Master Thesis Condensed Matter Theory - KOMET 337 ... · quantum Hall effect, which is still not fully understood [1]. This ex-empliﬁes the necessity of a theoretical understanding

E X A C T T E N S O R N E T W O R K S TAT E S F O R T H E K I TA E V

H O N E Y C O M B M O D E L

philipp schmoll

Master Thesis

Condensed Matter Theory - KOMET 337Institute of Physics

Johannes Gutenberg University Mainz

Philipp SchmollExact Tensor Network States for the Kitaev Honeycomb Model

supervisors

Jun.-Prof. Román Orús, Ph.D.Univ.-Prof. Dr. Peter van Loock

D E C L A R AT I O N

I hereby declare that I have completed the present thesis indepen-dently making use only of the specified literature and aids. Sentencesor parts of sentences quoted literally are marked as quotations; iden-tification of other references with regard to the statement and scopeof the work is quoted.

Mainz, May 2016

Philipp Schmoll

Philipp [email protected]

KOMET 337Institute of PhysicsJohannes Gutenberg UniversityStaudingerweg 755128 Mainz

A B S T R A C T

The Kitaev honeycomb model is a paradigm of exactly-solvablemodels. Introduced in the context of topological quantum computa-tion, it shows non-trivial physical properties such as topological quan-tum order, excitations in the form of abelian and non-abelian anyons,and chirality. Its solution is one of the most beautiful examples of theinterplay of different mathematical techniques in condensed matterphysics.In this thesis we focus on the construction of exact Tensor Network(TN) eigenstates of the spin-1/2 Kitaev honeycomb model in the ther-modynamic limit, in particular on the construction of the groundstate. We reverse the different transformations in the solution of themodel to represent them as a unitary quantum circuit building up themany-body wave function. The 3d TN structure emerging is numeri-cally contracted layer-by-layer, trying to approximate it at any step bya (fermionic) Projected Entangled Pair State (PEPS) with finite bonddimension.In terms of the exact TN structure we are able to generate the quan-tum state for up to 32 spins. As for the PEPS representation we aimto achieve larger lattices in order to verify analytically predictablephysical behaviour.

V

A C K N O W L E D G M E N T S

I would like to express my gratitude to my supervisor Román Orús,with whom it was always a pleasure to work. His encouragementand ideas helped me a lot during my thesis. I would like to thankMatteo Rizzi for supportive academic discussions especially aboutthe Jordan-Wigner transformation and Majorana fermions. Addition-ally I would like to thank Peter van Loock for being the co-advisor ofmy thesis.

I thank my colleagues in the group for the enjoyable time and forproofreading my thesis. Especially Serkan, Andreas and Charalampos- it was always a pleasure to come to the office and I acknowledge allof your help.

Finally I express my heartfelt gratitude towards my parents andSusanne, for their love and ongoing support.

VII

C O N T E N T S

i introduction and theory 1

1 introduction 3

2 basic concepts 5

2.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Schmidt Decomposition . . . . . . . . . . . . . . 6

2.1.2 Von Neumann Entropy . . . . . . . . . . . . . . 6

2.2 Singular Value Decomposition . . . . . . . . . . . . . . 7

3 tensor networks 9

3.1 The Idea of Tensor Networks . . . . . . . . . . . . . . . 9

3.2 Tensor Network Theory . . . . . . . . . . . . . . . . . . 10

3.3 Tensor Network Description of the Wave Function . . . 11

3.4 Entanglement Entropy and Area-Law . . . . . . . . . . 12

3.5 Important Families of Tensor Network States . . . . . . 14

3.6 Fermionic Tensor Networks . . . . . . . . . . . . . . . . 15

3.6.1 Fermionization Rules . . . . . . . . . . . . . . . 15

3.6.2 Basic Examples . . . . . . . . . . . . . . . . . . . 17

ii exact tensor network states 19

4 the spectral tensor network 21

4.1 Quantum Fast Fourier Transformation . . . . . . . . . . 21

4.2 Tensor Network Structure . . . . . . . . . . . . . . . . . 24

4.3 Spin Systems and Jordan-Wigner Transformation . . . 24

5 ising model with transverse field 27

5.1 Solution of the Model . . . . . . . . . . . . . . . . . . . 27

5.2 Representation of the Eigenstates . . . . . . . . . . . . . 30

IX

X contents

iii the kitaev honeycomb model 33

6 the kitaev honeycomb model 35

6.1 Fermionization of Spin-1/2 Operators . . . . . . . . . . 36

6.1.1 Quadratic Hamiltonian . . . . . . . . . . . . . . 37

6.1.2 The Ground State . . . . . . . . . . . . . . . . . . 39

6.2 Fermionization by a Jordan-Wigner Transformation . . 40

6.2.1 Diagonalization of the Hamiltonian . . . . . . . 44

6.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . 48

7 toric code limit of the honeycomb model 51

7.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . 52

7.2 PEPS Representation of the Ground State . . . . . . . . 53

7.3 Perturbative Limit of the Kitaev Model . . . . . . . . . 54

8 exact unitary 3D tensor network 59

8.1 Fermionic Vacuum and Bogoliubov Transformation . . 59

8.2 Fourier Transformation and Spectral Tensor Network . 62

8.3 Majorana Braiding Tensor Network . . . . . . . . . . . 65

8.3.1 Majorana Fermions as non-Abelian Anyons . . 65

8.3.2 Tensor Network of Majorana Braidings . . . . . 67

8.4 Jordan-Wigner Transformation . . . . . . . . . . . . . . 70

8.5 Arbitrary Vortex Sectors . . . . . . . . . . . . . . . . . . 72

8.6 Overall Tensor Network Structure . . . . . . . . . . . . 73

8.7 Ground State Fidelity and Topological Entropy . . . . . 74

9 reduction to a peps 79

iv conclusion and outlook 81

10 conclusion and future work 83

bibliography 85

L I S T O F F I G U R E S

Figure 2.1 Graphical Representation of the SVD . . . . . . 7

Figure 3.1 Area-Law for the Entanglement Entropy . . . . 9

Figure 3.2 Diagrammatic Notation of Basic Tensors . . . . 10

Figure 3.3 Basic Tensor Network Examples . . . . . . . . 10

Figure 3.4 Cyclic Property of the Trace . . . . . . . . . . . 11

Figure 3.5 Decomposition of the Wave Function . . . . . . 12

Figure 3.6 Area-Law of a Bipartite PEPS . . . . . . . . . . . 13

Figure 3.7 Exponentially Large Hilbert Space . . . . . . . 14

Figure 3.8 MPS with OBC and PBC . . . . . . . . . . . . . . 14

Figure 3.9 PEPS with OBC and PBC . . . . . . . . . . . . . . 15

Figure 3.10 Parity-Symmetric Tensor . . . . . . . . . . . . . 16

Figure 3.11 Fermionic Swap Tensor . . . . . . . . . . . . . . 16

Figure 3.12 Bosonic and Fermionic Jump Move . . . . . . . 17

Figure 3.13 Bosonic and Fermionic PEPS State . . . . . . . . 17

Figure 4.1 Unitary Circuit for the FT of eight Fermions . . 22

Figure 4.2 Unitary Circuit for the FT of 4× 4 Fermions . . 23

Figure 4.3 STN Ansatz for the Wave Function of 16 Fermions 24

Figure 4.4 STN with Bogoliubov Transformation . . . . . . 25

Figure 5.1 1d Ising Chain with PBC . . . . . . . . . . . . . 27

Figure 5.2 Ising Model - Construction of Eigenstates 1 . . 30



Figure 6.1 Honeycomb and Brickwall Lattice . . . . . . . 35

Figure 6.2 Elementary Plaquette Operators Bp . . . . . . 36

Figure 6.3 Majorana Hamiltonian and Gauge Fields . . . 38

Figure 6.4 Unit Cell and Basis Vectors . . . . . . . . . . . 40

Figure 6.5 Jordan-Wigner Path on the Brickwall Lattice . 42

Figure 6.6 Unit Cell on the Brickwall Lattice . . . . . . . . 43

Figure 6.7 Phase Diagram of the Kitaev Model . . . . . . 48

Figure 6.8 Dirac Cones in the Gapless B-Phase . . . . . . 48

Figure 7.1 Toric Code on the Square Lattice . . . . . . . . 51

Figure 7.2 Plaquette Operators and Star Operators . . . . 51

Figure 7.3 Nontrivial Loops on the Torus . . . . . . . . . 52

Figure 7.4 Toric Code Projector as PEPS Operator . . . . . 53

Figure 7.5 PEPS Construction of the TC Ground State . . . 53

Figure 7.6 PEPS Representation of the TC Ground State . . 54

Figure 7.7 Unit Cell Tensors of the TC . . . . . . . . . . . . 54

Figure 7.8 Effective Spin Mapping in the TC Limit . . . . 55

Figure 7.9 Square Plaquette of Effective Spins τ . . . . . . 55

Figure 7.10 Graphical Mapping of the Honeycomb Hamiltonianto the Toric Code Hamiltonian . . . . . . . . . 55

XI

Figure 7.11 Transformation of the Toric Code PEPS Tensors 56

Figure 7.12 SVD of Toric Code Tensors . . . . . . . . . . . . 57

Figure 7.13 PEPS Ground State of the Honeycomb Model . 57

Figure 8.1 Product State of Bogoliubov Modes . . . . . . 59

Figure 8.2 Sequence of the Bogoliubov Transformation . 60

Figure 8.3 TN of the Bogoliubov Transformation . . . . . 61

Figure 8.4 Planar TN of the Bogoliubov Transformation . 62

Figure 8.5 Four Mode Fourier Transformation . . . . . . . 63

Figure 8.7 Planar Implementation of the 4× 4 FT . . . . . 64

Figure 8.8 Interaction Terms at the Boundary . . . . . . . 65

Figure 8.9 Majorana Braiding Tensor . . . . . . . . . . . . 67

Figure 8.10 Majorana Braiding on the Honeycomb Lattice 68

Figure 8.11 Two-Body Gate as MPO . . . . . . . . . . . . . . 69

Figure 8.12 Planar Majorana Braiding Network . . . . . . . 69

Figure 8.13 Jordan-Wigner Path with Periodic Lattice . . . 71

Figure 8.14 Full Vortex Pattern . . . . . . . . . . . . . . . . 73

Figure 8.15 STN for the Eight Spin Honeycomb Lattice . . . 74

Figure 8.16 Trajectory in the Parameter Space . . . . . . . . 75

Figure 8.17 Ground State Fidelity . . . . . . . . . . . . . . . 75

Figure 8.18 Ground State Fidelity for 128 Spins . . . . . . . 76

Figure 8.19 Ground State Fidelity for 2048 Spins . . . . . . 76

Figure 8.20 Ground State Fidelity for 32 768 Spins . . . . . 77

Figure 8.21 Z2 Symmetry in a TN . . . . . . . . . . . . . . . 78

Figure 9.1 Parity Recovery of Tensors after the SVD . . . . 79

Figure 9.2 Eight Spin Honeycomb PEPS Network . . . . . 80

L I S T O F TA B L E S

Table 8.1 Conditions for sign(vk) = −1 . . . . . . . . . . 61

XII

A C R O N Y M S

CTM Corner Transfer Matrices

DMRG Density Matrix Renormalization Group

FT Fourier Transformation

FFT Fast Fourier Transformation

MERA Multiscale Entanglement Renormalization Ansatz

MPO Matrix Product Operator

MPS Matrix Product State

OBC Open Boundary Conditions

PBC Periodic Boundary Conditions

PEPO Projected Entangled Pair Operator

PEPS Projected Entangled Pair State

STN Spectral Tensor Network

SVD Singular Value Decomposition

SRG Second Renormalization Group

TEBD Time-Evolving Block Decimation

TFIM Transverse Field Ising Model

TC Toric Code

TEE Topological Entanglement Entropy

TN Tensor Network

TRG Tensor Renormalization Group

TTN Tree Tensor Network

XIII

Part I

I N T R O D U C T I O N A N D T H E O RY

1I N T R O D U C T I O N

The fascination of quantum many-body systems emerges from the in-terplay of many simple quantum objects. This often leads to new re-markable and complex phenomena, that the individual constituentsof the many-body system do not exhibit. It is this concept of emer-gence that leads to effects like superconductivity or superfluidity. De-spite the enormous evolution of computational power and resourcesin recent years, it is still a challenging task to compute and predictphysical properties of such systems. A good example is the fractionalquantum Hall effect, which is still not fully understood [1]. This ex-emplifies the necessity of a theoretical understanding of quantummany-body systems. Attaining a better understanding of such sys-tems would also benefit many related research fields, like condensedmatter physics and material science.The number of parameters necessary to exactly describe a physicalstate of a system grows exponentially in the number of particlestherein. Thus for larger system sizes it becomes increasingly diffi-cult to calculate macroscopic properties from the microscopic descrip-tion of the individual constituents. In some particular and rare casesthe model is exactly solvable and can be treated immediately in thethermodynamic limit. For the majority of the systems however, wehave to rely on numerical simulations to solve the model on a finitedimension. On account of this, it is indispensable to have an efficientand accurate description of the relevant physical behaviour.In recent years a new approach to describe the wave function forquantum many-body systems has been developed in the field of ten-sor networks. This representation is based on the quantum entangle-ment in the system, which describes correlations between its con-stituents. A TN can be seen as an entanglement representation of thequantum state which is able to describe physical states that are con-strained to live in a small subspace of the exponentially large Hilbertspace. In this low-entanglement region TNs are an efficient tool forthe simulation of quantum many-body systems.Well-known types of tensor networks are Matrix Product State (MPS)and Projected Entangled Pair States, serving as an ansatz to simulatethe wave function of 1d and 2d quantum systems respectively. Gen-erally TNs are particularly useful for ground states and low-energyeigenstates of gapped Hamiltonians with local interactions. Never-theless they strongly depend on the actual problem, so that there is avariety of different methods available.

3

4 introduction

In this thesis we examine a paradigm of exactly solvable models,the Kitaev model on a two-dimensional honeycomb lattice. This mo-del offers various physically non-trivial effects, such as excitationsin the form of abelian and non-abelian anyons and topological quan-tum order. It has been proposed in the context of topological quantumcomputation, which makes it not only theoretically but also conceptu-ally interesting. We will use TN methods to provide an exact descrip-tion of the model’s eigenstates. Therefore we revise basic concepts forquantum many-body systems and present the tensor network repre-sentation of wave functions in the first part of the thesis. In the secondpart we examine the general construction of eigenstates for quadraticHamiltonians that are exactly solvable. Here the 1d quantum Isingmodel with a transverse field serves to illustrate the procedure. Thethird part of the thesis examines the 2d Kitaev model on the honey-comb lattice. We revise two of the possible analytical solutions of themodel in order to build up exact eigenstates. Therefore an appropri-ate TN structure for all of the transformations is necessary, which willappear in reverse direction to the solution. The 3d structure of eigen-states is constructed for small lattices as an exact expression, disre-garding the advantages provided by the idea of tensor networks. Inorder to construct quantum states for larger systems, the unitary TN

structure is numerically contracted layer-by-layer, trying to approxi-mate it at any step by a (fermionic) PEPS with finite bond dimension.

2B A S I C C O N C E P T S

One of the most fundamental properties of quantum many-body sys-tems is the entanglement between the constituents. It is therefore use-ful to have a measure of entanglement, which we will describe in thisintroductory section.

2.1 entanglement

Entanglement describes non-local quantum correlations between twoor more particles. Due to these correlations the particles in the systemcan not be described individually, instead the quantum state of thewhole system needs to be considered. Entanglement is a key featureof quantum mechanics and it gives rise to interesting phenomena.Consider two quantum states |ψ1〉 ∈ H1 and |ψ2〉 ∈ H2 defined inseparate Hilbert spaces. A composite system is obtained by the tensorproduct of Hilbert spaces

|ψ3〉 ∈ H1 ⊗H2 . (2.1)

The dimension of the composite Hilbert space H3 = H1 ⊗H2 is theproduct of the dimensions of the individual Hilbert spaces, hencedim(H3) = dim(H1) · dim(H2). In terms of basis states the state |ψ3〉can be generally written as

|ψ3〉 =p∑

i1,i2=0

ci1,i2 |i1〉 ⊗ |i2〉 , (2.2)

where p is the physical dimension of the two subsystems. Given anarbitrary quantum state |ψ〉we say that the wave function is entangled,if it is not a product state of the wave functions of the subsystems.

|ψ〉 ∈ H1 ⊗H2 entangled ⇐⇒ |ψ〉 6= |ψ1〉 ⊗ |ψ2〉 (2.3)

Otherwise the state is separable. Assuming a two-level system (p = 2),we denote |0〉 , |1〉 as the eigenvectors of the Pauli σz operator. Weconstruct the two possible states

|ψ〉 = 1√2(|0〉 |0〉+ |0〉 |1〉) = |ψ1〉 |ψ2〉

|φ〉 = 1√2(|0〉 |0〉+ |1〉 |1〉) 6= |ψ1〉 |ψ2〉 .

(2.4)

According to the definition in Eq. (2.3) we see that |φ〉 is an entangledstate, whereas |ψ〉 is a separable state. |ψ〉 can be written as

|ψ〉 = 1√2(|0〉 |0〉+ |0〉 |1〉) = |0〉 1√

2(|0〉+ |1〉) = |0〉 |+〉 , (2.5)

5

6 basic concepts

where |+〉 , |−〉 with |±〉 = 1/√2(|0〉 ± |1〉) are the eigenvectors of the

Pauli σx operator. Today, entanglement is regarded as a resource andpromotes powerful applications like quantum teleportation, quantumencryption and ultimately quantum computation [2].

2.1.1 Schmidt Decomposition

The Schmidt Decomposition is a decomposition of a bipartite system,from which we are able to define a measure of entanglement. Con-sider a bipartite quantum state |ψAB〉 ∈ HA⊗HB, with Hilbert spacedimensions dim(HA) = dA, dim(HB) = dB. Generally we can write

|ψAB〉 =∑

i,j

ψij |i〉A |j〉B . (2.6)

TheoremThere exist orthonormal states |α〉A for system A and orthonormalstates |α〉B for system B such that

|ψAB〉 =χ∑

α=1

λα |α〉A |α〉B , (2.7)

with non-negative Schmidt coefficients λα satisfying∑α λα = 1. The

Schmidt rank χ is the number of non-zero Schmidt coefficients, it isalso given by χ = min(dA,dB). |α〉A and |α〉B are called the Schmidtvectors. This theorem can be proven using Singular Value Decomposi-tion (SVD) [3].

The Schmidt decomposition of the states in Eq. (2.4) yields

|ψ〉 : χ = 1 λ = 1

|φ〉 : χ = 2 λ = 1/√2 ,

(2.8)

where the product state |ψ〉 has a Schmidt rank of χ = 1. In fact thisis true for all separable states. A measure of bipartite entanglementis therefore readily available. By looking at the Schmidt rank χ it ispossible to quantify bipartite entanglement in the system, the largerχ the more entangled is the state.

2.1.2 Von Neumann Entropy

The Von Neumann entropy is the extension of classical entropy con-cepts to quantum mechanics. For a pure state |ψAB〉 ∈ HA ⊗HB of abipartite system with reduced density matrices

ρA = trB (|ψAB〉〈ψAB|) , ρB = trA (|ψAB〉〈ψAB|) (2.9)

the Von Neumann entropy is defined as

S (ρA) ≡ −tr (ρA ln ρA) = S (ρB) . (2.10)

2.2 singular value decomposition 7

The reduced density matrix can be computed from the Schmidt de-composition of the bipartite pure state |ψAB〉

|ψAB〉 =χ∑

α=1

λα |α〉A |α〉B

ρA =χ∑α=1

|λα|2 |α〉A A〈α|

ρB =χ∑α=1

|λα|2 |α〉B B〈α|

. (2.11)

The rank of ρA and ρB are the same as well as their eigenvalue spec-tra, however the eigenvectors |α〉A and |α〉B are different. InsertingρA and ρB in Eq. (2.10) yields

S (ρA) = S (ρB) = −

χ∑

α=1

|λα|2 · ln |λα|

2 . (2.12)

One important property of the Von Neumann entropy is that it isupper bounded by the Schmidt rank

S (ρA) 6 logχ. (2.13)

The Von Neumann entropy is a continuous measure of bipartite en-tanglement and is therefore also called entanglement entropy. In thetwo limiting cases of separable states and maximally entangled statesit yields S = 0 and S = logχ, respectively.

2.2 singular value decomposition

The SVD is a mathematical tool in linear algebra to factorize real andcomplex matrices [4].

TheoremAny rectangular matrix M ∈ Cm×n of arbitrary size can be decom-posed into

M = UΛV† , (2.14)

where U ∈ Cm×m and V† ∈ Cn×n are unitary matrices and Λ ∈Rm×n is a diagonal matrix containing the singular values.

M

=

U Λ V†

Figure 2.1: Graphical representation of the SVD of an m×n matrix.

Usually the singular values are arranged in descending order

8 basic concepts

λ1 > λ2 > . . . > λr, where r = min(m,n). In this case Λ is unique,whereas U and V† are not. The SVD can be used to perform theSchmidt decomposition of a bipartite system. Therefore it has usefulapplications in the field of tensor networks, for instance to generatethe so-called canonical form of an MPS [5].

3T E N S O R N E T W O R K S

The field of quantum many-body systems probably comprises themost challenging problems in modern condensed matter physics. In-vestigations and insights on these systems essentially rely on numer-ical methods, which are available for various purposes. New numer-ical and theoretical techniques to solve problems in quantum many-body physics have been developed in the field of TNs in recent years.Besides its application in condensed matter theory, the field of ten-sor networks is rapidly increasing with promising approaches in dif-ferent areas, for instance the AdS/CFT correspondence in quantumgravity [6].The review on tensor networks is based on Ref. [5], which provides anexcellent introduction to the topic, and on Ref. [7]. Extended reviewscan be found in Ref. [8] and Ref. [9].

3.1 the idea of tensor networks

The tensor network formalism is a new approach to represent wavefunctions of quantum many-body systems in a sophisticated way.Usually a quantum state is specified by a weighted superpositionof states in a given basis. The number of coefficients necessary to de-scribe the state grows exponentially in the system size, which makesthis representation ineffective [5].It is more convenient to describe the wave function using the entan-glement structure between the constituents. The coefficients of thewave function are distributed among the tensors, thereby the quan-tum correlations are readily available in the network. This entangle-ment based approach can be an efficient representation of the wavefunction, since most of the physically relevant states live in a smallcorner of the exponentially large Hilbert space. This is especially true

A B S ∝ δA

Figure 3.1: The area-law states that the entanglement entropy for abipartite system scales with the boundary.

for eigenstates of local, gapped Hamiltonians that obey the area-law.The area-law states that the entanglement entropy S between two sub-

9

10 tensor networks

systems scales with the boundary between them. This is visualized inFig. (3.1). Hamiltonians that obey the area-law tend to act locally, e.g.between nearest-neighbour and next-to-nearest-neighbour sites, andthe low-energy states are heavily constrained by locality [5].In this chapter we will give a useful diagrammatic notation and in-troduce important families of tensor network states, such as MPS andPEPS.

3.2 tensor network theory

For our purposes, a tensor is defined as a multi-dimensional array ofcomplex numbers, whose rank is the number of its indices. A rank-0tensor is therefore a scalar (y), a rank-1 tensor is a vector (νµ) and amatrix is a rank-2 tensor (Aµν). We can introduce a more convenientdiagrammatic representation that gives us a powerful visualizationfor tensor networks. Elementary tensors up to rank-3 are depicted inFig. (3.2). The ball represents the tensor, its legs represent the indices

scalar vector

matrix rank-3 tensor

Figure 3.2: Diagrammatic notation of basic tensors.

of the tensor. For simplicity we assume that every index can takeup to D different values. An index contraction is the sum over allpossible values of all recurrent indices in a set of tensors. Consider asan example the fictitious operation on the network of four tensors

Fµνρσ =

D∑

α,β,γ,δ,τ,ω=1

AµαβγBανδβCγτωρEτδσω. (3.1)

Indices that are not contracted are called open indices, they determinethe rank and the legs of the new tensor after the contraction.

A B

A B

C E

Figure 3.3: Visualization of a matrix multiplication and the contrac-tion of four tensors.

3.3 tensor network description of the wave function 11

A practical example is the matrix multiplication

Cµν =

D∑

ρ=1

AµρBρν. (3.2)

Using the diagrammatic notation introduced previously we can illus-trate the two operations in Fig. (3.3). A tensor network is the contrac-tion of a set of tensors, with basic examples given above. Instead ofwriting equations for the contractions it is more convenient to use thepictorial representation, that is able to expose inherent properties ofthe tensors. A good example is the cyclic property of the trace of aproduct of matrices, depicted in Fig. (3.4).

Figure 3.4: Cyclic property of the trace of a product of matrices.

3.3 tensor network description of the wave function

We pick up the basic idea of tensor networks to explain their represen-tation of the wave function. Consider a quantum many-body state ofN particles with p as the physical degree of freedom of each subsys-tem. For instance, in a system of spin-1/2 particles we have p = 2. Thewave function of such a quantum many-body systems can generallybe written as

|ψ〉 =p∑

i1,i2,...,iN

ci1,i2,...,iN |i1〉 ⊗ |i2〉 ⊗ . . .⊗ |iN〉 , (3.3)

where |ir〉 is a basis for the single particles r = 1, . . . ,N. In this repre-sentation the coefficients ci1,i2,...,iN are pN complex numbers that canbe seen as the entries of a tensor C with N indices i1, i2, . . . , iN. Eachof the indices can take up to p different values so that the tensor Cis of rank N and has O(pN) coefficients. This implies an exponentialscaling in the size of the system.With regard to computational cost this is a very ineffective descriptionof the quantum many-body system that limits calculations to small N.With TNs we aim to reduce the complexity in the representation of thequantum state |ψ〉 by providing an accurate description of the entan-glement properties therein. The overall tensor C is decomposed intosmaller tensors with smaller rank according to the physical problem(refer to Fig. (3.5)). The reduction of complexity is attended with the

12 tensor networks

Ci1,i2,...,iN

i1i2 i3 i4

· · ·iN

a)

b)

c)

Figure 3.5: Decomposition of the tensor C in different tensor net-works. a) MPS with open boundary conditions, b) PEPS with openboundary conditions, c) arbitrary tensor network [5].

introduction of new degrees of freedom, represented by connected in-dices between the tensors. These indices are referred to as bond indices,they can take values from 1 up to the bond dimension D. Bond indicesrepresent the structure of the many-body entanglement in the quan-tum state |ψ〉 and they give a quantitative measure of the amount ofquantum correlations in the system. Hence they have an importantphysical meaning.A TN with a trivial bond dimension D = 1 corresponds to a productstate. In order to describe entangled quantum states it is necessaryto allow for larger bond dimensions. Still the TN may be an approx-imation of the many-body wave function |ψ〉 with an accuracy de-termined by the bond dimension D. The TN representation usuallyscales polynomial in both the system size and the bond dimension ofthe tensor network, which is the maximal value of all bond dimensions.Therefore it can provide an effective description of quantum many-body systems [5].Fig. (3.5) illustrates the decomposition of the tensor C into two impor-tant families of TN states. MPS provide a reliable tool for the simula-tion of gapped 1d quantum many-body systems, PEPS are the naturalgeneralization to higher spatial dimension.

3.4 entanglement entropy and area-law

Tensor networks are able to accurately capture the physical proper-ties of systems that obey the area-law for the entanglement entropy,as stated in Sec. 3. To illustrate the adherence of the area-law weconsider the entanglement entropy of a PEPS. In order to form a bipar-tition the system is divided into a block of length L and the environ-ment. The TN indices along the boundary are combined into an index

3.4 entanglement entropy and area-law 13

α = (α1α2 . . . α4L). Assuming a uniform bond dimension D alongthe boundary indices α the combined index α can take values up toD4L. The wave function can be expressed as a product of the inner

L

L in

out

|in(α)〉

α13

α14

α15

α16

α8

α7

α6

α5

α12α11

α10α9

α1α2

α3α4

|out(α)〉

α13

α14

α15

α16

α8

α7

α6

α5

α12α11

α10α9

α1α2

α3α4

Figure 3.6: Bipartition of a 6× 6 PEPS state in |in(α)〉 and |out(α)〉 fora 4× 4 block.

and outer wave function according to

|ψ〉 =D4L∑

α=1

|in(α)〉 ⊗ |out(α)〉 . (3.4)

The reduced density matrix of the inner block ρin = trout(|ψ〉〈ψ|) canbe written as

ρin =∑

α,α ′χα,α ′ |in(α)〉〈in(α ′)| , (3.5)

where χα,α ′ = 〈out(α ′)|out(α)〉. The reduced density matrix has arank of at most D4L, the same result holds true when we consider theouter part. The entanglement entropy of the L× L block is then upperbounded by the rank of ρin, so that we get

S(L) 6 4L log(D). (3.6)

Since the size of the boundary in Fig. (3.6) is 4L, this result is an up-per bound of the area-law for the entanglement entropy in 2d. It alsoexposes the amount of entropy each broken bond contributes, that isat most log(D).The area-law is a strong constraint on the quantum states. In fact, itrestricts the states to live in an exponentially small region of the ex-ponentially large Hilbert space. However, many of the Hamiltoniansin nature tend to have local interaction between the different parti-cles (e.g. nearest or next-to-nearest neighbour interaction). One canprove that low-energy eigenstates of gapped Hamiltonians with lo-cal interaction obey the area-law [10]. TNs therefore provide a goodframework for theoretical and numerical study of various physicalsystems [5].

14 tensor networks

Product statesArea-law states

Many-bodyHilbert space

Figure 3.7: Many physically relevant quantum states obey the area-law, which restricts the state to live in an exponentially small cornerof the Hilbert space.

3.5 important families of tensor network states

In the following section we will introduce the two most famous typesof tensor networks, which are MPS and PEPS.

Matrix Product StatesMPS are used to simulate 1d quantum many-body systems with meth-ods like Density Matrix Renormalization Group (DMRG) [11–13] andTime-Evolving Block Decimation (TEBD) [14]. The network consistsof a one-dimensional array of tensors with Open Boundary Condi-tions (OBC) or Periodic Boundary Conditions (PBC), as depicted inFig. (3.8) . The bond indices between neighbouring sites can take up

a)

A[1]

i1

A[2]

i2

A[3]

i3

A[4]

i4

A[5]

i5

A[6]

i6

A[7]

i7

b)

A[1]

i1

A[2]

i2

A[3]

i3

A[4]

i4

A[5]

i5

A[6]

i6

A[7]

i7

Figure 3.8: Pictorial representation of a MPS with a) OBC and b) PBC.

to D values, the physical degrees of freedom ir (r = 1, . . . ,N) cor-respond to the local dimension of the Hilbert space and can take upto p values. Thus there is one tensor for each site in the quantummany-body system. MPS obey the area-law, the entanglement entropyhere is simply bounded by a constant, S(L) ∼ constant, because a bi-partition always cuts a constant number of bond indices regardless ofthe block size L.Here we disregard well-known properties of MPS such as the canoni-cal form, translational invariant infinite MPS or the exact calculation ofexpectation values due to the focus on other TNs and refer to Ref. [5]for more details.

Projected-Entangled Pair StatesPEPS are the natural generalization of MPS to higher spatial dimen-sions. However they are essentially used to simulate 2d quantummany-body systems with PEPS [9] and infinite-PEPS [15] algorithms

3.6 fermionic tensor networks 15

or methods based on Corner Transfer Matrices (CTM) [16]. For eachlattice site of the quantum many-body system there is one tensor withbond dimension D and physical dimension p. The rank depends onthe lattice structure, e.g. rank-5 tensors for a square lattice. Choosing

a) b)

Figure 3.9: Illustration of a 4× 4 PEPS with a) OBC and b) PBC.

all tensors to be different results in a PEPS that does not have transla-tional invariance. However it is also possible to choose a fundamentalunit cell of tensors which is repeated all over the 2d lattice to form atranslationally invariant, infinite PEPS. Projected Entangled Pair Statescan also be defined for other lattice structures, such as the honeycombor kagome lattice.Besides these two prominent states the field of TNs offers a varietyof other methods to simulate quantum many-body systems. There isthe Tree Tensor Network (TTN) [17], Branching Multiscale Entangle-ment Renormalization Ansatz (MERA) [18], Tensor RenormalizationGroup (TRG) [19] or Second Renormalization Group (SRG) [20], toname a few examples.

3.6 fermionic tensor networks

The accurate simulation of fermionic systems in 2d quantum sys-tems is a difficult problem in condensed matter theory, different ap-proaches suffer from different limitations. For instance, the simula-tion of systems in higher spatial dimensions using quantum MonteCarlo methods suffer from the so-called sign problem [21]. However,since many fundamental particles in the standard model are fermionsthere is an elementary interest and demand in efficient simulationmethods.TNs are able to simulate fermionic systems in any dimension effi-ciently with no increase in the leading computational cost [5]. Thisis possible due to additional rules, that are enjoined on the tensors.We follow Ref. [22] for the fermionization of PEPS.

3.6.1 Fermionization Rules

The fermionization procedure originates from the second quantiza-tion formalism for fermions and its consequences on TNs. Here wewill assume a lattice system with a complex vector space V of finite

16 tensor networks

dimension d = 2 for every site, that is associated to fermionic creationand annihilation operators. The anticommutation relation

ck , c†k ′

= ckc

†k ′ + c

†k ′ck = δk,k ′ (3.7)

motivates the necessary changes in the TN, for details refer to Ref. [22].Hamiltonians corresponding to fermionic systems preserve the parityof the fermionic particle number, which we refer to as parity. This meansthat fermions can only be created or annihilated in pairs. The singleparticle vector space V naturally decomposes into

V = V(+) ⊕ V(−) , (3.8)

a direct sum of an even parity subspace V(+) and an odd parity sub-space V(−). In order to transform a PEPS for bosonic systems to a PEPS

for fermionic systems, we need to apply the two following rules.

1. Parity-Symmetric TensorsThe Z2 symmetry of the overall parity of the fermions (even/odd) isincorporated into the tensors, this is visualized in Fig. (3.10).

TP

P P

P

P

=

T

Ti1,i2,...,iM= 0 if P(i1)P(i2) . . . P(iM) 6= 1

Figure 3.10: Introduction of parity-symmetric tensors, where P is arepresentation of the Z2 symmetry operator.

2. Swap Tensors for CrossingsEvery crossing in the planar representation of the fermionic TN getsreplaced by the two-body swap gate, that swaps the fermions andassures the anticommutation relation when two lines with odd parityget exchanged. This needs to be done for both physical and virtualbond indices, since they both represent fermionic degrees of freedom.

i1 i2

i2 i1

X

i1 i2

j1 j2

Xi1,i2,j1,j2 = δi1j2δi2j1S (P(i1)P(i2))

(P(i1)P(i2)) =

−1 if P(i1) = P(i2) = −1+1 otherwise

Figure 3.11: Introduction of a swap tensor for every crossing.

3.6 fermionic tensor networks 17

These two rules don’t enlarge the leading computational cost, so thatfermionic TNs can be efficiently contracted.

3.6.2 Basic Examples

As a simple example we consider the ambiguity of the graphical rep-resentation of TNs, depicted in Fig. (3.12). For bosonic systems thetensor network is unchanged when a line is dragged over a tensor.This jump move however alters the tensor network for fermionic sys-

a) b)

Figure 3.12: Ambiguousness in the TN representation illustrated bythe jump move for a) bosonic tensors and b) fermionic tensors.

tems, due to the need of additional swap gates. Another importantdifference exhibits the TN itself. Consider a wave function |ψ〉, whosecoefficients ci1,...,iN are arranged according to the 3 × 3 PEPS withopen boundary conditions in Fig. (3.13). In contrast to the bosonic

a) b)

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Figure 3.13: Representation of the wave function |ψ〉 ∈ V⊗9 in termsof a PEPS for a) bosonic and b) fermionic states.

state, the fermionic state depends on the labeling of the physical in-dices as well as on the perspective. This combined determines thenecessary swap gates that need to be included for every crossing offermionic lines.

General RemarkAll numerical calculations for the TNs described in this thesis wereperformed using MATLAB. We rely on the convenient routine ncon(),a tensor network contractor provided in Ref. [23].

Part II

E X A C T T E N S O R N E T W O R K S TAT E S

4T H E S P E C T R A L T E N S O R N E T W O R K

In this section we present a special type of tensor network, the so-called Spectral Tensor Network (STN). It was introduced by AndrewFerris in 2014 [24], however the idea appeared earlier in Ref. [25].Unlike the previously introduced MPS and PEPS, the STN is able torepresent quantum many-body systems with local entanglement thatis greater than the area-law, yet it is efficiently contractible. The STN

leverages the decomposability of the Fast Fourier Transformation (FFT)to represent translational invariant systems of free fermions in ar-bitrary dimensions and it covers one-dimensional systems that aresolved by the Jordan-Wigner transformation. The name originatesfrom the imitation of a spectral transformation, that includes thequantum fast Fourier transform as a trivial example.

4.1 quantum fast fourier transformation

The unitary circuit of the Fourier Transformation (FT) builds up theconnection between a quantum system in real space and in momen-tum space. It decomposes into a series of sparse operations. This me-thod is generally referred to as FFT which manifests a logarithmicdepth circuit and two-body gates at most. We start by denoting that afermionic FT over N sites can be decomposed into two parallel trans-formation over N/2 sites according to

c†k =

1√N

N−1∑

x=0

e2πikxN c†x

=1√N

N/2−1∑

x ′=0

e2πikx ′N/2 c

†2x ′ +

1√N

e2πikN

N/2−1∑

x ′=0

e2πikx ′N/2 c

†2x ′+1 .

(4.1)

The right-hand side is a FT over the subset of even and odd sites re-spectively. This sum is a linear operation, that can be implementedby only two and one-body gates. The decomposition for a unitaryquantum FT is shown in Fig. (4.1), where the overall transformation isdecomposed into smaller ones repeatedly until only two-body gatesand one-body phase gates are necessary. The permutation at the bot-tom of the circuit is given by a bit-reversal operation that restoresthe correct order of the modes. The entangling two-body gate de-picted above is an essential tensor and it represents the "circuit" for

21

22 the spectral tensor network

F8 =

ω08 ω1

8 ω28 ω3

8

ω04 ω1

4 ω04 ω1

4

Figure 4.1: Unitary circuit for the quantum Fourier transformationfor eight sites using only two-body "beam splitter" gates and "phasedelays". The part above the circuit corresponds to momentum space,the part below to the real space.

the Fourier transformation of only two modes. According to Eq. (4.1)the transformation of two modes is given by

c† ′0 =

1√2

(c†0 + c

†1

)c† ′1 =

1√2

(c†0 + eiπc†1

). (4.2)

We will analyze the action of this operation on the quantum state liv-ing in the Hilbert space, in which the rearrangement of the wave func-tion coefficients is described by the transformation A ′ = UA. There-fore we have

|ψ〉 =1∑

i,j=0

A ′ij(c† ′0

)i (c† ′1

)j|00〉 (4.3)

=

1∑

i,j=0

(UA)ij

[1√2

(c†0 + c

†1

)]i [ 1√2

(c†0 + eiπc†1

)]j|00〉 .

After an expansion of the last formula we get the action of the two-body gate on the quantum state. It is given by the 4× 4 unitary matrixU, which we will further denote as

F2 =

1 0 0 0

0 1/√2 1/

√2 0

0 1/√2 −1/

√2 0

0 0 0 −1

(4.4)

in the occupation number basis |00〉 , |01〉 , |10〉 , |11〉 for two fermions.The last entry accounts for the anticommutation relation and induces

4.1 quantum fast fourier transformation 23

a negative sign if two occupied modes are exchanged. The one-bodyphase delay includes the correct twiddle factor exp(2πik/N) and iswritten as ωkN = Z2k/N, with Z being the usual Pauli matrix. The cir-cuit in Fig. (4.1) is the Fourier transformation for a one-dimensionalsystem of eight sites. Using the self-similar structure it can be ex-tended to any number of sites N = 2n.Generally the procedure can be extended to two- and even higherdimensional systems. In such cases the Fourier transformation alongeach dimension is applied sequentially along parallel wires. To givean example, we consider the quantum Fourier transformation ona 4 × 4 square lattice. The circuit F4 is repeated four times alongthe x and y direction respectively. In order to maintain a clearerstructure, the twiddle factors are absorbed into the two-body gates,which results in the overall two-dimensional transformation shownin Fig. (4.2). Since the tensor network is used to describe fermionic

Figure 4.2: Tensor network for the Fourier transformation of fermionson the 4× 4 lattice, realized by separate transformations in x and ydirection and a bit-reversal permutation at the bottom.

systems, anticommutation relations have to be respected for all op-erations. In particular we have to follow the fermionization rules fortensor networks (refer to Sec. 3.6) and include swap tensors for everycrossing in the Fourier transformation.

24 the spectral tensor network

4.2 tensor network structure

With the fermionic Fourier transformation being the basis of the STN

it is convenient to consider a system of N = 2n fermions on a chain.The quantum state can be described by the second-quantization for-malism as the Fock state

|ψ〉 = |n1n2 . . . nN〉 =(c†1

)n1 (c†2

)n2. . .(c†N

)nN|0〉 . (4.5)

The spectral tensor network is the application of the inverse fermionicFourier transformation to an (unentangled) wave function in momen-tum space, resulting in a (highly entangled) wave function in realspace [24]. An example of a structure for sixteen sites can be seen inFig. (4.3), where the quantum state emerging is called spectral tensornetwork state. The structure relates to the amount of entanglement

Figure 4.3: Spectral tensor network structure for a system of sixteenoriginally unentangled fermions in momentum space, denoted by theblue circles. At each layer the two-body tensor F2 (including twiddlefactors) entangles two initially unentangled regions.

in the system, the initially unentangled blue momentum modes be-come more and more entangled the more layer of gates are applied.All twiddle factors have been absorbed into the two-body gates andfermionization rules have to be followed.

4.3 spin systems and jordan-wigner transformation

The STN is also useful to study 1d spin systems, which can be trans-formed to non-interacting fermionic models using the Jordan-Wignertransformation [26]. Consider a system of N spin-1/2 living on thesites of a one-dimensional lattice. We choose the σz basis, which isgiven by |↑〉 , |↓〉. Since there are only two possible spin alignments,we can map the spin system to a system of spinless fermions, alsoreferred to as hard-core bosons. They are described by the fermionicoperators c†, c in the basis |0〉 , |1〉. A naive choice for the mappingof the states would be

|↓〉 → |0〉 |↑〉 → |1〉 , (4.6)

4.3 spin systems and jordan-wigner transformation 25

which has a disadvantage. Despite the fact that it does introducefermionic anticommutation relation on-site, it introduces bosoniccommutation relation for the operators off-site. This would imply thenegligence of the ordering and is therefore not a correct mapping.We can make this intuitively clear. The spin state |↑↑〉 = σ+1 σ

+2 |00〉

does not depend on the ordering of the operators, because they com-mute. In contrast, the fermionic state |11〉 = c

†1c†2 |00〉 depends on a

preselected convention. Changing the order of the operators leads toa negative sign, due to the anticommuting behaviour. Therefore weneed to include a so-called Jordan-Wigner string into the transforma-tion, that introduces a negative sign for every site left to site i, that isoccupied. We shall see the application in the following section.The Jordan-Wigner transformation often leads to quadratic Hamilto-nians with anomalous terms, that require an additional Bogoliubovtransformation coupling opposite momentum modes to be solved [27].This needs to be included in the STN structure as a separate transfor-mation on top. The initial fermionic modes are incorporated into the

ω08 ω1

8 ω28 ω3

8

ω04 ω1

4 ω04 ω1

4

ω016 ω1

16 ω216 ω3

16 ω416 ω5

16 ω616 ω7

16

Figure 4.4: Spectral Tensor Network with Bogoliubov transformationon top to represent eigenstates of translationally invariant Hamiltoni-ans with anomalous terms. Additional half-integer momenta need tobe included after the FT.

gray tensors alongside with the Bogoliubov transformation itself. Ad-ditional half-integer momenta need to be included in the networkafter the FT has been applied [24]. This is accounted for by the lastrow of twiddle factors ωi2N.

5I S I N G M O D E L W I T H T R A N S V E R S E F I E L D

The one-dimensional Ising model in a transverse field (TFIM) is a well-known theoretical model, first solved by Pfeuty in 1970 [28]. It is oneof the simplest and most intuitive problems in quantum many-bodyphysics and will serve our purpose as a toy model, in a sense that weconstruct the eigenstates in terms of the STN. The construction will beextended to the Kitaev honeycomb model later on.

5.1 solution of the model

The one-dimensional quantum Ising model is a model of spin-1/2residing on a chain. Using periodic boundary conditions, the chainbends to a closed circle of particles. The spins couple through a fer-romagnetic exchange along the x axis. Further, we consider the Isingmodel with a transverse magnetic field in z direction, so that theHamiltonian is given by

H = −J

L∑

i=1

σxi σxi+1 − h

L∑

i=1

σzi , (5.1)

where σxL+1 = σx1. The model provides two limits, the strong fieldlimit and the zero field limit. In the strong field limit h J, the spinsalign in parallel along the z axis and the ground state is the fullypolarized state |↑↑ . . . ↑〉. For the complementary case h = 0, thatcorresponds to the absence of a magnetic field, the ground state is aparamagnet, with all the spins aligned along the x axis. The groundstate is two-fold degenerate and given by |→→ . . .→〉 or |←← . . .←〉.

6

5

4

3

2

1

L

Figure 5.1: Visualization of the one-dimensional spin model with PBC.

The Hilbert space for one spin-1/2 is of dimension two, the Paulimatrices σα for α = x,y, z satisfy anticommutation rules at one siteand commutation rules for different sites. In order to obtain a model

27

28 ising model with transverse field

with fully anticommuting spinless fermions, we have to use a non-local Jordan-Wigner transformation, given by

ci =∏

j<i

e+iπa†jaj(σxi − iσ

yi

)(5.2)

c†i =

(σxi + iσ

yi

)∏

j<i

e−iπa†jaj . (5.3)

The ladder operators are given by σ±i =(σxi ± iσ

yi

), so that we can

express σzi in terms of fermionic creation and annihilation operatorsas σzi = 2c

†ici − 1. In order to transform the Hamiltonian it is conve-

nient to rewrite the elements in the Jordan-Wigner string accordingto

exp(±iπc†jcj

)=(1− 2c†jcj

)= −σzj , (5.4)

which can be easily verified using series expansion of the exponentialfunction. The Jordan-Wigner strings in Eqs. (5.2) and (5.3) count thenumber of occupied sites to the left to site i in order to introducethe correct anticommuting behaviour. The phase therefore dependson whether the number of occupied sites is even or odd. The termsin the Hamiltonian become

σxi σxi+1 =

(c†i + ci

) i−1∏

j=1

e+iπa†jaj

(c†i+1 + ci+1

) i∏

j=1

e+iπa†jaj

=(c†i + ci

)e+iπa

†iai

(c†i+1 + ci+1

)

=(c†i − ci

)(c†i+1 + ci+1

)(5.5)

for 1 6 i 6 L− 1. Those terms include only nearest-neighbour interac-tions and the Jordan-Wigner string reduces to a simple form. For siteN, where the periodicity comes in, we have to pay attention becausehere the Jordan-Wigner transformation is non-local. The transforma-tion of the boundary term yields

σxLσx1 =

(c†L + cL

) L−1∏

j=1

e+iπa†jaj

(c†1 + c1

)

=(c†L + cL

) L−1∏

j=1

e+iπa†jaj

(e+iπa

†LaL

)2 (c†1 + c1

)

=(c†L − cL

) L∏

j=1

e+iπa†jaj

(c†1 + c1

)

= −(c†L − cL

)(c†1 + c1

) L∏

j=1

e+iπa†jaj .

(5.6)

5.1 solution of the model 29

We will define the last product that runs from site 1 to site L andintroduces a negative sign for each site that is occupied as an operator

P =

L∏

j=1

e+iπa†jaj . (5.7)

Since P2 = 1, this operator can only have the two eigenvalues ±1,corresponding to

P =

+1 for N even

−1 for N odd,(5.8)

where N =∑Lj=1 c

†jcj is the total number of fermions. The parity

operator P commutes with the Hamiltonian, so that the Hilbert spaceis a direct sum H = H+⊗H− of two subspaces that contain the stateswith P = +1 and P = −1 respectively. The full Hamiltonian now reads

H = −h

L∑

i=1

(2c†ici − 1

)− J

L∑

i=1

(c†i − ci

)(c†i+1 + ci+1

)

+J[(c†L − cL

)(c†1 + c1

)]L∏

j=1

e+iπa†jaj + 1

(5.9)

For large L we can omit the correction term in the second row so thatthe Hamiltonian is quadratic in the fermionic operators. To solve theHamiltonian we use a discrete Fourier transformation that exploitsthe translational invariance of H and takes the operators to momen-tum space. We define

cj =1√L

∑

k

e2πijkL ck (5.10)

c†j =

1√L

∑

k

e2πijkL c†-k , (5.11)

where the second equation follows from taking the adjoint of the firstone and the substitution k→ −k. The complete set of wave vectors kis given by

k = −(L− 1)/2, . . . ,−1/2,+1/2, . . . ,+(L− 1)/2 for L even

k = −L/2, . . . , 0, . . . ,+L/2 for L odd ,

but since we work with a discrete quantum Fourier transformation,we may as well restrict the analysis to system sizes of L = 2n, wheren ∈N. The Fourier transformation takes the Hamiltonian to momen-tum space where it is given by

H = hL− 2h∑

k

c†kck − J

∑

k

e2πikL

(c†k − c-k

)(c†-k + ck

). (5.12)


Due to the splitting of the exponential function in real and imagi-nary part one obtains the dispersion relation for the model and theHamiltonian can be written as

H =∑

k

[εkc†kck +

1

2i(∆kc

†kc†-k −∆kc-kck

)]+ hL ,

εk = −2J cos(2πk

L

)− 2h ,

∆k = −2J sin(2πk

L

).

(5.13)

Again it is easy to see that H is quadratic in the fermionic opera-tors, however it is not diagonal yet. A subsequent unitary Bogoliubovtransformation couples modes with opposite momentum to a newquasiparticle operator according to

γk = ukck + ivkc†-k

γ†k = ukc

†k − ivkc-k .

(5.14)

With the right choice of the parameters uk and vk the Hamiltoniangets into the diagonal form

H =∑

k

Ek

(γ†kγk −

1

2

), (5.15)

where Ek is the quasi-particle energy. The 1d Ising model with trans-verse field is now fully reduced to a quadratic model of free fermionsand therefore it is solved. The steps in this brief derivation will reoc-cur in chapters later on, where they will be worked out in details.

5.2 representation of the eigenstates

In the following section we want to present the tensor network forthe ground state of the one-dimensional Ising model. Alongside wegive the exact form and transformations in the Hamiltonian to checkthe intermediate steps during the reverse flow. Let us start with theproduct state in momentum space, which yields the eigenstate of thediagonal Hamiltonian. Here, all Bogoliubov modes γi are initialized

−3π4

−π4

+π4

+3π4

= |ψ〉Bogoliubov

Figure 5.2: Eigenstate of the diagonal Hamiltonian

with the vector |0〉 = (0, 1)> to generate the ground state. In this step

5.2 representation of the eigenstates 31

there is no entanglement at all, so the overall quantum state is givenby |ψ〉 = |0〉⊗L, this is an eigenstate of the Hamiltonian

H =∑

k

√ε2k +∆

2k

(γ†kγk −

1

2

). (5.16)

Each mode is associated with one momentum, however the quan-tum state does not yet depend on any actual value of k. In the nextstep the Bogoliubov transformation is applied, which creates entan-glement and brings the Hamiltonian back into a non-diagonal formin momentum space. The transformation on the states in the Hilbertspace is given by

Bk =

uk 0 0 vk

0 1 0 0

0 0 1 0

−v∗k 0 0 u∗k

. (5.17)

the matrix is reshaped into a two-body tensors coupling modes withopposite momentum. Therefore, the modes have to be reordered so

−3π4 −π

4 +π4 +3π

4

Bk Bk = |ψ〉Momentum Space

Figure 5.3: Eigenstates of the Hamiltonian in momentum space. Thenext step is to undo the quantum Fourier transformation.

that they form pairs. This is done by permuting them with respect tothe anticommutation relation, which we account for in the swap ten-sors that are included for every crossing. After the Bogoliubov trans-formation we have to restore the original ordering of the fermionicmodes. We checked that the quantum states are eigenstates of theHamiltonian in Eq. (5.13).The last step to obtain the ground state for the spin Hamiltonian isthe (reverse) fermionic Fourier transformation. Fig. (5.4) illustratesthe circuit for four modes, where only two-body Fourier gates andone-body phase gates are needed. Due to the Fourier transformationwe need to include a bit-reversal operation at the end of the tensor


|ψ〉Momentum Space

ω04 ω1

4

ω08 ω1

8 ω28 ω3

8

= |ψ〉Fermion

Figure 5.4: Tensor network to construct the eigenstates of the one-dimensional quantum Ising model. The bit-reversal operation at theend is necessary to restore the correct ordering of the fermionicmodes.

network to restore the correct mode ordering. After this step we havegenerated the ground state as well as all excited states in the chosenparity sector for the model. We have to use antiperiodic boundaryconditions to effectively simulate the model which was solved as ifit was infinite. Considering the Hamiltonian, a Jordan-Wigner trans-formation converts the fermionic Hamiltonian into the original spinHamiltonian. However, there is no action on the coefficients of thewave function, so that the eigenstates of the Ising model are given by

|ψ〉Spin = |ψ〉Fermion . (5.18)

Finally, in order to verify the tensor network eigenstates, we have toadapt the Hamiltonian for the spin state to our analysis. This is dueto the fact that the model was solved analytically as if it was infinite,but the tensor network structure is built up for a model with PBC.Therefore we have to include a Jordan-Wigner string for the boundaryterm, which then cancels out when actually performing the Jordan-Wigner transformation. This leads to the modified Hamiltonian

H = −J

L∑

i=1

σxi σxi+1 − Jσ

xLσx1

L∏

j=1

e+iπa†jaj − 1

− h

L∑

i=1

σzi , (5.19)

for which the TN construction is exact.

Part III

T H E K I TA E V H O N E Y C O M B M O D E L

6T H E K I TA E V H O N E Y C O M B M O D E L

The honeycomb model describes a frustrated spin system withspin-1/2 residing at the sites of a two-dimensional honeycomb lat-tice. It was originally introduced by Kitaev in 2006 in the context oftopological quantum computation [29]. The lattice has connectivitythree, which means that each site is connected to three neighbour-ing sites. Additionally, the lattice is bi-colorable, which can be seenin Fig. (6.1) and from the fact, that the honeycomb lattice is built fromtwo overlapping triangular sublattices. Equivalently to the hexagonalshape, the model can be defined on a brick wall lattice, which is atopologically equivalent deformation of the honeycomb lattice.

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Figure 6.1: Original honeycomb lattice and topologically equivalentbrick wall lattice. The nearest-neighbour spin-spin interaction de-pends on the orientation of the bonds.

Interaction between the spins is along three different bonds x, y andz, whose coupling strengths can be fully anisotropic. The model isthen described by the Hamiltonian

H = −Jx∑

x-links

σxi σxj − Jy

∑

y-links

σyi σyj − Jz

∑

z-links

σziσzj , (6.1)

with σαi as the α-Pauli matrix at site i. The physics is symmetric inthe permutation of the coupling strength parameters Jx, Jy and Jz.Even tough there are only nearest-neighbour interaction terms in theHamiltonian, the model offers a variety of interesting phenomena, in-cluding different topological phases supporting abelian, as well asnon-abelian anyons. Due to the anisotropic interaction the Kitaevmodel is a frustrated spin system, because a spin can not satisfy con-flicting demands of orientation from its three neighbouring sites. An-other important feature is the presence of infinitely many conservedquantities, which makes the model exactly solvable. Those quantitiesare associated with the plaquettes p of the lattice.

35

36 the kitaev honeycomb model

2

3

4

5

6

1

p

x

σx

z

σz

y

σy

x

σx

z

σz

y

σy

Figure 6.2: Illustration of the plaquettes of the honeycomb lattice. Theconserved quantity Bp is defined as a product of Pauli operatorsaround p, where the operator for every site is associated with theoutgoing bond.

For every plaquette there is an operator Bp, that commutes with theHamiltonian. In relation to Fig. (6.2) we define the operator as theproduct of Pauli operators around p, which is

Bp = σy1σz2σx3σy4σz5σx6 . (6.2)

These quantities play an important role for the analysis of the model.It is easy to verify that the operators commute with the Hamiltonianand among themselves, which implies that they are conserved. There-fore we have

[H,Bp] = 0 ∀p[Bp,Bp ′

]= 0 ∀p,p ′ .

(6.3)

Since B2p = 1, the eigenvalue of these plaquette operators can onlytake the values±1. The honeycomb model can be solved exactly usingdifferent techniques. Kitaev originally solved the model by express-ing the spin-1/2 operators in terms of four Majorana fermions andprojecting out redundant degrees of freedom. We will briefly presentthis solution in the following section. To get around the projection themodel can also be solved by a Jordan-Wigner transformation [30] ora mapping to hard-core bosons [31].

6.1 fermionization of spin-1/2 operators

We follow Kitaev’s solution and start the fermionization of the modelby assuming that every lattice site has two fermions, described bythe second quantization formalism via a†i , ai . The four-dimensionalFock space is spanned by the states |00〉 , |01〉 , |10〉 , |11〉, wherethe first position is the occupation number for fermion 1 and the

6.1 fermionization of spin-1/2 operators 37

second position the occupation number for fermion 2 respectively.For each site we construct four Majorana fermions according to

c =(c†1 + c1

)(6.4)

cx = i(c†1 − c1

)(6.5)

cy =(c†2 + c2

)(6.6)

cz = i(c†2 − c2

). (6.7)

For Majorana fermions we get the important relationscα, cβ

= 2δαβ cα = cα† α,β = x,y, z , (6.8)

which follow from the realness of Majorana fermions and the anti-commutation relation for complex fermions. Further, the original spinoperators σx, σy and σz are expressed using the four introduced Ma-jorana operators according to

σx = icxc (6.9)

σy = icyc (6.10)

σz = iczc . (6.11)

Any two of those operators σα, σβ anticommute for α 6= β, butthis does not complete the Pauli spin algebra. Due to the conditionσxσyσz = i we have the constraint cxcyczc = 1. We will further useD = cxcyczc, which can be written as

D =(2c†1c1 − 1

)(2c†2c2 − 1

)(6.12)

and has the eigenvalue +1 only for two of the four states, namely |00〉and |11〉. For the other two states |01〉 and |10〉 the eigenvalue is −1,corresponding to the two unphysical states, since they do not meetthe imposed constraint. Therefore we need a projector

Pi =1+Di2

, (6.13)

that projects out the two unphysical states and reduces the previouslyenlarged Hilbert space.

6.1.1 Quadratic Hamiltonian

The Hamiltonian can be expressed in terms of the four Majoranafermions, which have been introduced before. Inserting Eqs. (6.9) -(6.11) into the spin Hamiltonian yields

H = Jx∑

x-links

(icxi c

xj

)icicj + Jy

∑

y-links

(icyi cyj

)icicj

+Jz∑

z-links

(iczic

zj

)icicj ,

(6.14)


where site i and site j correspond to the even and odd sublattice re-spectively. At first sight the Hamiltonian is quartic in the Majoranafermion operators. However, each operator in parentheses commuteswith the Hamiltonian and among themselves. Therefore they repre-sent conserved quantities, denoted by

uαi,j = i(cαi c

αj

)for α = x,y, z . (6.15)

The Hamiltonian in Eq. (6.14) is therefore effectively quadratic in theMajorana operators. It is easy to verify that ui,j = −uj,i and u2i,j = 1,so that the eigenvalues of ui,j can only take the values ±1. We writethe Hamiltonian in terms of the conserved quantities according to

H = Jx∑

x-links

uxi,jicicj + Jy∑

y-links

uyi,jicicj + Jz

∑

z-links

uzi,jicicj . (6.16)

In this form we see that H describes a tight binding model of Majo-rana fermions hopping along the lattice, where the hopping matrixelements are coupled with gauge fields uαi,j on every bond.

ccxcy

cz

ccx cycz

uyi,j

i

j uxj,k

k

l

uzj,l

Figure 6.3: Graphical representation of the Hamiltonian in Eq. (6.16)with Majorana fermions and conserved quantities uαi,j.

For every plaquette p we define the vortex operator in the extendedHilbert space according to

Wp =∏

(i,j)∈∂puα(i,j)i,j where

i ∈ white sublattice

j ∈ black sublattice, (6.17)

so that ∂p denotes the boundary of the plaquette. This operator re-duces to the previously introduced form of the operator Bp in Eq. (6.2)in the physically relevant space, which can be seen by

Wp = ux12uy32u

z34u

x54u

y56u

z16

= −(i)6cx1cx2cy2cy3cz3cz4cx4cx5cy5cy6cz6cz1

= σy1D1σz2D2σ

x3D3σ

y4D4σ

z5D5σ

x6D6 .

(6.18)

6.1 fermionization of spin-1/2 operators 39

Here we used icxj cyj = −σzjDj and cyclic permutations. In the physi-

cal subspace with Dj = 1 for all j we recover the plaquette operatorBp. All Bp operators are gauge invariant objects, which means thatthey commute with the projection operator Pj. In contrast, the newconserved quantities ui,j do not commute with Pj.According to the choice of the set of operators ui,j along the differ-ent links, we get a distinct free fermion Hamiltonian H(u). A subse-quent diagonalization of H(u) leads to the spectrum and the groundstate energy for each distribution. The question arises for which con-figuration of u there is a global minimum leading to the actual low-est ground state energy for the model. This was answered in a the-orem by Lieb [32], that states that the ground state has to be in thesector where Bp = 1 for every plaquettes, achievable by fixing ui,j = 1for every link. Since the operators Bp are the only gauge invariant ob-jects, every fixing for u that leads to a certain configuration of Bpare equivalent.The ground state is easily obtained for the uniform choice ui,j = 1

in the extended space. In order to get the true ground state in thephysical space we have to project out the unphysical states. This isachieved by the global projector P

|ψ〉0 =∏

i

Pi |ψ〉extended =∏

i

(1+Di2

)|ψ〉extended . (6.19)

6.1.2 The Ground State

We now solve the Hamiltonian in Eq. (6.16) with the uniform choiceof ui,j = 1. This case corresponds to the absence of any vortices, hencethe ground state lies in the vortex free sector which was already statedby Lieb [32]. The Hamiltonian reduces to the translational invariantform

H = Jx∑

x-links

icicj + Jy∑

y-links

icicj + Jz∑

z-links

icicj . (6.20)

It is convenient to solve it in momentum space due to the translationinvariance. Therefore we define the Fourier transformation for theMajorana fermions according to

ci,λ =1√NxNy

∑

k

eikrck,λ , (6.21)

where λ denotes the sublattice index and Nx and Ny is the numberof unit cells along the n1 and n2 direction respectively. Here we haven1 = (+12 ,

√32 ) and n2 = (−12 ,

√32 ) in xy-coordinates.

Note that the property c†i = ci leads to ck = c†-k. The possible valuesfor k = (kx,ky)> are given by

ki =2πniNi

ni = −Ni − 1

2, . . . ,+

Ni − 1

2. (6.22)


unitcell

n1n2

Figure 6.4: Unit cell and basis vectors of the honeycomb lattice.

Inserting the Fourier transformation of the operators into Eq. (6.20)leads to the Hamiltonian in momentum space, which now has theform

H =∑

k

(c†k,a c

†k,b

)( 0 ifk

−if∗k

)(ck,a

ck,b

). (6.23)

In order to diagonalize the Hamiltonian we have to diagonalize theoff-diagonal square matrix, that contains the spectral function

fk = 2(Jxe−ikx + Jye−iky + Jz

). (6.24)

This can be easily achieved by a suitable unitary transformation, thatmaps the Majorana operators to new quasi-particle operators γk. TheHamiltonian then reads

H =∑

k

Ek

(γ†kγk −

1

2

), (6.25)

with the energy Ek = |fk| to create one quasi-particle excitation.Studying the dispersion relation one can come up with the phasediagram of the model, as done originally by Kitaev [29]. We will givea review over the phase diagram in Sec. 6.3 after presenting a secondsolution of the model.

6.2 fermionization by a jordan-wigner transformation

In the following section we will present a solution of Kitaev’s modelon the honeycomb lattice based on the Jordan-Wigner transforma-tion. This transformation expresses spin operators in terms of newfermionic operators in a generally non-local way. The procedure isstill often applicable for one-dimensional spin systems involving lo-cal spin-spin interaction, which results in Hamiltonians that are easyto solve. A generalization of the Jordan-Wigner transformation tohigher dimensions has been suggested [33]. However the Kitaev hon-eycomb model represents a special model due to its topology. In the

6.2 fermionization by a jordan-wigner transformation 41

form of the topologically equivalent brick wall lattice it is built up byhorizontal chains connected by vertical bonds. The topology and thepreservation of XX, YY and ZZ type bonds allow us to use the one-dimensional Jordan-Wigner transformation to solve the model on thetwo-dimensional lattice.The original method by Kitaev expressed the spin at a given site usingfour Majorana operators or equivalently two complex fermions. Thisleads to a local Fock space dimension of four instead of two for theoriginal spin. The Jordan-Wigner transformation does not have thisdisadvantage, it maps the Hilbert space of spins to the Hilbert spaceof spinless complex fermions, also called hard-core bosons. Recallingthe definition in Sec. 6 the original Hamiltonian is given by

H = −Jx∑

x-links

σxi σxj − Jy

∑

y-links

σyi σyj − Jz

∑

z-links

σziσzj . (6.26)

Note that the Pauli spin matrices anticommute at the same site butcommute at different sites. We define the following Jordan-Wignertransformation, that acts on all spins along each row to span thewhole two-dimensional lattice.

σ+i,j = 2

∏

j ′<j

∏

i ′σzi ′,j ′

[∏

i ′<i

σzi ′,j

]a†i,j

σ−i,j = 2

∏

j ′<j

∏

i ′σzi ′,j ′

[∏

i ′<i

σzi ′,j

]ai,j

σzi,j = 2a†i,jai,j − 1

(6.27)

Here we use the two spin ladder operators σ± = (σx ± iσy) and theindices label row and column (refer to Fig. (6.5)). With this transfor-mation defined the individual terms in the Hamiltonian transformaccording to the following expressions.

σxi,jσxi+1,j =

∏

i ′<i

σzi ′,j

(a†i,j + ai,j

) ∏

i ′<i+1

σzi ′,j

(a†i+1,j + ai+1,j

)

=(a†i,j + ai,j

)σzi,j

(a†i+1,j + ai+1,j

)

= −(a†i,j − ai,j

)(a†i+1,j + ai+1,j

)

σyi−1,jσ

yi,j = −

∏

i ′<i−1

σzi ′,j

(a†i−1,j − ai−1,j

)∏

i ′<i

σzi ′,j

(a†i,j − ai,j

)

= −(a†i−1,j − ai−1,j

)σzi−1,j

(a†i,j − ai,j

)

=(a†i−1,j + ai−1,j

)(a†i,j − ai,j

)

σzi,jσzi,j+1 =

(2a†i,jai,j − 1

)(2a†i,j+1ai,j+1 − 1

)(6.28)


The one-dimensional Jordan-Wigner transformation can be performedalong each row of the honeycomb lattice, as indicated in Fig. (6.5).Therefore it only affects XX− YY chains and all expressions remain

Figure 6.5: Path of the one-dimensional Jordan-Wigner transforma-tion to span the whole two-dimensional honeycomb lattice. Boundaryterms can be neglected due to the infinite dilatation.

interaction between nearest neighbours, so that the resulting termsare local. The σzσz interaction is local anyway, hence the HamiltonianH has the form

H =+ Jx∑

x-links

(a†i,j − ai,j

)(a†i+1,j + ai+1,j

)

− Jy∑

y-links

(a†i−1,j + ai−1,j

)(a†i,j − ai,j

)

− Jz∑

z-links

(2a†i,jai,j − 1

)(2a†i,j+1ai,j+1 − 1

),

(6.29)

where i+ j is even. The Jx and Jy terms give quadratic interaction inthe spinless Dirac fermions, which is desirable for a feasible solutionof the model. The Jz term, however, gives some density-density inter-actions which are quartic in the fermionic modes. Due to the presenceof the conserved quantities Bp these terms can be further simplified.We introduce two fermionic Majorana operators ci,j and di,j, where

ci,j = i(a†i,j − ai,j

)di,j = a

†i,j + ai,j i+ j = even ≡

ci,j = a†i,j + ai,j di,j = i

(a†i,j − ai,j

)i+ j = odd ≡

(6.30)

The complex fermions on each site can be reconstructed using theseMajorana operators, that obey the commutation relations

c2i,j = d2i,j = 1

ci,j, ci ′,j ′=di,j,di ′,j ′

= 2δii ′δjj ′

ci,j,di ′,j ′

= 0 .

(6.31)


While the expressions for σxi σxj and σyi σ

yj follow easily, the term σziσ

zj

needs some reformulation. It can be written as

σzi,j+1σzi,j =

(2a†i,j+1ai,j+1 − 1

)(2a†i,jai,j − 1

)

= i(idi,j+1di,j

)ci,j+1ci,j .

(6.32)

In order to maintain a clearer structure we will use pictorial indicesthat correspond to the even and odd lattice sites respectively. The fullHamiltonian then reads

H = −iJx∑

x-links

c c + iJy∑

y-links

c c − iJz∑

z-links

(id d

)c c . (6.33)

For the subsequent diagonalization we first rewrite the Hamiltonianonce again in terms of lattice vectors r1 and r2 and vectors r, thatlabel the unit cells. This unit cell consists of one even and one oddlattice site, the plane vectors connect them to span the whole lattice.The geometric setting can be seen in Fig. (6.6). The sums over all

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z

z

z

z

z

z

~r1 ~r2

Figure 6.6: Brickwall lattice with unit cell and plane vectors

x, y and z links are translated into a sum over all unit cells in theHamiltonian, so that we get

H = i∑

r

[Jxc ,r c ,r+r1

+ Jyc ,r c ,r+r2− Jz

(id ,r d ,r

)c ,r c ,r

]

(6.34)

The new operator αr = (id ,r d ,r) is defined on each z bond of thelattice labeled by the unit cell vector r. We can take them as goodquantum numbers since they commute with the Hamiltonian. This iseasy to verify for the first two terms, because the anticommutationrelation for single fermions leads to commuting pairs of fermions.This is also true for the third term and all lattice sites except site r,for which we have

[d d c c , d d

]r= d d c c d d − d d d d c c

= d d[c c , d d

]

= 0 .

(6.35)


Therefore, operators H and αr can be diagonalized in the same basis.Once a configuration of eigenvalues for all operators αr is fixed, theHamiltonian again describes free Majorana fermions in a static Z2

gauge field. Compared to the Hamiltonian of the first solution we seethat Eq. (6.34) is of the same form as Eq. (6.16) with a uniform gaugefixing of ui,j = 1 along x and y bonds.The good quantum numbers αr are related to the plaquette operatorsBp. From Fig. (6.2) we recall

Bp = σy1σz2σx3σy4σz5σx6 , (6.36)

which are conserved quantities of the original Kitaev model. Apply-ing the Jordan-Wigner transformation and some reformulation yields

Bp =1

i

(a†1 − a1

)σz2

(a†3 + a3

)σz1σ

z2 ·1

i

(a†4 − a4

)σz5σ

z6σz5

(a†6 + a6

)

= i(a†1 + a1

)(a†3 + a3

)· i1i

(a†4 − a4

) 1i

(a†6 − a6

)

=(id ,1 d ,3

)(id ,4 d ,6

)

= α61α43 . (6.37)

Therefore each hexagon or plaquette p has one independent link vari-able, that is related to the vortex. A vortex configuration of the pla-quettes corresponds therefore to the fixing of one αr for each row ofthe brick wall lattice [30].

Since the conserved quantities αr live on the z links of the latticewe also introduce a Dirac fermion on each z link, defined by theMajorana operators for both even and odd lattice sites.

dr =1

2

(c ,r − ic ,r

)

d†r =1

2

(c ,r + ic ,r

) (6.38)

This yields a model of fermions residing on a square lattice withside-dependent chemical potential. The original Majorana modes willbe replaced by the inverse transformation which leads to the newHamiltonian

H =∑

r

[Jx

(d†r + dr

)(d†r+r1

− dr+r1

)

+Jy

(d†r + dr

)(d†r+r2

− dr+r2

)+Jzαr

(2d†rdr − 1

)].

(6.39)

In order to solve the Hamiltonian in the ground state sector we needto fix a set for α corresponding to Bp = 1 for all p.

6.2.1 Diagonalization of the Hamiltonian

The uniform choice of αr = 1 for all r is the intuitive choice in thevortex-free sector, nevertheless all configurations leading to the same


configuration are equivalent. The Hamiltonian is translational invari-ant and can be transformed to momentum space in order to be solved.Therefore we define the Fourier transformation for the Dirac fermionsaccording to

dr =1√N

∑

k

e+ikrdk

d†r =1√N

∑

k

e−ikrd†k .(6.40)

The substitution k→ −k in the definition of d†r simplifies the calcula-tions. The terms in the Hamiltonian of Eq. (6.39) become

Jx : Jx1

N

∑

r

∑

k,k ′eikreik

′(r+r1)(d†-k + dk

)(d†-k ′ − dk ′

)

= Jx∑

k

[−2 cos (k1)d

†kdk +

1

22i sin (k1)

(d†kd†-k − h.c.

)] (6.41)

Jy : Jy1

N

∑

r

∑

k,k ′eikreik

′(r+r2)(d†-k + dk

)(d†-k ′ − dk ′

)

= Jy∑

k

[−2 cos (k2)d

†kdk +

1

22i sin (k2)

(d†kd†-k − h.c.

)] (6.42)

Jz : Jz1

N

∑

r

∑

k,k ′eikreik

′r(2d†-kdk ′ − 1

)

= Jz∑

k

2d†kdk − JzN .

(6.43)

Here N is the total number of sites on the lattice and we used theorthogonality relation of the Fourier transformation

∑

r

ei(k+k ′)r = N · δ(k+ k ′

). (6.44)

With use of all previous transformations we obtain a Hamiltonian H,that is quadratic in the Dirac fermions living on each of the unit cellsalong the z links. It can be written as

H =∑

k

[εkd

†kdk +

1

2

(i∆kd

†kd†-k − i∆kd-kdk

)]− JzN

εk = 2[Jz − Jx cos (k1) − Jy cos (k2)

]

∆k = 2[Jx sin (k1) + Jy sin (k2)

].

(6.45)


Using the symmetric property εk = ε-k we can represent it in theform

H =∑

k

1

2

(d†k d-k

)( εk i∆k

−i∆∗k −εk

)(dk

d†-k

). (6.46)

In order to diagonalize the Hamiltonian we need to write it in a differ-ent basis, which is achieved by the unitary Bogoliubov transformation

γk = ukdk + vkd†-k

γ†-k = −v∗kdk + u∗kd

†-k ,

(6.47)

where uk and vk are complex coefficients satisfying the relation|uk|

2 + |vk|2 = 1. The new Hamiltonian is then given by

H =∑

k

1

2

(γ†k γ-k

)( uk vk

−v∗k u∗k

)(εk i∆k

−i∆∗k −εk

)(u∗k −vk

v∗k uk

)(γk

γ†-k

),

(6.48)

expressed by quasi-particle operators γk, γ†-k. The product of thethree quadratic matrices H ′k = T†HkT becomes

(H ′k)11

= εk(|uk|

2 − |vk|2)+ i∆kukv

∗k − i∆ku

∗kvk(

H ′k)12

= −2εkukvk + i∆ku2k + i∆kv

2k(

H ′k)21

= −2εku∗kv∗k − i∆ku

∗2k − i∆kv

∗2k(

H ′k)22

= −(εk(|uk|

2 − |vk|2)+ i∆kukv

∗k − i∆ku

∗kvk

).

(6.49)

To obtain a diagonal form of H we can impose that the non-diagonalterms in H ′k vanish, and derive further results. Due to the normaliza-tion condition |uk|

2 + |vk|2 = 1 it is convenient to express uk and vk

as

uk = eiφ1 cos (θk/2) , vk = eiφ2 sin (θk/2) . (6.50)

It is sufficient to use only one condition for vanishing off-diagonalmatrix elements, so that we get

−2εkukvk + i∆ku2k − (−i∆k)v

2k = 0 . (6.51)

Inserting the trigonometric expressions in Eq. (6.50) leads to

−2εk cos(θk2

)sin(θk2

)ei(φ1+φ2) +∆k cos2

(θk2

)ei(2φ1+

π2 )

−∆k sin2(θk2

)ei(2φ2−

π2 ) = 0 ,

(6.52)


where the relation φ1+φ2 = 2φ1+π/2 = 2φ2−π/2must be fulfilled.Without the loss of generality we can choose φ1 = 0 and φ2 = π/2,so that the condition above can be expressed as

tan (θk) =∆k

εk. (6.53)

It is now possible to derive the required expressions for the parame-ters uk and vk in the Bogoliubov transformation using

|uk|2 − |vk|

2 = cos (θk) =1√

1+ tan2 (θk)=εkEk

, (6.54)

ukvk =i

2sin (θk) =

i

2tan (θk) cos (θk) =

i

2

∆k

Ek, (6.55)

where Ek =√ε2k +∆2k is the quasi-particle excitation energy taken as

the positive square root. The coefficients become

uk =

√1

2

(1+

εkEk

)vk = i

√1

2

(1−

εkEk

). (6.56)

Additional properties follow from Eq. (6.55). Using the relationsE-k = Ek and ∆-k = −∆k we see that

u-kv-k = −i

2

∆k

Ek. (6.57)

Therefore one of the Bogoliubov coefficients needs to be symmetricwhile the other one needs to be antisymmetric under k → −k. Bothchoices are similar, here we take

u-k = uk v-k = −vk . (6.58)

This property becomes particularly important for the actual imple-mentation of the Bogoliubov transformation for finite-size systemsand will be discussed later on. Finally, the obtained coefficients canbe used to express the remaining terms in the Hamiltonian, whichnow has the diagonal form

H =∑

k

1

2

(γ†k γ-k

)(Ek 0

0 −Ek

)(γk

γ†-k

)

=∑

k

Ek

(γ†kγk −

1

2

).

(6.59)

This completes the analysis of the Kitaev honeycomb lattice on theinfinite plane. We are now left with a diagonal Hamiltonian, wherethe ground state is the Bogoliubov vacuum with no quasi-particleexcitations.


6.3 phase diagram

We want to give a brief overview about the phase diagram and itsmost important features. The phase diagram can be derived from thedispersion relations in Eq. (6.24) or in Eq. (6.45), equivalently. Sum-marized in Fig. (6.7), the energy spectrum can be gapless if and onlyif the triangle inequalities

|Jx| 6 |Jy|+ |Jz| ,

|Jy| 6 |Jx|+ |Jz| ,

|Jz| 6 |Jx|+ |Jy|

(6.60)

are satisfied. The gapless B-phase, defined by having strict inequali-ties, is shown as the shaded region of Fig. (6.7). In this phase there

Jz = 1

Jx = 1 Jy = 1

APhase

APhase

APhase

PhaseB

Figure 6.7: Phase diagram for the Kitaev model in the coupling pa-rameter space. The three sides correspond to Jx = 0, Jy = 0 andJz = 0 respectively, the three corners represent the opposite limits.

are exactly two Fermi points, k = ±q∗, one in each half of the firstBrillouin zone. The spectrum shows conic dispersion relations at thesepoints, the so-called "Dirac Cones". This means that quasiparticles inthis regime are massless and relativistic. Additionally this phase has

δkx

δky

εk

Figure 6.8: Dirac cones in the gapless B-phase.

a non-zero Chern number (ν = 1) and is therefore gapless, chiral andtopologically-ordered (time-reversal symmetry being spontaneously

6.3 phase diagram 49

broken) [29]. Here we introduced some of the most important andinteresting concepts in quantum many-body physics, which we willbriefly explain.

Topological OrderPhases of ordinary matter are usually classified by to the Landau the-ory according their different orders, which are connected to differentsymmetries. As for the case of quantum matter at zero temperature,this classification is not sufficient any more due to the emergenceof new kind of order. This new kind of order was named topologicalorder [34]. Topological ordered systems come along with long-rangeentanglement pervading the system as well as topology-dependentground state degeneracy, that is robust under local perturbations.

Chiral States of Matter and the Chern NumberChiral topological states are states that spontaneously break the time-reversal symmetry. They can be distinguished from non-chiral topo-logical phases by the spectral Chern number, which is an importanttopological quantity to characterize non-interacting systems [29].

The B-phase acquires a gap in the presence of an external magneticfield, which can be written as an effective three-spin interaction in aperturbative approximation. The three-spin terms maintain the exactsolvability of the model [29]. The gapped quasiparticles are then non-abelian Ising anyons.In the gapped regions of the phase diagram in Fig. (6.7) one coup-ling parameter predominates over the other two. Here the modelis in the A-phase, which has non-chiral Z2 topological order andtherefore a spectral Chern number of zero. The quasiparticles corre-spond to excitations of the plaquettes Bp, which are abelian anyons.The Hamiltonian can be perturbatively mapped onto the Toric CodeHamiltonian, as we will show in the following section.

7T O R I C C O D E L I M I T O F T H E H O N E Y C O M B M O D E L

Besides the honeycomb model, Kitaev introduced another topologi-cal interesting model, the Toric Code (TC) [35]. This model describesa system of spin-1/2 residing on the edges of a two-dimensional, in-finite square lattice, as illustrated in Fig. (7.1). There are plaquettes p,

p

s

Figure 7.1: Toric Code model defined on the square lattice with pla-quettes p and stars s.

that go around one square and stars s, that go around one vertex. Wedefine the plaquette operator Bp and the star operator As accordingto

X

X

X X = As ≡∏j∈s

σxj

Z

Z

Z Z = Bp ≡∏j∈p

σzj

Figure 7.2: Definition and visualization of the plaquette and star op-erators.

where σx and σz denote the spin-1/2 Pauli matrices. We now definethe Toric Code Hamiltonian as

H = −∑

s

As −∑

p

Bp , (7.1)

that is interesting for several reasons. First of all, this model is exactlysolvable and it is a Z2-lattice gauge theory. Further it inherits topo-logical order, so that the degenerate ground state defines a pair oftopologically protected qubits on a torus.

51

52 toric code limit of the honeycomb model

7.1 exact solution

In the following sections we will present the exact solution to themodel as well as the representation of the eigenstates with tensornetworks. We start with the two important properties

∏

s

As =∏

p

Bp = I

[As,Bp] = 0 ∀s,p .(7.2)

The first condition is a global constraint, so that not all As and Bpoperators are independent. The second condition can be easily veri-fied. The operators overlap in either zero or two sites and due to theanticommutation relation for the Pauli matrices they commute. Sim-ilarly we have [As,As ′ ] =

[Bp,Bp ′

]= 0 for all plaquettes and stars.

In order to write the ground state explicitly we start with the pro-duct state |0 . . . 0〉, that is the ground state of the second term in theHamiltonian, hence

−∑

p

Bp |0 . . . 0〉 = − |0 . . . 0〉 . (7.3)

Due to the second condition in Eq. (7.2) the overall ground state willbe the projection of |0 . . . 0〉 into the ground state of the first part inthe Hamiltonian. It is therefore given by

|0, 0〉 =∏

s

(I +As√

2

)|0 . . . 0〉 (7.4)

with the projector given as the operator in parentheses. The labelling|0, 0〉 is due to the four-fold degeneracy if the Toric Code model is putonto a torus using periodic boundary conditions. We will briefly out-line how to construct the four ground states for this case. On a toruswe have two non-trivial loops, that can not be contracted to a point, asshown in Fig. (7.3). All other non-trivial loops can be topologically de-

γ1γ2

Figure 7.3: Torus geometry for the Toric Code model with periodicboundary conditions, here the ground state is four-fold degenerate.Every non-trivial loop can be deformed to either γ1 or γ2.

formed into either γ1 or γ2. The non-trivial loop operator acts alonga path γ and is given by

Wx(γ) =∏

j∈γσxj . (7.5)

7.2 peps representation of the ground state 53

For both elementary loops Wx(γ) will always go through either zeroor two sites of a plaquette or star respectively. This means that bothloop operators commute with the Hamiltonian, so that we have[Wx(γ1),H] = [Wx(γ2),H] = 0. The ground state is therefore four-fold degenerate, it can be written as

|j,k〉 = [Wx(γ1)]j [Wx(γ2)]

k|0, 0〉 , (7.6)

with the state |0, 0〉 defined as in Eq. (7.4). The ground state can beinterpreted as two qubits, which are topologically protected due tothe properties of the model. This means that the quantum state storedin those qubits can not be destroyed by local perturbations [35].

7.2 peps representation of the ground state

The ground state for the Toric Code model can be expressed as aPEPS with bond dimension two. We return to the thermodynamiclimit and analyze the model on the infinite square lattice. Again, westart with the unentangled product state |0 . . . 0〉 of all spins. In orderto construct the ground state PEPS, we need to express the operator1/√2(I+As) as a Projected Entangled Pair Operator (PEPO), which is

simply given by the expressions in Fig. (7.4).

1

1

= 2−18I 2

2

= 2−18σx

Figure 7.4: PEPS operator for the ground state projector with explicitexpressions for the tensor entries, that are non-zero.

According to Eq. (7.4) the ground state is given by the product of allprojectors over all stars s acting on the unentangled product state. Inthis way each spin is effected by two operators, however the orderingdoes not matter since all As operators commute with each other. Thepattern to apply the operators of Eq. (7.4) is shown in Fig. (7.5). Green

Figure 7.5: Projectors applied to the unentangled product state ofspins. Because of the commutation of all As operators the order doesnot matter, however the pattern must be maintained.

tensors on top represent the product state |0 . . . 0〉. The contraction of


all indices yields the PEPS representation for the Toric Code groundstate. Due to the arrangement of the projectors there are two different

A

A

B

B

A

A

B

B

A

A

B

B

A

A

B

B

Figure 7.6: PEPS for the Toric Code ground state with a two tensorunit cell.

tensors A and B, each of them appearing twice. The tensor pairs withthe same entries are graphically related by a rotation around π, asdemonstrated in Fig. (7.7). One tensor of A and B each build the unitcell to construct the infinite PEPS.

|0〉

=

|0〉

=

A

|0〉

=

|0〉

=

B

Figure 7.7: The contraction of the top tensor with the PEPS operatorsleads to two different tensors A and B.

7.3 perturbative limit of the kitaev model

The Kitaev model can be mapped to the Toric Code Hamiltonian inthe perturbative limit, where the interaction in one direction is muchstronger than the interaction in the other two directions [29]. Here wetake Jz Jx, Jy and write the Hamiltonian as

H = H0 +H1 = −Jz∑

z-links

σziσzj +

−Jx

∑

x-links

σxi σxj − Jy

∑

y-links

σyi σyj ,

(7.7)

with the x and y terms as the perturbation H1. The z bonds can betreated as isolated dimers due to the strong Jz interaction in thislimit, so that only the two states |↑↑〉 and |↓↓〉 out of the four pos-sible spin states contribute to the ground state subspace of H0. Theground state degeneracy is therefore 2Nz , where Nz is the numberof z bonds. The effective Hamiltonian taking the perturbation into ac-count is calculated using perturbation theory, which yields the first

7.3 perturbative limit of the kitaev model 55

z =

|↑↑〉|↓↓〉

Figure 7.8: The two spins of the isolated dimers on the z bonds canbe mapped to an effective spin in the perturbative limit Jz Jx, Jy.

non-trivial expressions in fourth order. Without further details, wegive the Hamiltonian obtained in [29], it is given by

Heff = −J2xJ2y

16|Jz|3

∑

p

Qp Qp = τyi τzj τykτzl . (7.8)

Here τ denotes the effective spin operator in the ground state sub-space of each z bond, as illustrated in Fig. (7.8). With the transfor-

-

τyi

τzj

τyk

τzl

p

Figure 7.9: Plaquettes p of the Toric Code Hamiltonian with effectivespins τ.

mation above, the honeycomb lattice changes into a square lattice,where the effective spins reside on the vertices, illustrated in Fig. (7.9).This effective spin lattice can be transformed into the previously intro-duced lattice of the Toric Code, using a side-to-bond and bond-to-sidetransformation, as shown in Fig. (7.10). Due to this transformation the

Figure 7.10: The two spins on the z links of the honeycomb lattice in(a) get mapped to an effective spin on one site of the square lattice in(b). The application of a side-to-bond and bond-to-side transforma-tion yields the square lattice of the Toric Code in (c).

plaquettes of the effective spin lattice becomes stars and plaquettes ofthe new lattice. Therefore the sum over all plaquettes in Eq. (7.8) can


be written as a sum over all plaquettes and stars of the lattice inFig. (7.1). The Hamiltonian takes the form

Heff = −Jeff

(∑

s

Qs +∑

p

Qp

), with Jeff =

J2xJ2y

16|Jz|3. (7.9)

In order to achieve the original form of the Toric Code Hamiltonian,we define a unitary transformation along the horizontal and verticallinks according to

U =∏ 1√

2(σx + σy) ·

∏ 1√2(σx + σy) · 1√

2(σy + σz) . (7.10)

The Hamiltonian now takes the form

H ′eff = UHeffU† = −Jeff

(∑

s

As +∑

p

Bp

), (7.11)

where As and Bp are defined in terms of the effective spin operatorsτ on the square lattice as in Fig. (7.1).

Since the tensor network expression for the Toric Code ground stateis known, we might as well express the ground state for the honey-comb model using a similar PEPS. However this approach can onlywork in the perturbative limit of the Hamiltonian in Eq. (7.7). In or-der to obtain the tensors, we have to undo the transformations intro-duced previously. This includes the the unitary transformation U as

|0〉

Y†

W

=

A ′

|0〉

X†

W

=

B ′

Figure 7.11: Transformations in reverse order to transform the ToricCode PEPS into the ground state of the honeycomb lattice in the per-turbative limit.

well as the mapping of two spins into one site, applied directly to thetwo tensors A and B of the Toric Code ground state. U is decomposedinto the horizontal transformation X and the vertical transformationY. From Fig. (7.6) it can be seen that vertical links only affect tensors Aand horizontal transformations only affect tensors B. Here the isom-etry W extends the Hilbert space again, which was reduced due to

7.3 perturbative limit of the kitaev model 57

the spin mapping before. The tensor is given by Wijk = δijk, whicheffectively copies the spin state that enters.The resulting spin state still has the form of a PEPS, however eachsite is occupied by two spins. In order to regain the honeycomb lat-tice structure, the tensors have to be split up using a singular valuedecomposition. The procedure is shown in Fig. (7.12). Reshaping ofthe tensor leads to a matrix with dimension (ijk)× (klm) on whichwe perform the SVD. The gray diagonal matrix of singular values isincorporated into the two matrices U and V† with S1/2 each. It is

k

j

i

lm n

Reshape (k, l,n)(i, j,m)SVD

U V†S

(k, l,n)(i, j,m) Reshape

k

j

i

lm

n

Figure 7.12: The Toric Code PEPS tensors are split up in order to regainthe honeycomb lattice structure. This is done by reshaping the arraysand applying a singular value decomposition.

easy to see that a final reshaping of the two matrices into rank-4 ten-sors recovers the honeycomb or brick wall lattice structure. This isvisualized in Fig. (7.13)

Figure 7.13: PEPS ground state of the honeycomb model in the pertur-bative limit emerging from the TC ground state.

8E X A C T U N I TA RY 3D T E N S O R N E T W O R K

In this section we will present the exact eigenstates of the Kitaev hon-eycomb model, in particular in the vortex-free ground state sector, interms of a 3d unitary TN. We shall see how this works for relativelysmall systems. We will then argue that the construction scales wellup to the thermodynamic limit, supported by intermediate numericalchecks for finite-size systems of up to 32 spins on a honeycomb latticewith PBC. The construction could follow the flow of the model’s so-lution in Sec. 6.2. However we reverse the direction in our approach,going from a product state in the fermionic Bogoliubov modes to thespins on the honeycomb lattice. This way the overall TN has an in-teresting interpretation in terms of a unitary circuit building up thequantum many-body wave function.

8.1 fermionic vacuum and bogoliubov transformation

We start in momentum space, where we have a diagonal Hamiltonianand hence unentangled momentum modes. The possible momentak = (kx,ky)> for a finite-size Nx ×Ny system are given by

ki =2πniNi

ni = −Ni − 1

2, . . . ,+

Ni − 1

2, (8.1)

with Ni as the size of the system along the direction i = x,y. Whentaking the thermodynamic limit, the discrete series becomes a con-tinuum of momentum modes. In what follows we describe the pro-cedure for the example of an 8 spin and 32 spin honeycomb lattice,which at this stage amounts respectively to 2× 2 and 4× 4 squarelattices of free fermions in momentum space.The first step to build the TN is the unentangled quantum state rep-resented in Fig. (8.1). This is a product of state vectors and each ten-

(−π

2 ,−π2

)

(−π

2 ,+π2

)(+π

2 ,−π2

)

(+π

2 ,+π2

)

Figure 8.1: Product state in momentum space corresponding to theBogoliubov vacuum on a 2× 2 square lattice.

sor denotes one fermionic mode with physical dimension two. The

59

60 exact unitary 3d tensor network

values of the physical index correspond to the fermionic occupationnumber of the Bogoliubov momentum modes. If the physical indexis 0 the corresponding mode is unoccupied, whereas it is occupiedfor a physical index of 1. In the case of the ground state each tensorhas a non-zero component only for the unoccupied fermionic mode,denoted by |0〉. The quantum state is then simply the Bogoliubov va-cuum.In order to couple the modes with opposite momentum we have toundo the Bogoliubov transformation that diagonalized theHamiltonian previously. This can be achieved using only two-bodygates, so that for a number of N fermionic modes we need N/2 ofsuch gates. The correct way to perform the Bogoliubov transforma-tion is shown in Fig. (8.2), where all momenta along the gray arrowenter the transformation and get coupled with their opposite momen-tum mode. The sequence in which the modes are coupled, indicated

kx

ky

(−3π

4 ,−3π4

)

(−3π

4 ,−1π4

)

(−3π

4 ,+1π4

)

(−3π

4 ,+3π4

)

(−1π

4 ,−3π4

)

(−1π

4 ,−1π4

)

(−1π

4 ,+1π4

)

(−1π

4 ,+3π4

)

(+1π

4 ,−3π4

)

(+1π

4 ,−1π4

)

(+1π

4 ,+1π4

)

(+1π

4 ,+3π4

)

(+3π

4 ,−3π4

)

(+3π

4 ,−1π4

)

(+3π

4 ,+1π4

)

(+3π

4 ,+3π4

)

Figure 8.2: The fermions along the gray arrow get coupled with theones on the opposite site carrying opposite momentum. The valuesof k along the sequence also determine the input parameters for theBogoliubov transformation.

by the arrow, also persists for larger lattices. The actual Bogoliubovtransformation is achieved by a sequence of two-body gates actingon pairs of fermions with opposite momentum. These gates are de-scribed by the unitary matrix

Bk =

uk 0 0 vk

0 1 0 0

0 0 1 0

−v∗k 0 0 u∗k

. (8.2)

Besides the momentum dependency, both uk and vk depend inher-ently on Jx, Jy and Jz. Due to intrinsic properties of the Bogoliubovtransformation (refer to Eq. (6.57)) the coefficients must satisfy

u−k = uk v−k = −vk . (8.3)

Additionally we observed that the sign of vk needs to be chosen ac-cording to the signs of Jx and Jy to generate the correct transfor-

8.1 fermionic vacuum and bogoliubov transformation 61

mations in all eight octants of the model. Numerical checks indi-cate that the correct convention is the one in Table 8.1, where thecondition for an overall negative transformation sign of vk is speci-fied. This Bogoliubov transformation works smoothly for quadratic

Octant Classification Condition

|Jx| > |Jy|

1 Jx > 0, Jy > 0 kx < 0

2 Jx > 0, Jy < 0 kx < 0

3 Jx < 0, Jy > 0 kx > 0

4 Jx < 0, Jy < 0 kx > 0

|Jx| < |Jy|

5 Jx > 0, Jy > 0 ky < 0

6 Jx > 0, Jy < 0 ky > 0

7 Jx < 0, Jy > 0 ky < 0

8 Jx < 0, Jy < 0 ky > 0

Table 8.1: Conditions for sign(vk) = −1 of the Bogoliubov transfor-mation in all octants.

lattices with an even number of sites in both directions, which can beconveniently scaled up to the thermodynamic limit. For rectangularlattices with Nx 6= Ny the convention can differ and it needs to bechosen for each case individually. Applying the transformation to theBogoliubov vacuum we get the entangled state in momentum spaceshown in Fig. (8.3).

(−π

2 ,−π2

)

(−π

2 ,+π2

)(+π

2 ,−π2

)

(+π

2 ,+π2

)

Bk

Bk

Figure 8.3: Ground state of the Kitaev honeycomb model after ap-plying the Bogoliubov transformation, this state now shows entangle-ment between the fermions.

The TN we are constructing has a 3d structure. The TN diagrams, how-ever, will be projections of this network in the 2d plane for whichwe have to apply fermionization rules as presented in Sec. 3.6.1. Dif-


ferent projections will produce different patterns of fermionic swapgates, which in turn correspond to different orderings of fermionicmodes in second quantization [22]. The resulting planar structure isshown in Fig. (8.4). Using the intermediate fermionic Hamiltonian in

(−π

2

−π2

) (−π

2

+π2

) (+π

2

−π2

) (+π

2

+π2

)

Bk Bk

Figure 8.4: Redrawn tensor network without crossings of fermioniclines and gates. To satisfy the anticommutation relations a swap ten-sor is introduced each time two fermions are exchanged.

Eq. (6.46) we verified that the tensor network generates the correcteigenstates for up to 4× 4 lattice sites.

8.2 fourier transformation and spectral tensor net-work

The next step is to undo the Fourier transformation of the fermions,bringing the modes from momentum space back to real space. Forconvenience we assume that the number of modes is a power of two.The Fourier transformation of fermions will be represented as a spec-tral tensor network, which was presented in Sec. 4 and originallyproposed in Ref. [24]. We will briefly recap the important properties.The Fourier transformation over N sites (regardless of the lattice)can be decomposed recursively, until ending up eventually with two-site transformations and one-body gates accounting for the relativephases. An example is the one-dimensional transformation F4 overfour modes, depicted in Fig. (8.5). This procedure can be extendedto higher spatial dimensions, where the Fourier transformation is ap-plied sequentially along each direction.

8.2 fourier transformation and spectral tensor network 63

F4 = ω04 ω1

4 =ω0

4 ω14

Figure 8.5: Unitary circuit to perform the Fourier transformation overfour modes. Black tensors represent fermionic swap gates.

We will proceed our construction of the eigenstates of the Kitaevhoneycomb model. Since the BCS-type Hamiltonian contains anoma-lous terms in the fermionic creation and annihilation operators, ad-ditional "half-integer" momenta need to be included in the STN [24].This is implemented by the one-body phase rotations after the Fouriergates. The TN structure for the 2×2 square lattice is shown in Fig. (8.6).Numerical contraction of the TN up to this level and 4× 4 sites verifies

(−π

2 ,−π2

)

(−π

2 ,+π2

)(+π

2 ,−π2

)

(+π

2 ,+π2

)

F2

F2

F2

F2

ω04

ω14

ω14

ω24

Bk

Bk

Figure 8.6: Eigenstates of the fermionic Hamiltonian in real space.The STN incorporates additional phases indicating half-integer mo-menta, which is necessary due to the Bogoliubov transformation.

that the quantum state is an eigenstate of the intermediate fermionicHamiltonian in real space given in Eq. (6.39). There are however some


subtleties coming from finite-size effects of the lattice. First, for a fea-sible contraction of the TN it is worth drawing the Fourier transfor-mation as in Fig. (8.7), where the labeling of the modes is explicit.The four-side Fourier transformation F4 is the one in Fig. (8.5) and in-

F4 F4 F4 F4

Mode Ordering Tensor

F4 F4 F4 F4

Mode Ordering Tensor†

0

0

0

0

0

1

2

4

8

2

2

1

8

4

1

3

3

12

12

3

4

4

2

2

8

5

6

6

10

10

6

5

10

6

9

7

7

14

14

11

8

8

1

1

4

9

10

5

9

6

10

9

9

5

5

11

11

13

13

7

12

12

3

3

12

13

14

7

11

14

14

13

11

7

13

15

15

15

15

15

Figure 8.7: Planar implementation of the Fourier transformation of4×4 sites. All crossings in the tensors F4 as well as the mode orderingtensors need to be realized by fermionic swap gates.

cludes the bit-reversal operation. The mode ordering tensors changethe direction of the transformation, for instance from x to y, whichhas to be undone once the FT in y direction has been applied. Allcrossings in these tensors need to be accounted for by fermionic swapgates. Second, in order to verify the quantum state the Hamiltonianneeds to be constructed carefully. This is because, due to the finite-size, the interaction terms at the boundary contribute with a negativesign to the Hamiltonian, unlike the bulk interaction terms. Thereforewe need to work with antiperiodic boundary conditions which is clearfrom the fact that

cNi+1 =1√Ni

∑

ki

e2πikiNiNi e

2πikiNi cki

= −1√Ni

∑

ki

e2πikiNi cki = −c1

(8.4)

for both spatial dimensions and all possible values of the momentum.The links that are affected are shown in Fig. (8.8) for the example of a4× 4 lattice. This enables us to construct the finite-size Hamiltonian,of which the intermediate TN in Fig. (8.6) is an exact eigenstate. Theboundary effects become irrelevant when scaling up the system tothe thermodynamic limit.

8.3 majorana braiding tensor network 65

~r1

~r2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Figure 8.8: Distinction of the interaction terms in the Hamiltonian inboundary terms (blue) and bulk terms (black).

8.3 majorana braiding tensor network

In this section we will present the TN structure that deals with thenecessary Majorana transformations. The Dirac fermions on the sitesof the square lattice are split into two Majorana modes, the samething is done for the Dirac fermions that represent the conservedquantities of the model. Following the split-up, the different types ofMajorana fermions need to be recombined into Dirac fermions livingon the sites of the honeycomb lattice. Algebraically these steps areclear, as we reviewed (in reverse) in Sec. 6.2. In terms of a TN structurehowever, these steps lead to a non-trivial construction. We should beable to split up and exchange Majorana fermions in order to fusethem back together into different combinations. This necessarily leadsto braiding of Majorana fermions, for which we know that they behavelike non-abelian anyons [36].

8.3.1 Majorana Fermions as non-Abelian Anyons

The fact that we have to treat the Majorana fermions as non-abeliananyons leads to specific consequences for the TN construction. To ana-lyze this, consider a system of two Dirac fermions with creation oper-ators f†1 and f†2. We express both Dirac fermions as Majorana fermionsγ1,γ2,γ3 and γ4 according to

f1 =1

2(γ1 + iγ2) f

†1 =

1

2(γ1 − iγ2) , (8.5)

f2 =1

2(γ3 + iγ4) f

†2 =

1

2(γ3 − iγ4) . (8.6)

Complementary, the Majorana fermions correspond to the real andimaginary part of the original Dirac fermions, since

γ1 = f†1 + f1 γ2 = i

(f†1 − f1

), (8.7)

γ3 = f†2 + f2 γ4 = i

(f†2 − f2

). (8.8)

Majorana fermions obey the anticommutation relation γi,γj = 2δij,which implies γ2i = 1. This leads to major differences compared to


Dirac fermions and the Pauli exclusion principle does not hold forMajorana fermions. In fact, since γ†i = γi one can not even define anumber operator ni = γ

†iγi , so that an occupation number represen-

tation is not appropriate.

Clockwise and anticlockwise braidings of two Majorana fermionsγi and γj can be accounted for by the operators

B =1√2

(1+ γiγj

)(clockwise) ,

B =1√2

(1− γiγj

)(anticlockwise) .

(8.9)

Those operators act on the Fock space of the Dirac fermions. If γiand γj correspond to the Majorana modes of one complex fermion,the operators are effectively one-body gates that implement "internal"braidings. However, if γi and γj correspond to Majorana modes of dif-ferent complex fermions, the operators implement "external" braidingby exchanging one Majorana mode of each Dirac fermion. This leadsto an action of the Fock space of both Dirac fermions and the opera-tors B and B are effectively two-body gates. First let us consider theaction within one Dirac fermion f1, where we get

B =1√2

(1+ i(f1f

†1 − f

†1f1)

), (8.10)

B =1√2

(1− i(f1f

†1 − f

†1f1)

). (8.11)

In the occupation number basis of the Dirac fermion |0〉 , |1〉 we canexpress both of the operators as a unitary matrix

B =1√2

(1+ i 0

0 1− i

)B =

1√2

(1− i 0

0 1+ i

), (8.12)

acting on the one-particle Hilbert space. The first row/column corre-sponds to the state |0〉, the second one to |1〉. In the case of a braidingof Majorana fermions belonging to different complex fermions, forexample the exchange of γ2 from f1 and γ3 from f2, the operatorsbecome

B =1√2

(1+ i(f†1f

†2 + f

†1f2 − f1f

†2 − f1f2)

),

B =1√2

(1− i(f†1f

†2 + f

†1f2 − f1f

†2 − f1f2)

).

(8.13)

They act on the two-particle Hilbert space |00〉 , |01〉 , |10〉 , |11〉 as atwo-body gate, the matrix elements are given by

B =1√2

1 0 0 −i

0 1 −i 0

0 +i 1 0

+i 0 0 1

B =1√2

1 0 0 +i

0 1 +i 0

0 −i 1 0

−i 0 0 1

, (8.14)


where the rows and columns follow the labeling of the basis states inthe Hilbert space. As expected, all the Majorana braiding operatorspreserve the fermionic parity, which is important for the construc-tion of the fermionic TN. Furthermore, the braiding of other Majoranafermions, such as γ1 and γ4, can be constructed by a concatenationof the operators presented above.

8.3.2 Tensor Network of Majorana Braidings

The way in which braiding of Majorana fermions has to be accountedfor in the language of TN is now explicit. It can be incorporated bythe action of the corresponding unitary operators in the Fock spaceof the Dirac fermions. Therefore we can proceed the construction ofthe eigenstates.The model hosts a large number of conserved quantities - representedby the occupation number of the "vortex" Dirac fermions. They haveto be reintroduced in the flow of the TN, restoring the correct numberof original spins on the honeycomb lattice. The procedure is shown inFig. (8.9), where one part of the fermion out of the Fourier transforma-tion gets combined with one part of the fermion that represents theconserved quantity. The different transformations can be merged into

FT

c c

a ,a†

dc

CQ

dd

a ,a†

d c

=

B34

B34

B23

FT CQ

a ,a†

a ,a†

Tαβγδ

α β

γ δ

Figure 8.9: Tensor network to combine the conserved quantities thatspecify the vortex sector of the honeycomb model with the fermionsout of the Fourier transformation. The transformation T contains thenecessary operations in order to respect the Majorana fermion’s any-onic behaviour.

an overall two-body tensor T . Numerical observations show that the


structure of clockwise and anticlockwise braidings depends on theconserved quantities. We can perform the two-body "external" braid-ing in an anticlockwise rotation, corresponding to B23. However thedirection of the one-body "internal" braiding depends on whetherthe conserved vortex fermion is occupied or not, corresponding toαr = −1 or αr = +1, respectively. Since the Kitaev model has a Z2

symmetry we will recover the ground state vortex sector for fully oc-cupied and fully unoccupied conserved modes. This is clear from therelation Bp = α61α43 (refer to Eq. (6.37)), that links the original Bpoperator to the two conserved quantities living on both of the plaque-ttes’ z links. For the one-body braiding we have found the rules

αr =

+1 −→ |CQ〉 = |0〉 −→ anticlockwise

−1 −→ |CQ〉 = |1〉 −→ clockwise. (8.15)

Both of these choices lead to the absence of vortices and hence theground state vortex sector for a uniform choice. The correct choice ofbraiding matrices is also important for other eigenstates apart fromthe ground state.

The overall Majorana braiding tensor T acts locally on the z linksof the honeycomb lattice, as shown in Fig. (8.10). We set fermion 3,5, 7 and 8 to be the conserved quantities that specify the vortex sec-tor. The other four fermions are the ones coming out of the Fourier

T

T

T

T

f1

f2

f3

f4

f5

f6

f7

f8

Figure 8.10: Fermionic TN to reintroduce the conserved quantitiesand braid the Majorana fermions. The transformation T is given inFig. (8.9) for the case of unoccupied vortex fermions.


transformation. For the implementation of the TN it has to be redrawnin such a way that no fermionic wire crosses a gate. One way to dothis is to express the tensor T as a two-site fermionic Matrix ProductOperator (MPO) with bond dimension χ = 4, as shown in Fig. (8.11).This would result in crossings of wires only. Another possibility is

T

χ

Figure 8.11: Two-body gate expressed as a two-side fermionic MPO

with bond dimension χ = 4, e.g. by performing a SVD.

the rearrangement of the fermionic modes with a different projectiononto the plane. It is important however to ensure the correct orderingof the modes at the bottom of the network. The diagram in Fig. (8.10)is reconstructed in Fig. (8.12), and represents the overall network toundo all the Majorana transformations in the Hamiltonian for an 8-side honeycomb lattice. This procedure can be extended to lattices of

FourierTransformation

ConservedQuantities

T T T T

f1 f2 f4 f6f3 f5 f7 f8

Figure 8.12: Planar implementation of the 3d tensor network inFig. (8.10) where wires do not cross any gate.

any size. Again, we have numerically verified that the TN structureup to this point generates the correct eigenstates of the intermediateHamiltonian.


8.4 jordan-wigner transformation

In the last step of the construction the fermions on the honeycomblattice need to be mapped back to spins. The reverse procedure wasperformed by the Jordan-Wigner transformation in the solution ofthe model. Surprisingly this transformation only changes the inter-pretation of the network, and not the values of the overall coefficientobtained by the contraction. This fact has already been noticed [25]and can be straightforwardly shown. A general quantum state of ar-bitrary number of spin-1/2 particles is given by

|ψ〉 =∑

n1n2...

cn1n2... |n1n2 . . .〉 , (8.16)

where ni = 0, 1 corresponds to the spin down/up state at site i. Interms of spin ladder operators it can be written as

|ψ〉 =∑

n1n2...

cn1n2...(σ+1)n1 (σ+2

)n2 . . . |Ω〉 , (8.17)

with |Ω〉 = |00 . . .〉 as the state with all spins down. Applying theJordan-Wigner transformation yields the fermionic state

|ψ〉 =∑

n1n2...

cn1n2...

(a†1

)n1S1

(a†2

)n2S2 . . . |Ω〉 . (8.18)

Here Si stands for the Jordan-Wigner string attached to the creationoperator a†i+1 and it is given by

Si =

i∏

j=1

eiπa†jaj =

i∏

j=1

(1− 2a†jaj

). (8.19)

This can be easily checked by a Taylor expansion of the exponential.The state |Ω〉 in Eq. (8.18) is reinterpreted as the fermionic vacuum. Itis also easy to verify that

a†i , eiπa

†jaj

= 0 for i = j

[a†i , eiπa

†jaj

]= 0 for i 6= j .

(8.20)

Using these relations and the fact, that eiπn = (−1)n for the occu-pation number n = a†a = 0, 1 for fermions, we can write the stateas

|ψ〉 =∑

n1n2...

cn1n2...(−1)n1(−1)n1+n2 . . . S1S2 . . .×

(a†1

)n1 (a†2

)n2. . . |Ω〉 .

(8.21)

The phases accumulate due to the anticommutation relations men-tioned above. Notice that this state is again the original state |ψ〉,

8.4 jordan-wigner transformation 71

which becomes clear when we write out the Jordan-Wigner stringscorresponding to

|ψ〉 =∑

n1n2...

cn1n2...(−1)n1(−1)n1+n2 . . . (−1)n1(−1)n1+n2 . . .×

(a†1

)n1 (a†2

)n2. . . |Ω〉 .

(8.22)

The phases cancel out exactly, so that the spin state of the Kitaevhoneycomb model is directly the fermionic state with exactly thesame coefficients. Therefore the Jordan-Wigner transformation doesnot need to be incorporated into the tensor network construction. Forour purpose it is sufficient to regard the wires as bosonic, or spin de-grees of freedom from this point on. Formally this corresponds to amapping of the Fock space for a spinless fermion to the Hilbert spacefor a spin-1/2.

The Jordan-Wigner transformation works fine in the thermodyna-mic limit where we do not need to consider boundary terms. How-ever if we assume PBC in order to approximate an infinite lattice, weget some subtleties in the Hamiltonian. First of all, the path of theJordan-Wigner transformation is not unique and the labelling of lat-tice sites is arbitrary. If we take the infinite lattice and cut out thesmaller lattice we want to realize, we end up with the model inFig. (8.13). We see that the path connects sites, which are not nearest-

1

2

3

4

5

6

7

8

Figure 8.13: Jordan-Wigner path for the infinite lattice applied to theeight-side honeycomb lattice.

neighbours if the lattice is put onto a torus. It is intuitive that thoseboundary terms become less and less important the larger the latticebecomes. In the end the bulk properties prevail, but in the case ofthe eight-side lattice both sorts contribute with equal quantity. In theHamiltonian we have the contributions

x links : [1, 5], [2, 3], [4, 8], [6, 7]

y links : [3, 4], [5, 6], [7, 1], [8, 2]

z links : [3, 1], [5, 2], [7, 4], [8, 6] .

(8.23)


In order to account for finite-size boundary effects, we have to intro-duce the Jordan-Wigner string separately into the boundary terms ofthe spin Hamiltonian. In this way they get eliminated again whenperforming the Jordan-Wigner transformation. This ensures the cor-rect form for the σxσx and σyσy coupling terms, as if they were inthe bulk and not on the boundary. The four boundary terms as theyhave to be included in the spin Hamiltonian are

σx1σx5 −→ −σx1σ

x5

4∏

j=2

σzjJW−−−−−→ +

(a†1 − a1

)(a†5 + a5

),

σx4σx8 −→ −σx4σ

x8

7∏

j=5

σzjJW−−−−−→ +

(a†4 − a4

)(a†8 + a8

),

σy7σy1 −→ +σy7σ

y1

7∏

j=1

σzjJW−−−−−→ −

(a†7 + a7

)(a†1 − a1

),

σy8σy2 −→ +σy8σ

y2

8∏

j=2

σzjJW−−−−−→ −

(a†8 + a8

)(a†2 − a2

).

(8.24)

Note that those four terms show the opposite sign compared to thebulk terms in Eq. (6.29), which contribute as

σxi σxj

JW−−−−−→ −(a†i − ai

)(a†j + aj

),

σyi σyj

JW−−−−−→ +(a†i + ai

)(a†j − aj

),

(8.25)

where (i, j) denote neighbouring sites. So if we include the strings ofσz into the boundary terms beforehand they will cancel out and wemay perform all subsequent steps to obtain a diagonal Hamiltonian.With all boundary terms correctly included the Hamiltonian has thecorrect spectrum.

8.5 arbitrary vortex sectors

So far we have considered the ground state vortex sector, the sec-tor with no excitations of the plaquettes. The problems when dealingwith arbitrary vortex patterns, periodic or not, is that the Hamiltonianis no longer translationally invariant. This can be exemplified in thecase of the full vortex sector. Since we want to induce a magnetic fluxfor every plaquette p, we need to have an eigenvalue of Bp = −1

everywhere. This is not achievable with a uniform choice of the Z2

gauge fields, in fact we need to introduce alternating signs of the con-served quantities on the z bonds. This is visualized in Fig. (8.14). Inthe case of a uniform gauge fixing the Hamiltonian is translationallyinvariant right away. In contrast, for a more general vortex configura-tion we can generate translational invariant Hamiltonians only by en-larging the elementary unit cell. For the full vortex sector in Fig. (8.14)

8.6 overall tensor network structure 73

-1 -1 -1

-1 -1

+ −

+ − +

Figure 8.14: Gauge fixing to generate the full vortex sector.

for instance, we have to choose a unit cell that consists of two z linksin either x or y direction. Compared to the labeling of the unit cellsusing one vector r for the vortex-free sector, the new lattice needs tobe labeled with (r, λ). Here λ = 1, 2 denotes the z link within the newunit cell at position r.

Our tensor network construction can not be used to generate eigen-states of the Kitaev model in arbitrary vortex sectors, since thoseHamiltonians can not be solved by a Fourier transformation. How-ever, even without translational invariance they are still quadratic inthe fermionic creation and annihilation operators. It is well knownthat such Hamiltonians are always classically solvable in polynomialtime. On account of this there exists an efficient classical circuit with apolynomial number of gates for all eigenstates. Promoted to the quan-tum regime, there is always a reversible, unitary quantum circuit thatsolves the Hamiltonian at polynomial cost, the same holds for the con-struction of the eigenstates. In terms of the TN structure, this circuitneeds to replace the network of the Fourier and Bogoliubov transfor-mation.

8.6 overall tensor network structure

Summarizing the construction of the eigenstates, the spectral tensornetwork for the small honeycomb lattice of eight spins and PBC is visu-alized in Fig. (8.15). This is the smallest reasonable quantum state ofthe Kitaev model on the honeycomb lattice. It is important to note thatthe TN is a fermionic network, for which fermionization rules have tobe applied. Finally the fermionic degrees of freedom get mapped tospin (bosonic) degrees of freedom at the bottom of the STN. The net-work in Fig. (8.15) generates the ground state as well as all excitedstates in the vortex-free sector. This was numerically checked usingthe original spin Hamiltonian with modified boundary terms, as de-scribed in Sec. 8.4. The whole procedure can be extended to largerlattices and the thermodynamic limit.


(−π

2,−π

2

)

(−π

2,+π

2

)

(+π

2,−π

2

)

(+π

2,+π

2

)

|0〉|0〉

|0〉|0〉

F2

F2

F2F2

ω04

ω14

ω14

ω24

T

T

T

T

Bk

Bk

Figure 8.15: Holistic construction of the spectral tensor network forthe 8-spin honeycomb lattice with PBC.

In the next chapter we focus on two important physical propertiesof the model, which can be conveniently analyzed in the context ofour tensor network representation.

8.7 ground state fidelity and topological entropy

In the thermodynamic limit the ground state fidelity per lattice site isa well defined quantity, which can be used to characterize the phasediagram of the model [37]. For the Kitaev honeycomb model we want

8.7 ground state fidelity and topological entropy 75

to uncover the phase transition between the gapped A-phase andthe gapless B-phase. Consider a finite lattice system described by theHamiltonian H(Jz), where Jz is the only control parameter. In the hon-eycomb Hamiltonian the other coupling parameters are then given byJx = Jy = 0.5 · (1− Jz). The ground state fidelity is defined as

F(Jz, J′z) = | 〈ψ(Jz)|ψ(J′z)〉 |2 , (8.26)

where |ψ(Jz)〉 and |ψ(J′z)〉 correspond to ground states of two differ-ent values of the control parameter Jz. The fidelity asymptoticallyscales like F(Jz, J′z) ∼ d(Jz, J′z)N. Here N is the total number of spinsand d(Jz, J′z) is the scaling parameter, which can be physically seenas the averaged fidelity per lattice site [37]. Referring to the phase di-agram in Fig. (8.16) the phase transition is expected to occur at thepoint (Jx, Jy, Jz) = (0.25, 0.25, 0.5). The green line indicates the tra-jectory in the parameter space. Using the tensor network structure to

Jz = 1

Jx = 1 Jy = 1

APhase

APhase

APhase

PhaseB

Jx = Jy

Jz ∈ [0, 1]

Figure 8.16: Trajectory in the parameter space to uncover the phasetransition between the gapped A-phase and the gapless B-phase.

calculate the overlap for different ground states has a great advantage.Due to the unitarity of all the gates involved, the TN overlap reducesto a form where only the Bogoliubov transformation remains. All

∏k

Bk

B†k

|0〉 |0〉

〈0| 〈0|

≡ ∏k

Bk |00〉

〈00| B†k

Figure 8.17: The ground state fidelity is given by a product of theoverlap of the quantum states after the Bogoliubov transformation.

fermionic swap gates cancel as well. The fidelity is now simply givenby a product over all terms 〈00|B†kBk|00〉, illustrated in Fig. (8.17).Here Bk and B

†k represent the Bogoliubov tensor for the same mo-


mentum k, but different coupling parameters. The fidelity for a hon-eycomb lattice of 128 and 2048 spins, corresponding to an 8× 8 and32× 32 square lattice in momentum space respectively, is shown inFig. (8.18) and Fig. (8.19). The transition between the A- and the B-

Figure 8.18: Ground state fidelity for an 8× 8 square lattice in mo-mentum space, corresponding to 128 spins on the honeycomb lattice.

phase is clearly visible. The gapped A-phase shows a high fidelity,indicating that the states are very similar. In contrast, the states inthe gapless B-phase only show small areas of high fidelity and strongfinite-size effects. We see a shift of the transition line towards the ex-

Figure 8.19: Ground state fidelity for a 32× 32 square lattice in mo-mentum space, corresponding to 2048 spins on the honeycomb lat-tice.

pected value of Jz = 0.5 for larger lattices and a decreasing area of

8.7 ground state fidelity and topological entropy 77

high fidelity. This is clearly visible in Fig. (8.20) for a 128× 128 squarelattice. Here the states in the gapless B-phase are essentially differentfrom each other, resulting in very little overlap apart from the diago-nal line. As pointed out in Ref. [38], the phase transition of the Kitaev

Figure 8.20: Ground state fidelity for a 128 × 128 square lattice inmomentum space, corresponding to 32 768 spins on the honeycomblattice.

honeycomb model is a second-order transition. However the sharpedges of the fidelity in Fig. (8.20) indicate that the phase transition isof first-order [37]. This could be due to the nature of the transition be-tween a gapped and a gapless phase, which is possibly very sudden.

The second important property we want to expose in the TN struc-ture is the entanglement entropy. It was calculated in Ref. [39] to be

S = (α+ log 2)L− log 2 . (8.27)

For the specific model Stopo = − log 2 is the Topological Entangle-ment Entropy (TEE), a constant contribution that only depends onthe topological order. For a generic fermionic Hamiltonian howeverit could be something else, because not all fermionic states are topo-logically ordered. This topological contribution is incorporated as theZ2 fermionic parity symmetry in the TN representation. In order tosee this we consider a bipartition of the quantum state, that can bewritten as

|ψ〉 =χ∑

α=1

|in(α)〉 ⊗ |out(α)〉 , (8.28)

where α is the combined index of all indices along the boundary. Theentanglement entropy is then bounded by S(L) 6 logχ. Though theexact scaling of χ with respect to L is not known for our TN structure,


we can assume an average bond dimension Dav for the m bonds thatare cut. This leads to χ ≈ Dmav . However, the incorporation of a Z2

symmetry operator P leaves the overall TN structure unchanged. Thisis demonstrated in Fig. (8.21), also refer to Sec. 3.6. Therefore the

=ZZ

Z Z

ZZ

Z Z

ZZ

Z Z

Figure 8.21: The incorporation of a Z2 symmetry operator leaves theTN unchanged. This leads to a constant contribution to the entangle-ment entropy, known as the topological entropy.

fermionic parity reduces the rank of the reduced density matrix, sothat χ is given by

χ ≈ Dmav/2 (8.29)

in the thermodynamic limit. We now calculate the entanglement en-tropy of a block of size L, where the exact scaling with respect to L isnot important. Instead,

S(L) = m · logDav︸︷︷︸grows linearly with L

− log 2︸︷︷︸TEE

(8.30)

exposes the TEE as a constant for the whole honeycomb model. Ac-cording to Ref. [39], the first expression needs to grow linearly withthe size of the block.

These two examples show the usefulness of the TN representationof the exact eigenstates. Other interesting physical properties mightbe readily available, too.

9R E D U C T I O N T O A P E P S

The feasibility of reducing the structure of the spectral tensor net-work to that of a PEPS is interesting for several reasons. First, it wouldleverage the underlying idea of TNs to represent the wave function inan efficient way, scaling polynomially in the size as well as the bonddimension of the system. Second, a PEPS for the Kitaev honeycombmodel would be very relevant, due to the chirality of the model inthe gapless, topologically-ordered B-phase.The question arising is, whether this PEPS representation of the eigen-states is still exact. It should be possible to reduce the STN structureto a PEPS with finite bond dimension of 2 (or 4 due to PBC) in theToric Code limit of the Kitaev model. However in the gapless phasewe might only achieve approximate eigenstates.

A PEPS representation of the wave functions would give direct ac-cess to important properties such as the number of relevant bond in-dices. In this one could clearly see the convergence towards the ToricCode PEPS with D = 4 in the perturbative limit of the model. As anexample we consider the collapse of the STN in Fig. (8.15). Thereforewe need to express all the two-body gates generating the Bogoliubovtransformation and the Fourier transformation as two-side MPOs, ac-cording to Fig. (8.11). It is important to notice that the SVD, that splits

a) 0

0

=00

b) 0

1

=00

c) 1

0

=00

d) 1

1

=00

Figure 9.1: The overall parity needs to be even, so that the bond indexneeds to have even parity for a) and d) and odd parity for b) and c).Rectangular blocks denote values which are allowed to be differentfrom zero.

up the tensors, does not preserve the fermionic parity for the bondindex. It has to be recuperated into the tensors by gathering the cor-

79

80 reduction to a peps

rect entries for each fermionic sector. This is visualized in Fig. (9.1).Input and output mode 0 corresponds to even parity, whereas 1 corre-sponds to odd parity. The parity of the bond index needs to be chosenaccording to the overall fermionic parity, which has to be even. There-fore the elements of the 4× 1 bond index vector need to be permutedin order to get the configuration in Fig. (9.1). The rectangular blocksdenote values which are allowed to be different from zero.The PEPS network for the 8-spin honeycomb lattice is shown in Fig. (9.2),where each physical index denotes two spin degrees of freedom. Un-

1 2 3 4

Figure 9.2: 2 × 2 fermionic PEPS network for the eigenstates of the8-spin honeycomb model. Each physical index denotes two spin de-grees of freedom, hence the physical dimension is 4.

fortunately we can not see the reduction to a maximal number ofD = 4 in the TC limit of the model due to the inherent small systemsize. For the lattice of eight spins the maximum bond dimension isexactly D = 4. It is therefore necessary to move to larger system sizes,for which the construction of the state becomes more challenging.Not only is it necessary to include a large amount of swap gates, nei-ther there is a systematic way to perform the Bogoliubov transforma-tion. The construction of PEPS states for larger lattices is still work inprogress. Here the contraction should be performed layer after layer,with truncation of the bond dimension in between, if necessary.

Part IV

C O N C L U S I O N A N D O U T L O O K

10C O N C L U S I O N A N D F U T U R E W O R K

In this project we investigated the Kitaev model on the 2d honey-comb lattice with regard to the construction of exact eigenstates. Weanalyzed the necessary transformations to solve the model by a reduc-tion to free fermions in order to construct a unitary tensor networkrepresentation thereof. Combining all parts successively, the overallstructure can be interpreted as a quantum circuit that transforms aninitially unentangled quantum state of fermions to a strongly corre-lated quantum state of spins. This construction is exact up to smallcorrections on the boundary terms, which are introduced due to theassumption of antiperiodic boundary conditions. For each step ofthe construction the scaling to the thermodynamic limit is known, inwhich the contribution of the boundary terms disappear. This proce-dure was numerically checked for the vortex-free sector of the modelon small lattices of up to 32 spins.Besides the interpretation in terms of a quantum circuit to generateeigenstates of the honeycomb model, the STN we constructed maycontribute to the research on chiral PEPS. If it is possible to contractthe STN in a particular way to generate a PEPS on the honeycomb lat-tice and if this PEPS representation is exact in the gapless B-phase,we would have constructed a chiral PEPS with conic dispersion re-lation. As to our knowledge this would be the first one of its kind,so far only chiral PEPS with quadratic dispersion relations have beenengineered. Additionally the Kitaev honeycomb model presents a re-markable model, due to its ability for topological quantum computa-tion [29]. Thus even an approximate PEPS of the model would be veryinteresting.The PEPS representation of the eigenstates, exact or as a reasonableapproximation, benefits the analysis of the physical behaviour due toits enhanced complexity in terms of computational cost. It would beinteresting to study and reproduce the phase transition in the model,as well as long-range correlations for instance. These considerationswill be subject to future work.

The findings of this project are summarized in a paper, that is atthe final stage of preparation and will be submitted for publicationsoon.

83

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Master Thesis Condensed Matter Theory - KOMET 337 ... · quantum Hall effect, which is still not fully understood [1]. This ex-empliﬁes the necessity of a theoretical understanding