresponds to counterclockwise permutation. The bracket polynomial of a knot induces a “trace” function on AlgT L via braid closure:

tr(α) ≡ hˆ αi,

where α ∈ AlgT L. For example tr( ∪∩ ) = h

(4.2.2)

i = d and tr(||) = h

i = d2 .

Together with a star-structure on AlgT L given by horizontal reflection, the former π trace becomes a positive semi-definite bilinear form at d = 2cos( k+2 ) and we denote

it by h·|·id [5]. This trace is often called the Markov’s trace.

4.3

Jones-Wenzl Projector

π ), we can define for each level of When d is “special” i.e. when d = 2cos( k+2

AlgT L an element fn−1 ∈ AlgT Ln : hJ(fn−1 )|AlgT Ln id = 0, where J is the ideal closure. By the ideal closure in this context is meant, the smallest subset of AlgT Ln 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 AlgT L. They share the

following properties: 1. fi2 = fi . 2. fi Uj = Uj fi = 0 for j ≥ i.

24

  x 3. hfˆn−1 i = ∆n (−A2 ), where ∆n (x) = 

n+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 1 + + = − 2 d − 1 ∩ ∩ d2 − 1

4.4

+

 .

Recoupling Theory

As pointed out in section 3.4, coassociativity of fusion is stated up to an isomorphism 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 AlgT L and the Jones-Wenzl projectors to pro-

Figure 4.5: 3-vertex are the building blocks of diagrams in the q-6j-symbol transformation. 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 AlgT L 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) 3

The angular momentum j associated to a given projector a is given by: j = a/2, the reason is 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 theorem 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

=

X

αi

.

(4.4.2)

i

{b,c,i}∈ADMq {a,d,i}∈ADMq

4

It 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), constitute 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 vector space 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 transformation 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 ≡ π 2js , the former constraint implies that the path F F F ˆ ⊗3)) ˆ ⊗4 ˆ −→ ˆ ⊗3) ˆ ⊗4) ˆ −→ ˆ ⊗(3 ˆ ⊗4)) ˆ ˆ ⊗(3 ˆ ⊗4), ˆ (1⊗(2 1⊗((2 1⊗(2 −→ (1⊗2)

(4.4.3)

is equivalent to the alternative path, F

F

ˆ ⊗3)) ˆ ⊗4 ˆ −→ ((1⊗2) ˆ ⊗3) ˆ ⊗4 ˆ −→ (1⊗2) ˆ ⊗(3 ˆ ⊗4). ˆ (1⊗(2

(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 action 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 physically 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

σ1 σ2 σ1 = σ2 σ1 σ2 ,

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 quasitriangular 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 σi2 = 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 5

Decay, 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)/2 A[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 6

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

32

a the pairwise action of R will be denoted as Rbc with diagonal elements Rbc . 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 nondiagonal 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 to describe 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:

R13 F213 R12 = F231 R1b F123

33

(4.5.3)

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 and recoupling. former equation reduces to:

c a R13 (F213 )ca R12 =

X b

34

4 (F231 )cb R1b (F123 )ba .

(4.5.4)

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 computation. 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 AlgT L 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+ 5 2

(“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(H0 V¯2 ) = tr(

)=h

i ≡ Θ,

(5.0.2)

where ∗(V2 ) = V¯2 . 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 H2 V¯2 , where

≡ H2 . We then have,

tr(H2 V¯2 ) = h

i ≡ T.

(5.0.3)

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

tr(H0 V¯0 ) = h

i = ∆,

(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)

∆ , d Θ T = (d − d−1 )2 (d2 − 2) − 2 . 2 Θ = (d − d−1 )2 −

(5.0.6) (5.0.7)

We can solve Eqs.(5.0.1) by performing the following projections: h∗d V¯0 i and h∗d V¯2 i, where ∗ are the elements at each side of the corresponding equation. This leads to the net equations:

=w

=x

−→ ∆ = w ∆2 ⇒ w =

−→ Θ = x

Θ2 ∆

⇒ x=

1 , ∆

(5.0.8)

∆ , Θ

(5.0.9)

=y

−→ Θ = y ∆2 ⇒ y =

Θ , ∆2

(5.0.10)

=z

−→ T = z

Θ2 ∆

T∆ . Θ2

(5.0.11)

⇒ z=

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

That is

,

= A8





,

−4 0   −A R= . 8 0 A

(5.0.12)

(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, 







∆/Θ   w x   1/∆ F ≡ . = Θ/∆2 T ∆/Θ2 y z

(5.0.14)

Then,

F2 ≡ 1 √

=⇒ (F 2 )11 =

(5.0.15)

1 1 1± 5 + 2 = 1 ⇐⇒ ∆ = . ∆ ∆ 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 → α2 Θ. The corresponding value of F takes the form:   F =

 1/∆ α2 Θ/∆2

2

∆/α Θ  . T ∆/Θ2

(5.0.16)

In particular, if α2 = ∆3/2 /Θ then F becomes symmetric:  √  1/∆ 1/ ∆  F −→  √ . 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+ : q r = −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 F RF i.e. {id, R, F RF } 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. 1

This procedure follows after noticing that r is meant to be the smallest integer such that the Chebischev 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 recoupling 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 universal 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 , S and ∆ called counit, antipode and coproduct. 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  of A is an algebra map from A to C, or equivalently, a one-dimensional representation of A, satisfiying

( ⊗ id)∆ = (id ⊗ )∆.

(.0.2)

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

µ(S ⊗ id)∆(a) = µ(id ⊗ S)∆(a) = (a)1,

(.0.3)

for each a ∈ A. Whenever the coassociativity condition is weakened up to a nontrivial isomorphism the algebra is called a quasi-Hopf algebra. Together with braiding they formed a quasi-triangular Hopf algebra or weak quasi-quantum group. 43

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