A lattice model for topological phases

A lattice model for topological phases

Jonatan Andersson

Fakulteten för hälsa, natur- och teknikvetenskap Fysik

Examensarbete 15 hp, kandidatnivå Jürgen Fuchs, examinator

A lattice model for topological phases

Jonatan Andersson

Contents

1 The lattice 7

2 The operators 10

2.1 Triangle operators . . . 10

2.2 Ribbon operators . . . 12

2.2.1 Ribbon operators, definition . . . 12

2.2.2 Ribbon operators, Composition and Commutation relations . . . 14

2.2.3 Ribbon operators in Tensor form . . . 16

2.3 Local operators . . . 17

3 The Hamiltonian and the space of states 20 3.1 The Hamiltonian and Ground state Definition . . . 20

3.2 Excited states, Definition . . . 21

3.3 Obtaining the ground state . . . 22

Abstract

Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau’s symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.

Abstrakt p˚a Svenska.

Materia existerar i m˚anga olika faser, till exempel fast eller flytande fas. Det finns ocks˚a faser d¨ar atomerna sitter likadant, men som skiljer sig p˚a n˚agot annat s¨att, till exempel ferromagnetiska och paramagnetiska faser. Landau’s symmetribrottsteori s¨ager att varje fas¨overg˚ang h¨anger ihop med en symmetribrottsprocess. Fast material har diskret translationssymmetri medans materia i flytande fas har kontinuerlig trans-lationssymmetri. Men det har visat sig att det existerar fas¨overg˚angar som kan ske utan ett symmetribrott. Detta fenomen kallas Topologisk ordning. I den h¨ar uppsatsen studerar vi en teoretisk modell som ¨ar konstruerad p˚a ett tv˚adimensionellt gitter d¨ar vi erh˚aller Topologisk ordning.

Introduction

There are many different forms of matter; liquid, solid, ferromagnetic, are some of them. The reason why there are so many forms of matter is not due to different ingredients of the mate-rial, since all matter is built of electrons, protons and neutrons. The reason we get different materials and properties depends on the way the constituent particles are arranged. This is called principle of emergence in solid state physics. But what is characteristic for different arrangements? Landau’s symmetry breaking theory gives the answer to this question. Dif-ferent symmetries of the arrangement of the constituent particles give rise to difDif-ferent phases. For example, a liquid phase has continuous translational symmetry, a solid (crystal) mate-rial has discrete translational symmetry. The phase transition from liquid to crystal phase corresponds to a symmetry breaking process. According to Landau’s symmetry breaking theory, every phase transition corresponds to a symmetry breaking process.

Liquid phase for example occurs only at high enough temperatures, but there exist dif-ferent phases of matter at zero temperature. Examples are crystal, ferromagnetic, supercon-ducting, superfluid, etc. These states are called quantum states (or zero temperature states). Important properties for some of them need to be explained by quantum mechanics.

But, when one obtained the fractional Hall effect in the 80’s for example, one found out that there exist different quantum phases, both with the same symmetry. As a result, the Landau symmetry breaking theory is not complete. These new states of matter are called Topologically ordered states [3].

Topologically ordered states have some exotic properties which characterize them. These properties can only be explained by the theory of topological order, and show the impor-tance of topological order. Some of these properties are; ground state degeneracy, fractional statistics, abelian and non-abelian anyonic excitations [3].

There are many examples of systems which show topological order, both concrete exper-imentally accessible systems and theoretical models. For example fractional quantum Hall effect, string net condensation, high temperature superconductors [3].

In [3] they describe topological order as “Patterns of long range quantum entanglements (for entanglement see below). They also try to give it an intuitive picture. They describe it as a collective dance of a group where every member dance with everybody else, by following some local rules. Every member follow the same local rules and it give rise to a global dance pattern. The topological order for the system refers to this global dance pattern.

This description has some similarities with the Fractional Quantum Hall (FQH) effect. (FQH) states are obtained when placing a two dimensional electron gas in a strong magnetic field perpendicular to the ”gas” at low temperatures. The electrons in the gas move in small ”loops” due to the Lorentz force, but since all electrons move in loops they have to do it in some organized way. The Topological order refers to this organized way in which every

electron move. One find out that the Hall coefficient RH is a rational number times _eh2 [3].

Different topological states refers to different rational Hall coefficients or different surface topology for the two dimensional electron gas, (sphere, torus,...).

Fractional Quantum Hall states are topologically ordered states. Besides that there exists different FQH states with the same symmetry, another property that show this is the existence of Abelian anyons. The existence of Abelian anyons in FQH states has been confirmed from experiments that satisfies the theoretical proposals [1]. For a description of anyons see below.

The local rules tell you in the example of FQH effect how two electrons move (dance) around each other. Since the local rules are the local (dance) rule for every pair of electrons, it tells you in which way all electron move (dance collectively). Different local (dance) rules give rise to different movements (collective dance) for all electrons, i.e. different topological orders.

Topologically ordered states can be obtained in two or three spatial dimensions. For ease we consider the two dimensional case here.

A theoretical model describing topological order is given by suitable spin lattice models. That is a lattice with a finite number of vertices, edges and faces. To some parts (vertices or edges) of the lattice one associate degrees of freedom corresponding to quantum-mechanical spin variables. One possibility is to take such variables to be labelled by the elements of some finite group G. The state space for the whole system is the tensor product between every individual spinor Hilbert spaces, so the total Hilbert space for our model is finite-dimensional. For the specific model one define the Hamilton operator. The Hamilton operator is important because it gives the time evolution for the system (as long as no measurements are made), and states of definite energy, in particular the ground state, are eigenstates of the Hamiltonian.

The precise form of the lattice is not important, it can be constructed rather arbitrary. If the state vector deviates from the ground state just at a particular place in the lattice, we have an excited state. We say that there exists a quasi particle at that particular position. In the model one can create such excited states, quasi particles by applying some creation operator to the ground state. The quasi particles are in general called anyons. To char-acterize anyons, first recall that the total wave function does not change by interchanging two bosons. If they are fermions we get a phase factor eiπ = −1 to the wave function. By interchanging two anyons the wave function aquire an arbitrary phase factor. In that case they are called abelian anyons. Interchanging two non-abelian anyons corresponds to a more complicated transformation of the state. The existence of anyonic excitations are properties of topological order. Another important property of topological order is that there exists an energy gap between the ground state and the first excited state, which also remains in the thermodynamic limit.

There are many theoretical models in this subject, for example the Kitaev model and the toric code [5] [3]. In the theoretical models there are proposals for possible ways to obtain quantum memories and to perform quantum computation. Different ways in different models. That is one of the reasons why it is interesting to study topological order.

Entanglement. Consider a quantum system, for which we have a Hamiltonian H with energy eigenvalues and energy eigenstates. The set of eigenstates of H forms a basis for the state space, i.e. a possible state for the system can be written as a linear combination of the eigenstates. If we have two separate systems with state spaces V1 and V2, respectively, we

can consider these states together as the tensor product

V = V1⊗ V2,

of the two spaces. A basis of this space is the set of base vectors {e1n⊗e2n} where e1n belongs

to V1 and e2n belongs to V2. A general state, that must be written as a superposition of

these base vectors instead of v1⊗ v2, where v1 belongs to V1 and v2 belongs to V2, is called

an entangled state.

This thesis follows an article written by H. Bombin and M.A. Martin-Delgado [2]. The model studied in [2] is a generalization of the Kitaev model [5]. In this thesis the definitions, necessary calculations and proofs are included in the main text in contrast to [2], where the definitions and calculations are written in the appendix. The reason for this is to make the thesis more readable. We include some of the intermediate steps in our calculations to show how we get the results. For example how the ribbon operators affect our state (26) − (29) and how we determined the commutation relations for different types of ribbon operators (42) − (46).

In section one we define the lattice and the geometrical objects, triangles ribbons etc. In section two we associate operators to the geometrical objects and show how they affect the state for our system. Triangle operators in section 2.1 Ribbon operators in section 2.2 and Local operators in section 2.3.

In section 3.1 we define the Hamiltonian and the ground state for our system. In section 3.2 we define excited states.

In section 3.3 we obtain the ground state for the lattice in Figure 1 and we apply some ribbon operators to it.

1

The lattice

To describe a quantum mechanical system we need a state space and observables. We model the system by a lattice on an arbitrary two dimensional surface. Every edge is oriented arbitrarily and joins two different vertices. Every vertex is connected to at least three edges. A face together with a vertex, that is contained in the boundary of that face, defines a site. We can draw the sites as dotted lines connecting the corresponding vertex and face. The orientations of the edges in the dual lattice (dual edges) are defined to cross the direct edges from right to left. This is shown in Figure 1.

Figure 1: The lattice with its vertices, edges, faces, sites and the edge orientations.

As mentioned in the introduction, as a first step to specify our model we have to provide the degrees of freedom of the system. We associate the degrees of freedom to the edges of our lattice, where we consider them as the elements of a finite group G which is fixed once and for all. Choosing a different group gives you a different model. We associate a |G| dimensional Hilbert space to each edge. This construction gives us some kind of spin particle at each edge that can take |G| different values. We call it a qudit (or spin particle). For the Hilbert

space at each edge we construct the basis {|gi} where g ∈ G. This construction gives us the possibilities to perform group operations to obtain different states. As an example, we can make a left translation in the group to get another state ket |gi → |hgi where h, g, hg ∈ G. The state for our total system is the tensor product of the states of each spin particle.

In our lattice we define a direct triangle τn as a direct edge, two sites connecting the

vertices at the ends of the edge, and one of the faces next to it. The n in τn is the label

for the edge. We define a dual triangle δn as a dual edge and two of its neighbouring sites

so that they connect the two faces at the ends of the dual edge with one of the vertices that is located between the faces. We define orientations for the triangles. If we count the sides of the triangle (counter clock wise for direct triangles and clock wise for dual) we get site1, site2, edgen and say that the triangle begins at site1 and it ends at site2. The

orientation of the triangle can either match the orientation of the edge or not. The orientation of triangles are shown in Figure 2.

Figure 2: Direct and dual triangles with their orientations.

If the orientation of a triangle matches the orientation of it’s edge we write τ, δ. We write ¯

τ , ¯δ if the orientations doesn’t match. For the beginning and end of the triangles we write ∂0τ, ∂0δ and ∂1τ, ∂1δ. We say that two triangles overlap if they share part of the area, this is

possible in two different ways. The two triangles either begin or end at the same site. If the triangles begin together, the triangle direction matches the edge orientation for either both triangles or none of them. If the triangles end together, only one of their directions match.

“head to tail” to each other and none of the triangles overlap with any of the other ones. The orientation of the ribbon is the same as the orientation of the triangles. We write ρ = (τ1, τ2, ¯δ5, τ6, δ9) for example. Here we see the constituent triangles of the ribbon and if

the orientation of the edges matches. We can attach ribbons ”head to tail” to each other and get longer ones. We write ρ = (ρ1, ρ2) for example. For the beginning and end of a ribbon

we write ∂0ρ and ∂1ρ. A ribbon is closed if it begins and ends at the same site. In that case

we write σ for the ribbon and ∂σ for its single end. Two ribbons can overlap in different ways. If two ribbons coincide in their beginning and then split into two, at their intersection we have ∂0τn = ∂0δn (or ∂0τ¯n = ∂0δ¯n) and we write (ρ1, ρ2)≺ where δn or ( ¯δn) belongs to

ρ1. If two ribbons coincide at their ends instead but start at different places, we have at

their intersection point ∂1τn = ∂1δ¯n (or ∂1τ¯n = ∂1δn) and we write (ρ1, ρ2) where δn or ( ¯δn)

belongs to ρ1. If two ribbons coincide both at their beginning and end but not in the middle

we have a combination of the two previous cases and we write (ρ1, ρ2)≺, where the dual

triangles in both intersections belongs to ρ1. For closed ribbons we can have two ribbons

that make the same loop but have different end sites. In that case we write (σ1, σ2)◦. We

write σ1. σ2 for the ribbon that starts at ∂σ1 and ends at ∂σ2. We have σ1 = (σ1. σ2, σ2. σ1)

and σ2 = (σ2. σ1, σ1 . σ2). This is illustrated in Figure 3.

2

The operators

2.1

Triangle operators

We obtain excited states from the ground state by applying some kind of creation operators. They are called ribbon operators because of their appearance. These operators are compli-cated, but all operators in this thesis are different compositions of simpler operators. Fore ease we start by introducing the simplest kind of operator, that is the operator for a single direct or dual edge, called triangle operator. If the direction of the triangle is the same as the direction of the edge we define for direct triangles τ and dual triangles δ the triangle operator to be T_τg := |gihg|, (1) and Lh_δ :=X g∈G |hgihg|. (2)

So T is a projection operator and L is a left translation operator in the group. The mapping for the qudit at respective edge become

T_τg : |g1i 7→ δg,g1|gi, (3)

and

Lh_δ : |gi 7→ |hgi. (4) For triangle operators we have

T r(T_τg) = T r(|gihg|) = hg|gi = 1, (5) in the direct case. We see that the trace is independent of the group element g. In the dual case we have T r(Lh_δ) = T r(X g∈G |hgihg|) =X g∈G hg|hgi = δh,1|G|. (6)

If on the other hand the direction of the triangle is opposite to the direction of the edge to which the triangle is attained, we transform the qudit at that edge to its inverse before we apply the operator, and afterwards transform it back. The operator that takes the qudit to its inverse is for both τ and δ

I :=X

g∈G

|¯gihg|. (7)

The square of I equals unity.

I2 = X g,h∈G |¯gihg||¯hihh| = X g,h∈G δ_g,¯h|¯gihh| =X h∈G |hihh| = 1op. (8)

The mapping for the qudit at respective edge when applying the inverse operator becomes

|gi 7→ |¯gi, (¯g = g).¯ (9) When applying a triangle operator for triangles with edges whose orientation don’t match the orientation of the triangle, the mapping at respective qudit becomes for τ

T_τ¯g : |g1i 7→ δg, ¯g1|¯gi = δg, ¯g1|g1i, (10)

and for δ

Lh_δ¯ : |gi 7→ |g¯hi. (11)

So there is a connection between reversing the orientation of an edge and taking the inverse group element for that state.

For the composition of triangle operators in the direct case we get

T_τgT_τg0 = |gihg||g0ihg0| = δg,g0|gihg| = δ_g,g0Tg

τ, (12)

And in the dual case

Lh_δLh_δ0|g1i = Lhδ|h 0

g1i = |hh0g1i = Lhh

0

δ |g1i. (13)

When taking the Hermitian conjugation we get in the direct case

T_τg†= |gihg|†= |gihg| = T_τg. (14)

The identity operator for direct triangles can be expressed as

1op=

X

g∈G

T_τg. (15)

In the dual case, taking the hermitian conjugation of the triangle operators we get

Lh_δ†|g1i = X g∈G |gihhg||g1i = X g∈G δhg,g1|gi = X g∈G δ_g,¯hg1|gi = |¯hg1i = L ¯ h δ|g1i, (16)

And the identity operator for dual triangles is simply given by

1op= L1δ. (17)

We now determine the commutation relations for the composition of direct and dual triangle operators. For the operators to be composable we need that the triangles overlap. When ∂0δ = ∂0τ we get

T_τg0Lh_δ0|g1i = Tg 0 τ |h 0_g 1i = δh0_g 1,g0|g 0_{i = δ} h0_g 1,g0|h 0_g 1i. (19)

For these two operations to be the same we need that h0 = h and g0 = hg, which gives

Lh_δT_τg = T_τhgLh_δ. (20)

In a similar way we get

Lh¯_δTτg = T g¯h τ L h ¯ δ. (21) when ∂1δ = ∂¯ 1τ .

If we have neither ∂0δ = ∂0τ nor ∂1δ = ∂1τ then the triangles don’t overlap and their

triangle operators commute.

2.2

Ribbon operators

2.2.1 Ribbon operators, definition

Now we know everything about triangle operators, so lets combine triangles to each other one at the time to get what we call a ribbon ρ. The ribbon operator is a global operator because of its extension over many qudits. A ribbon operator for a ribbon that consists of just one triangle is defined using the triangle operators. For direct triangles τ we define the ribbon operator by

F_τh,g := T_τg. (22) For dual triangles δ we define it by

F_δh,g := δ1,gLhδ. (23)

We can combine two ribbons ρ1 and ρ2 to a single one, ρ if the second ribbon starts where

the first one ends. In this way we build up the ribbon operator, one triangle at the time. We define this “glueing process” as follows:

F_ρh,g :=X k∈G F_ρh,k 1 F ¯ khk,¯kg ρ2 . (24)

What happens with the states for the qudits where the ribbon operator operate? We start by constructing a ribbon out of just two triangles because we know the “glueing formula” and the ribbon operators that operates on single triangles. If the triangles in the ribbon are ρ = (τ, δ) with the orientation of the edges in the same direction as the orientation of the ribbon, then the ribbon operator becomes

F_ρh,g =X

k∈G

F_τh,kF_δ¯khk,¯kg =X

k∈G

This operator acts on |g1, g2i as follows:

F_ρh,g|g1, g2i =

X

k∈G

δk,g1δ1,¯kg|k, ¯khkg2i = δg,g1|g1, ¯g1hg1g2i. (26)

If the directions of the ribbon would not agree with the directions of the edges, their qudits transform in a slightly different way, compare (26). The calculation is similar to those above. The transformations for the qudits become

ρ = (¯τ , δ) : F_ρh,g|g1, g2i = δg, ¯g1|g1, g1h ¯g1g2i, (27)

ρ = (τ, ¯δ) : F_ρh,g|g1, g2i = δg,g1|g1, g2g¯1¯hg1i, (28)

ρ = (¯τ , ¯δ) : F_ρh,g|g1, g2i = δg, ¯g1|g1, g2g1¯h ¯g1i. (29)

We can then build longer ribbon operators out of shorter ones. It is just to add triangles, one at a time or putting together ribbon operators for more than one triangle. The important thing is that the second starts where the first one ends and that they don’t overlap. The “glueing process” also needs to be associative, so that we can split the ribbon in shorter ones in any way we want. Let ρ = ρ1ρ2ρ3. We need to have

F_ρh,g =X

k∈G

F_ρh,k_1ρ2F_ρkhk,¯¯₃ kg =X

k∈G

F_ρh,k₁ F_ρ¯khk,¯_2ρ3kg. (30)

To show this we just split ρ1ρ2 into (ρ1, ρ2) and ρ2ρ3 into (ρ2, ρ3) by using (24) and show

that the two expressions are equivalent. For simplicity and sufficiently we only show that it works when ρ2 = τ2 and when ρ2 = δ2. Starting with ρ2 dual we get the two expressions

Fh,g ρ = X k∈G  X k0_∈G F_ρh,k0 1 F ¯ k0_hk0_{, ¯k}0_k ρ2  F_ρ¯khk,¯kg 3 = X k,k0_∈G F_ρh,k0 1 δ1, ¯k0kL ¯ k0_hk0 δ2 F ¯ khk,¯kg ρ3 =X k∈G F_ρh,k 1 L ¯ khk δ2 F ¯ khk,¯kg ρ3 , (31) and F_ρh,g =X k∈G F_ρh,k₁  X k00_∈G F_ρkhk,k¯₂ 00F_ρk₃¯00khkk¯ 00, ¯k00kg¯  = X k,k00_∈G F_ρh,k₁ δ1,k00L ¯ khk δ2 F ¯ k00¯_khkk00_{, ¯k}00¯_kg ρ3 =X k∈G F_ρh,k₁ L¯khk_δ2 F_ρ¯khk,¯₃ kg. (32)

We see that the two expressions for Fh,g

ρ are identical. We now consider the case when ρ2 is

direct. The two expressions become

F_ρh,g =X k∈G  X k0_∈G F_ρh,k₁ 0F_ρk¯₂0hk0, ¯k0kF_ρ¯khk,¯₃ kg = X k,k0_∈G F_ρh,k₁ 0T_τk¯₂0kF_ρ¯khk,¯₃ kg, (33)

and F_ρh,g =X k∈G F_ρh,k 1  X k00_∈G F_ρ¯khk,k00 2 F ¯ k00¯_khkk00_{, ¯k}00¯_kg ρ3  = X k,k00_∈G F_ρh,k 1 T k00 τ2 F ¯ k00¯_khkk00_{, ¯k}00¯_kg ρ3 . (34)

These two expressions are not identical, but when we let the two operators operate on our state, the transformation becomes identical because of the Kronecker delta that appears. The transformation for our state becomes

X k,k0_∈G T_τk¯₂0k|..., g2, ...i = X k,k0_∈G δ_{g2, ¯k}0_k|..., g₂, ...i, (35) and X k,k00_∈G T_τk₂00|..., g2, ...i = X k,k00_∈G δg2,k00|..., g2, ...i. (36)

This completes the proof of the associativity property of the “glueing formula”. We can now subdivide a ribbon operator into shorter ones in any way we want using that formula. Consider a ribbon of the form ρ = (ρ1, ρ2, ..., ρn),

Fh,g ρ1−n = X k∈G F_ρh,k 1−(n−1)F ¯ khk,¯kg ρn = X k,k0_∈G F_ρh,k0 1−(n−2)F ¯ k0_hk0_{, ¯k}0_k ρn−1 F ¯ khk,¯kg ρn = X k,k0_,k00_∈G F_ρh,k00 1−(n−3)F ¯ k00_hk00_{, ¯k}00_k0 ρn−2 F ¯ k0_hk0_{, ¯k}0_k ρn−1 F ¯ khk,¯kg ρn . (37)

After subdividing the ribbon n − 1 times, it becomes

X

k,k0_,k00_,...,kn−2_∈G

F_ρh,k₁ n−2F_ρk₂n−3hkn−3,kn−3kn−2· · · F_ρ¯khk,¯_n kg. (38)

The Hermitian conjugate of a ribbon operator is again a ribbon operator. To show this in general it is enough to show it for ρ = (τ, δ). In that case we get

F_ρh,g†= X k∈G T_τkδ_1,¯kgL¯khk_δ  † =X k∈G T_τk†δ_1,¯kgL¯khk_δ †=X k∈G T_τkδ_1,¯kgL_δk¯¯hk = F_ρ¯h,g. (39)

The unit operator among the ribbon operators is the one that didn’t do any translation on the qudits or give us the nullket. That is

1op=

X

g∈G

F_ρ1,g. (40)

2.2.2 Ribbon operators, Composition and Commutation relations

For the composition of ribbon operators we have to distinguish many different situations, depending on how the ribbons overlap. The simplest case is when we compose two ribbon

operators for the same ribbon. It is enough to consider a ribbon of the type ρ = (τ1, δ2). In

that case we get

F_ρh,gF_ρh0,g0 = X k,k0_∈G F_τh,k₁ F_δ¯khk,¯kg 2 F h0_,k0 τ1 F ¯ k0_h0_k0_{, ¯k}0_g0 δ2 = X k,k0_∈G T_τk₁T_τk₁0δ_1,¯kgδ_{1, ¯k}0_g0L ¯ khk δ2 L ¯ k0_h0_k0 δ2 = X k,k0_∈G δk,k0T_τk 1δ1,¯kgδ1, ¯k0g0L ¯ khk ¯k0_h0_k0 δ2 = X k,k0_∈G δk,k0δ_g,g0T_τk 1δ1,¯kgL ¯ khh0k δ2 =X k∈G δg,g0Fhh 0_,k τ1 F ¯ khh0k,¯kg δ2 = δg,g0F hh0,g ρ . (41)

If the ribbons form a left joint (ρ1, ρ2)≺ we have ∂0δ = ∂0τ for some dual triangle in ρ1

and some direct triangle in ρ2. When computing the commutation relations between the

operators Fh,g

ρ1 and F

j,l

ρ2 we construct two proper ribbons ρ1 = (δ1, τ2) and ρ2 = (τ1, δ3) which

form a left joint. In that case we get

F_ρh,g 1 F j,l ρ2 = X k,k0_∈G F_δh,k 1 F ¯ khk,¯kg τ2 F j,k0 τ1 F ¯ k0_jk0_{, ¯k}0_l δ3 = X k,k0_∈G δ1,kLhδ1T ¯ kg τ2 T k0 τ1δ1, ¯k0lL ¯ k0_jk δ3 . (42)

The transformation for the qudits becomes

|g1, g2, g3i 7→ δg,g2δl,g1|hg1, g2, ¯g1jg1g3i = δg,g2δl,g1|hg1, g2, ¯ljlg3i. (43)

If we reverse the order of the operators, then

F_ρj₂0,l0F_ρh₁0,g0 = X t,t0_∈G F_τj₁0,tF_δ¯tj0t,¯tl0 3 F h0_,t0 δ1 F ¯ t0_h0_t0_{, ¯t}0_g0 τ2 = X t,t0_∈G T_τt₁δ1,¯tl0L ¯ tj0_t δ3 δ1,t0L h0 δ1T ¯ t0_g0 τ2 . (44)

The transformation for the qudits in this case becomes

|g1, g2, g3i 7→ δl0_,h0_g 1δg0,g2|h 0 g1, g2, ¯l0j0l0g3i = δl0_,h0_g 1δg0,g2|h 0 g1, g2, ¯g1h¯0j0h0g1g3i. (45)

For these operators and transformations to be the same we need that h0 = h, g0 = g, j0 = hj¯h and l0 = hl, so we get

F_ρh,g₁ F_ρj,l₂ = F_ρhj¯₂h,hlF_ρh,g₁ . (46) Similarly we get that

F_ρh,g₁ F_ρj,l₂ = F_ρj,l¯₂g¯hgF_ρh,g₁ , (47) if (ρ1, ρ2),

F_ρh,g₁ F_ρj,l₂ = F_ρhj¯₂h,hl¯g¯hgF_ρh,g₁ , (48) if (ρ1, ρ2)≺ and

if (σ1, σ2)◦.

From (40) and (49) we get that

F_σh,g₁ =X

l∈G

F_σ1,l_1.σ2F_σh,g₁ =X

l∈G

F_σh¯₁g¯_.σhg,l₂ F_σ¯lhl,¯₂ lgl. (50)

For ribbon operators we have

T r(F_ρh,g†F_ρh0,g0) = δg,g0T r(F¯hh 0_,g

ρ ). (51)

To compute this we consider two different ribbons, ρ = (ρ0, τ ) and ρ = (ρ0, δ). In the first case we have F_ρh,g =X k∈G F_ρh,k0 T ¯ kg τ , (52) and here T r(F_ρh,g) = X k∈G T r(F_ρh,k0 )T r(T ¯ kg τ ) = X k∈G T r(F_ρh,k0 ) = |G|T r(F h,k ρ0 ). (53)

Here we used (5) and (24). In the second case we have

F_ρh,g =X k∈G F_ρh,k0 δ_1,¯kgL ¯ khk δ , (54) and here T r(F_ρh,g) = X k∈G T r(F_ρh,k0 )δ_1,¯kgT r(L ¯ khk δ ) = T r(F h,g ρ0 )δ_1,h|G|. (55)

Here we used (6) and (24). These equations show us that every time we take away one triangle from the ribbon we get a factor of |G| to the trace of the total ribbon operator, and in the dual case we also get the constraint δh,1. Let’s use this together with (51), we get

T r(F_ρh,g†F_ρh0,g0) = δg,g0δ_1,¯hh0|G|l−1, (56) or equivalently |G|T r(Fh,g ρ † F_ρh0,g0) = δg,g0δ_h,h0|G|l. (57)

where l is the number of triangles in the ribbon.

2.2.3 Ribbon operators in Tensor form

The composition of two ribbon operators can be written in tensor form by using the tensor Λ(h_(h,g)1,g1),(h2,g2). In this case we get

F_ρh1,g1F_ρh2,g2 = X

h,g∈G

For this composition to be the same as the previous one we need that

Λ(h_(h,g)1,g1),(h2,g2) = δ_h,h1_h2δ_g,g1δ_g,g2. (59)

In [5] this operation is called multiplication in F -space. We can write our “glueing formula” (24) for ρ = (ρ1, ρ2) in tensor form too. Here we use the tensor Ω

(h,g) (h1_,g1_),(h2_,g2₎, F_ρh,g = X h1_,h2_,g1_,g2_∈G Ω(h,g)_(h1_,g1_),(h2_,g2₎F h1_,g1 ρ1 F h2_,g2 ρ2 . (60)

For this operation to be the same as (24) we need that g1 _{= k and}

Ω(h,g)_(h1_,g1_),(h2_,g2₎= δh1_,hδ_h2_{, ¯g}1_h1_g1δ_g2_{, ¯g}1_g. (61)

In [5] this operation is called comultiplication in F -space.

2.3

Local operators

If we have a dual or direct ribbon, we name them αs,s0 or β_s,s0 where s and s0 are the sites

where the ribbon begins and ends. If the ribbon is closed we simply name it αs or βs, where

s is the site common for the beginning and the end. For a dual closed ribbon we define the corresponding ribbon operator

αs: Ahs := F h,1

αs , (62)

For a direct closed ribbon we define it

βs: Bsg := F 1,¯g

βs . (63)

A dual closed ribbon encloses a single vertex v. For that vertex we define the vertex operator Av in terms of Ahs, here s = (v, f ) is one of the sites next to v,

Av := 1 |G| X h∈G Ah_s. (64)

A direct closed ribbon encloses a single face f, and for that face we define the face operator Bf in terms of Bsg. As in the previous case s = (v, f ) is one of the sites that belongs to f ,

Bf := Bs1. (65)

We show later why the vertex and face operators (64) and (65) only depends on the vertex or face and not on a particular site that belongs to it, (as in Ah_s or B_sg).

The operators Ah_s, Bg_s or different compositions of them are our local operators. One important property for them is that they commute with ribbon operators if the sites at

the ends of the ribbon do not belong to the corresponding vertex or face. To show this we construct a ribbon and a closed dual (or direct) ribbon that overlap, but not at the end of the open ribbon. For example ρ = (δ1, τ4, δ2, τ5, δ3), αs = (δ6, ¯δ4, δ2, δ5). We have

Fh,g ρ = X k...k3_∈G δk3_,1Lh_δ 1T ¯ k3_k2 τ4 δ1, ¯k2k0L ¯ k2_hk2 δ2 T ¯ k0_k τ5 δ1,¯kgL ¯ khk δ3 = X k0_∈G L_δh₁T_τk₄0Lk_δ¯₂0hk0T_τk¯₅0gL¯ghg_δ 3 , (66) and Av = 1 |G| X l∈G F_αl,1_s = 1 |G| X l,t2_,t0_,t∈G δ1,t2Ll_δ 6δ1, ¯t2t0L ¯ t2_lt2 ¯ δ4 δ1, ¯t0tL ¯ t0_lt0 δ2 δ1,¯t1L ¯ tlt δ5 = 1 |G| X l∈G Ll_δ 6L l ¯ δ4L l δ2L l δ5. (67)

In that case we get

Fh,g ρ Av|g1, g2, g3, g4, g5, g6i = 1 |G| X l,k0_∈G δ_g4¯_l,k0δ_lg 5, ¯k0g |hg1, ¯k0hk0lg2, ¯ghgg3, g4¯l, lg5, lg6i, (68) and AvFρh,g|g1, g2, g3, g4, g5, g6i = 1 |G| X l,k0_∈G δk0_,g 4δk¯0_g,g 5|hg1, l ¯k 0_hk0 g2, ¯ghgg3, g4¯l, lg5, lg6i. (69)

Both these operators transform the state equivalently

|...i 7→ 1 |G|

X

l,∈G

δg,g4g5|hg1, l ¯g4hg4g2, ¯g5g¯4hg4g5g3, g4¯l, lg5, lg6i. (70)

The calculations is similar for face operators or different types of ribbons.

The case is not the same if the face or vertex for the local operator overlaps with some of the ends of the ribbon. Consider for example that we have a local operator that begins at the same site (s0) as the ribbon. In that case we have (ρ, βs0)≺ and (αs0, ρ)≺, which give us

F_ρh,gB_sk₀ = F_ρh,gF_β1,¯k s0 = F h1¯h,h¯k β_s0 F h,g ρ = B k¯h s0F h,g ρ ⇔ B k s0F h,g ρ = F h,g ρ B kh s0, (71) Ak_s0F_ρh,g = F_αk,1 s0F h,g ρ = F kh¯k,kg ρ F k,1 α_s0 = F kh¯k,kg ρ A k s0. (72)

If they overlap at the last site of the ribbon (s1), we have (ρ, βs1) and (αs1, ρ), which gives

us F_ρh,gB_sk 1 = F h,g ρ F 1,¯k β_s1 = F 1,¯k¯g¯hg β_s1 F h,g ρ = B ¯ ghgk s1 F h,g ρ ⇔ B k s1F h,g ρ = F h,g ρ B ¯ g¯hgk s1 , (73)

Ak_s 1F h,g ρ = F k,1 α_s1F h,g ρ = F h,g¯1¯k1 ρ F k,1 α_s1 = F h,g¯k ρ A k s1. (74)

Another interesting case is the commutation relation between a dual and a direct local operator. In this case we have (αs, βs)≺ which gives us

Ah_sB_sg = F_αh,1_s F_β1,¯_sg = F_β1,h¯_s g¯hF_αh,1_s = B_shg¯hAh_s. (75) For proving the site invariance of the vertex operator (64), consider two sites s and s0 next to the same vertex v. We have (αs, αs0)_◦ and from (50) we get

Ah_s = F_αh,1_s =X l∈G F_αh¯_s1¯_.αh1,l s0F ¯ lhl,¯l1l α_s0 = X l∈G F_α1,l s,s0F ¯ lhl,1 α_s0 . (76)

Since αs,s0 is a dual ribbon we get a δ_1,lwhen applying the ribbon operator to our state. The

only meaningful operator among the sum is then obtained when l equals 1, which give us

Ah_s = F_αh,1

s0 = A

h

s0. (77)

The site invariance for face operators (65) is proved easier. Consider two sites s and s0 in the same face f . We have (βs, βs0)_◦ and from (50) we get

B1_s = F_β1,1_s =X l∈G F_β1¯_s1¯_.β11,l s0F ¯ l1l,¯l1l βs0 = X l∈G F_β1,l s,s0F 1,1 βs0 = F 1,1 βs0 = B 1 s0. (78)

The vertex and face operators (64) and (65) commute with each other. The face operators are only projectors that don’t translate the qudits in question, so that case is trivial. The vertex operators commute even if they share some edge. That is because of the edge ori-entations in the lattice. For the common edge one vertex operator makes a left translation and the other makes a right translation. For a vertex and face that overlap we get their commutation easily from (75) when g equals 1.

We construct the operator

Dh,g_s := B_shAg_s. (79) Since both h and g belong to G we can write B_sh, Ag_s instead of B_sg, Ah_s as was introduced in (62) and (63). The reason for this will become clear later. In terms of D we have

Ag_s =X

h∈G

D_sh,g, (80)

and

B_sh = Dh,1_s . (81) For the composition of two D-operators we get

D_sh1,g1Dh_s2,g2 = B_sh1Ag_s1B_sh2Ag_s2 = B_sh1B_sg1h2g¯1Ag_s1Ag_s2 = δ_h1_,g1_h2_g¯1Bh 1 s Ag 1_g2 s = δh1_,g1_h2_g¯1Dh 1_,g1_g2 s . (82)

The hermitian conjugate of a D-operator is again a D-operator

D_sh,g†= Ag_s†B_sh†= A¯g_sBh_s = Bghg_s¯ A¯g_s = Dghg,¯_s¯ g. (83)

The unit operator among the D-operators are

1op=

X

h∈G

Dh,1_s . (84)

The composition of D-operators can be written in tensor form, we use the tensor Ω0(h,g)_(h1_,g1_),(h2_,g2₎,

Dh_s1,g1D_sh2,g2 = X

h,g∈G

Ω0(h,g)_(h1_,g1_),(h2_,g2₎D

h,g

s . (85)

For this composition to be the same as the previous one we need that

Ω0(h,g)_(h1_,g1_),(h2_,g2₎= δ_h1_,g1_h2_g¯1δ_h,h1δ_g,g1_g2. (86)

In [5] this operation is called multiplication in D-space. We see that Ω0 = Ω.

3

The Hamiltonian and the space of states

3.1

The Hamiltonian and Ground state Definition

For our model we define the Hamiltonian H0 as the negative sum of all vertex and face

operators for the lattice we are studying:

H0 := − X v∈V Av − X f ∈F Bf. (87)

Here Av and Bf are defined in (64) and (65). The operator Av is a projector onto the states

with are “gauge invariant” at the vertex v [2]. This means that if we simultaneously perform a left or right translation on all the spins next to that vertex, the state is unchanged. The face operator Bf is a projector to trivial flux for the face f . This means that the product of

the group elements that label all states, counted counter clockwise, around the face f equals the unit element. If the direction of some or all edges around the face do not point counter clockwise we take the inverse element for them.

This Hamiltonian was introduced in [5]. It is a summation of commutating local opera-tors, so it has eigenstates which are at the same time eigenstates for the local operators (64) and (65).

The size of the lattice makes the Hamiltonian a complicated operator in general. But since it is composed of simpler operators it is possible to diagonalize it. The ground state

for this Hamiltonian is the eigenstate which doesn’t transform by the action of any vertex or face operator. We can obtain the ground state by applying all face and vertex operators to the state |1i. Here |1i is the tensor product of the states |1i labelled by the unit element 1 ∈ G on all edges, |ψGi = Y v∈V Av Y f ∈F Bf|1i. (88)

The state with unit elements only has trivial flux at each face, so we actually don’t need the face operators when obtaining the ground state this way. We can write

|ψGi =

Y

v∈V

Av|1i. (89)

The spin particles are attached to the edges, the projectors onto the ground state are attached to the vertices and faces. Thus, in particular when obtaining the ground state this way, we have different amounts of projector operators for the same number of spin particles depending on the topology of the surface. On the sphere for example we have V + F − E = 2, but on the torus we have V + F − E = 0. This corresponds to the Euler number 2 − 2g for a general surface with genus g, (for our purposes a plane can be considered as a sphere with very large radius). From these properties we have a four-fold ground state degeneracy on the torus (g = 1) but no ground state degeneracy on the sphere (g = 0). According to Kitaev [5] the ground state degeneracy is equal to 4g. The ground states has the same symmetry on all surfaces, but one quantum number differs, the ground state degeneracy. The ground state degeneracy only depends on the overall topology for the surface. This is a typical property of topological ordered states.

3.2

Excited states, Definition

If an eigenstate |ψi does not satisfy Av|ψi = |ψi and Bf|ψi = |ψi for all vertices or faces,

|ψi is not the ground state. We have an excited state with higher energy. If two constraints are violated, the energy is 2 over the ground state energy, in some unit. We say that we have an excitation at the respective vertex or face (or both vertex and face). Since ribbon operators commute with vertex and face operators except at their ends, (68) and (69), we see that when applying a ribbon operator to the ground state, the only possible locations for excitations to be produced are at the ends of the ribbon.

We can thereby apply ribbon operators to the ground state to obtain two excitations, one at each end of the ribbon. According to [2] we get from (57) that the |G|2 states obtained by ribbon operators applied to the ground state form a basis for all excitations at these positions.

Two interesting versions of the unit operator are 1op = |G| X g∈G F_ρ1,gAvFρ1,g, (90) and 1op= X h,g,g0_∈G F_ρ¯h,gBfFh,g 0 ρ . (91)

The vertex and face in these operators are the vertex and face corresponding to the end site of the ribbon. Either the beginning or end. To show why these operators are unit operators, take the version (90) and the vertex for the beginning site of the ribbon as an example. We get |G|X g∈G F_ρ1,gAvFρ1,g = |G| |G| X g,k∈G F_ρ1,gAk_s0F_ρ1,g = X g,k∈G F_ρ1,gF_ρ1,kgAk_s0 = X g,k∈G δk,1Fρ1,gA k s0 = 1op. (92)

The other three cases are similar.

The property that make these operators interesting is that if we consider a general state |ψi with vertex excitations at s0 and s1, we have

|ψi = |G|X g∈G F_ρ1,g|ψs1i, (93) where |ψs1i = AvF 1,g ρ |ψi. (94)

But |ψs1i can not have a vertex excitation at s0 because Av is a projector onto states with no

excitation there. From here we see the important properties of our local and global operators and different compositions of them. Let us look at (90). We begin with a state with two excitations at sites s0 and s1 far away from each other. We annihilate the excitation at s0

by applying AvFρ1,g. Then we put the excitation back again by applying ribbon operators

between the sites. From processes like this we can create, annihilate, change and move excitations by applying operators to the ground state.

3.3

Obtaining the ground state

Let us consider a simple example where we obtain the ground state and apply different ribbon operators to it. We start with the state that is labeled with the unit element everywhere. At vertex n we write the index n on that vertex operator:

Avn = 1 |G| X hn_∈G Ah_sn. (95)

Consider the edge pointing from vertex n to vertex m. After applying both vertex operators n and m, the state for that qudit becomes

AvnA_vm|1i = 1 |G|2 X hn_,hm_∈G |hn_h¯m_{i. (96)}

This construction works analogously for more than one edge. For the lattice in Figure 1 the ground state is |ψGSi = 1 |G|8 X h1_−h8_∈G |h2h¯1_{, h}3_h¯1_{, h}4_h¯2_{, h}5_h¯2_{, h}3_h¯4_{, h}5_h¯4_{, h}6_h¯3_{, h}7_h¯4_{, h}5_h¯7_{, h}6_h¯7_{, h}6_h¯8_{, h}7_h¯8_i. (97) We see that this is an entangled state. If calculating the flux at the vertices, one always obtain the identity element. Now consider the ribbon ρ1 = ( ¯δ5, ¯τ8, ¯δ10) and the corresponding

ribbon operator Fk1,k2

ρ1 . We let it operate on our ground state and obtain the state

Fk1,k2 ρ1 |ψGSi = 1 |G|8 X h1_−h8_∈G δ_k¯_2,h7_h¯4|₁₋₄, h3h¯4k¯₁,₆₋₉, h6h¯7h7h¯4k¯₁h4h¯7,₁₁₋₁₂i. (98)

Writing1−4 means that the state of these qudits are the same as before. When counting for

the Dirac delta that appears, the state obtained is

Fk1,k2 ρ1 |ψGSi = 1 |G|8 X h1_−h6_,h8_∈G |1−4, h3h¯4k¯1,6−7, ¯k2, h5h¯4k2, h6h¯4k¯1k2,11−12i. (99)

We see that the eighth qudit has the state ¯k2 for every term in the sum. When calculating

the flux at the face where the ribbon begins one obtain ¯k1, which is not unity in general.

So this is not the ground state. By slightly changing the ribbon to ρ2 = ( ¯δ5, ¯τ8, ¯δ10, τ12), we

obtain by similar calculations:

Fk1,k2 ρ2 |ψGSi = 1 |G|8 X h1_−h3_,h5_−h8_∈G |1−4, h3h¯8k¯2, h5h¯8k¯2,7, h7h¯8k¯2,9, h6h¯8k¯2k¯1k2h8h¯7,11−12i. (100) Here we have summation of every group member for the states at every qudit. This shows the importance of the length of the ribbon when obtaining excited states. It would be nice to write it out and see what it looks like. But for a ribbon consisting of four triangles, when it operates on the ground state, we obtain a summation of |G|4 terms. The smallest nonabelian group (permutation group of three elements) has six elements so it becomes a summation of 1296 terms. Too many to consider here. Instead one uses other properties of groups when calculating what kind of quasi particle excitation one got. See [2] [5].

In the introduction we talked about what happens to the overall wave function when interchanging two anyons. How is this attained in the theoretical model? Consider that we

have two cars, one at each end of a street. If they simultaneously drive along the street they have interchanged their positions. Now they do it once more. The two interchanging processes can be generalized to one process when one of the cars drive one revolution counter clockwise around the other car. In theoretical models, we can obtain two successive inter-changing processes by dragging one anyon once around another anyon. This is obtained in models like this [5], by applying a closed ribbon operator that encloses a single excitation, to another excitation far away, see Figure 4. From the result of processes like this one can calculate what kind of quasi particles one got. But in this thesis as in the previous case, one obtain summations of many terms. Too many to include here.

Figure 4: A closed ribbon, that encloses an excitation, is applied to another excitation far away.

4

Summary

In this thesis we define an orientated lattice with certain degrees of freedom. We define local and global operators through the concept of ribbon (and triangle) operators. We show how to

obtain the ground state and how we obtain excited states by applying ribbon operators. This model shows properties for Topological ordered states, for example ground state degeneracy and quasi particle excitations [3].

References

[1] Sanghun An, P Jiang, H Choi, W Kang, SH Simon, LN Pfeiffer, KW West, and KW Bald-win. Braiding of abelian and non-abelian anyons in the fractional quantum hall effect. arXiv preprint arXiv:1112.3400, 2011.

[2] H Bombin and MA Martin-Delgado. Family of non-abelian kitaev models on a lattice: Topological condensation and confinement. Physical Review B, 78(11):115421, 2008.

[3] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Physical Review B, 82(15):155138, 2010.

[4] Mattias Flygare. Topological color codes. Internal report, Karlstad University 2012.

[5] A Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2–30, 2003.

[6] A Micheli, GK Brennen, and P Zoller. A toolbox for lattice spin models with polar molecules. Nature Physics, 2:341, 2006.

[7] Xiao-Gang Wen. An introduction of topological orders. Available at http://dao.mit.edu/ wen/articles.html.