Beschrijving van de rand en defecten in systemen met topologische orde door middel van Matrix Product Operators

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Characterizing Boundaries and Defects for

Systems with Topological Order in a Matrix

Product Operator Formalism

Marco Deweirdt

Promotor: prof. dr. Frank Verstraete

Supervisor: Laurens Lootens

Submitted in partial fulfillment of the requirements for the degree of Master of Science in Physics and Astronomy

Department of Theoretical and Mathematical Physics Academic Year 2019 - 2020

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The author grants permission to make this master’s thesis available for consultation and to copy parts of the master’s thesis for personal use. Any other use falls under the limitations of the copyright law, in particular with regard to the obligation to explicitly mention the source when citing results from this master’s thesis.

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Acknowledgements

The past academic year marked one of the most turbulent years in my young life and also marked the time I spent on the work provided in this thesis. Therefore I would like to take the time to thank a few people without whom I would not be where I am today. First and foremost I want to thank my supervisor Laurens Lootens for his excellent guidance and useful discussions. Thanks to your help I was able to acquire the skills necessary to solve the problems presented in this thesis. The input you provided was invaluable. I would also like to thank my promotor Frank Verstraete, for making every-thing possible in the first place and for providing a dynamic research environment where people with a passion for physics can truly blossom up.

The five years I studied physics were five of the best of my life because of the amazing friends I met throughout these years. Therefore I want to thank them and in particular Lander. Partly for proofreading some chapters of this thesis, but more importantly for the continuous support I got throughout the year and for the clear words that gave me the confidence to move on ahead.

Last but definitely not least I want to thank my family and especially my mother for supporting me in any way she can. The strength you possess will always inspire me to keep going.

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Abstract

Topological order, the order associated with new states of matter at very low temper-atures, is invaluable in the theory of topological quantum computation. This theory uses the quasiparticle excitations or anyons present in these topological orders to store quantum information and exploits the inherent robustness of such orders to naturally increase the fault tolerance. Driven by the idea of engineering such topological quan-tum codes, it is paramount to search for extensions of these topological orders to include boundaries and defects, since boundaries provide a scalable solution to store big amounts of information, while defects in turn have the potential to increase the computational power of these codes.

In this master’s thesis we provide an extension to include such boundaries and defects in a formalism that is able to describe systems with topological order through the language of tensor networks. Starting from a description of boundaries, domain walls and defects in the string-net model, we use the underlying mathematical framework of tensor fusion categories, module categories and bimodule categories as a guideline. With matrix prod-uct operators (MPOs) and projected entangled pair states (PEPS) as our tools we obtain an efficient and intuitive framework to understand topological order through projector matrix product operators (PMPOs) and MPO-injective PEPS. This formalism allows us to characterize topological order and the associated quasiparticle excitations or anyons, while naturally allowing for extensions towards boundaries and defects through the bi-module categories mentioned before. These bibi-module categories are present on multiple levels in this PMPO formalism and are the crucial piece to provide an efficient and in-tuitive tensor network representation for topologically ordered systems with boundaries and defects. The results obtained are in agreement with known results for the toric code model.

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Nederlandstalige samenvatting

Topologische orde, de orde die wordt geassocieerd met nieuwe toestanden van materie bij zeer lage temperaturen, is van onschatbare waarde in de theorie van topologische kwan-tumberekeningen. Deze theorie maakt gebruik van de quasideeltjes of anyonen aanwezig in deze topologische ordes om kwantuminformatie op te slaan en benut de inherente robuustheid van dergelijke ordes om op natuurlijke wijze de fouttolerantie te vergroten. Gedreven door het idee om dergelijke topologische kwantumcodes te ontwerpen, is het van het grootste belang om te zoeken naar uitbreidingen van deze topologische ordes die randen en defecten kunnen beschrijven, aangezien randen een schaalbare oplossing bieden om grote hoeveelheden informatie op te slaan, terwijl defecten op hun beurt de rekenkracht van deze codes kunnen opdrijven.

In deze masterproef voorzien we een uitbreiding tot dergelijke randen en defecten in een formalisme dat in staat is om systemen met topologische orde te beschrijven door middel van de taal van tensornetwerken. Vertrekkend van een beschrijving van randen, wanden en defecten in het string-net model, gebruiken we het onderliggende wiskundige raamwerk van tensorfusie-categorieën, module-categorieën en bimodule-categorieën als richtlijn. Door middel van matrixproductoperatoren (MPO’s) en geprojecteerde ver-strengelde paartoestanden (PEPS1) verkrijgen we een efficiënt en intu¨ıtief raamwerk om de topologische orde te begrijpen met behulp van projectormatrixproductoperatoren (PMPO’s) en MPO-injectieve PEPS. Dit formalisme stelt ons in staat om de topolo-gische orde en de bijbehorende quasideeltjes of anyonen te karakteriseren, terwijl we op natuurlijke wijze uitbreidingen naar randen en defecten kunnen toevoegen via de eerder genoemde bimodule-categorieën. Deze bimodule-categorieën zijn aanwezig op meerdere niveaus in dit PMPO-formalisme en zijn het cruciale puzzelstuk om een efficiënte en intu¨ıtieve tensornetwerkrepresentatie te bekomen voor topologisch geordende systemen met randen en defecten. De verkregen resultaten zijn in overeenstemming met gekende resultaten voor het toric code model.

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Introduction and Outline

Topological order and defects. Both discoveries provided a fundamental change in our understanding of nature. Topological order as it is known today has its origins in the experimental realization of the two-dimensional electron gas. This gas revealed new states of matter at low temperatures which could not be explained by the conventional theories at the time. These states are strongly correlated but also very sensitive to quantum fluctuations, which means that they cannot be classified by the four conven-tional states of matter, namely solid, liquid, gas and plasma. Due to the strong local correlations however they still possess an internal global pattern and hence are dubbed topological phases or topological orders, since we obtain global orders from strong local interactions. Similarly the existence of defects has proven to be invaluable to explain macroscopic properties of condensed matter systems. The first appearance of so-called topological defects, which essentially combines these two discoveries, was in the context of topological quantum field theories [1], which provide an effective low-energy descrip-tion of condensed matter systems, characterizing the topological order and the associated defects, also called quasiparticle excitations or anyons.

The reason why we are so interested in these systems is hidden in the theory of quantum computation. The birth of quantum computation [2, 3] meant the search for procedures to protect the fragile quantum information encoded in qubits by using so-called quantum error correcting codes [4]. One way to implement such procedures is through topological quantum computation, introduced by Kitaev [5]. By exploiting the properties of systems with topological order we can store the quantum information non-locally in the fusion state of anyons and use the inherent robustness of the ground state of such systems to protect that information from local perturbations, reducing the error rate of quantum computations. The simplest model, the toric code [5], which has abelian anyons [6], is defined on a torus. The amount of information we can store depends on the hole in this torus and is thus limited by the genus of such surfaces in general. This is detrimental for the scaling of such systems, since it is practically impossible to construct such models designed for storing large amounts of information. We can tackle this issue however by including boundaries in the model [7], which makes it scalable and thus approachable from an engineering point of view. Another issue is the fact that we currently can only achieve universal computation through non-abelian anyons [8], which are harder to deal with. Defects in models with abelian anyons however can increase the computational

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power of such models, since the defects themselves behave as non-abelian anyons. This opens up the possibility to attain universal quantum computation through abelian anyons [9]. We eventually need an efficient description of boundaries and defects in topologically ordered systems, leading us to the main subject of this master’s thesis. We start with a brief introduction to tensor networks [10], since they are the main tool used in this thesis and since they provide an efficient description of systems with topological order [11, 12]. In Chapter 2 we give a short introduction to topological order through the Fractional Quantum Hall Effect and provide the main properties of topo-logically ordered systems. We follow up with a detailed description of the toric code and show the topological order present in this model. We also extend the model to include boundaries and defects. These defects obey non-abelian fusion rules and are interesting from a computational point of view. In Chapter 4 we generalize the toric code to handle arbitrary systems with topological order to so-called string-nets [13]. These string-nets single out tensor fusion categories as the underlying mathematical framework to study topological order. We also provide a tensor network representation for the ground state of these string-nets using projected entangled pair states or PEPS for short. Through an extension of these tensor fusion categories we provide a description for boundaries, domain walls and defects using so called module and bimodule categories [14]. The main chapter of this thesis is dedicated to the PMPO formalism [11, 12], which is a gener-alized tensor network description of topological order. In this chapter we explain how matrix product operators (MPOs) and PEPS can characterize topological order and the associated quasiparticle excitations or anyons. We show the similarity to tensor fusion categories and explain how we can recreate the description for boundaries, domain walls and defects in the string-net picture by extending this formalism with module and bi-module categories as well. Finally we apply this formalism to the toric code and show that the results are in agreement with the known results for boundaries, domain walls and defects in the toric code.

The research presented in this thesis is performed in collaboration with Laurens Lootens and is based on results yet to be published [15].

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Chapter 1 Tensor Networks

Over the past two decades, tensor networks [10, 16, 17] have proven to be invaluable in numerous fields of physics and other sciences. Since they will also be the main tool used in this thesis, we will briefly motivate their use and discuss some important tensor networks for the study of strongly correlated quantum systems given by Matrix Product States (MPS), Projected Entangled Pair States (PEPS) and Matrix Product Operators (MPOs).

1.1 Lifting the Curse of Dimensionality

When we want to study the physical properties of quantum-many-body systems, we quickly bump into the obstacle of dimensionality or thus the exponential growth of the Hilbert space of such systems. The fact that this Hilbert space scales exponentially bears the sad truth that an exact description of quantum-many-body systems will be impossible. To illustrate this we can consider a spin 1/2-system with N particles. The Hilbert space of such a spin system scales as 2N and thus exponential in terms of the number of particles. If we make the reasonable assumption that all computational power in the world is estimated to be around 1020 _{instructions per second, we would only be} able to simulate systems of at most 64 particles. Given that system sizes in nature typ-ically consist of around 1023particles, we can immediately conclude that the amount of computational power necessary for the description of such Hilbert spaces will be insur-mountable, even in the distant future.

Luckily nature is on our side and most physics only take place in a small corner of this huge Hilbert space. Most interactions in nature are local and can be described by gapped Hamiltonians, by which we mean that the first excited state is separated from the ground state by some energy gap ∆E. This means that our physical systems of interest possess extra structure which translates itself into an area law for the entangle-ment entropy of these systems. This basically tells us that the entropy of some region scales proportional to its boundary and not proportional to its volume, which would be the case for purely random states. So to effectively describe our quantum-many-body

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1.2. MPS - MATRIX PRODUCT STATES

systems, we only need to take into account states that satisfy this area law. For the specific case of systems with topological order, which are the systems we are interested in, we also expect a correction to this area law given by some topological entanglement entropy [18]. That said, states satisfying some area law for the entanglement entropy naturally lead us to tensor network states, since these states directly target this small corner in the Hilbert space [19, 20].

1.2 MPS - Matrix Product States

Our main goal is to obtain some wave function |Ψi given by

|Ψi = d X

i1i2...iN

Ci1i2...iN|i1i2...iNi , (1.1)

which lives in some Hilbert space that scales as dN. Since all information is contained within this rank N tensor Ci1i2...iN or thus by d

N _{coefficients, we can identify our wave} function by this tensor and we have graphically

...

i1i2i3 iN

C

i1i2i3...iN

|Ψi =

(1.2) We know want to restructure this rank N tensor such that we only describe the physically relevant corner of this huge Hilbert space. One way of doing this is by isolating one leg of this tensor and grouping together all other legs, which results in a rank 2 tensor or matrix. Applying a singular value decomposition, such that C = AΛC0, we obtain the singular values or Schmidt coefficients given by the diagonal matrix Λ. These coefficients quantify our entanglement and carry the relevant information to truncate our Hilbert space or thus to reduce it to the physically relevant corner. We can absorb them in the matrix A and introduce a virtual leg to connect our two blocks, resulting in

i1 Ai1 ... i2i3 iN

C

_i0₂_i₃_...i_N D |Ψi = (1.3) The virtual leg, labeled by the bond dimension D, quantifies the amount of entangle-ment between our two subsystems. The bond dimension D is equal to the number of singular values different from zero. Repeating this procedure for all N legs we obtain the Matrix Product State or MPS. Imposing periodic boundary conditions and translational invariance results in an expression for the wave function given by

|Ψi = X

i1i2...iN

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1.3. PEPS - PROJECTED ENTANGLED PAIR STATES

This MPS consists of a chain of rank 3 tensors, where one leg corresponds to the phys-ical degree of freedom present in the original rank N tensor and the other two legs to virtual degrees of freedom, important for the entanglement structure. This MPS can be represented graphically as D i1 i2 i3 iN ... Ai1 A_i AiN 3 Ai2 |Ψi = (1.5) The bond dimension D acts as an upper limit for the entanglement between two MPS tensors and provides control over the truncation of the Hilbert space and over possible approximations made in numerical simulations. We will briefly mention two important theorems of MPS, since they are relevant in descriptions of topological order.

• The fundamental theorem of MPS

Two translationally invariant MPS of the same length which are equal to each other have tensors related by a gauge transformation.

• The theorem of MPS-injectivity

An MPS is injective, or thus spans its whole space, if and only if the corresponding transfer matrix has a unique largest eigenvalue, which is, up to some rescaling, equal to one. This implies that the ground state of an injective MPS is unique.

1.3 PEPS - Projected Entangled Pair States

A second way of obtaining MPS is given by the Projected Entangled Pair construc-tion, which is more intuitive from a physical point of view and which allows for higher-dimensional generalizations. Thereto we start at the virtual level and consider N qudit pairs each taking values in {0, 1, ..., D}. Then we maximally entangle qudits in these pairs with their neighbors, with the maximally entangled states given by

X a

|aai =

(1.6) and which results in a chain of maximally entangled states

... D

(1.7) By mapping our initial pairs to the physical level, we eventually obtain Projected En-tangled Pair States or PEPS

... i1i2 iN D i1 i2 i3 i3 iN

C

i1i2i3...iN _... ... D i1 i2 i3 iN D Ai1 A_i AiN 3 Ai2 i11 i12 ... i11 i12 D _... ... .. . ... ... i1N (1.8) 5

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1.4. MPOS - MATRIX PRODUCT OPERATORS

In one dimension this construction is completely equivalent to MPS when we identify the blue cells with the MPS tensors Ai. PEPS however can be defined for arbitrary coordi-nation numbers and can easily be generalized to higher dimensions. A two-dimensional PEPS or just PEPS as illustrated below, will be one of the key components to charac-terize two-dimensional topological order.

... i1i2 iN D i1 i2 i3 i3 iN

C

i1i2i3...iN _... ... D i1 i2 i3 iN D Ai1 A_i AiN 3 Ai2 i11 i12 ... i11 i12 D _... ... ... .. . ... ... i1N (1.9)

1.4 MPOs - Matrix Product Operators

So far we gave two equivalent descriptions to obtain tensor network representations for states of some quantum mechanical system. A natural extension to MPS provides us with Matrix Product Operators or MPOs [21], defined as

O = X i1i2...iN X j1j2...jN tr Bi1 j1B i2 j2...B iN jN |i1i2...iNi hj1j2...jN| . (1.10) or graphically ... i1 i2 i3 iN jN j3 j2 j1 B B B B O = (1.11) From their structure we can basically see that they act as matrices mapping some physi-cal label j to some other physiphysi-cal label i and thus provide a tensor network description for operators. We can act with MPOs on MPS, which will be useful to characterize bound-aries later on, but in this thesis, MPOs will mainly appear in the context of so-called MPO symmetries at the virtual level, treated thoroughly in Chapter 5.

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Chapter 2 Topological Order

When confronted with advanced condensed matter physics, we quickly get thrown in the world of quantum orders and quantum phase transitions. A subclass of these quantum orders is coined topological order and is crucial in our understanding of new states or topological phases of matter. In this section we will provide a criminally short introduc-tion to this vast area of research [22].

2.1 Fractional Quantum Hall Effect

The experimental realization of the two-dimensional electron gas marked the beginning of a whole new set of exotic phases, which could not be captured by Landau’s theory of symmetry breaking [23]. By cooling down this two-dimensional electron gas to tem-peratures close to the absolute minimum, the interactions between the electrons start to dominate the system, resulting in so-called Fractional Quantum Hall states or FQH states. These states are strongly correlated but are, due to the light mass of the electrons, very sensitive to quantum fluctuations, preventing the formation of crystal structures. This implies that a FQH state is some special kind of liquid also known as a quantum liquid. The study of such quantum Hall liquids lead to a lot of new and amazing prop-erties such as the incompressibilty of quantum liquids for example. The most surprising property however is the electron density in terms of the filling factor ν, defined as

ν = nhc eB =

electron density

fluxon density , (2.1)

where we interpret the fluxon as some kind of magnetic charge. These filling factors are exactly given by rational numbers as illustrated in figure 2.1 [24, 25]. A filling factor 1/3 for example signifies that we have three fluxons for every electron and we can see that such a density configuration results in a resistance dropping to zero. The fact that this happens for rational filling factors makes it the Fractional Quantum Hall Effect, which, in contrast to the Integer Quantum Hall Effect, cannot be explained by Landau’s theory of symmetry breaking. From a theoretical point of view we expect some internal pattern for states associated to these specific filling factors. We thus have some new

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2.1. FRACTIONAL QUANTUM HALL EFFECT

Figure 2.1: Experimental evidence for the Fractional Quantum Hall Effect in terms of the Hall resistance R and the magnetic field B. The dashed line gives predictions for the classical Hall Effect and overlaps with the experimental results. The steps in these results, indicating some drop in the resistance, are caused by the Quantum Hall effect. This effect cannot be explained by the classical theory and thus we need a quantum mechanical extension. The Integer Quantum Hall Effect can be explained by Landau’s conventional theory of symmetry breaking and explains the steps for integer filling factors. The steps for rational filling factors at high values of the magnetic field however, are not covered by this theory and demand a new theoretical framework to explain this Fractional Quantum Hall Effect.

kind of internal order, called topological order, which we cannot describe by a symmetry breaking of some Hamiltonian. We can gain some intuitive understanding of this topo-logical order by comparing it with the order present in crystals. Atoms in crystals have a fixed position, which is determined by some order specifying this position relative to the other atoms in the crystal. A FQH liquid is no crystal, but it is still fairly reasonable to assume that the electrons follow some correlated motion representing some internal pattern in the liquid and thus not move randomly. Electrons in a magnetic field always move in circles. The wave property of these electrons makes sure that each circle consists of an integer number of electron wavelengths, which implies that the electrons move in integer steps. We also need to take into account that these electrons can move to circles of other neighboring electrons. Due to the Pauli exclusion principle these movements are restricted and result in the odd denominators for the filling factors in the Fractional Quantum Hall Effect, corresponding to an odd number of steps for an electron to move

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2.1. FRACTIONAL QUANTUM HALL EFFECT

to another circle. So our notion of order here is given by the movements of electrons in a magnetic field, while also avoiding other electrons due to the strong Coulomb re-pulsion and the exclusion principle. Since this behavior applies for all electrons, we get some collective global order and hence topological order in some FQH state. This also means that different FQH states can be characterized by different topological orders. The simplest FQH states we can consider are Laughlin states [26], given by a filling factor of the form ν = 1/m, where m quantifies the (odd) number of steps an electron needs to take to go around another electron. We can easily explain the incompressiblity now. A compression of some FQH liquid to break the order requires some finite energy, since we always need to maintain our m steps, which implies that we cannot continuously reduce the distance between electrons because we cannot continuously reduce m. The main question we ask ourselves eventually is how to measure topological order. This order is not associated to some symmetries in the Hamiltonian and thus has no local order parameter to characterize it, implying that we need to find some other way. The first remarkable property is given by the fact that the ground state degeneracy depends on the topology of the underlying surface. This ground state degeneracy is not determined by symmetries of the Hamiltonian and only depends on the genus g of such surfaces. This means that we can use the ground state degeneracy as one of our quantum numbers to characterize systems with topological order. Orders in crystals automatically characterize defects in these crystals as well. In a similar way we can thus identify topological orders by characterizing their defects. These defects are called quasiparticle excitations or anyons and have very unusual properties like fractional elec-tric charges and fractional particle statistics [27], such that we cannot classify them as bosons or fermions. These unusual properties are measurable and thus contain relevant information about the topological order present. For completeness we also mention the gapless edge modes, which actually provide the most practical measurements, but which are beyond the scope of this thesis.

Our discussion is not restricted to FQH states only, which implies that topological or-der is a feature of most systems at low temperatures, characterized by the following properties

• A ground state degeneracy dependent on the topology of the underlying surface • The abscence of a local order parameter to distinguish between the different ground

states

• The presence of quasiparticle excitations or anyons • Gapless edge modes

For models describing this topological order we can choose some symmetries characteriz-ing this global, internal pattern and inspect the properties. The simplest of such models is given by a Z2-symmetry and will be discussed in the next chapter.

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Chapter 3 Toric Code

We will illustrate this topological order with a very simple model, namely the toric code [5][28], introduced by Kitaev. The toric code was initially conceived as a way to use the properties of systems with topological order to construct quantum error correcting codes. The main idea behind this kind of topological quantum computing is to store the information non-locally in a fusion state of a system of anyons and use the inher-ent robustness of topologically ordered systems against local perturbations to naturally increase the fault tolerance of such codes.

3.1 Hamiltonian and Ground State

Z Z Z Z X X X X Av Bp

Figure 3.1: A lattice with periodic boundary conditions in both directions, representing the topology of a torus. The spins are indicated with dots and live on the edges. The stabilizer operators are also shown, with the vertex operator Avdepicted in blue and the plaquette operator Bp depicted in red, where X and Z represent the Pauli operators σxand σz respectively.

We consider a Hamiltonian on a N ×N square lattice with periodic boundary conditions, giving it the topology of a torus, and with spins living on the edges labeled by {0, 1},

HT C = − X v Av− X p Bp, (3.1)

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3.1. HAMILTONIAN AND GROUND STATE

with the so-called stabilizer operators Av and Bp defined as

Av = 4 Y i=1 σ_iz Bp = 4 Y j=1 σx_j (3.2)

which we call the vertex operator and the plaquette operator respectively and which are depicted in figure 3.1. These operators are expressed1 in terms of the Pauli operators σx and σz, and are therefore Hermitian. Because these stabilizer operators share an even number of edges (0 or 2), all Av and Bp commute, since {σx, σz} = 0, so that

[Av, Bp] = [Av, Av0] = [B_p, B_p0] = 0. (3.3)

This means that all stabilizer operators can be diagonalized simultaneously, and that the toric code ground state can therefore be characterized in terms of the eigenstates with eigenvalue +1 of all stabilizer operators. We will denote the ground state of the system by |ψi. The other states, which have eigenvalue -1 for at least one stabilizer operator, are penalized by the Hamiltonian and correspond to excited states of the sys-tem, separated from the ground state by an energy gap ∆E ≥ 2. For qubits living on a generic surface, that is a surface with some generic topology, the degeneracy of this ground state is given by 4g, where g is the genus of the surface. This means that the ground state of the toric code, which has the topology of a torus, is fourfold degenerate, since the torus has one hole or equivalently g = 1. The different ground states are given in figure 3.2 in terms of the possible non-contractible loops on the torus.

Figure 3.2: Ground state degeneracy on the torus. There exist two non-contractible loops on the torus, one in the horizontal direction and one in the vertical direction. The different ground states are characterized in terms of the possible combinations of these two non-contractible loops.

The reason why we only consider non-contractible loops follows from the stabilizer op-erators or more precisely from the stabilizer subspace, the space preserved by these operators. The operator Av implements the constraint that we can only have closed loops of σx _{in the ground state and so A}

v|ψi = |ψi. The operator Av is violated if we act with some open string of σx on the ground state, for which the endpoints on the lattice indicate that the string is not closed. This means that configurations with an open string represent an excited state of the system. The operator Bp in turn allows us to deform loops in the ground state and provides dynamics, since acting with Bp on

1

Usually the vertex operators are written in terms of σxand the plaquette operators in terms of σz, however it will be more convenient to use the definition given above. The different Hamiltonians are related to each other by a unitary transformation H0 = U HU†, where the operator U is the Hadamard operator. So both conventions are completely equivalent.

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3.2. ANYONS

some loop configuration results in shrinking or growing the loop with one tile without altering the ground state, that is Bp|ψi = |ψi. By taking into account these constraints, the ground state is thus the superposition of all possible closed loop configurations. We can now classify states in this stabilizer subspace by introducing contractible and non-contractible loops of σx or loops of σz on the dual lattice respectively. Contractible loops of σx acting on the ground state can always be expressed as a product of Bp, which means we can always reduce these loops to a single point, such that their ac-tion is trivial. Non-contractible loops however cannot be reduced, since they enclose the hole of the torus, which means that they cannot be smoothly deformed to a point. Although these non-contractible loops still commute with the Hamiltonian and preserve the ground state subspace, they act non-trivially within this subspace. The operators associated with these non-contractible loops, denoted by X1 and X2 for the horizontal and vertical direction respectively, have eigenvalues ±1. These eigenvalues are not fixed by the Hamiltonian, resulting in four different ground states determined by the possible combinations of eigenvalues of X1 and X2. The action of such an operator basically transforms degenerate ground states into one another. This is a non-local operation, since such operators act on the whole system, which in turn implies that we cannot distinguish between the different ground states locally or thus cannot define a local or-der parameter to make this distinction. To illustrate this, consior-der the application of some state of the art measurement device. We assume this device measures some local quantity, meaning that its action only involves a limited number of connected sites on the lattice. In this case we can always deform an existing non-contractible loop to move it away from the relevant sites, so that the result of this measurement cannot distinguish between the different ground states.

This same property provides robustness against local perturbations, treated in quan-tum error correction. Suppose we have a local perturbation or equivalently the creation of a pair of excitations, then it takes a finite amount of time, associated to the Lieb-Robinson bounds [29], before this pair actually completes a non-contractible loop on the torus and causes an error. This operation scales with the system size and hence for sufficiently large systems, we can intervene and fix this error before it can cause harm.

3.2 Anyons

So far we have described the ground state degeneracy and the absence of a local order parameter when investigating the topological order present in the toric code. A third property was the presence of anyonic excitations, which we will consider now. To intro-duce excitations in the lattice, we make use of the following string operators along some paths L1 and L2 in the lattice and dual lattice respectively

S_Lx₁ = L1 Y i σx_i Sz_L₂ = L2 Y j σ_jz (3.4)

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3.2. ANYONS

These string operators commute with all stabilizer operators, except for those at the endpoints. At the endpoints S_Lx

1 violates the stabilizer operator Av and S

z

L2 violates

Bp. As was mentioned before, such string operators thus create excitations in the toric code. Suppose we now have two strings of the form Sx. If we consider the commutator of those two strings, we can see that it is zero, since all σx commute. This means that we can exchange Sx-strings, or equivalently Sx-particles freely. Particles created by these string operators are thus bosonic when we consider the same kind of string. If we however consider two different kind of strings, exchanging particles results in a minus sign, coming from the fact that σxand σzanticommute. The mutual statistics are bosonic with respect to particles of the same type, but fermionic with respect to particles of the other type. This means that these particles are neither bosons nor fermions and thus correspond to anyons. In the toric code we have four types of anyons written as {1, e, m, }, denoting the vacuum sector with 1, the electric charge with e, the magnetic charge or flux with m and the bound e-m pair with . The e- and m-particles have the same energy gap and the same particle statistics resulting in a so-called electromagnetic duality. The distinction between them is made by our assignment of e and m. For the remainder of this chapter electric charges will always be represented by the endpoints of a red string, while magnetic charges will always be represented by the endpoints of a blue string. The bound e-m pair or fermion, since it has fermionic statistics with respect to itself, is an excitation which violates both stabilizer operators and so it always has two strings, namely a red one and a blue one. We can also represent this as a string of σy_{, since [σ}x_,σz_{] = 2σ}x_σz _{= 2iσ}y _{and so σ}x_σz _{∼ σ}y_{. For a graphical representation of} these anyons we refer to figure 3.3.

X X X X X e e e e m m m m Z Z Z Z Z Z Z Z Z Y Y Y Y Y

Figure 3.3: Anyons in the toric code, apart from the identity sector. On the left we have an electric charge e, indicated by a red string of σx. In the middle we have a magnetic charge m, indicated by a blue string of σz. On the right we have both blue and red strings, indicating a fermion . Note how σy acts on the spins where we act with σx and σz simultaneously. Exchanging σxand σz at a site will result in a minus sign.

The fusion of these anyons, imposing which anyons can combine or fuse into one another, are fairly straightforward and can be summarized in fusion rules given by

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3.3. BOUNDARIES

Note that the ground state degeneracy was fourfold and we also have four different anyons, which is in agreement with the fact that the number of anyons and the ground state degeneracy are in a one-to-one correspondence, which is crucial for the presence of topological order [30].

3.3 Boundaries

In practical applications it is impossible to consider systems living on tori, and so it would be useful if we can consider the same system on a finite (square) lattice. We immediately see that we have a clear notion of a boundary now, which we will need to incorporate in the original toric code. This adds an extra layer of complexity by considering what happens when we try to describe the anyons living at these boundaries. The name toric code becomes a bit contradictory en we will from now on use the term planar code to refer to this lattice configuration. For the planar code there are two topologically distinct choices of boundary, namely a smooth boundary and a rough boundary [7], depending on whether the lattice ends in a set of vertices or in a set of edges respectively. This lattice is presented in figure 3.4, where we have smooth boundaries at the top and bottom and rough boundaries at the left and right.

Z Z Z Z Z Z Z X X X X X X X Z Z Z Z Z Z Z X X X X X X X

Figure 3.4: A finite lattice with qubits living on the edges. The top and bottom of the lattice have smooth boundaries and the left and right of the lattice have rough boundaries. The corresponding stabilizer operators are also shown and are modified at the boundary when necessary. Additionally we also show some excitations represented by a string of σx _{or σ}z _for the electric and magnetic charges respectively.

Having to deal with boundaries in the system also means we have to modify the stabilizer operators such that they are consistent with these boundaries.

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3.4. DEFECT LINES AND DEFECT POINTS

At the boundaries we have the following stabilizer operators Asmooth_v = 3 Y i=1 σz_i B_prough= 3 Y j=1 σx_j, (3.6)

which we can also see in figure 3.4.

For topological quantum error correcting codes it is important to understand what hap-pens with the anyons when they approach a boundary. Again referring to figure 3.4, we see that for some cases the stabilizer operators are no longer violated despite the presence of a string end, which means that some anyons have disappeared at these boundaries. To be more concrete, an electric charge e, which violates the stabilizer operator Av, still violates this stabilizer operator when approaching a smooth boundary and so cannot disappear at this boundary. If this charge approaches a rough boundary however, Av is no longer violated, which means this e has disappeared. For a magnetic charge m the dual holds, meaning that a magnetic charge m that moves to a smooth boundary no longer violates the stabilizer operator Bp and hence disappears at this boundary. If however this m approaches a rough boundary, then this charge gets stuck there, since Bp is still violated. So different choices of boundary act as sinks and sources for different types of anyons. More explicitly, at a rough boundary electric charges come and go and magnetic charges get stuck, while at a smooth boundary magnetic charges come and go and electric charges get stuck. For completeness we can now also consider what happens to the fermion , but since this anyon is made out of an e and an m, we see that this fermion still gets stuck everywhere on the lattice. To be more precise, if an approaches a rough boundary, its e disappears, but its m does not, and when an moves to a smooth boundary, its m disappears, but its e still gets stuck. We can summarize these results formally in what we we will call a condensation table

Bulk Smooth Rough

Anyons 1, e, m, 1s, es 1r, mr using a notation based on the one in [31]

1s= 1 ⊕ m es= e ⊕ 1r= 1 ⊕ e mr= m ⊕ . (3.7)

3.4 Defect Lines and Defect Points

The next step is to study defect lines in the toric code. To do this, we symmetrize the Hamiltonian such that the magnetic excitations are explicitly realized on the physical lattice. We can do this by doubling the amount of plaquettes in the system and placing them on a bipartite lattice. Eventually the Hamiltonian obtained by conjugating a sublattice with respect to the Hadamard transformation and hence related by a unitary transformation to the original one [9, 32], becomes

H_{T C}0 = −X k

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with the stabilizer operator Ak given by

Ak= σkxσzk+iσxk+i+jσk+jz (3.9)

where we denote the unit vectors along the x-axis and y-axis by i and j respectively. In figure 3.5 we see that for this lattice configuration, the stabilizer operator act the same on every plaquette. If Ak = 1, the plaquette is in the ground state and if Ak = -1, the plaquette is excited. Z X X Z Z Z Z Z Z X X X X Z Z Z Z Z Z X X Z X Z X X X Z Z Z Z Z X X X X X X X Y Z Z X X X X X X Z Z

Figure 3.5: A bipartite lattice with spins living on the vertices. The stabilizer operators are given in purple and are the same for both plaquette colors. The anyon e is described by a string living on the black plaquettes only, while the anyon m is described by a string living on the white plaquettes only. The operators σx _{and σ}z _{now determine how the strings move. Both types of} strings together can also be seen as a fermion .

We also see that σxand σzare no longer in correspondence with the electric and magnetic charges respectively. We now use σx to move a string to the upper left vertex or the lower right vertex and use σz to move a string to the upper right or lower left vertex. So using this lattice we can clearly see which excitation we have by checking on which color of plaquette the string lives. In exchange we mixed up the meanings of σx and σz, as they are now present in both strings. The magnetic charges are represented by a string living on white plaquettes only and the electric charges by a string living on black plaquettes only. The choice of assigning black or white is arbitrary and is a manifestation of the underlying electromagnetic duality present in the toric code. So

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both anyons are now explicitly present on the real lattice. A fermion is described by two neighboring strings living on white and black plaquettes respectively. The fact that the fermion anticommutes is recovered by considering the crossing of two strings on different plaquette colors. When those strings cross, a minus signs arises, due to the anticommutativity of σx and σz. Z X X Z Z Z Z Z Z X X X X Z Z Z Z Z Z Z X X Z X X Z X X X Z Z Z Z Z X X X X X X X Y Z Z X X X X X Z X X Z Z

Figure 3.6: The same bipartite lattice with a defect line. On the defect line, the stabilizer operators work exactly the same. The pentagon is necessary to keep the bipartite lattice in check and has a new associated stabilizer operator Qd. Strings moving through a defect line change plaquette colors and hence e becomes m and m becomes e. A fermion stays a fermion but with neighboring strings swapped.

We now introduce a defect line in this lattice by skipping the alternating pattern of colors once. A defect line in this lattice then becomes a mismatch in plaquette colors. On the defect line the stabilizer operators above still work, so there is no need to modify them here. At the endpoints however, we need to add a new stabilizer operator Qdthat commutes with the Ak, resulting in

Qd= σdxσzd+jσxd+i+jσ y d+iσ

z

d+2i, (3.10)

since we can only add σy to maintain commutativity. With this in mind we arrive at the lattice configuration in figure 3.6. Note that we can introduce boundaries by fixing the plaquette colors, such that a smooth boundary corresponds to a bulk of black plaquettes only and a rough boundary corresponds to a bulk of white plaquettes only, since the color assignment of strings fixes which excitations survive. Now that we have introduced this defect line, we can take a look at what happens when anyons approach. Consider for example an electric charge e, represented by a red string living on black plaquettes

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initially. When it moves through the defect line or domain wall, it can only move fur-ther to white plaquettes, since these are the only available plaquettes by going through the upper vertices. This means that the string now lives on white plaquettes and the electric charge has become a magnetic charge m, represented by a blue string living on white plaquettes. The dual holds for an incoming magnetic charge m. This means that the defect line implements the electromagnetic duality transformation between e and m. The fermion is invariant under this duality in the sense that it stays a fermion when moving through a defect line, however the two neighboring strings are swapped when this happens. Z X X Z Z Z Z Z Z X X X X Z Z Z Z Z Z Z X X Z X X Z X X X Z Z Z Z Z X X X X X X X Y Z Z X X X X X Z X X Z Z

Figure 3.7: Sinks and sources for fermions. On the left we have an incoming fermion of which one string winds around a defect point and connects with the other string, resulting in the annihilation of the incoming fermion. On the right we have the creation of an e¯e-pair of which one winds around the defect point, resulting in the creation of a fermion .

When we look at what happens at regions around the defect points or endpoints of a defect line, also known as twists [9], things become even more interesting. The intro-duction of these defect points adds new topological degrees of freedom, since the new stabilizer operator Qd adds two new eigenstates in the system, denoted by σ±. Topo-logically speaking, the most important feature of twists is that they act as sinks and sources for the fermions . This is presented in figure 3.7. A fermion approaches the neighborhood of a defect point and one of its strings winds around the defect point, resulting in the annihilation of this fermion. The opposite can also happen, namely the anyons e and m can be created in pairs from the identity sector and one anyon in those pairs can wind around a defect point and change its color, thereby creating a fermion, as shown explicitly in figure 3.7. An e¯e - pair is created and one e goes through the defect line, becoming an m, and returns to its original position, such that we have now an e and an m at neighboring sites, equivalent to a fermion . We also stress that in the neighborhood of a defect point, the assignment of strings becomes ambiguous and so e and m are two realisations of the same sector, since the same string around this defect point can represent both excitations. Taking into account this fermion condensation, as we would call it intuitively, but which is just some extended boson condensation, we

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have the following table, indicating a region near defects Bulk Defects Anyons 1, e, m, 1d, emd with assignments

1d= 1 ⊕ emd= e ⊕ m. (3.11)

The fusion rules for these new degrees of freedom can be recovered by considering two separate twists which each have a string wrapped around them twice. In the neighbor-hood of these defects, strings can only be distinguished through their orientation or thus how the strings cross over themselves. By encircling both defects with the blue and red strings, we can determine how these defects combine, taking into account the orientation of the defect strings. The fusion rules obtained by Bombin [9] are given by

σ±× σ±= 1 + σ±× σ∓= e + m. (3.12)

We will calculate these fusion processes in Chapter 5, where we introduce a formalism that is particularly suitable for computing fusion of anyons. If we now consider the subset {1, σ+, }, these anyons show a lot of similarity to Ising anyons, but they are not true Ising anyons, since we cannot directly compare the braiding of around σ+ and the braiding of around 1. The main point however is that using these defect points, we can construct non-abelian anyons from abelian anyons [8]. This gives rise to an in-teresting procedure to potentially obtain computationally powerful anyon models from rather simple anyon models.

So far we have described a model with topological order and went over the topologi-cal properties of this model. That is the topology-dependent ground state degeneracy, the absence of a local order parameter and the presence of anyonic excitations. Unfor-tunately such simple models are not that computationally powerful. However, we also discussed the modifications induced by introducing boundaries and defects and have concluded that these bring new possibilities to increase the computational power of such simple models. They also provide practical implementations from the perspective of engineering such materials with topological order. By introducing an alternating pat-tern of boundaries [7] for example, we are no longer dependent on the impractical genus scaling of the ground state degeneracy, since we no longer need to complete full loops around different holes in such surfaces to encode qubits. We will now consider a general-ized framework for these models, which explains intuitively how these topological phases arise and most importantly, what the underlying mathematical framework is. This will eventually provide a general formalism to tackle systems with topological order.

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Chapter 4 String-nets

Up to now we have only considered stabilizer operators to study the topological order present in systems. A more general framework to study topological order is provided by the string-net formalism, introduced by Levin and Wen [13]. The term stems from the fact that the microscopic degrees of freedom in certain topologically ordered systems organize themselves in a network of extended objects or string-nets. We can describe these extended objects through the local microscopic degrees of freedom, which allows us to characterize the topological order present in the system.

4.1 Formalism

For simplicity we consider the string-nets to be trivalent graphs, that is each node has three strings attached to it. These strings have a set of labels or types {1, ..., N } and an orientation, denoting the direction of the string. The dual of a label i, i∗, is then defined as that same string with an opposite orientation. Furthermore we need to specify a certain set of rules, called branching rules N_ijk, imposing which combinations of three strings are allowed to meet at a node or branching point. If the combination is allowed Nk

ij is 1 and if the combination is not allowed Nijk = 0. In short, a string-net is thus a network of oriented strings, for which each string has a type from a set of labels {1, ..., N } and these strings need to satisfy the branching rules N_ijk, imposing which strings can combine together and which cannot. We now consider the following Hamiltonian for such an extended object

HSN = tHt+ U HU, (4.1)

consisting of some kinetic term Ht and some potential HU with couplings t and U respectively, where HU corresponds to some kind of string tension, penalizing strings proportional to their length. In analogy with statistical mechanics, we can now realize topological phases from a quantum phase transition. If we first take U t, then the string tension dominates the system, hence there are few string-nets in the ground state. Suppose U t now, then the kinetic energy obviously dominates the system and we have a lot of large fluctuating string-nets. This means that there is a phase transition

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4.1. FORMALISM

at some critical ratio t/U close to one, going from a regime with high string tension to a regime with high kinetic energy as illustrated in figure 4.1. This toy model provides some physical intuition for the realization of topological order in condensed matter systems. The microscopic degrees of freedom organize themselves into string-nets described by some string tension and kinetic energy. If this kinetic energy becomes sufficiently large, the string-nets condense and, depending on the different structures and condensation mechanisms of the string-nets, give rise to different topological phases.

t/U 1

Figure 4.1: String-net condensation visualized. For large U , the string tension dominates and few structures are formed. For large t however, the kinetic energy dominates and large, fluctuating structures are formed. We thus have two different regimes of structure formation and expect a phase transition for t/U of order unity between the trivial phase on the left and the string-net condensed phase on the right.

4.1.1 Mathematical Framework

To put things in a mathematical framework, we now consider some fixed point wave func-tion for all string-net condensed states. This means that in order to characterize each string-net state, we only need to describe the fixed point, since this fixed point should contain all relevant information or equivalently all universal properties that characterize the corresponding topological phases. To motivate this some more, suppose we have some string-net configuration with two different string-net condensed states belonging to the same phase, hence two different wave functions. By making use of the Renor-malization Group (RG) [33, 34] and taking a look at the RG flow, these two different wave functions will flow to the same fixed point wave function. This means that the long-range behavior of these two states is the same and hence, is also the universal

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be-4.1. FORMALISM

havior of the corresponding topological phase. The reason why they differ initially is due to short-range properties, which are irrelevant when studying the topological behavior. This RG flow is presented in figure 4.2.

Φ

1

Φ

4

Φ

3

Φ

2

Figure 4.2: RG flow of the system to some fixed point wave functions, which capture the universal, long-range behavior of the associated topological phases.

Now that we have motivated the use of fixed point wave functions, we will explicitly construct them. In the string-net condensed phase, the kinetic energy dominates and we basically try to find states |Φi which minimize the energy when acted upon by Ht. With this fixed point picture in mind, |Φi needs to have the following properties

Φ

=

(F_lijk)n_m

Φ

n

Σ

Φ

=

_Φ

Φ

=

d

i

_Φ

Φ

=

δij

_Φ

i j i i i i i i i j k l m n k k k l l l j (4.2)

Φ

=

(F_lijk)n_m

Φ

n

Σ

Φ

=

_Φ

Φ

=

d

i

_Φ

Φ

=

δij

_Φ

i j i i i i i i i j k l m n k k k l l l j (4.3)

Φ

=

(F_lijk)n_m

Φ

n

Σ

Φ

=

_Φ

Φ

=

d

i

_Φ

Φ

=

δij

_Φ

i j i i i i i i i j k l m _n k k k l l l j (4.4)

Φ

=

(F_lijk)n_m

Φ

n

Σ

Φ

=

_Φ

Φ

=

d

i

_Φ

Φ

=

δij

_Φ

i j i i i i i i i j k l m n k k k l l l j _(4.5)

which basically implement the long-range behavior associated to topological order. First we want |Φi to be invariant under continuous deformations of strings. Stated otherwise, the fixed point is insensitive to such deformations. The second equation represents scale

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4.1. FORMALISM

invariance, since the fixed point in RG is scale invariant and hence string configurations with closed loops are proportional to configurations without loops up to some scale factor di. This factor can be thought of as the local dimension associated to some string type i and thus dubbed the quantum dimension, since the Hilbert space for such a string i scales as dN_i . The third equation follows a similar line of reasoning, since at large scales the bubble disappears and hence i = j, which implies that this must hold at smaller scales as well. The last equation, characterized by a 6-index1 symbol F_lijkn_m, is the simplest local constraint such that the ground state is uniquely determined, given the properties above. This constraint basically implements the fact that the wave function is invariant under retriangulations given by some unitary transformation F . We thus arrive at a fixed point ground state determined by the objects F_lijkn_m and di, having in general entries in C and R+0 respectively. The most important property of these F -symbols, is what is called an F -move, which makes it possible to recouple string configurations

=

f b a e d

Σ

f (Fbad e ) f c c e d b a (4.6) For well-defined string-nets, these 6-index objects or F-symbols need to satisfy what is called the pentagon equation, namely

F_ef cdh_g F_eabhi_f =X j

F_gabcj_f F_eajdi_g F_ibcdh_j. (4.7)

This equation basically encodes the fact that associativity allows to obtain the same result in two different, equivalent ways, as depicted in figure 4.3. From a mathematical point of view, solutions of this pentagon equation correspond to tensor fusion categories [35] and hence a complete classification of string-net condensed phases is entirely cap-tured by this mathematical framework, on which we elaborate a bit in Appendix A. Unfortunately finding solutions of the pentagon equation is far from trivial, however we can always find a solution for a given group G. Thereto we set the string types equal to the irreducible representations or irreps of that group and their fusion rules provide us with branching rules. The quantum dimensions are then equal to the dimensions of these irreps and the F -symbol is given by the 6j-symbol of that group.

4.1.2 Hamiltonian and Ground State

So far we have found a fixed point wave function for the ground state containing all relevant information with respect to its long-range behavior. The next step is to find the fixed point Hamiltonians which have these wave functions as eigenstates, therefore realizing all so-called (2 + 1)-dimensional doubled topological phases [36]. Thus given

1

In more general treatments, we need to add the multiplicities at the nodes as well, which would actually make this a 10-index symbol.

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4.1. FORMALISM F F F F F e e e e e f f a a a a a g g h h i i j j b b b b b c c c c c d d d d d

Figure 4.3: The pentagon equation illustrates how we obtain the same result by applying associativity in two different ways.

solutions of the pentagon equation, we can always construct an exactly soluble Hamil-tonian. Let us make this explicit by considering a honeycomb lattice with spin degrees of freedom living on the edges, where each spin takes values in {0, 1, ..., N } and every edge has an arbitrary orientation. We added the label 0 to the set to denote null strings which can be thought of as the vacuum. In analogy with the stabilizer operators in the toric code, we arrive at the Levin-Wen Hamiltonian

HLW = − X v Qv− X p Bp Bp= N X i=0 diBpi d2_tot (4.8)

where the sums in HLW run over the vertices and plaquettes respectively. As was the case for the toric code, the vertex operator Qv measures now some generalized electric charge and simply implements the branching rules at every vertex, namely

Qv|ijki = vivj vk F_iijj∗ ∗ 0 k≡ N k ij|ijki (4.9)

denoting an arbitrary vertex with |ijki and introducing some normalization vi = √

di. The second term in the Hamiltonian then measures a generalized magnetic charge or flux and favors states with no flux. The operator Bp is a linear combination of all different string types and acts on the 12 spins adjacent to a plaquette, only changing the values of the inner 6 spins. These operators are illustrated in figure 4.4. The operator B_pi adds a loop of string type i when all vertices are stable or thus satisfy the branching rules. By

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4.1. FORMALISM

using the F -moves we can fuse this loop with the plaquette edges to obtain the ground state configuration with modified labels on these edges. Therefore the operators B_pi can be written as a product of 6 F -symbols. By slightly abusing notation we can write such an operator B_pi as B_pi|jklmnoi = X j0_k0_l0_m0_n0_o0 6 Y s F_si|j0k0l0m0n0o0i , (4.10)

where we denote the plaquette edges as |jklmnoi and where the F -symbols contain additional indices mapping |j0k0l0m0n0o0i to |jklmnoi. This action of Bi

p will be key to obtain a tensor network representation for these string-net states. Additionally, given the consistency equations, Qv and Bp commute. The coefficient for Bip was chosen such that we have a smooth continuum limit with a ground state wave function satisfying the fixed point properties given above. Taking these properties into account we can see that Qv and Bp act as projectors onto ground state and thus as constraints, such that violating these Qv and Bp results in excited states.

Qv

Bp

Figure 4.4: String-nets defined on a honeycomb lattice with oriented edges. The vertex operator Qvchecks whether a vertex is allowed, given the labels and orientation of the incoming strings. If this is not the case, Qvis violated and measures some generalized electric charge. The plaquette operator Bpacts on a hexagonal plaquette and its surrounding edges and creates a superposition of loops of all possible string types i, leaving this plaquette invariant if the plaquette does not contain a magnetic charge.

4.1.3 Quasiparticle Excitations or Anyons

Since we have now defined the ground state properly, we can take a look at quasiparticle excitations or anyons, which correspond to violations of the operators Qv and Bp. In

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4.2. BOUNDARIES, DOMAIN WALLS AND DEFECTS

analogy with the toric code, we can only detect endpoints of strings moving through the hexagonal lattice. This means that the possible anyons are always created in pairs and depend on the different string types present. To be able to describe all excitations, it suffices to determine all possible closed string operators in the string-net model, since these are just a special case of strings where the two endpoints coincide and hence the corresponding anyons annihilate. We can thus describe all anyons by considering string operators that commute with the Hamiltonian. We will start by considering simple string operators, which just add a loop of type i to the vacuum, in correspondence with the operator B_pi. A general string operator Wi(P ) along some path P then takes the following form Wi= N Y s ωsFsi (4.11)

with the product running over all relevant edges of the string path P . We thus have an expression similar to that of Bi_pin terms of the F -symbols but with additional factors ωs (and their complex conjugates ωs) with elements in C to characterize all strings. Note that for Bi

p we have QN

s ωs= 1. The fact that the string operator should commute with the Hamiltonian, since it is a closed string, sets constraints on these ωs and Fsi resulting in the different quasiparticle excitations. A graphical representation of open and closed string operators is given in figure 4.5.

For abelian groups all string types are simple and we can make a classification according to the values of i and ωs. If i = 0 and ωs = 1 we simply have the null string which is equivalent to the vacuum. On the other hand if i = 0 and ωs6= 1 then we have generalized magnetic charges. Generalized electric charges are classified through i 6= 0 and ωs= 1. Finally, i 6= 0 and ωs 6= 1 results in bound electric charge-magnetic charge pairs. An extension to non-abelian anyons then basically comes down to considering non-simple strings given by some matrix Ωis

s as a generalization of ωs. General string operators will then be reducible and solutions will be given by the set of irreducible {Ωα,Ωα}, denoted by a Greek subscript. With these string operators in mind we now have all necessary information to compute the topological properties of the corresponding phase. Fusion is given by considering the combining two strings such that WαWβ = hγ_αβWγ according to some fusion constant hγ_αβ. Furthermore we can also calculate other universal properties such as the topological spin θα and the S-matrix for the scattering of these anyons, but for the cases at hand, this will be beyond the scope of this thesis.

4.2 Boundaries, Domain Walls and Defects

To extend the string-nets to boundaries and domain walls, we will summarize the for-malism in a more categorical language first, for which we also refer to Appendix A. To determine the ground state wave function |Φi of some topological phase, we needed an F -symbol 0F , denoting the bulk with front superscript 0, and quantum dimensions di, captured by the mathematical framework of (unitary) tensor fusion categories. We thus

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4.2. BOUNDARIES, DOMAIN WALLS AND DEFECTS W1 W2 ¯ a a

Figure 4.5: Some string operators W1 and W2. W1 is an open string operator and hence its endpoints correspond to quasiparticle excitations or anyons, created in pairs aa. W2 is a closed string operator and does not violate any constraints. By characterizing all possible string operators we can describe all different anyons of the corresponding topological phase.

have the bulk of our topological phase which is described by some tensor fusion category C, equipped with a tensor product and consisting of some finite set of string types or sim-ple objects {i} and a set of morphisms. The main point of these tensor categories is that we consider all equivalent configurations up to some isomorphisms. These equivalences of morphisms manifest themselves as the triangle equation and the pentagon equation. From a physical point of view tensor categories are thus just a bunch of arrows equipped with a tensor product structure and we can use these tools to describe all physically relevant properties. Our main starting point will be a tensor fusion category C as input category with anyons given by some braided output category Z(C), the monoidal center of C.

4.2.1 Module Categories as Boundaries

To include boundaries, domain walls and defects, Kitaev and Kong [14] introduced mod-ule categories and bimodmod-ule categories M, equipped with a tensor product such that for a left C-module2 for example, objects in C and M can be composed such that, abusing the notation for the tensor product, C ⊗ M 7→ M. An important sidenote is that these modules are not necessarily monoidal themselves, or thus in general we do not have a

2

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4.2. BOUNDARIES, DOMAIN WALLS AND DEFECTS M C C C C C C C

Figure 4.6: A module category M acting as a boundary in a string-net model based on the tensor category C.

tensor product such that M ⊗ M 7→ M. Another important thing to keep in mind is that we cannot change orientations arbitrarily and hence once we choose an orien-tation, it stays fixed for the relevant category M. We can however always consider a dual category Mop for which the orientation is reversed. This will play an important role when we consider two tensor categories C and C_M∗ , the dual of C with respect to the module M. The modules M and Mopthen act as interfaces, defining a so-called Morita context [37] with objects C, C_M∗ , M and Mop, used later on. Let us now go into some more detail about these modules. We can construct a boundary on the right of C by considering a left C-module M. By choosing the orientation upwards for M, we arrive at the string-net configuration depicted in figure 4.6. Of great importance is that we need a new F -symbol 1_{F , to implement the associativity constraint when considering} fusing objects in C and M. We now define both bulk and boundary F -symbols as

=

f b a e d

Σ

_c (0Fdabe) f c c e d b a

=

_Σ

c (1FAabB) C c C B A B A a b b a c (4.12) with small letters as bulk labels or objects in C and capital letters as boundary labels or objects in M. These 1F s satisfy a new pentagon equation, namely

1 F_Bf cAC g 1 F_BabCD f = X j 0 F_gabcj f 1 F_BajAD g 1 F_DbcAC j . (4.13)

4.2.2 Module Functors or Boundary Excitations

To describe the boundary excitations, we refer to [14, 31] for a rigorous mathematical treatment. We can basically view them as unitary C-module functors denoted by Θ such that Θ : M 7→ M. The reason why is made partially clear when considering a

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4.2. BOUNDARIES, DOMAIN WALLS AND DEFECTS

string operator in the string-net model acting on a boundary as illustrated in figure 4.7. This string operator is then partially fused with the boundary and the resulting objects,

M C C C C C Θ Θ M C C C C C WΘ M C C C C C WΘ

Figure 4.7: Equivalence between string operators WΘ and functors Θ. By fusing a string operator with the boundary we naturally arrive a the notion of functors to describe boundary excitations.

attached to the boundary, are completely equivalent with a set of functors Θ and Θ. This also introduces a natural isomorphism 2F , given by

=

_Σ

C (2FaBΘ A ) D C D B A a Θ C B A a Θ (4.14) which obeys another pentagon equation

2

F_Bf AΘD_C 1F_BabDE_f =X F

1

F_CabAF_f 2F_BaF ΘE_C 2F_EbAΘD_F. (4.15) To capture all boundary excitations, we consider the category of these unitary C-module functors, written as FunC(M, M) or CM∗ with its simple objects representing boundary excitations and the tensor product given in the opposite order. This category again has some bulk and boundary associativity constraints, allowing us to introduce two final F -symbols, namely3F for the boundary and4F for the bulk

=

_Σ

C (3FBΘΨ A ) Ξ C C B A Θ Ψ B A Ξ Ψ Θ

=

Σ

Γ Γ (4FΘΛΩ Ψ ) Ξ Γ Ξ Λ Ω Θ Ψ _Ψ Λ Ω Θ (4.16) introducing Greek capital letters to denote objects in C_M∗ . These satisfy a final set of pentagon equations 3 F_BCΘΓΞ_D 2F_BaAΞE_C =X F 2 F_DaAΘF_C 2F_BaF ΓE_D 3F_EAΘΓΞ_F (4.17)

Beschrijving van de rand en defecten in systemen met topologische orde door middel van Matrix Product Operators

Characterizing Boundaries and Defects for

Systems with Topological Order in a Matrix

Product Operator Formalism

Marco Deweirdt

Promotor: prof. dr. Frank Verstraete

Supervisor: Laurens Lootens

Acknowledgements

Abstract

Nederlandstalige samenvatting

Contents

Introduction and Outline

Chapter 1

Tensor Networks

1.1

Lifting the Curse of Dimensionality

1.2

MPS - Matrix Product States

C

C

1.3

PEPS - Projected Entangled Pair States

C

C

1.4

MPOs - Matrix Product Operators

Chapter 2

Topological Order

2.1

Fractional Quantum Hall Effect

Chapter 3

Toric Code

3.1

Hamiltonian and Ground State

3.2

Anyons

3.3

Boundaries

3.4

Defect Lines and Defect Points

Chapter 4

String-nets

4.1

Formalism

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