Homology theory and MBL in supersymmetric systems

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Homology theory and MBL in supersymmetric

systems

Joris van der Hijden

August 18, 2019

Bachelor thesis Mathematics and Physics & Astronomy Supervisor: prof. dr. Sergey Shadrin, prof. dr. Kareljan Schoutens

Institute of Physics

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

This thesis studies a supersymmetric model based on partial symmetries. It was first defined by Padmanabhan et al[Pad+17] and studied based on the suspicion that many body localisation occurs in this model. In this thesis the entanglement entropy of a specific density state is found. The result is in line with the expectation of Padmanab-han et al. The entanglement entropy is linked to a another concept in quantum theory, the out of order time correlator, which could speed up future calculations of the en-tanglement entropy. Lastly the Witten index and number of groundstates of the model are determined. This is done using homology theory, the Poincar´e polynomial and the homological pertubation lemma[Cra04].

Title: Homology theory and MBL in supersymmetric systems Authors: Joris van der Hijden, jvanderhijden@gmail.com, 11306955 Supervisor: prof. dr. Sergey Shadrin, prof. dr. Kareljan Schoutens, Second grader: dr. Raf Bocklandt, dr. Ben Freivogel,

End date: August 18, 2019 Institute of Physics

University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1 Introduction

Fifty years ago, P.W. Anderson found the first rigorous proof for absence of diffusion in strongly disordered theoretical models. His research was inspired by experimental results that showed that in highly disordered crystals, single electrons could be trapped. He designed a model on a Zd lattice where something very similar occurred. Ever since, researchers have been trying to find a similar result in models with multiple interacting particles. More specifically, models in which the eigenstate thermalisation hypothesis (ETH) does not hold[AP17]. The ETH states that in isolated systems, the individual excited eigenstates have expectation values of physical observables, which can be predicted by statistical physics, with the Gibbs and micro canonical ensembles. Models with multiple interacting particles that do not satisfy the ETH are called many body localised models (MBL).

In this thesis, one such model is studied, namely the model introduced by Padman-abhan et al[Pad+17]. It is based on partial symmetries and symmetrical inverse semi-groups (SIS). The main model examined is based on a SIS with nine elements, partial bijections of {1,2,..,n} to itself. In the case n = 3, which is most studied, the elements can be represented by arrows between {1, 2, 3}. The multiplication is defined by

com-Figure 1.1: The model S₁3 with ten elements, the last one being zero.

position. This thesis starts with building the necessary prerequisites in physics (chapter 2) and mathematics (chapter 3) to study the model. Then, in chapter 4, the model is introduced formally. The model studied is 2-fold supersymmetric, which yield us tools to study the properties of the model.

In chapter 5 and 6 two properties of the model are examined. In chapter 5, the Wit-ten index and the number of groundstates of the model are determined using homology theory. In the derivation the homological pertubation lemma[Cra04] is used. The calcu-lation is based on an article by La et al [LSS18], where the same procedure was used to find the number of groundstates in the Z2 Nicolai model, an extensively studied model

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in MBL theory.

In chapter 6, the entanglement entropy (EE) is used to indicate the occurrence of MBL in our model. Furthermore, a theorem is proven that links the entanglement entropy to the out of time order correlator (OTOC). This is useful, because the EE is often hard to calculate whereas the OTOC is a well understood object in MBL theory.

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2 Physics prerequisites

This chapter will introduce the reader to some key concepts in quantum physics used throughout this thesis. The main two are super symmetry and the Renyi entropy.

2.1 Super symmetry

All models in this thesis have a few noteworthy properties, first of all, all of them are 2-fold super symmetrical.

Definition 2.1.1 (2-Fold super symmetry). Let H be a Hilbert space and the state space of our model, with Hamiltonian H. H, H is called super symmetric if there exists an linear operator Q on H such that

Q2= Q†2= 0, H =nQ, Q†o= QQ†+ Q†Q =Q + Q†2

Remark 2.1.2. Note that the eigenvalues of H are positive, since for all eigenvectors |ψi ∈ H hψ| H |ψi = hψ|Q + Q†2|ψi =DQ + Q†ψ Q + Q†ψE = |Q |ψi|2+ Q †_|ψi 2 ≥ 0, because of the positive definiteness of the norm.

This also implies that |ψi ∈ H is a ground state if and only if Q |ψi = Q†|ψi = 0 The reason this is called 2 fold super symmetry is because there exists a two fold degen-eracy in the energy spectrum. By construction of the Hamiltonian both Q, Q†commute with the Hamiltonian and must thus conserve the energy. Thus we can arrange all elements with nonzero energy into doublets {|ψi , Q†|ψi}, which are sometimes called superpartners. This observation does not hold for ground state (a state with energy zero), since for a ground state Q |ψi = Q†|ψi = 0. This gives us a way to differentiate between states with zero and nonzero energies that will be instrumental to chapter 5.

A second important operator in 2-Fold supersymmetric is the fermion number operator F . It is an operator that commutes with the Hamiltonian and we call the eigenvectors of H fermionic if their eigenvalue for F is odd and bosonic if the eigenvalue is even. It is used to grade the Hilbert space and for the purpose of this thesis, its is constructed in a way that that guaranties that the parity of |ψi and Q†|ψi are different. This implies that the superpartners have different parity.

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2.2 R´

enyi entropy

In the next section, we will define the Renyi entropy, before we do so, the reader will be introduced to the basic quantum theoretical machinery used in this definition. In the later chapters the entropy is used to find indications for MBL systems.

The first is the concept of a trace of an operator.

Definition 2.2.1 (Trace of an operator). Let {|ii} be a orthonormal basis of a Hilbert space H and let P be an operator on H.

Tr(P ) =X

i

hi|P |ii

Note that the trace does not dependent on the basis, a basis independent definition can be given by using the characteristic polynomial of the operator.

Using the definition of the trace, we can define the concept of a density operator. Definition 2.2.2 (Density operator). An operator ρ on H is a density operator if

1) tr(ρ) = 1

2) ρ is a positive operator i.e. if |ψi ∈ H, hψ|ρ|ψi ≥ 0

The density operator is sometimes called a state, but remember that it is not a part of the state space H. We can prove that every density operator is of the following form. Remark 2.2.3 (Characterisation of density operators). Let ρ be a density operator then there is a set {pi, ψi|pi ∈ [0, 1], ψi ∈ H}, where i is an index set, and

P ipi = 1 such that ρ =X i pi|ψiihψi| .

Here pi can be interpreted as the probability associated with |ψii . Secondly, every

oper-ator constructed using a set of this form is a density operoper-ator.

Proof. Assume an operator ρ is a density operator. Then there exists a spectral decom-position

ρ =X

k

λk|ψkihψk| ,

with orthonormal vectors |ψki and real, non-negative eigenvalues λk. All eigenvalues

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density operator, the eigenvalues must sum to one, thus there exists a set {pk, ψk|pk ∈

[0, 1], ψk∈ H}, such that

ρ =X

k

pk|ψkihψk| .

Conversely, assume {pi, ψi|pi ∈ [0, 1], ψi ∈ H} is a set where i is an index set and

P

ipi = 1. Define the operator

ρ =X

i

pi|ψiihψi| .

Then Tr{ρ} = P

ipi = 1, thus the first condition for a density operator is satisfied.

Moreover, let |ψi ∈ H, then

hψ|ρ|ψi =X i pihψ|ψii hψi|ψi =X i pi| hψ|ψii |2≥ 0

This guaranties that ρ is positive thus satisfies both conditions for a density operator and concludes the proof.

This result gives an easier and more specific form for all density operators which will greatly simplify later proofs in this chapter. It is important to note that the set of states and probabilities associated with a density operator is not unique, thus there is no one to one correspondence between states and density operators. If it is possible to write ρ = |ψihψ|, for some |psii ∈ H, the density operator is called a pure state. If this is not possible, ρ is called a mixed state.

Remark 2.2.4. A general operator ρ is a pure state if and only if Tr ρ2 = 1.

Proof. Assume ρ is a pure state, then ρ = |ψihψ|, for some ψ ∈ H. So ρ2 = |ψi hψ|ψi hψ| = |ψihψ|, hence Tr ρ2_{= Tr(ρ) = 1.}

Assume T r(ρ2) = 1 and ρ =P

ici|iihi| is a mixed state, so ci is nonzero for more then

one i. Then T r(ρ2) =P

ic2i < 1 since for all nonzero ci, c2i < ci, which is a contradiction,

so ρ must be pure.

This co¨efficient Tr ρ2 is sometimes called the purity of ρ. It measures how mixed a state is. Then state with minimal purity is called the maximally mixed state. Its purity is 1/d, where d is the dimension of the system. Before introducing the final definition of this section, the second R´enyi entropy, we will introduce the partial trace as a final stepping stone.

A partial trace is defined on a Hilbert space H, such that H = A ⊗ B, where A, B are both also Hilbert spaces. The goal of the partial trace over B is to construct a intuitive map from operators on H to operators on A. For a separable operator O, i.e. if there

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are operators M, N on A, B respectively, such that O = M ⊗ N , this can be done by defining

TrB(M ⊗ N ) = Tr(N )M

The idea will be to extend this definition, using the fact that every element in the tensor product H is a finite sum of element of the form ai⊗bi, with ai, bifrom A, B respectively.

By this method it is also possible to deconstruct all operators on H, thus we can extend our definiton of the partial trace to a general operator on H.

Definition 2.2.5 (Partial trace). Let A, B be Hilbert spaces and H = A ⊗ B. Then the partial trace over B of an density operator ρ in H, with ρ = P

iMi⊗ Ni, where where

Mi, Ni are operators on A, B respectively and the index set i is finite, TrB(ρ) is given by

TrB(ρ) = TrB X i Mi⊗ Ni ! =X i Tr(Ni)Mi

Most often only the matrix elements of the resulting operator interest us. Then a more workable definition can be given by

hβ| Tr_B(ρ) |αi =X

i

hβ, i| ρ |α, ii (2.1)

Definition 2.2.6 (Reny´ı entropy). Let H = A ⊗ B, where A, B are Hilbert spaces. The second order Renyi entropy of a density operator, also known as the entanglement entropy, is given by SA(ρ) = − log Tr(ρA)2 = − logTr(TrBρ)2 .

To indicate why this definition is useful, we will show that the R´enyi entropy can be used to identify separable states.

Remark 2.2.7. A state ρ is separable, i.e. it can be written as a product of a density operators on A and B, if and only if T r(T rB(ρ)2) = 1.

To prove this, we will need a result from quantum information theory, the Schmidt decomposition[Mic10].

Theorem 2.2.8 (Schimdt decomposition). For all |ψi ∈ H = A ⊗ B, where A, B are complex vector spaces, there exist orthonormal states |iAi ∈ A, |iBi ∈ B such that

|ψi =X

i

λi|iAi ⊗ |iBi .

where λi are non-negative real numbers satisfying P_iλ2_i = 1. The size of the index set

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Proof remark 2.2.7. Assume T r(T rB(ρ)2) = 1, then T rB(ρ) is a pure state, i.e. T rB(ρ) =

|ψi hψ| for some |ψi ∈ A. In general it holds that ρ =P

iMi⊗ Ni, where where Mi, Ni

are operators on A, B respectively and the index set i is finite and assume i is minimal. Then it holds that

TrB(ρ) = TrB X i Mi⊗ Ni ! =X i Tr(Ni)Mi= |ψi hψ| =⇒ i = 1

Since different Mi can not be linear combinations of each other, since i is minimal, thus

ρ is separable.

Now assume ρ is separable, ρ = M ⊗N , where M, N are operators on A, B respectively. Now observe that the Witten index of a state is preserved by separable operators, which implies that M , must be pure, thus TrB(ρ2) = 1.

Informally, this gives rise to the interpretation of the Renyi entropy as a measurement of the level of entanglement between two subspaces. It is zero when a density operator is separable and increases when the state is more entangled.

For the proof of theorem 6.1.1 a final concept is introduced, a complete set of operators. In general this is defined differently, but to minimise confusion we will only introduce it by the most relevant characterisation.

Definition 2.2.9 (Complete set of operators). Let H be a Hilbert space, a set C is a set complete set of operators on H if it holds that

X

ˆ M ∈C

MijMlm= δimδjl

In the following chapters the necessary mathematical background will be defined, together with our model of interest to be able to apply all concepts of this chapter in chapter 6.

2.3 Witten index

A second important tool in analysing super symmetric systems is the Witten index. It is an useful concept in all of QFT and statistical mechanics, but because the models in this thesis are supersymmetric, we can simplify the general definition. In general the witten index ∆ of a system with statespace H and Hamiltonian H is given by

∆ = Tr

(−1)Fe−βH

.

Where F is the fermion number operator and β the inverse temperature.

From the introduction of super symmetry in the beginning of this chapter, we know that all states with non-zero energy form doublets, of which one is fermionic and one

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bosonic. This implies that all terms with nonzero energy cancel out, thus only terms with energy zero contribute to the trace over H. Thus for supersymmetric systems the Witten index reduces to

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3 Mathematical prerequesites

3.1 Chain modules

Definition 3.1.1 (chaincomplex). Let R be a ring. A chain complex M• consists of

1. for all n ∈ Z a R-module Mn

2. for all n ∈ Z a map dn: Mn→ Mn−1

such that dn−1◦ dn= 0

Later in this thesis this definition will mostly be used on vector spaces. Note that a vector space is nothing more than a module over a field.

To be able to compare two chain complexes we define a morphism between two chain complexes using morphisms between the modules.

Definition 3.1.2 (Morphism between chain complexes). A morphism between two chain complexes N•, M• consists of fn : Nn → Mn for all n ∈ Z, such that the following

diagram commutes.

If there exists an f−1 such that (f−1)n= (fn)−1 the two chain complexes are

isomor-phic.

Definition 3.1.3 (Homology module). Let M• be a chain complex of R-modules and

n ∈ Z. The nth homology module is defined as

Hn(M•) =

ker dn

im dn+1

and is an R-module.

Remark 3.1.4. Because the homology modules themself are R-modules again, a mor-phism bewteen two chain complexes induces a mormor-phism between their respective homol-ogy modules. The chain complex of homolhomol-ogy modules is denoted by H•(M•).

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Definition 3.1.5 (quasi-morphism). A morphism f : N• → M• between two chain

complexes is a quasi-morphism if the induced morphism between Hn(N•) → Hn(M•)

is an isomorphism of R modules for all n ∈ Z

Lastly we introduce the notion of an homotopy between two morphisms. This is the final piece before we can formulate the Homological perturbation lemma.

Definition 3.1.6 (Homotopy between two morphisms). Let N•, M• be chain complexes

and f, g : N• → M• morphisms. A homotopy between f, g is a set of maps {hn|n ∈ Z},

such that

gn− fn= d0n+1hn+ hn−1dn

We call f, g homotopic if there exists a homotopy between f, g.

3.2 Homological perturbation lemma

The homological pertubation lemma is concerned with the following setup:

Definition 3.2.1 (HE data). A homotopy equivalence data (HE Data) consists of

(N, d) (M, d0) h p

i

1. two chain complexes (N, d), (M, d0) with quasi i, p isomorphisms between them 2. a homotopy h between ip and 1

Using this, we formulate the main theorem of this section, the homological pertubation theorem. It will not be proven in this thesis, but used extensively in chapter 5.

Theorem 3.2.2 (Homological pertubation lemma). Consider the following HE data:

(N, d) (M, d0) h p

i

Let δ be a map on the chain M of the same degree as d0, such that (d0+ δ)2 = 0, i.e. (M, d0+ δ) forms a chain complex, then δ is called a smal perturbation. Furthermore it must hold that (1 − δh) is invertible. Now define

A = (1 − δh)−1δ

i1= i + hAi, p1 = p+pAh, h1 = h + hAh, d1 = d + pAi

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(N, d1) (M, d0+ δ) h1

p1

i1

is a HE data.

As mentioned, a proof of this theorem is beyond the scope of this thesis and can be found in [Cra04]. The consequence of theorem 3.2.2 we will use most is as follows. Result 3.2.3. If the conditions of theorem 3.2.2 are met, it holds that

H•(N, d1) ∼= H•(M, d0+ δ)

These concepts will be applied in chapter 5. To simplify the notation the concept of a graded vector space is introduced.

Definition 3.2.4 (Graded vector space). Let Vn with n ∈ Z be vector spaces, then

V =M

n∈Z

Vn

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4 The S

₁

n

model

In this chapter the S3₁ model is introduced. It is the main focus of this thesis, but fits in a larger family of models, denoted by S₁n. It is a family of supersymmetric models that based on symmetric inverse semigroups (SIS). A concept that will not be explored no great extend, but yields an easy way to construct the Hilbert space that forms the state space of our model. Later on we define supercharge operators on this Hilbert space and explore the consequences of our definitions. Not all concepts in this chapter will be formally defined, since they are only used in the construction and will not be used elsewhere in the thesis.

4.1 Symmetrical inverse semigroup

Definition 4.1.1 (Inverse semigroup). An pair (S, ∗) of a set S and a binairy operation ∗ on S is an inverse semigroup if

∀a, b, c ∈ S (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀x ∈ S ∃!y ∈ S y ∗ x ∗ y = y & x ∗ y ∗ x = x.

In this setting y is still called the inverse of x. Note that this defines something that looks like a group, but without an identity. The concept of an inverse is adapted to still fit in this context. This is reasonable, because the inverse in an inverse semigroup is still unique. Note that this is not guarantied without demanding the uniqueness of the inverse in the definition.

The S₁n model We will define an inverse semigroup S₁n using partial bijections on Sn= {1, 2, 3, . . . , n}. A partial bijection from X to Y is an injective map from a subset of X to Y . In this case we examine maps where the subsets have one element i.e. a pair of elements in Sn. In the case of n = 3 we can represent these elements as in figure 4.1

To make an inverse semigroup out of these element, define the multiplication as the composition

xijxkl= δjkxil

or as combining the arrows. Note many multiplications of elements are zero. Moreover the operation in associative and every element has an inverse given by

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An inverse semigroup formed by this construction is called a symmetric inverse semi-group (SIS). To avoid confusion, observe that this does not mean the multiplication is commutative.

Hilbert space based on S₁n This construct can be turned into a Hilbert space by making a vector space over the elements in S₁n over C with the inner product naturally defined by

hx_ij|x_kli = δ_ikδjl.

This vector or Hilbert space is denoted by Hn₁ We can use the underling multiplication of S₁nto define operators on the Hilbert space H constructed in this section.

Before we do so, it is important to note that in physics not all elements of this Hilbert space are studied. An element x ∈ Hn₁\{0} is only called a state if it is normalised, i.e. hx|xi = 1.

4.2 Supersymmetry

In this section, a supersymmetric operator on the Hilbert space H will be defined using the SIS structure. To understand how this works, we will find the adjoint or dagger of an operator from S₁n on Hn₁. For xab∈ Hn1, xij ∈ S1n c ∈ C

hcxijxkl|xpqi = hxkl|c∗xjixpqi =⇒ (cxij)†= c∗xji

Define the operator q on the Hilbert space defined by Sn

1 over C as q = n X j=2 1 √ n(x1,j) , q †₌ n X j=2 1 √ n(xj,i) so q 2 _{= (q}†₎2 _{= 0}

These operators are supercharges by definition 2.1.1 and give rise to the following Hamiltonian.

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H =nq, q†o= x11+ 1 n n X i,j=2 xij,

n=3 From now on, we will almost exclusively examine the case where n = 3. We will explore its structure more extensively in this paragraph. On Hn₁ the supercharge and Hamiltonian are given by

q =√2 (x12+ x13) , H = x11+

1

2(x22+ x23+ x32+ x33) . (4.1) This splits the basis elements of the state space H3₁ into two parts. Either qxij = 0 or

q†xij = 0 . All elements where i = 2, 3 are in the former group, while elements with i = 1

are in the latter. Furthermore, q, q† are maps between the two groups in the following way. Where we call the left group bosons and the right group fermions. We introduce

Figure 4.1: The 2-graded structure of H3₁, with a fermionic subspace Hf and a bosonic

subspace Hb, such that H31 = Hb⊕ Hf.

the operator F to find if a state is fermionic or bosonic. It is given by the projection on H_f.

F = x22+ x33

A state x ∈ H3₁ is fermionic if F x = x and bosonic if F x = 0.

It is also useful to determine the eigenstates and eigenvalues (energies) of the Hamil-tonian H. There are three types, three (fermionic) ground states and six states with energy one. The three ground states are given by

zi = √1

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and are fermionic. There are three more fermionic eigenstates.

fi = √1

2(|x2,ii + |x3,ii) with i ∈ {1, 2, 3}. (4.3) All these three states have energy one. Finally there are three bosonic eigenstates given by

bi = x1i with i ∈ {1, 2, 3} (4.4)

These eigenstates will be used quite frequently in chapter 5. Something useful to note is that all non-groundstates have a superpartner, i.e.

qfi = bi & q†bi = fi,

while the ground states do not. This will enable the construction described in chapter 5.

This description of the eigenstates spilt by their there fermion number gives us a new way to write the Hamiltionion, which follows naturally from the odd shape of the Hamiltonion defined in equation 4.1.

H = x11+ 1 2(x22+ x23+ x32+ x33) = M + P with M = x11, P = 1 2(x22+ x23+ x32+ x33) and M2= M, P2= P

Here M, P are projections on Hb, Hf respectively. We will refer to them as the bosonic

and fermionic part of the Hamiltonian. Now that we have established the terminology and definitions of this section, we can explore different systems based on H3₁.

4.3 A chain of Hilbert spaces

Until now, we have only looked at a system with one particle. This section is dedicated to using the terminology of the previous two sections to describe a chain of particles with interaction. The state space for our multi-particle setup is simply H =NN

i=1H31,

where H₁3 is a copy of the Hilbert space defined in the previous sections and N is odd. In the remainder of this chapter, we will indicate the site the operator R is working on with Ri, where i is the index of the site.

Remark 4.3.1 (Sign rule). An important note is that for this setup to work, usually we must guaranty that operators working on different sites anti-commute, this can be done in two ways, first as done in the article by Padmanabhan [Pad+17], a coefficient can

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be added to the local supercharges q, q† that counts the number of fermions the opera-tor ”passes through” before reaching the site it works on. This can be achieved by the coefficient

Y

1≥g<i

(−1)Fj_,

if the operator acts on site i. The second way is to encode this into the structure of the tensor product by means of the Koszul sign rule. For almost all steps of the calculations in this thesis, it has no influence, since N is odd. Only at the end of chapter 5 it will make an appearance, thus we have chosen to suppress this relation for all places where it serves no purpose. But note that to fully formalise this model, it is important to consider.

Choosing the supercharge After defining the state space we can choose multiple supercharges, two useful ones are defined by

Qlr = N Y i=1 ciqi & Qsr = N −2 X i=1 ciqiqi+1qi+2. (4.5)

One with only short range (sr) interaction and one with interaction over the entire chain. The effect of this can be seen in the corresponding Hamiltonians.

Hlr= {Qlr, Q†_lr} = N Y i=1 ciMi+ N Y i=1 ciPi Hsr= {Qsr, Q†sr} = N −2 X i=1 ciMiMi+2Mi+3+ N −2 X i=1 ciPiPi+2Pi+3

In the former Hamiltonian the energy of a state is influenced by all other sites. If one site has energy zero, the total state has. In the latter the influence of a site on the energy is only through its own and neighbouring sites. In the next chapters the coefficients will all be 1 for simplicity. Lastly the fermion operator F in this context is simply given by the sum of the fermion operators on a single site. Now that the model of interest is defined, we will explore a few important properties of the model in the next two chapters.

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5 Ground state generating function

The goal of this chapter will be to calculate both the number of groundstates and the Witten index for an N site system HN with the local interaction Hamiltonian Hsr, based

on the supercharges

Qsr= N −2

X

i=1

ciqiqi+1qi+2 & Q†sr = N −2 X i=1 ciq † iq † i+1q † i+2. (5.1)

This reason this is useful, is that the Witten index has been linked to the breaking of supersymmetry. To calculate both quantities, we will use two important results from the last chapters

1. All nonzero energy states form doublets {|ψi , Q†|ψi} and ground states do not. 2. Given HE data

(N, d) (M, d0) h p

i

there exists an isomorphism of Homology modules, Hn(N, d1) ∼= Hn(M, d0+ δ),

combining definition 3.1.5 and the main perturbation lemma 3.2.2.

To combine these two results a ground state generating function is used. After its introduction it will become clear what is left to prove in the remainder of this chapter.

Before we introduce the ground state generating function, observe that HN =NNi=1H31

is a graded vector space, since it is possible to split states based on fermion number. If we define Vn, n ∈ N as all elements of H with fermion number n. Then Vn is a vector

space for all n and

H_N =

N

M

n=0

Vn.

Now recall that Q† lowers the fermion number by three. We will use this to construct something analogous to a chain module out of HN, since (Q†)2 = 0. Thus it is possible

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Now the ground state generator function of Hn, with Hamiltonian Hsr is defined by

PN(z) =

X

i∈N

dim(Hi(HN, Q))zi,

which is the Poincair´e polynomial of the homology of Q†. Since the maximal fermion number in an N -site system is N , all terms with i > N are zero. To explain why this is called the ground state generating function, recall that that all non-groundstates form doublets {|ψi , Q†|ψi} and groundstates do not. Moreover, Q† applied to any ground state yields 0. Thus if V_i0 are all groundstates with fermion number i and V_i6=0 all non-ground states with fermion number i, with Vi∼= Vi0⊕ V

6=0 i , then ker(Q|Vi) ∼= V 0 i ⊕ Q†V 6=0 i+3 im(Q|Vi+3) = Q †_V6=0 i+3

Which implies that

dim(Hi(HN, Q)) = #groundstates with fermion number i.

This gives a way to relate the polynomial PN(z) to the number of groundstates and the

Witten index, using the fact that a state with an odd fermion number is fermionic and with an even fermion number is bosonic.

PN(1) = N X i=0 dim(Hi(HN, Q)) = #groundstates PN(−1) = N X i=0

dim(Hi(HN, Q))(−1)i = #bosonic groundstates − #fermionic groundstates = ∆

Hence PN(z) is called the ground state generating function. The remainder of this

chapter is used to proof the following theorem.

Theorem 5.0.1 (Ground state generating function). The recursion of the ground state generating function is given by

PN(z) = zPN −1+ (z2+ 3z)PN −2(z) + (z3+ z2+ 2z)PN −2(z)

The proof will calculate dim(Hi(HN, Q)), by choosing proper HE data, such that the

perturbed HE data are convenient for further calculations. Before the start of the proof, we make two remarks to avoid over complicated notation. First of all, note that all eigenstates of the local Hamiltonian fi, bi, zi, which are used extensively in this proof, behave independently of their index i. We will thus suppress the index in this proof. Secondly, many times in this proof, indices used to denote which object in the chain is meant are suppressed, since it is obvious from context and removes clutter in the notation.

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5.1 Proof of the ground state generating function

Proof. Define dN, d1, d2 in the following way:

d(N )= N −2 X i=1 q_i†q_i+1† q_i+2† = q_n−2† q_n−1† q_n† + N −3 X i=1 q_i†q†_i+1q_i+2† = d1+ d2

Observe that d2₁ = d2₂ = 0, thus can be both be used as differentials. We will split H_N into three sub-spaces, depending on the effect of d1. Let |ψi ∈ HN, if |ψi =

|b_NbN −1bN −1. . .i, then d1|ψi = |fNfN −1fN −1. . .i, if not then d1|ψi = 0. We will split

HN accordingly, HN = V1⊕ V2⊕ V0 = |bNbN −1bN −2i ⊗ HN −3 ⊕ |fNfN −1fN −2i ⊗ HN −3 M k,l,m |kNlN −1mN −2i ⊗ HN −3,

where k, l, m ∈ {b, f, z}3\ {(b, b, b), (f, f, f )}. Note that the tensor product is purely for notation sake. We can conclude that all elements in the kernel of d1 are in V0, V2 and

all elements in the image of d1 are in V0, i.e.

H•(HN, d1) ∼= V0

This allows us to define a natural inclusion i : H•(HN, d1) → HN, since V0 ⊂ HN.

Moreover we can also define a projection π : HN → H•(HN, d1), with kernel V1⊕ V2

and a map h : HN → HN given by qnqn−1qn−2 Note that the map iπ : HN → HN is

(H•(HN, d1), 0) (HN, d1) h

π i

Figure 5.1: HE-data

the projection onto V0 ⊂ HN. Since idHN− iπ is the projection with kernel V0 and the

projection on V1, V2 are given by hd1, d1h respectively, it holds that

idHN − iπ = d1h + hd1.

Thus h forms a homotopy between iπ and idHN.

To apply the main perturbation theorem, it must hold that (1 − d2h) is invertible. If

the inverse exists, it is equal toP∞

k=0(d2h)k. Observe that hd2h = qnqn−1qn−2 n−3 X i=1 q_i†q†_i+1q_i+2† ! qnqn−1qn−2 = n−5 X i=1 q†_iq_i+1† q†_i+2 =⇒ hd2hd2 = 0,

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both since (q†)2 = q2 = 0. Thus (1 − d2h) is invertible and (1 − d2h)−1 = 1 + d2h.

Since we have now proved that the data in figure 5.1 forms HE data and d2 is a small

perturbation, in line with result 3.2.3, it holds that

H•(H•(HN, d1), π(1 − d2h)−1d2i) ∼= H•(HN, d1+ d2) = H•(HN, Q).

Since we know the inverse of (1 − d2h) explicitly, we can simplify π(1 − d2h)−1d2i to

πd2i+πd2hd2i. In order to better understand the action of πd2i+πd2hd2i on H•(HN, d1),

it is useful to further examine H•(HN, d1). Since the inclusion map i is always applied

first, it is enough to examine V0 ⊂ HN. Recall that Lk,l,m|kNlN −1mN −2i ⊗ HN −3,

where k, l, m ∈ {b, f, z}3\ {(b, b, b), (f, f, f )}. Let us look at these 25 elements in more closely.

|zN . . . .i 9

|bNzN −1. . .i 3

|f_NzN −1. . .i 3

|. . . z_{N −2}i 4 and six more without z 6

25

Remark 5.1.1 (Homology of direct sum and product). Let A, B vector spaces and V = A ⊕ B, d : V → V such that d2 _{= 0 and d(A) ⊂ A, d(B) ⊂ B, then}

H•(V, d) ∼= H•(A, d|A) ⊕ H•(B, d|_B),

since the homology functor commutes with direct sums.

This remark will form the basis of the calculation of H•(H•(HN, d1), π(1 − d2h)−1d2i).

To show how, observe that

(πd2+ πd2hd2)(|zNi ⊗ HN −1) = πd2(|zNi ⊗ HN −1) = |zNi ⊗ d(N −1)(HN −1).

Where the first equality holds because qN|zNi = 0 and thus the second term maps all

to zero. In the second equality is used that the action of πd2 does not depend on |zNi

apart from the fact that we know that π acts trivially, since the resulting state is always in V0. Note that the injection map i is applied beforehand such that our notation of

|zNi ⊗ HN −1holds.

We can also conclude now that

H•(|zNi ⊗ HN −1, π(1 − d2h)−1d2i)) ∼= |zNi ⊗ H•

H_{N −1}, d(N −1).

For all ten other subspaces of V0 that have a z in one of their last three positions a very

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H•(|...zN −mi ⊗ HN −m, π(1 − d2h)−1d2i) ∼= |...zN −mi ⊗ H•

H_{N −m}, d(N −m).

All six other terms must be examined in more detail, since the d2, h operators can change

the first three elements into something that is affected by h this time around. Let us list all six and explore elements from what subspace can be mapped where. To do so, remember that the projection map π has kernel V1⊕V2, so even though it is possible that

d2 maps an element to either of those subspaces, it have any influence on the homology.

Futhermore, the map d2only changes fermions in sites N − 1, N − 2 and non of the other

elements in the first three sites.

(1) |bNbN −1fN −2i ⊗ HN −3 (2) |bNfN −1bN −2i ⊗ HN −3 (3) |fNbN −1bN −2i ⊗ HN −3

(4) |bNfN −1fN −2i ⊗ HN −3 (5) |fNbN −1fN −2i ⊗ HN −3 (6) |fNfN −1bN −2i ⊗ HN −3

For some subspaces, the effect of πd2+ πd2hd2 is simple. Elements in (2) can be mapped

to (2),(4), while elements in (4) are mapped to elements in (4). Note that the map d2

on (4) could change fN −1, fN −2 to bN −1, bN −2, but then the resulting element is in the

kernel of h, π. Using this, we can draw the conclusion that H•((2) ⊕ (4), πd2+ πd2hd2) ∼= |bN, fN −1i ⊗ H•

H_{N −2}, d(N −2).

Analogous this the same can be concluded for (3),(5). Elements in (6) are simply mapped to (6), since the element in position N − 2 is bosonic, the map πd2+ πd2hd2 constricted

to (6) is dN −3. The same holds for (1), which allows us to conclude that:

H• HN, d(N ) ∼= |zNi ⊗ H• HN −1, d(N −1) ⊕ |bN, zN −1i ⊗ H• HN −2, d(N −2) |f_N, bN −1i ⊗ H• H_{N −2}, d(N −2) ⊕ |b_N, fN −1i ⊗ H• H_{N −2}, d(N −2) |fN, zN −1i ⊗ H• HN −2, d(N −2) ⊕ |bN, bN −1, zN −2i ⊗ H• HN −3, d(N −3) |f_N, fN −1, zN −2i ⊗ H• H_{N −3}, d(N −3) ⊕ |b_N, bN −1, fN −2i ⊗ H• H_{N −3}, d(N −3) |fN, fN −1, bN −2i ⊗ H• HN −3, d(N −3) Recall that PN(z) = N X i=0 dim Hi HN, d(N ) zi

Because it holds that dim(V ⊕ W ) = dim(V ) + dim(W ), we can take the dimension for all terms separately. For example

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dim|z_Ni ⊗ H_iH_{N −1}, d(N −1)= dimHi H_{N −1}, d(N −1) =⇒ N X i=0 dimHi |z_Ni ⊗ H_iH_{N −1}, d(N −1)zi= PN −1

Here Remark 4.3.1 is of vital importance. During this entire calculation we have ignored that all operators working on HN −1, HN −2, HN −3 pick up a −1 for each fermion in the

terms ψN −1, ψN −2, ψN −3. This can be corrected by multiplying the resulting poincare

polynomial with a factor z for each fermion in the leading terms [LSS18]. This yields a recursion for PN(z) given by

PN(z) = zPN −1+ (z2+ 3z)PN −2(z) + (z3+ z2+ 2z)PN −3(z)

which concludes the proof of theorem 5.0.1

5.2 Conclusion

The result proven in the last section, allows us to calculate both the Witten index and the number of groundstates of the model HN with Hamiltonian Hsr. To do so three initial

values are needed. The number of groundstates and the Witten index of H1, H2, H3.

Note that our original definition of the Hamiltonian is not defined for N < 3, since only interactions between three elements can contribute to the Hamiltonian, thus all states in H1, H2 have energy zero.

For N=1 All three states are groundstates, two of which are fermionic, thus the Witten index is −1.

For N=2 All nine states are groundstates. he Witten index is a little harder to calculate, but is equal to −1, since all groundstates that are not of the form |z2z1i have

either an f or a b, which can be interchanged to change a boson into a fermion and vice versa. Thus those terms must cancel out.

For N=3 A state is a ground state if one of its three elements is z, thus there are 27 − 23= 19 groundstates. Similar to the reasoning in the case N = 2, again, the Witten index is equal to −1

Thus, for the number of groundstates of the N site model, #GN, it holds that the

recursion is given by PN(−1), i.e.

#GN = PN(1) = PN −1(1) + 4PN −2(1) + 4PN −3(1)

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with #G1 = 3, #G2 = 9, #G2 = 19. This recursion is enough to find the the number

of groundstates for all N . Unfortunately we have not been able to find a closed form for the recursion.

For the Witten index holds that

∆N = PN(−1) = −PN −1(1) − 2PN −2(1) − 3PN −3(1)

= −∆N −1− 2∆N −2− 3∆N −3,

with ∆1= −1, ∆2 = −1, ∆3 = −1. Again, we were not able to find a closed formula.

A very important observation is that in this chapter, we suppressed the difference between z1, z2, z3 and the same for f, b, since their behaviour is identical. This does not affect the calculation, but to find the actual number of groundstates and the Witten index, one needs to multiply ∆N, #GN by 3N to correct for this.

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6 OTOC-EE

Something that has been known to be able to indicate whether or not MBL occurs is the entanglement entropy. ALso called the second order R´enyi entropy as defined in definition 2.2.6. In MBL systems an logarithmic growth of the EE occurs[Den+17]. Unfortunately we are not able to reverse the implication, but an logarithmic increase of the EE is an indication for MBL. In practice it turns out that this quantity is not always easy to calculate. The goal of this chapter is to link the EE to a different concept from the study of MBL, the Out of Time Order Correlator (OTOC). It is easier to determine and already studied in relation to the rate of scrambling of information[HMY17]. We will start by formulating the main theorem that links the OTOCs and EE. Before proving an example is explored to get an intuition as to why the theorem holds and to motivate its use. The chapter finishes with conclusions about the occurrence of MBL in the model defined in chapter 4.

6.1 Formulation and example of OTOC-EE theorem

The Out-of-Time-Order Correlation Entanglement Entropy theory for β = 0, where β is the inverse temperature, states the following:

Theorem 6.1.1 (OTOC-EE). Let S_A(2) be the second R´enyi entropy of A, CBa complete

set of operators on B and O a quench operator with T r(OO†) = 1. Then the following equality holds:

exp−S_A(2)OO†(t)= X

M ∈CB

TrM†(t)OO†M (t)OO†. (6.1)

Note that this theorem is very general and does not depend on choices made earlier in this thesis. It does not even depend on super symmetry. To illustrate this theorem and to get familiar with the models introduced, we will explore an example.

Example 6.1.2 (Otoc EE for Hlr). Let H =

NN

i=1H 3

1 be the state space with the

Hamiltonian Hlr(4.5). Let the quench operator be

O = r

3

4 · 9N(1 + q1)

Lastly we split the state space into H = HA⊗ HB, where the B-subsystem is any single

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complete set on B as {(xkl)j}3kl=1. Now it holds that

exp−S_A(2)OO†(t)= X

M ∈CB

TrM†(t)OO†M (t)OO†. (6.2)

In this chapter we will not always specify the domain and co-domain of operators, but it is clear from context what is meant. To show that the conditions for theorem 6.1.1 are met, we will show that Tr O O† = 1 and {(xkl)j}3kl=1forms a complete set of operators.

Firstly it holds that

TrO O†= 3 4 · 9N Tr 1 + q1+ q † 1+ q1q † 1 .

Since the trace is linear, we can separately calculate the trace of each element of the sum. The trace of the identity is simply the dimension over which the trace is taken. In our case, this is 9N. To calculate the other terms, it is useful to specify a basis which is used to calculate the trace. The basis we will use to calculate the trace are the combinations of local eigenvectors of the local Hamiltionian, i.e. elements of the form

n

O

l=1

|ψ_li_l,

with ψlbeing one of the eigenvectors defined in equations 4.2-4.4. If not stated otherwise,

this basis will be used throughout this chapter. An important observation is that neither q nor q† has eigenvectors, since both operators are nilpotent. Thus, their trace must be equal to zero. For the last term q1q₁†= M1, expand the trace to

Tr q1q1† = X ψl∈{bi,zi,fi} n O l=1 hψl|lq1q†1 n O l=1 |ψlil

Where all terms with ψ 6= bi do not contribute, thus Tr

q1q₁†

= 3 · 9N −1. Because there are 3 · 9N −1 basis states with a bosonic first element and all have eigenvalue 1 with M1.

This allows us to calculate the trace of O O†. Tr

O O†= 3 4 · 9N 9

N_{+ 3 · 9}N −1_{= 1}

To show that {(xkl)j}3_kl=1 forms a complete set. Note that

X

M ∈{(xkl)j}3_kl=1

MwzMxy = δwyδxz,

by the multiplication defined on S₁3.

We will prove that this theorem holds in example 6.1.2, something that has been done in [Pad+17] without much explanation. This proof will fill all gaps left in their proof.

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Proof. The EE of the described system is − log

Tr(TrBρ)2

, so the left hand side of equation 6.1.1 reduces to

(Tr(TrBρ)2)−1.

Recall that ρ(t) = O O†(t) = e−iHlrt_{O O}†_eiHlrt_{. To find ρ(t) we will first examine the}

building blocks of ρ(t), namely q1(t). The time evolution of q1 is given by

q1(t) = e−iHlrtq1eiHlrt= ∞ X n=0 1 n!(−iHlrt) n_{· q} 1· ∞ X m=0 1 m!(−iHlrt) m_.

Thus it is useful to calculate the commutator of Hlr and q1, where we use that [Mi, qi] =

qi and [Pi, qi] = −qi. Hlrq1 = N Y k=1 Mk+ N Y k=1 Pk ! q1= M1q1 N Y k=2 Mk+ P1q1 N Y k=2 Pk = q1(Mi+ 1) N Y k=2 Mk+ q1(Pi− 1) N Y k=2 Pk = q1Hlr+ q1 N Y k=2 Mk− q1 N Y k=2 Pk

Furthermore, observe that Hlr is a projection map, because Mi, Pi are also projection

maps. Thus Hlr is idempotent, which we use to simplify e−iHlrtq1. ∞ X n=0 1 n!(−iHlrt) n q1 = ∞ X n=0 1 n!(−it) n Hlrq1 = ∞ X n=0 1 n!(−it) n_q 1 Hlr+ N Y k=2 Mk− N Y k=2 Pk ! = q1 e−iHlrt+ e−it N Y k=2 Mk+ eit N Y k=2 Pk !

If we substitute this result in the definition of q1(t), the first term will cancel out.

q1(t) = q1 e−iHlrt+ e−it N Y k=2 Mk+ eit N Y k=2 Pk ! eiHlrt = q1 1 + e−it N Y k=2 Mk· eiHlrt+ eit N Y k=2 Pk· eiHlrt !

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Examine the second and third term individually and use that Mi, Pi are projectors to see that N Y k=2 Mk· eiHlrt= N Y k=2 Mk ∞ X m=0 1 m!(it) m N Y k=1 Mk+ N Y k=1 Pk ! = (1 − e−it) N Y k=2 Mk.

Something similar holds for for the last term, from which we conclude that

q1(t) = q1+ eit− 1 q1 N Y k=2 Mk+ e−it− 1 q1 N Y k=2 Pk. (6.3)

Since this is quite a long derivation, we will not expand all steps in later sections. If we recall the definition of O we can now rewrite the density operator ρ to

ρ = e−iHlrt_{O O}†_eiHlrt_{= e}−iHlrt r 3 4 · 9N(1 + q1) r 3 4 · 9N(1 + q † 1)eiHlrt = 3 4 · 9Ne −iHlrt_{(1 + q} 1+ q†+ q1q†1)eiHlrt = 3 4 · 9N(1 + q1(t) + q † 1(t) + (q1q1†)(t))

The first three terms are known, since taking the ad-joint of equation 6.3 yields the third term. By making the observation that

q1(t)q†₁(t) = e−iHlrtq1eiHlrte−iHlrtq₁†eiHlrt= q1q₁†(t),

it is also possible to calculate the final term. This results in the following conclusion. O O†(t) = 3 4 · 9N 1 + q1Ξi(t) + Ξ†i(t)q † 1+ q1Ξi(t)Ξ†i(t)q † 1 , where Ξi(t) = 1 + eit− 1 QN k=2Mk+ e−it− 1 QN

k=2Pk. The last term can be greatly

simplified using the fact that Mk, Pk are projections to disjoint subsets.

Ξi(t)Ξ†_i(t) = 1 + N Y k=2 Mk+ N Y k=2 Pk !

(eit− 1)(e−it− 1) + (eit− 1) + (e−it− 1) (6.4) = 1 + N Y k=2 Mk+ N Y k=2 Pk ! (2 − 2 cos(t) + 2 cos(t) − 2) = 1 (6.5)

This yields the simplified form of the density operator used in the rest of the calculations. O O†(t) = 3 4 · 9N 1 + q1Ξi(t) + Ξ † i(t)q † 1+ q1q † 1 , (6.6)

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Remember that the goal is to calculate the trace over HB of this operator. The

conve-nient basis in this cases is given by the eigenvectors of the local Hamiltonian Mj + Pj

on B. Recall that in that case the eigenvectors are |x_1,ki_k=1,2,3 & √1

2|x2,k± x3,kik=1,2,3.

We will use definition from equation 2.1 to calculate the partial trace over HB.

ha1| TrB(O O†(t)) |a2i = 3 X l=1 ha1, x1,l| ρ |a2, x1,li + 1 √ 2ha1, x2,l± x3,l| ρ |a2, x2,l± x3,li !

where a1,2 ∈ HA. In this thesis we will not cover every single term, but a few are done

explicitly. For example, the second term in equation 6.6. The operator q1 does not work

on HB, i.e. the jth site, and the effect of Ξi(t) is similar to the local Hamiltonian Hj.

Thus all groundstates do not contribute in the trace.

ha1| TrB(q1Ξi(t) |a2i = 3 X l=1 ha1, x1,l| q1Ξi(t) |a2, x1,li + 1 2ha1, x2,l± x3,l| q1Ξi(t) |a2, x2,l± x3,li = ha1| 3q1  1 + (e−it− 1) Y k6=1,j Mk  + 3 2q1 2 + (e it_{− 1)} Y k6=1,j Pk !! |a₂i

The factors of three appear since there are three choices of l, but they behave the same under the effect of the density operator. The next step is to do this for all terms in equation 6.6 and use the linearity of the trace to find the following result.

ha₁| Tr_B(ρ(t)) |a2i = 3 4 · 9N ha1| 9 + 3q1  1 + (e−it− 1) Y k6=1,j Mk  + 3 2q1 2 + (e it_{− 1)} Y k6=1,j Pk ! + 3q₁†  1 + (eit− 1) Y k6=1,j Mk  + 3 2q † 1 2 + (e −it_{− 1)} Y k6=1,j Pk ! + q1q†₁ ! |a2i

It is now possible to calculate (TrB(ρ))2.

(TrB(ρ))2= 3 4 · 9N 2 9 3 4 · 9N −1 TrBρ(t) + 9q1q₁† Y k6=1,j Mk+ 9 4q1q † 1 Y k6=1,j Pk(2 cos(t) − 2) + q1q†₁ !

Using a very similar argument to 6.4 to cancel out many terms by using the fact that q2 = (q†)2 = 0. The final step is to take the trace over H of (TrB(ρ))2. Again, by the

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Something to note beforehand is that all terms with either q1 or q†1 will not appear in

the final result, since they are both nil potent and must have trace zero.

The basis we will use to calculate the trace are the combinations of local eigenvectors of the local Hamiltionian, i.e. elements of the form

O

l6=j

|ψi_l,

with ψ being one of the eigenvectors defined in equations 4.2-4.4. There is a distinction to be made between the first and all other sites. On the first site only q1qi† = (x1,1)1 = M1

acts, thus the elements with |ψi_l being either zi, fi, the fermionic eigenstates, do not contribute to the final answer. Secondly, if a site is not affected by an operator, it simply adds a factor of nine for each site, for the dimension of that subspace.

Tr (TrB(ρ))2 = 3 4 · 9N 2 9 3 4 · 9N −1 Tr (TrBρ(t)) + 9 Tr  q1q1† Y k6=1,j Mk  + 9 4Tr  q1q1† Y k6=1,j Pk(2 cos(t) − 2)  + Tr q1q1† ! = 3 4 · 9N 2 9 3 4 · 9N −1 3 4 · 9N 9N + 3 · 9N −1 + 9 · 3N −1+9 4 · 3 · 3 N −2_{(2 cos(t) − 2) + 3 · 9}N −2 ! = 3 N +3 25_{· 3}4N(cos(t) − 1) + 3 4 · 9N 2 32N + 32N +1+ 3N +1+ 32N −3 =2−5 1 33N −3 (cos(t) − 1) + 3 4 · 9N 2 32N + 32N +1+ 3N +1+ 32N −3

This does not correspond completely with the answer found in [Pad+17], it is of by a factor 25 and a constant. The mistake in the derivation has not been found, but the behaviour of both answers in time is the same.

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To compute the OTOC, we have chosen the set {(xkl)j}3k,l=1 as a complete set of

operators on HB, where the j indicates the site on which the operator works. Recall

that j is fixed in this derivation. This leaves us with the following expression:

3 X p,q=1 Tr(xp,q)j(t) O O†(xq,p)j(t) O O† .

It will turn out to be useful to split the operators (xp,q(t))j based on whether both p, q

are 1, either one is equal to 1 or neither. In the case both are one we quickly find that (x11)j(t) = qjqj†(t), thus, by equation 6.3, (x11)j(t) = (x11)j. This yields the following

result: Tr (x11)j(t) O O†(x11)j(t) O O† = Tr (x11)jO O†(x11)jO O† = Tr (x11)j O O†2 = 3 4 · 9N 2 Tr(x11)j(1 + q1+ q†+ q1q1†) 2 = 3 4 · 9N 2 Tr(x11)j(1 + q1+ 2q1†+ 3q1q1†)

Since q1, q†1 are nilpotent, this again reduces to

3 4 · 9N 2 Tr (x11)j(1 + 3q1q†1) = 3 4 · 9N 2 9N −1(1 + 3) = 1 4 · 32N.

For all other cases, so in the cases that not both p, q are equal to one, we rewrite the equation to calculate the OTOC, using the fact that a trace is cyclic.

Tr (xp,q)j(t) O O†(xq,p)j(t) O O† = Tr e−iHlrt_(x p,q)jeiHlrtO O†e−iHlrt(xq,p)jeiHlrtO O† = Tr(xp,q)jeiHlrtO O†e−iHlrt(xq,p)jeiHlrtO O†e−iHlrt = Tr(xp,q)jO O†(−t)(xq,p)jO O†(−t)

And now all parts are know, thus the only task left is to evaluate this expression. By the definition of Ξi(t) it holds that Ξi(−t) = Ξi(t)†, which is used to simplify the expression

for O O†(−t). Combining all these, results in the conclusion that [Pad+17]

Tr (x11)j(t) O O†= 1 4 · 32N(x11)j(t) O O † = 1 4 · 32N Tr(x21)j(t) O O†(x12)j(t) O O† = Tr(x12)j(t) O O†(x21)j(t) O O† = Tr(x31)j(t) O O†(x13)j(t) O O† = Tr(x13)j(t) O O†(x31)j(t) O O† = cos(t) − 1 16 · 33N −3 + 1 2 · 32N Tr(x22)j(t) O O†(x22)j(t) O O† = Tr(x22)j(t) O O†(x22)j(t) O O† = Tr(x32)j(t) O O†(x23)j(t) O O† = Tr(x23)j(t) O O†(x32)j(t) O O† = cos(t) − 1 16 · 33N −3 + 1 2 · 32N.

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Not all terms have been checked by hand, the last two are taken directly form [Pad+17]. Combining all terms does lead to the conclusion that

3 X p,q=1 Tr (xp,q)j(t) O O†(xq,p)j(t) O O† = exp −S_A(2)OO†(t) .

This calculation has proven the usefulness of a theorem that links the EE to OTOCs, by showing that it is much easier to calculate the OTOC, since it does not require any partial traces. The second thing this proofs, is that the EE grows as

− log 1 33N −3(cos t − 1) + 1 2 · 32N −3 .

Which is a logarithmic growth, for early time t. That is an important indication that MBL occurres[Den+17].

6.2 Proof Otoc-EE

This section will be completely dedicated to the proof of the Otoc-EE theorem. To more easily understand why this theorem holds a graphical notation is introduced and the proof is thus illustrated by 6.1. This visual proof was first given in a paper by R. Fan[Fan+16].

Proof Otoc-EE. This proof will be given by first expanding both sides of equation 6.1.1 and then reformulating the RHS and is based on the proof in [Fan+16].

We will start with introducing the schematic notation and the LHS. Denote the basis of the full system by using the bases of A, B as {|ii_A⊗ |ii_B}. Now an general operator can be described as Q =P

ijQij(|iiA⊗ |iiB)(hj|A⊗ hj|B), with complex constants Qij.

Note that there are constrains on these constant, not every object of this form is an operator. The partial trace over B can be easily calculated if the operator is in this form:

TrBQ =

X

ij

Qij|ii_Ahi|_A.

We represent this schematically in figure 6.1 as (a1), (a2). We can represent the LHS of equation 6.1.1 in the same way. The reduced density is given by

TrB

U (t)OO†U (t)†

, (b), so the full LHS is equal to (c).

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Figure 6.1: Diagram for the proof of the otoc-EE theorem

The RHS can be represented in the same way. We will first rewrite the RHS slightly using the fact that the trace is cyclic and that M is an observable.

X M ∈CB Tr M†(t)OO†M (t)OO† = X M ∈CB TrM†U (t)OO†U†(t)M U OO†U†(t) = X M ∈CB Tr U (t)OO†U†(t)M†U (t)OO†U†(t)M . (d)

We should remark the following, if CB is a complete set of operators on B, then it holds

that,

X

M ∈CB

M†QM = TrBQ ⊗ IB.

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proof that X M ∈CB TrU (t)OO†U†(t)M U (t)OO†U†(t)M = Tr   U (t)OO†U†(t) X M ∈CB M†U (t)OO†U†(t)M   = Tr U (t)OO†U†(t) TrB U (t)OO†U†(t) ⊗ IB = TrB U (t)OO†U†(t) 2

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7 Conclusion

In this thesis, we have been able to find recurrence relations for the both the number of groundstates as the Witten index of the model Hn with Hamiltonian Hsr. The number

of groundstates is given by 3N#GN, where #GN can be determined by the recursion

relation

#GN = #GN −1+ 4#GN −2+ 4#GN −3,

with #G1 = 3, #G2 = 9, #G2 = 19. Unfortunately we have not been able to find a

closed form for the recursion.

Furthermore, the Witten index is equal to 3N∆N. ∆N can be determined using the

recursion relation

∆N = −∆N −1− 2∆N −2− 3∆N −3,

with ∆1= ∆2 = ∆3 = −1. Again, we were not able to find a closed formula.

Secondly, we have proven that the OTOC-EE theorem which, can be used to find indications for MBL in model and states the following:

Theorem 6.1.1 (OTOC-EE). Let S_A(2) be the second R´enyi entropy of A, CBa complete

set of operators on B and O a quench operator with T r(OO†) = 1. Then the following equality holds: exp −S_A(2)OO†(t) = X M ∈CB Tr M†(t)OO†M (t)OO† . (6.1)

This theorem can be used to link the often hard to calculate EE, to the well studied OTOC. The EE can be used to verify that MBL occurs in a model. We have also shown that for the model Hn with Hamiltonian Hlr the EE grows logarithmic for early times

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Bibliography

[1] Dmitry A. Abanin and Zlatko Papi´c. “Recent progress in many-body localiza-tion”. In: Annalen der Physik 529.7 (July 2017), p. 1700169. doi: 10.1002/andp. 201700169. arXiv: 1705.09103 [cond-mat.dis-nn].

[2] P. W. Anderson. “Absence of Diffusion in Certain Random Lattices”. In: Phys. Rev. 109 (5 Mar. 1958), pp. 1492–1505. doi: 10.1103/PhysRev.109.1492. url: https://link.aps.org/doi/10.1103/PhysRev.109.1492.

[3] M. Crainic. “On the perturbation lemma, and deformations”. In: arXiv Mathe-matics e-prints, math/0403266 (Mar. 2004), math/0403266. arXiv: math/0403266 [math.AT].

[4] Dong-Ling Deng et al. “Logarithmic entanglement lightcone in many-body localized systems”. In: prb 95.2, 024202 (Jan. 2017), p. 024202. doi: 10.1103/PhysRevB.95. 024202. arXiv: 1607.08611 [cond-mat.dis-nn].

[5] Ruihua Fan et al. “Out-of-Time-Order Correlation for Many-Body Localization”. In: arXiv e-prints, arXiv:1608.01914 (Aug. 2016), arXiv:1608.01914. arXiv: 1608.01914 [cond-mat.quant-gas].

[6] Koji Hashimoto, Keiju Murata, and Ryosuke Yoshii. “Out-of-time-order correlators in quantum mechanics”. In: Journal of High Energy Physics 2017.10, 138 (Oct. 2017), p. 138. doi: 10.1007/JHEP10(2017)138. arXiv: 1703.09435 [hep-th]. [7] Ruben La, Kareljan Schoutens, and Sergey Shadrin. “Ground states of Nicolai and

Z2 Nicolai models”. In: Journal of Physics A: Mathematical and Theoretical 52.2

(Dec. 2018), 02LT01. doi: 10.1088/1751-8121/aaf181. url: https://doi.org/ 10.1088%2F1751-8121%2Faaf181.

[8] Isaac L. Chuang Michael A. Nielsen. Quantum Computation and Quantum Infor-mation. Cambridge University Press, 2010.

[9] Pramod Padmanabhan et al. “Supersymmetric many-body systems from partial symmetries — integrability, localization and scrambling”. In: Journal of High En-ergy Physics 2017.5, 136 (May 2017), p. 136. doi: 10 . 1007 / JHEP05(2017 ) 136. arXiv: 1702.02091 [hep-th].

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Populaire samenvatting

Anderson lokalisatie en many body localisation Kristallen zijn structuren met veel symmetrie en bestaan uit dicht op een gepakte moleculen in een regelmatig patroon.

Figure 6.1: Theoretische weergave NaCl kristal

Een voorbeeld is een zoutkristal (NaCl), waarvan een ab-stracte weergave te zien is in figuur 6.1. In werkelijkheid zijn kristallen niet perfect regelmatig en zijn er onzuiverheden in de vorm van andere moleculen die de symmetrie in het rooster verbreken. Halverwege de vorige eeuw werd er on-derzoek gedaan naar kristalstructuren door er elektronen op af te vuren en te bekijken hoe het elektron er weer uit komt. Dit werd gedaan om een idee te krijgen van de microstruc-tuur van het kristal. Al snel viel op dat de tijd die het duurt voordat het elektron weer uit het kristal komt, langer wordt als er meer onzuiverheden in het kristal aanwezig zijn. Dit is ook in lijn der verwachting: het elektron wordt telkens verstrooid door botsing met de verschillende moleculen en verandert dus steeds van richting. Bij experimenten waarin de wanorde, een maat voor hoeveelheid onzuiverheden,

kun-stmatig werd verhoogd, viel iets op. Als het kristal onzuiver genoeg is, ontsnapt het elek-tron er helemaal niet meer uit. Intu¨ıtief kan dit verklaard worden doordat de golffunctie van het elektron sterk verstrooid wordt, en negatief interfereert buiten het kristal.

Om dit beter te begrijpen zijn er eenvoudige theoretische modellen gemaakt waarin hetzelfde gebeurde. Als eerste door P.W. Anderson[And58], die liet zien dat op een eendimensionale keten met een willekeurige potentiaal geen diffusie plaatsvindt. Dit is voor te stellen als een kralenketting waarop een deeltje kan bewegen. Op elke kraal waar het deeltje zich kan bevinden ken je een energie toe. De potentiaal is hoeveel energie het ”kost” om ergens te zijn. De kans dat het deeltje vervolgens springt tussen twee kralen, is afhankelijk van het verschil tussen de energie van de kralen en het deeltje. Daarbij is de kans om over te springen groter als de sprong de energie van het deeltje verlaagt. Wat Anderson bewees is dat, gegeven dat de spreiding van de energie¨en groot genoeg is, het deeltje uiteindelijk in een gebied vast komt te zitten. Het zal nooit naar rechts (of links) blijven bewegen. Het fenomeen dat deeltjes vast blijven zitten terwijl ze er wel uit zouden kunnen, heet Anderson lokalisatie.

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Als snel probeerde men dit resultaat te herhalen voor meerdere deeltjes met onderlinge interactie, maar dit bleek een stuk lastiger. In het geval van lokalisatie van meerdere deeltjes met onderlinge interactie spreekt men van many body localisation (MBL). In deze scriptie wordt een indicator van MBL onderzocht. Deze indicator heet de verstren-gelingsentropie (engels: entanglement entropy, EE). Entropie is een maat voor de snel-heid waarmee de wanorde in een systeem toeneemt. De verstrengelingsentropie meet hoe snel twee gebieden elkaar be¨ınvloeden. Als deeltjes vast komen te zitten in een gebied, zal de entropie dan ook langzaam toenemen. De verstrengelingsentropie neemt in An-derson gelokaliseerde modellen neemt dus niet toe. Als er interactie in het spel is tussen verschillende deeltjes komen de deeltjes niet vast te zitten, maar spreiden de deeltjes zich langzamer uit dan verwacht. Daarom neemt de verstrengelingsentropie langzamer toe dan in een model zonder MBL. Dus kan een traag stijgende verstrengelingsentropie een indicatie zijn dat MBL optreedt.

Het S3

1 model Het fenomeen MBL wordt in deze scriptie bestudeerd aan de hand

van een eenvoudig model (Hoofdstuk 4), waarin we verwachten dat MBL optreedt.

Figure 6.3: Het model S₁3 met negen elementen

Versimpeld weergegeven bestaat het model uit negen ”deeltjes” die we voorstellen met pijlen tussen de getallen {1, 2, 3}. We kunnen de ”interactie” tussen de deelt-jes bepalen door de pijlen te volgen, dus x11· x12 = x12, maar x12· x11 = 0, want

de pijlen zijn niet te combineren.

Vervolgens kunnen we elk deeltje ook een energie geven en loslaten op de kralen-ketting uit afbeelding 6.2. Het eerste doel van de scriptie is het bepalen van de ver-strengelingsentropie. Daarnaast bewijzen we dat de verstrengelingsentropie traag groeit (hoofdstuk 6). Het is daarmee niet bewezen dat er MBL optreedt, maar het is, mede vanwege andere redenen, wel aannemelijk te maken.

Een tweede deel van de scriptie (hoofdstuk 5) ging over het bepalen van het aantal deeltjes met energie nul, ook wel grondtoestanden genoemd. Hiervoor werd een andere eigenschap van het model gebruikt. Het is mogelijk om de deeltjes in twee groepen op te delen, waartussen een afbeelding is die de energie niet verandert. Dit wordt supersymmetrie genoemd. Uit eerder onderzoek blijkt dat alle deeltjes een superpartner in de andere groep hebben met dezelfde energie, behalve de deeltjes met energie nul. Dit geeft ons een manier om de deeltjes met energie nul te scheiden van de andere. Daarmee hebben wij in deze scriptie het aantal grondtoestanden bepaald.

De verstrengelingsentropie en het aantal grondtoestanden van het model, geven een beschrijving van de tijdevolutie van het model. Daarmee zouden andere indicatoren van MBL kunnen worden gevonden.

Homology theory and MBL in supersymmetric systems