MARIANNE THERESIA VIKTORIA SCHNEIDER

Fock Space Localization of Many-Body States in the Tilted Bose-Hubbard Model

BACHELOROF SCIENCE ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG

SUPERVISEDBY: Alberto Rodríguez González

30 July 2019

Abstract

In the present thesis, we study the localization properties in Fock space of the tilted Bose-Hubbard model, which describes a one-dimensional lattice with an additional linear potential in which a fixed number of bosons is allowed to tunnel to next-neighbouring sites and to interact pairwise when they are on the same site. In order to characterize Fock space localization, we make use of the form- alism of multifractal analysis, namely of the finite-size generalized fractal dimensions D˜ q . This approach allows us to unveil the development of the localization properties of states undergoing an avoided crossing, and more prominently, to identify avoided crossings in the parametric evolution of D˜ q . We identify a certain manifold of states with special localization prop- erties which are strongly localized on Fock states with all bosons on one lattice site, and we observe that there is no other manifold with compar- able characteristics. We study the parametric evolution of the localization properties and demonstrate that the manifold survives in regimes where all system parameters (hopping, interaction, and tilt strengths) have the same order of magnitude, i.e., when no individual term in the model’s Hamilto- nian dominates. Moreover, we see, that the stability of the robust manifold is independent of the number of lattice sites when fixing the number of bosons.

i Zusammenfassung

In der vorliegenden Arbeit untersuchen wir die Lokalisationsseigen- schaften im Fock-Raum des geneigten Bose-Hubbard-Modells, welches ein eindimensionales Gitter mit addiertem, linearen Potential beschreibt, in dem sich eine konstante Anzahl an Bosonen befindet. Letztere dürfen in die nächstanliegenden Gitterstellen tunneln, sowie paarweise miteinander interagieren, falls sie sich auf der selben Gitterstelle befinden. Um die Lokalisation im Fock-Raum zu charakterisieren, verwenden wir den Formalismus der Multifraktionalanalyse, genauer gesagt die generalis- ierten fraktalen Dimensionen D˜ q endlichdimensionale Fock-Räume. Dieser Ansatz ermöglicht uns, die Entwicklung der Lokalisationseigenschaften von Zuständen zu ergründen, welche eine vermiedene Kreuzung (auch “avoided crossing”) erfahren. Gleichermaßen können wir vermiedene Kreuzungen in der parametrischen Entwicklung von D˜ q identifizieren. Wir können eine bestimmte Mannigfaltigkeit von Zuständen mit speziellen Lokalisationsei- genschaften erkennen, welche stark auf Fock-Zuständen lokalisiert ist, die alle Bosonen auf einer Gitterstelle aufweisen. Gleichzeitig beobachten wir, dass es keine weitere Gruppe von Zuständen gibt, die ähnliche Eigenschaften besitzen. Wir erforschen die parametrische Entwicklung der Lokalisation- seigenschaften der Mannigfaltigkeit und erläutern deren Fortbestehen in Systemen, in denen alle Systemparameter (Tunnel-, Interaktions- und Nei- gungsstärke) in gleicher Größenordnung auftreten, d. h. keiner der drei Systemgrößen im entsprechenden Hamiltonoperator überwiegt. Darüber hinaus beobachten wir, dass die Stabilität der robusten Mannigfaltigkeit unabhängig von der Anzahl an Gitterstellen im Gitter ist, sofern wir die Zahl der Boson konstant halten.

ii Acknowledgements

Dear Andreas, thank you for enabling me to write my bachelor thesis in your group. You succeeded in creating a friendly and productive atmosphere to work in and made it enjoyable to come up to the 9th floor every morning. I honestly appreciate your efforts concerning the personal development of the members of the group. Dear Alberto, do you remember the moment when we configured the printer from my laptop, I wanted to name it “qscolour” and you waited until I discovered the typo? I agree with you, it was funny. If I had not been too tired, I would have definitely joined your laughter. Throughout the past three months, it has been a great pleasure working with you and benefiting from your skills. I have learned an enormous amount of new things in many respects, caused by your patience, your high demands and your attention to detail. You deserve my thanks and utmost respect. Dear members of the 9th floor, it was a huge pleasure to get to know you. I enjoyed your integration endeavours, the curving sessions, the coffee break conversations, the politischer Stammtisch, and amusing football matches. Edo, Nico and Slava, thanks for sharing the office with me and being the first contact point for my problems. A special thanks goes to Severin for his Latex support, and to Janine (I will miss you, Greenly) for her Mathematica help.

A Part of this work was performed on the computational resource bw- UniCluster funded by the Ministry of Science, Research and the Arts Baden- Württemberg and the Universities of the State of Baden-Württemberg, Ger- many, within the framework program bwHPC. Liebste WG, herzlichen Dank für die vielen gesprächigen Nudel-mit- Pesto-Abende. Mella, jedes Mal wenn wir uns sehen, geht mir das Herz auf. Tim, danke für nächtliche Diskussionen und Skateabende. Und zuletzt ein Danke an meine Familie, Mampa, Chrissi und Verena. Ihr wart mir immer ein sicherer Hafen und werdet auch immer einer bleiben.

iii

CONTENTS

Contentsv

1 Introduction1

2 Formalism3 2.1 The Tilted Bose-Hubbard Hamiltonian...... 3 2.2 Multifractality ...... 6

3 Features of the System for M N 3 11 = = 3.1 Spectrum of Eigenenergies...... 11 3.1.1 E/U versus F /U for different coupling J/U ...... 12 3.1.2 E/F versus U/F for different coupling J/F ...... 13 3.1.3 E/J versus F /J for different interaction U/J ...... 13 3.2 Avoided Crossings in the Energy Spectrum ...... 13 3.3 Structure of Eigenstates in terms of Generalized Fractal Dimensions 19 3.3.1 Behaviour of D˜ q versus J/U for different tilt F /U ...... 19 3.3.2 Avoided crossings exposed by D˜ q ...... 21 3.3.3 Behaviour of D˜ q versus F /U for fixed coupling J/U . . . . . 24

4 Localization and Robust Eigenstate Structures in Fock Space 27 4.1 “Solitonic States” in Tilted Lattices ...... 27 4.2 Parameter Dependence of Eigenstate Structure...... 28 4.2.1 System with M N 4...... 29 = = Features of the eigenstates for weak hopping J/U ...... 29 Stability of localization as a function of J/U ...... 33 4.2.2 Localization in Fock space for larger M ...... 36 Localized manifolds for comparable values of J/U and F /U 36 Robust states and threshold value for the tilt strength . . . 38

5 Conclusion and Outlook 41

v Contents

Bibliography 43

vi CHAPTER 1

INTRODUCTION

One of the challenging problems in modern quantum mechanics is the under- standing and control of the dynamics in many-particle systems. When increasing the number of particles, the size of the underlying Hilbert space grows exponen- tially and thereby also the complexity of the system. Such complexity manifests itself, e.g., in the emergence of spectral chaos [1–5], which to a certain degree entails a high sensitivity of the many-particle dynamics on the initial conditions. Complexity also carries over to the Hilbert space structure of eigenstates, which can exhibit unusual statistical properties in the form of multifractality [6,7]. Eigenstate multifractality was first observed at the Anderson transition (dis- order-induce localization-delocalization transition) in single particle Hamiltoni- ans [8], and it also appears in random matrix models [9, 10] or quantum maps [11, 12]. The method of multifractal analysis happens to be a very useful tool to characterize the complexity of eigenstates in many-body systems [13–17]. A paradigmatic many-particle Hamiltonian is the Bose-Hubbard model [18, 19] which was originally considered to study the phase diagram and phase trans- itions of short-range interacting bosons at zero temperature. This Hamiltonian can be experimentally implemented with ultracold atoms in optical lattices (see Refs. [20–23] and references therein), which nowadays provide a powerful play- ground to investigate the physics of interacting many-body quantum systems. Interestingly, a recent study of the Bose-Hubbard model in the presence of an external static field (the so called tilted Bose-Hubbard model) reported the exist- ence of eigenstates with remarkably robust properties [24, 25]. The dynamical robustness of the states found is not a mere consequence of energy separation (they exist within the bulk of the energy spectrum), but rather of a strong localiza- tion in Fock space. This characteristic renders them excellent candidates, e.g., for the preparation of stable states in few-body systems in parameter regimes where the global majority of eigenstates is extremely sensitive to parametric perturba- tions.

1 Chapter 1. Introduction

In this thesis, we will study the eigenstate structure of the tilted Bose-Hubbard Hamiltonian in different parameter regimes using the tools provided by multi- fractal analysis. One of our aims is to give a detailed description of localization and potential robust eigenstate structures in Fock space. The organization of this thesis is as follows. We introduce the tilted Bose- Hubbard Hamiltonian as well as the concept of multifractality in Chapter 2. In Chapter 3, we investigate the features of a system with three bosons located on a lattice with three sites. There, we consider how the spectrum of eigenenergies evolves when different system parameters, such as the on-site interaction, the hopping strength and the tilt, change. The observation of avoided crossings in the spectrum and their relation to the structure of eigenstates is explained in detail. In Chapter 4, we discuss Fock space localization as well as robust eigenstate structures of certain states in the tilted Bose-Hubbard system. We first revise previous considerations of such states presented in the literature. Then, we scrutinize the parameter dependence of the eigenstate’s structure in a system consisting of four bosons on a lattice with four, eight and twelve sites. A final discussion is given in Chapter 5.

2 CHAPTER 2

FORMALISM

In this chapter, we introduce the theoretical background. We present the central model studied in this thesis, the tilted Bose-Hubbard Hamiltonian, and discuss its general features. Afterwards, we explain a method of studying the properties of the eigenstates of the tilted Bose-Hubbard Hamiltonian, namely the multifractal formalism. The latter mathematical concept as well as its applicability to quantum states is discussed.

2.1 The Tilted Bose-Hubbard Hamiltonian

The Bose-Hubbard Hamiltonian (BHH) provides an effective description of a system that is composed of a one-dimensional lattice with M sites in which one puts N indistinguishable bosons [18, 19]. The bosons are allowed to tunnel to next-neighbouring sites with tunnelling strength J, and interact with each other if they are located on the same site. The pairwise interaction strength is denoted by U. Such Hamiltonian can be faithfully experimentally realized using ultracold quantum gases in optical lattices and it has been extensively studied theoretically and experimentally (see Refs. [20–23, 26–28] and references therein). A detailed derivation of the BHH can be found, e.g., in Ref. [16]. The BHH can be extended by including an external potential. The easiest way to do this is by adding a static tilt to the system which causes an energy difference F between sites l and l 1, as depicted in Fig. 2.1. Physically, this can + be interpreted as a potential due to, e.g., the gravitational field. The resulting Hamiltonian corresponds to the tilted BHH, which can be written as

M 1 ³ ´ U M M ˆ X− † † X X ˜ H J aˆl 1aˆl aˆl aˆl 1 nˆl (nˆl 1) F lnˆl . (2.1) = − l 1 + + + + 2 l 1 − + l 1 | = {z } | = {z } | ={z } tunnelling of individual particles on-site interaction static tilt

3 Chapter 2. Formalism

+U F J l+1 J l l-1

Figure 2.1: Schema of the tilted Bose-Hubbard model. The tunnel- ling strength J is associated with one-boson tunnelling events between neighbouring sites. When several bosons are on the same site, they couple pairwise with interaction strength U. The static tilt leads to an energy difference F between neighbouring sites.

† The operator aˆl (aˆl ) is the annihilation (creation) operator that annihilates (cre- ates) one boson at the lth site. The number operator is denoted by nˆ aˆ†aˆ and l = l l counts the number of bosons being located at the lth site. We consider a symmet- ric tilt around the centre of the lattice, which means that l˜ l (M 1)/2. We also = − + assume hard wall boundary conditions (HWBC), so tunnelling between first and last sites is forbidden. The tilted BHH is a paradigmatic model for investigating the phenomenology of Bloch oscillations (see, e.g., Refs. [29,30]). The Bose-Hubbard Hamiltonian [Eq. (2.1) with F 0] can be solved analyt- = ically in the limits J/U and J/U 0, which lead to two different natural → ∞ → basis systems: the Bloch basis and the Wannier basis, respectively. For a detailed discussion of these two bases we refer the reader to Ref. [24]. Regarding the Wannier basis, one constructs the underlying Fock space H with dimension à ! N M 1 N + − (2.2) = N

out of Hilbert spaces H j for each site via tensor products,

M O H H j . (2.3) = j 1 = ©¯ ®ª As a basis for each H j we choose ¯n j , i.e., the eigenstates of the number operator, where n N is the number of bosons located at the jth site. The basis j ∈ of H is therefore ( M ) O¯ ® ¯n j { n1,n2,...,nM } { n }. (2.4) j 1 = | 〉 ≡ | 〉 = 4 Chapter 2. Formalism

The Fock states n in the Wannier representation thus follow from the set of | 〉 all possibilities of distributing N bosons on M lattice sites. The actions of the annihilation, creation and number operators on the Fock states read

¯ ® aˆ n ,...,n pn ¯n ,...,n 1,...,n , (2.5) j | 1 M 〉 = j 1 j − M † q ¯ ® aˆ n ,...,n n 1¯n ,...,n 1,...,n , (2.6) j | 1 M 〉 = j + 1 j + M ¯ ® ¯ ® nˆ ¯n ,...,n ,...,n n ¯n ,...,n ,...,n . (2.7) j 1 j M = j 1 j M In the Fock basis, the interaction and tilt terms of the Hamiltonian (2.1) are diagonal, since they only depend on the on-site number operators nˆ j . On the other hand, the hopping term J couples different Fock states and hence it is responsible for the off-diagonal elements in the matrix representation of Hˆ . In the limit J 0, the eigenenergies of the system depend linearly on U and F , = and are given by

U X 2 X M 1 U E n nl F ln j + FN N. (2.8) | 〉 = 2 l + l − 2 − 2

From the latter expression, one can straightforwardly see that certain different Fock states can be degenerate for all values of U and F . For example, all Fock states that exhibit mirror symmetry about the middle of the lattice and that are related by a permutation of the on-site densities will have a common eigenenergy. Additionally, notable degeneracies between Fock states can occur for specific values of the ration F /U. If we consider the simplest case of all Fock states that differ only in two on-site densities placed l sites apart, we can see that degeneracies occur for1 F 1 2 N 1 , ,..., − , (2.9) U = l l l where we have assumed U 0 and F 0. Hence, in the absence of hopping, the > > spectrum of the system as a function of F /U will exhibit crossings of the energy levels for specific rational and integer values of this ratio. In the presence of non-vanishing J all these degeneracies will be lifted, as we discuss in detail in the subsequent part of this thesis. In the opposite limit, for U F 0 and finite J, the energy of the many-particle = = eigenstate results from the sum over a single-particle spectrum. The eigenenergies εj of one particle in a one-dimensional (untilted) lattice with M sites and HWBC read [31] µ jπ ¶ εj 2J cos , j 1,...,M. (2.10) = − M 1 = + 1 The derivation can be done straightforwardly by setting the energies E n and E n˜ of two | 〉 | 〉 different Fock states n ,...,n and n˜ ,...,n˜ whose on-site densities differ at sites i and i l | 1 M 〉 | 1 M 〉 + only, to be the same. This leads to F /U (ni n˜i l )/l. The maximum value in the numerator = − + occurs when ni is maximal and n˜i l minimal, i.e., N and 1, respectively. +

5 Chapter 2. Formalism

In a system with M sites and n j bosons being in the jth single-particle level, the state of the whole system can be represented by

Ψ n ,...,n } (2.11) | 〉 = | 1 M which we refer to as Fock states in the Bloch basis,2 denoted by } in order to dis- |· tinguish them from the Fock states in the Wannier basis. In this limit degeneracies also exist which split up by turning U and F on. Our basis of choice in this thesis is the Fock basis in the Wannier representation since it provides a direct description of the bosonic density in real space which is suitable for our purpose.

2.2 Multifractality

An important property of an object is its dimension. The algebraic definition, the cardinal number of the object’s basis (i.e., the number of coordinates needed to specify a position within the object) is very intuitive. A straight line has dimension one, a plane has dimension two, et cetera. But there is another way of defining the dimension D, namely by observing a proportionality between the generalized volume V and the object’s linear size L as

V LD . (2.12) ∝ One can see that this definition leads to the same result as regarded in the first approach if one considers simple geometries such as straight lines, planes or (hyper-)cubes. For the straight line, the generalized volume V corresponds to its length; for the plane it is the surface and for the (hyper-)cube it is the (hyper-) volume. Up to this point only positive integer numbers occurred for D, but for complex geometries, such as the Sierpinski carpet shown in Fig. 2.2, the dimension D in Eq. (2.12) may have non-integer values. Objects with an non-integer dimension are called fractals. If an object is characterized by several different fractal dimen- sions (e.g., depending on what part of the object one is looking at) it is called a multifractal. Multifractality not only occurs in geometrical forms, but it can also charac- terize the complexity of data sets, such as, e.g., the probability distribution of a quantum state in Hilbert space. This was shown, e.g., in Refs. [13, 14] where it was proved that the ground state of different one-dimensional quantum spin-1/2 chain models is a multifractal in the natural spin basis. The multifractal proper- ties of the ground state of the Bose-Hubbard model were also demonstrated in

2Although we do not have translational symmetry in our system, for simplicity we use the term Bloch basis to refer to the discrete Fourier transform of the Wannier basis in the presence of HWBC.

6 Chapter 2. Formalism

Figure 2.2: Figure taken from Ref. [32]. Initial stages of the build-up of the Sierpinski carpet. Empty squares are shaded. At each step of the iteration, the linear dimension L is enlarged by a factor 3 and the mass (generalized volume) V by a factor 8, since each occupied square is replaced by a 3 3 array of nine squares × of which the centre square is empty. One finds a dimension D log8/log3 1.89. = ≈

Refs. [15–17]. How multifractality is described specifically in quantum states will be presented in the following. Let us consider a normalized state

N X ¯ ® Ψ ψj ¯j (2.13) | 〉 = j 1 = ©¯ ®ª in a Hilbert space H with dimension N and basis ¯j . The coefficients of Ψ | 〉 in this basis are given by ψj . We define the singularity strength αj to be as follows,

¯ ¯2 α ¯ψ ¯ N − j . (2.14) j = This definition makes it easier to compare properties of wavefunctions in Hilbert spaces with different sizes. Due to the normalization of Ψ , the average value of ¯ ¯2 | 〉 the occupation probability ¯ψj ¯ av decreases when enlarging the Hilbert space, ¯ ¯2 1 ¯ψ ¯ N − . In contrast, the corresponding singularity strength α does not j av = j,av depend on the Hilbert space dimension, α 1. j,av = Consider now a certain interval [α,α ∆α] (for ∆α 0) and let us count the + → number Nα of αj that lie within. If for large N the relation

N N f (α) (2.15) α ∝ is observed for non-integer exponents f (α) that depend on α, one calls the un- derlying state multifractal. With the previous approach to fractality as given in Eq. (2.12), one may correlate Nα with the generalized volume V and the Hilbert

7 Chapter 2. Formalism

space dimension N with the length scale L. In this way, f (α) would be the fractal dimension of the set of intensities corresponding to the singularity strength α. The maximum allowed value of f (α) is naturally 1. ¯ ¯2 One can now define the q-moments Rq of the distribution of ¯ψj ¯ to be

N X ¯ ¯2q Rq ¯ψj ¯ . (2.16) = j 1 = Sometimes they are also called generalized inverse participation ratios, since for q 2 one recovers the standard inverse participation ratio (IPR). The inverse of = R2, i.e., the participation ratio (PR),

à ! 1 − 1 X¯ ¯4 PR R2− ¯ψj ¯ (2.17) = = j

quantifies how much a state sees of Hilbert space. A fully extended state with ¯ ¯2 1 ¯ψj ¯ N − has PR N ; it sees its whole underlying space. On the other hand an = = ¯ ¯2 extremely localized state ¯ψ ¯ δ has PR 1 and sees only one basis element. j = j,j0 = Effectively, the choice of q in Eq. (2.16) varies the range of αj which has the ¯ ¯2 largest impact on the qth moment: Large positive q amplify the highest ¯ψj ¯ ¯ ¯2 values, whereas large negative q cause small values of ¯ψj ¯ to receive greater ¯ ¯2 importance in Rq . For a multifractal distribution ¯ψj ¯ , the Rq obey the scaling law τq R N − for N , (2.18) q ∝ → ∞ where the exponent τq must depend non-linearly on q. The exponents τq and the multifractal spectrum f (α) can be seen to be related by a Legendre Trans- formation [7]. Note that for q 1, the moment R is equal to 1 because of the = 1 normalization of Ψ , one finds R 1 N 0 for any N and thus τ 0. It is there- | 〉 1 = = 1 = fore convenient to define the so called generalized fractal dimensions (GFDs) Dq to obey τ (q 1)D . (2.19) q = − q Since any N -dependent prefactor h(N ) on the right-hand side of the scaling law (2.18) must become irrelevant as N , the GFDs can be obtained as → ∞

(q 1)Dq R h(N )N − − q = logRq 1 logh(N ) 1 Dq ⇒ logN · 1 q = logN ·1 q + − | {z } − lim 0 N → →∞ 1 logRq Dq lim lim D˜ q , (2.20) ⇒ = N 1 q logN ≡ N →∞ − →∞ 8 Chapter 2. Formalism

where the definition of D˜ q is meaningful for each finite value N of the dimen- sionality of Hilbert space. It can be shown that D D and D˜ D˜ for q q0 q ≥ q0 q ≥ q0 < [33]. For positive q, the dimensions Dq and D˜ q may vary between 0 and 1. Among the GFDs we single out the cases q 1,2 and . Due to the connection = ∞ to the Shannon information entropy P p logp of a probability mass function − j j j p j , the GFD D1 is know as the information dimension. The dimension D2 governs the scaling of R2 and thus of the participation ratio, as discussed earlier. In the ¯ ¯2 case q only the maximum value of the intensities ¯ψ ¯ matters. For finite = ∞ j max N one finds that De1, De2 and De are given as ∞

NP ¯ ¯2 ¯ ¯2 ¯ψj ¯ log¯ψj ¯ j 1 = De1 lim D˜ q , (2.21) = q 1 = − log → N NP ¯ ¯4 log ¯ψj ¯ j 1 = De2 , (2.22) = − logN ³¯ ¯2 ´ log ¯ψj ¯max De . (2.23) ∞ = − logN

The calculation and analysis of Dq permits us to discriminate among different localization properties in Hilbert space of the state Ψ : | 〉 ¯ ¯2 - A homogeneously extended state has an occupation probability ¯ψj ¯ 1 ∝ N − for all j. One finds

N X ¯ ¯2q (q 1) Rq ¯ψj ¯ N − − (2.24) = j 1 ∝ = D 1 for all q. (2.25) ⇒ q = In the literature, this kind of state is sometimes referred to as an extended ergodic state.

- A localized state covers only a finite region of Hilbert space. Thus, when enlarging the size of Hilbert space, the q-moments become independent of N . This leads to D 0 for all q 0. (2.26) q = >

- For multifractal states, one finds that Dq changes with q,

1 D 0. (2.27) ≥ q ≥

Although the GFDs are strictly defined as scaling exponents in the limit N , the finite size dimensions D˜ are still helpful when analysing states → ∞ q 9 Chapter 2. Formalism

1.0

0.8 2 ψj

˜ 0.6 Dq 0.4

2 0.2 ψj

0.0 0 1 2 3 4 5 6 7 50 100 150 200 250 q j

Figure 2.3: (Left) Visualisation of the generalized fractal dimensions D˜ versus q in a finite Hilbert space of size N 286. The q = four lines shown correspond to a homogeneously extended state (blue), a fully localized state (black) and two eigenstates of the tilted BHH, the ground state (red) and the 100th excited state (green). The parameters of the Hamiltonian were chosen to be M 11, N 3, U 1, J 0.631, F 0.5. (Right) Illustration = = = = = of the coefficients of the corresponding eigenstates in the Fock basis.

in finite Hilbert spaces. They provide a way of quantifying localization properties. In Fig. 2.3, one can see an illustration of the behaviour of D˜ q versus q for two eigenstates of the Bose-Hubbard model. The coefficients of the two eigenstates are portrayed next to their corresponding D˜ q representation. Qualitatively, one can see differences in their localization properties, and how these are reflected in the values of D˜ q , ranging from 0 (extreme localization) to 1 (fully extended).

10 CHAPTER 3

FEATURES OF THE SYSTEMFOR M N 3 = =

In order to understand tilted Bose-Hubbard systems with large Hilbert space dimensions, it is convenient to look at a small system first. We will discuss in detail the tilted BHH with M N 3 that has a Hilbert space dimension N 10. = = = We are first interested in the development of the spectrum of eigenvalues for different choices of the parameters J, U and F . Afterwards, we explain the avoided crossings in the energy spectra in detail, and we study the properties of the eigenstates in terms of the generalized fractal dimensions D˜ q .

3.1 Spectrum of Eigenenergies

Let us now consider a system with three bosons and three sites, i.e., M N 3. = = The dimension of Hilbert space is then given by

à ! 3 3 1 N + − 10. (3.1) = 3 =

The Fock basis (in the Wannier representation) can be written as

{ 3,0,0 , 0,3,0 , 0,0,3 , 2,1,0 , 2,0,1 , | 〉 | 〉 | 〉 | 〉 | 〉 (3.2) { 1,2,0 , 0,2,1 , 0,1,2 , 1,0,2 , 1,1,1 }. | 〉 | 〉 | 〉 | 〉 | 〉

The tilted Bose-Hubbard Hamiltonian Hˆ can be expressed in this basis via the following matrix,

11 Chapter 3. Features of the System for M N 3 = =

3U 3F 0 0 p3J 0 0 0 0 0 0  − − 0 3U 0 0 0 p3J p3J 0 0 0 − −  0 0 3U 3F 0 0 0 0 p3J 0 0   + −   p3J 0 0 U 2F J 2J 0 0 0 0   − − − −   0 0 0 JU F 0 0 0 0 p2J Hˆ − − − .  0 p3J 0 2J 0 U F 0 0 0 p2J =  − − − −   0 p3J 0 0 0 0 U F 2J 0 p2J  − + − −   0 0 p3J 0 0 0 2JU 2F J 0  − − + −  0 0 0 0 0 0 0 JU F p2J − + − 0 0 0 0 p2J p2J p2J 0 p2J 0 − − − −

(3.3)

By setting one of the parameters J, U or F to set the energy scale, i.e., by normalizing the eigenenergies by one of them, we can look at the three following cases: E F J 1) versus for different , (3.4) U U U E U J 2) versus for different , (3.5) F F F E F U 3) versus for different . (3.6) J J J Whereas these representations of the spectrum are obviously pairwise equivalent (e.g., 1 and 3), they still provide different perspectives. The suitability of one of them will be determined by the ranges of interest of the ratios of the parameters J, U and F .

3.1.1 E/U versus F /U for different coupling J/U The spectrum for this normalization choice is depicted in Fig. 3.1. There, one can see the spectrum of eigenvalues E/U versus F /U [0,2.2] for values J/U ∈ ∈ {0.0,0.1,0.2,0.5}. The case J/U 0 can be understood straightforwardly. As it was explained = in Sec. 2.1, one can observe several straight lines that cross each other. The eigenenergies are given by Ei F αi βi (3.7) U = + U for certain combinations of α {0,1,3} and β { 3, 2, 1,0,1,2,3} as follows i ∈ i ∈ − − − from Eq. (2.8) on page5. Each eigenenergy is associated with a Fock state. As discussed in Sec. 2.1, crossings of the energy levels (degeneracies) can be seen for

12 Chapter 3. Features of the System for M N 3 = =

F /U 1/2,1,2. Additionally, one observes two degeneracies for any value of F /U = coming from states 2,0,1 and 1,2,0 , as well as from 0,2,1 and 1,0,2 . | 〉 | 〉 | 〉 | 〉 When increasing J/U, we note that all real crossings become avoided crossings of different widths. The mechanism that enforces the lines to avoid each other will be discussed in the next section.

3.1.2 E/F versus U/F for different coupling J/F We consider now the eigenspectrum E/F versus U/F [0,3] for different J/F ∈ ∈ {0.0,0.1,0.2,0.5} as depicted in Fig. 3.2. We observe straight lines crossing each other at U/F 1/2,1,2 in the absence = of hopping J/F 0 given by = Ei U αi βi (3.8) F = F + for certain combinations of α {0,1,3} and β { 3, 2, 1,0,1,2,3}. Each of i ∈ i ∈ − − − these trajectories is associated with a given Fock state, as indicated in Fig. 3.2. For non-vanishing J/F , all crossing become avoided crossings.

3.1.3 E/J versus F /J for different interaction U/J The eigenenergies E/J versus F /J [0,4] for different U/J {0.0,0.2,1.0,3.0} are ∈ ∈ plotted in Fig. 3.3. We concern ourselves first with the non-interacting situation, U/J 0. As = explained in Sec. 2.1, in the limit F /U 0, the eigenstates of the system corres- → pond to the Fock states in the Bloch basis, and the eigenenergies follow from the addition of the single-particle energies εj given in Eq. (2.10), and degeneracies are possible. For very large F /J the spectrum is dominated by the tilt and the energy levels approach the asymptotes

Ei F βi , βi { 3, 2, 1,0,1,2,3}. (3.9) J = J ∈ − − − For increasing U/J, the degeneracies are lifted, and avoided crossings become visible.

3.2 Avoided Crossings in the Energy Spectrum

Avoided crossings are crucial in this system, hence we will discuss their appear- ance in the energy spectrum in detail. Let us consider a Hamiltonian of the form H H V (3.10) = 0 + where H0 is diagonal and V is responsible for all off-diagonal elements of H. For the tilted BHH, H0 includes the interaction and tilt terms, U and F , whereas

13 Chapter 3. Features of the System for M N 3 = = 2. 1.5 F / U 1. 0.5 0.1 = 0.5 = J J U U 2. | 0,2,1 〉 | 1,0,2 〉 , | 1,2,0 〉 | 2,0,1 〉 , 1.5 | 0,1,2 〉 | 2,1,0 〉 F / U 1. 0.5 | 0,3,0 〉 0.0 = 0.2 = J J U U | 0,0,3 〉 | 3,0,0 〉 | 1,1,1 〉 0. 6. 4. 2. 0. 4. 2. 0. - 2. - 2. E E U U

Figure 3.1: Eigenspectrum E/U versus F /U of the tilted Bose- Hubbard Hamiltonian with M N 3 and four different = = choices of J/U. One can observe how the crossings at J/U 0 = become avoided crossings as soon as J/U becomes non-zero. For J/U 0, the spectral lines are associated with the corresponding = Fock state. Vertical dotted lines highlight the position of real crossings at F /U 1/2,1,2 for J/U 0. 14 = = Chapter 3. Features of the System for M N 3 = = 3. 0.1 = 0.5 = 2.5 J J F F 2. 1.5 U / F 1. 0.5 | 1,1,1 〉 | 3,0,0 〉 0.0 = 0.2 = 2.5 J J F F | 0,2,1 〉 | 1,0,2 〉 , | 1,2,0 〉 | 2,0,1 〉 , 2. | 0,1,2 〉 | 0,3,0 〉 1.5 U / F | 2,1,0 〉 1. | 0,0,3 〉 0.5 0. 6. 4. 2. 0. 4. 2. 0. - 2. - 2. F F E E

Figure 3.2: Eigenspectrum E/F versus U/F of the tilted Bose-Hubbard Hamiltonian with M N 3 and four different choices of J/F . = = One can observe how the crossings at J/F 0 become avoided = crossings as soon as J/F becomes non-zero. For J/F 0, the = spectral lines are associated with the corresponding Fock state. Vertical dotted lines highlight the position of real crossings at U/F 1/2,1,2 for J/F 0. = = 15 Chapter 3. Features of the System for M N 3 = = 4. 3. 2. F / J 1. 0.1 = 2.0 = J J U U 3. 2. F / J 1. 0.0 = 0.2 = J J U U | 0,0,3 〉 | 0,1,2 〉 | 2,1,0 〉 | 3,0,0 〉 | 1,0,2 〉 | 0,2,1 〉 , | 1,1,1 〉 | 0,3,0 〉 , | 2,0,1 〉 | 1,2,0 } , 0. 5 0 5 0 10 - 5 - 5 J J E E

Figure 3.3: Eigenspectrum E/J versus F /J of the tilted Bose-Hubbard Hamiltonian with M N 3 and four different choices of U/J. = = For U/J 0, the Fock states in the Bloch basis [see discussion = around Eq. (2.11) on page6] which are eigenstates in the limit F /J 0, are indicated. =

16 Chapter 3. Features of the System for M N 3 = =

V corresponds to the hopping term J. As discussed, the eigenenergies of H0 exhibit real crossings when they evolve as a function of U or F . In the presence of V (J 0), the real crossings turn into avoided crossings. To understand the 6= underlying mechanism, we first only consider sufficiently small J, so that in order ¯ ® to describe an avoided crossing between two states ¯Φ1,2 , one only has to take ¯ ® the subspace spanned by ¯Φ1,2 into account. Describing the latter is possible with the Hamiltonian of the subspace,

µ ² Φ V Φ ¶ H S 1 〈 1| | 2〉 , (3.11) = Φ V Φ ² 〈 2| | 1〉 2 ­ ¯ ¯ ® ¯ ® where ² Φ ¯ H ¯Φ are the energies of the eigenstates ¯Φ of the un- 1,2 = 1,2 0 1,2 1,2 perturbed system. Therefore, we consider a 2 2 hermitian matrix with three × remaining degrees of freedom that depend on the system parameters U, J, and F , and define the parameters a, b and c such that

µa b c ¶ H S + , a,b,c R, (3.12) = c a b ∈ − c c(J), a a(U,F ), b b(U,F ). (3.13) = = = Off-diagonal elements come only from hopping/tunnelling J, while diagonal elements are due to the on-site interaction U or the tilt F . When looking at the case c 0, the eigenenergies are given as =

E (c 0) a b, (3.14) 1,2 = = ± which exhibit a real crossing as a function of b (see Fig. 3.4). For c 0, the 6= eigenvalues have the form

p E a b2 c2. (3.15) 1,2 = ± + Now, what we see in Fig. 3.4 is an avoided crossing of minimal gap ∆E 2 c at min = | | b 0. One can estimate the length of the avoided crossing as ∆b 4 c .1 Within = ≈ | | this interval, the curvature of the eigenvalue trajectory is rather large. The eigenvectors of the 2 2 matrix read ×

Ψ cos(θ/2) Φ sin(θ/2) Φ , (3.16) | 1〉 = | 1〉 + | 2〉 Ψ sin(θ/2) Φ cos(θ/2) Φ , (3.17) | 2〉 = | 1〉 − | 2〉 where tan(θ) c/b. (3.18) = 1This corresponds to twice the length of the semi-minor axis of the hyperbola defined by the spectrum of eigenenergies.

17 Chapter 3. Features of the System for M N 3 = =

1.0

c=0.4 0.5

c=0.0 ΔEmin 0.0 E1/2-a =2|c|

-0.5

Δb≈4|c|

-1.0 -1.0 -0.5 0.0 0.5 1.0 b

Figure 3.4: Schema of an avoided crossing. Solid lines: Plot of the ei- genvalues from Eq. (3.14). Dashed lines: Plot of the eigenvalues from Eq. (3.15) for c 0.4. The coupling c induces an avoided = crossing with a minimal gap ∆E 2 c and width ∆b 4 c . min = | | ≈ | |

For fixed c 0, the evolution of the eigenvectors is as follows, 6= b Ψ1 Φ2 , 0 θ π− | 〉 = | 〉 (3.19) c ¿ ⇒ → ⇒ Ψ2 Φ1 , | 〉 = | 〉 ¯ ® 1 b 0 θ π/2 ¯Ψ1,2 [ Φ1 Φ2 ], (3.20) = ⇒ → ⇒ = p2 | 〉 ± | 〉 b Ψ1 Φ1 , 0 θ 0+ | 〉 = | 〉 − (3.21) c À ⇒ → ⇒ Ψ2 Φ2 . | 〉 = −| 〉 ¯ ® The evolution of ¯Ψ1,2 versus b is simply given by a clockwise rotation of the ¯ ® unperturbed states ¯Φ . In the limit of very large b/c , the new eigenstates 1,2 | | become proportional to the eigenstates of the unperturbed matrix. For lower b/c , a mixing of the unperturbed basis takes place. The smaller c , the faster | | | | (when continuously changing b) this happens. With this theoretical background, we are able to simulate, e.g., the avoided crossing between two Fock states of the form

Φ 0,...,0,N,0,...,0 , Φ 0,...,0,N 1,1,0,...,0 (3.22) | 1〉 = | 〉 | 2〉 = | − 〉 where only one boson tunnels. The Hamiltonian H S as in Eq. (3.11) is then given by µ ² pNJ¶ H S 1 − , (3.23) = pNJ ² − 2 where only the energy difference ² ² (N 1)U F is relevant. Therefore, 1 − 2 = − − the crossing for J 0 happens at F (N 1)U. When turning J on, an avoided = = − 18 Chapter 3. Features of the System for M N 3 = = crossing with minimal gap ∆² 2pNJ and width ∆F 4pNJ or ∆U 4pNJ, min = = = depending on which parameter is changed, is expected. The gap and width of the avoided crossings will depend on the magnitude of the off-diagonal element, which depends on the type of Fock states involved in the crossing. The two- level treatment presented is not applicable to describe the emergence of avoided crossings involving Fock states which differ in two or more particle hoppings. For example, in order to understand the avoided crossing involving the Fock states Φ 0,...,0,N,0,...,0 and Φ 0,...,0,N 1,0,1,0,...,0 , one would need a | 1〉 = | 〉 | 2〉 = | − 〉 three level approach including also the state Φ 0,...,0,N 1,1,0,...,0 . Such | 3〉 = | − 〉 a crossing requires transitions of higher order in J (in this case of second order, i.e., two hopping processes). For fixed small J, the largest and widest avoided crossings are due to the first order hopping transitions between Fock states with a highly localized bosonic density [as in Eq. (3.22)], as can be checked in Fig. 3.1. We also note that for large J, the approximation of considering a reduced subspace of two states may not be justified. In this case, several Fock states can highly mix in a non-trivial way such that the former avoided crossing is no longer recognizable (see Fig. 3.1 for J/U 0.5 at F /U 2). = =

3.3 Structure of Eigenstates in terms of GFDs

In this section, we analyse the properties of the eigenstates of the M N 3 = = system using the generalized fractal dimensions D˜ q , introduced in Sec. 2.2. Recall that the dimensions D˜ q quantify the localization of a state in the con- sidered basis. A value of D˜ q close to 0 indicates very strong localization, whereas D˜ q close to 1 corresponds to fully extended states.

3.3.1 Behaviour of D˜ q versus J/U for different tilt F /U

In Fig. 3.5, one can see plots of D˜ 1, D˜ 2 and D˜ versus J/U for tilt F /U 0 in the ∞ = M N 3 system. The numbers next to the trajectories order the states according = = to their eigenenergy, 1 being the state with smallest eigenenergy, 10 being the one with highest eigenenergy. When comparing the different D˜ q plots, we observe the same qualitative behaviour. Except for states 4 and 5, one can observe a continuous evolution. In general, the lines change little for small J/U and then have a transition interval around J/U 0.3 to larger values of D˜ . The relation between the change of the ≈ q ground state’s D˜ q versus J/U and the Mott insulator to superfluid transition of the BHH was studied in Refs. [16, 17]. Most interesting is the discontinuity of states 4 and 5 that appears for any D˜ q . In order to understand this observation better, one needs to look at the spectrum of eigenenergies. This is done in Fig. 3.6. In the following, we will restrict the discussion to D˜ 2 for simplicity. We expect the same qualitative behaviour for D˜ 1 and D˜ . At F 0 one observes a real crossing of states 4 and 5 in the energy ∞ = 19 Chapter 3. Features of the System for M N 3 = =

1.0 1.0

0.8 0.8

˜ 0.6 ˜ 0.6 D1 D∞ 0.4 0.4

0.2 0.2

0.0 0.0 10-1 1 101 10-1 1 101 J/U J/U

1.0

7 0.8 6 2 0.6 3 ˜ 5 D2 4 0.4 8 9 0.2 1 F/U=0.0

10 0.0 10-1 1 101 J/U

Figure 3.5: D˜ 1, (top) and D˜ 2 (bottom) versus J/U at F /U 0 for ∞ = the eigenstates of the system with M N 3. The 10 eigenstates = = are ordered with respect to their eigenenergy (from low to high). States 4 and 5, which undergo a real crossing, are highlighted by dashed lines.

spectrum. Hence, the discontinuity on the trajectories of D˜ 2 for states 4 and 5 is due to the reordering of the eigenenergies. However, for F 0, the real 6= crossing in the energy becomes an avoided crossing, and as a result, the spurious discontinuity of the D˜ 2 trajectories turns into a continuous change, reflecting the mixing of the former eigenstates at the avoided crossing. The values of D˜ 2 at the former discontinuity exhibit now a sharp minimum. The widths of the avoided crossing in terms of D˜ 2 are larger than when considering the energy spectrum. Here one can see a benefit of the representation with GFDs: Avoided crossings,

20 Chapter 3. Features of the System for M N 3 = =

Figure 3.6: Correlation of the behaviour of the energy spectum E/U 5 (left) and D˜ 2 (right) versus J/U for F /U 0 (top), F /U 5 10− 4 = = × (centre) and F /U 10− (bottom). The 4th and 5th states are = marked by dashed lines as in Fig. 3.5.

even if they are narrow in energy, can potentially be better resolved via D˜ q . The phenomenology of an avoided crossing in terms of D˜ q will be discussed in the next subsection.

3.3.2 Avoided crossings exposed by D˜ q

Let us first discuss the behaviour of D˜ 2 when considering two states that undergo an avoided crossing. As in Sec. 3.2, we describe the avoided crossing of two states ¯ ® ¯ ® ¯Ψ H with respect to the unperturbed states ¯Φ by the driving parameter 1,2 ∈ 1,2 21 Chapter 3. Features of the System for M N 3 = =

0 θ π as, in Eq. (3.18), via ≤ ≤ Ψ cos(θ/2) Φ sin(θ/2) Φ , (3.24) | 1〉 = | 1〉 + | 2〉 Ψ sin(θ/2) Φ cos(θ/2) Φ . (3.25) | 2〉 = | 1〉 − | 2〉

Since we want to calculate D˜ 2 in the Fock basis, we need to know the expansion ¯ ® of ¯Ψ1,2 in terms of Fock states. We will consider two exemplary cases. In the first case we assume that

Φ n , (3.26a) | 1〉 = | 〉 1 Φ2 { m δ o }, δ R, (3.26b) | 〉 = p1 δ2 | 〉 + | 〉 ∈ + where n , m and o denote orthogonal Fock states. These states can be inserted | 〉 | 〉 | 〉 ¯ ® in Eq. (3.24) and (3.25) to find the coefficients of ¯Ψ1,2 with respect to the given basis. The IPR can be calculated explicitly from Eq. (2.16),

4 4 4 4 sin (θ/2) δ sin (θ/2) R2 ( Ψ1 ) cos (θ/2) 2 2 , (3.27) | 〉 = + ¡1 δ2¢ + ¡1 δ2¢ + + 4 4 4 4 cos (θ/2) δ cos (θ/2) R2 ( Ψ2 ) sin (θ/2) 2 2 . (3.28) | 〉 = + ¡1 δ2¢ + ¡1 δ2¢ + + The corresponding GFDs are obtained as

£ ¡¯ ®¢¤ D˜ log R ¯Ψ /log[N ], (3.29) 2 = 2 1,2 when N is the total dimension of the underlying Hilbert space. For θ π/2, i.e, at ¯ ® = the centre of the avoided crossing, ¯Ψ1,2 have the form

1 · 1 δ ¸ Ψ1 n m o , (3.30) | 〉 = p2 | 〉 + p1 δ2 | 〉 + p1 δ2 | 〉 1 · 1+ δ+ ¸ Ψ2 n m o . (3.31) | 〉 = p2 | 〉 − p1 δ2 | 〉 − p1 δ2 | 〉 + + Comparing these states with the states far away from the crossing in Eqs. (3.26), one observes that the states have more non-zero Fock coefficients and thus are less localized. Correspondingly, the values of D˜ 2 are enhanced when the states are going through the avoided crossing, as shown in the upper two panels of Fig. 3.7. When δ 0, both states exhibit the same D˜ value for any θ, and hence only one = 2 trajectory is visible. When δ 0, however, we can distinguish them. 6= In a second case, we assume that 1 Φ1 { n m δ o }, δ R, (3.32a) | 〉 = p2 δ2 | 〉 − | 〉 + | 〉 ∈ 1 + Φ2 { n m }. (3.32b) | 〉 = p2 | 〉 + | 〉

22 Chapter 3. Features of the System for M N 3 = =

0.4 δ=0 δ=0.6

0.3 ˜ Dq 0.2

0.1

0.4 δ=0 δ=0.6

0.3 ˜ Dq 0.2

0.1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

θ/2 θ/2

Figure 3.7: Exemplary behaviours of the evolution of D˜ q through an avoided crossing [q 1 (red), q 2 (blue), q (green)]. = = = ∞ The parameter θ is defined as in Eq. (3.18) and controls the evolution through the avoided crossing. The upper and lower panels belong to the cases considered in Eqs. (3.26) and (3.32), respectively [ Ψ (dashed) and Ψ (dotted)], for δ 0 (left) | 1〉 | 2〉 = and δ 0 (right). For the numerical determination of the GFDs, 6= we set N 10. =

As above, the IPRs in the given basis can be calculated as

µcos(θ/2) sin(θ/2)¶4 µcos(θ/2) sin(θ/2)¶4 R2 ( Ψ1 ) | 〉 = p2 δ2 + p2 + p2 δ2 − p2 + + µδcos(θ/2)¶4 , (3.33) + p2 δ2 + µsin(θ/2) cos(θ/2)¶4 µsin(θ/2) cos(θ/2)¶4 R2 ( Ψ2 ) | 〉 = p2 δ2 − p2 + p2 δ2 + p2 + + µδsin(θ/2)¶4 . (3.34) + p2 δ2 + 23 Chapter 3. Features of the System for M N 3 = =

We consider again how the states mix at the centre of the avoided crossing:

1 ·µ 1 1 ¶ µ 1 1 ¶ δ ¸ Ψ1 n m o , (3.35) | 〉 = p2 p2 δ2 + p2 | 〉 − p2 δ2 − p2 | 〉 + p2 δ2 | 〉 1 ·µ 1+ 1 ¶ µ 1+ 1 ¶ δ+ ¸ Ψ2 n m o . (3.36) | 〉 = p2 p2 δ2 − p2 | 〉 − p2 δ2 + p2 | 〉 + p2 δ2 | 〉 + + + In both states, one of the coefficients in n or m gets suppressed when com- | 〉 | 〉 pared with the expansion in Eqs. (3.32), corresponding to the states far away from the crossing. Hence, in this case localization is enhanced when going through the avoided crossing. This can be recognized in the lower two panels of Fig. 3.7, where D˜ 2 is plotted versus θ. As above, the trajectories are indistinguishable when δ 0 and decouple for δ 0. = 6= In Fig. 3.7 one can also see plots of D˜ 1 (red) and D˜ (green) versus θ for the ∞ discussed avoided crossings. As one can see, the qualitative behaviour of D˜ 1, D˜ 2, and D˜ is the same. ∞ This simple analysis enables us to describe and identify different behaviours of the GFDs, and correlate them with avoided crossings in the energy spectrum. Furthermore, we receive information about the underlying Fock structure of the states involved in the avoided crossing.

3.3.3 Behaviour of D˜ q versus F /U for fixed coupling J/U Now, we study the behaviour of D˜ as a function of F /U for fixed J/U 0.1 2 = as illustrated in Fig. 3.8 for the M N 3 system. The corresponding energy = = spectrum is also shown. The general behaviour in the D˜ 2 plot might seem a bit chaotic at first. We can identify four positions, namely around F /U 0.1,0.5,1.0,2.0, where a large = number of peaks can be observed. The widths, however, differ strongly. In the representation of Fig. 3.8, one can immediately identify peaks in the D˜ 2 plot with avoided crossings in the energy spectrum. One can also correlate the width of the peak with the width of the corresponding avoided crossing. Let us first have a closer look at the wide peak at F /U 2 where four lines = can be discovered to participate. As one sees immediately, the shape and height of the peak correspond to the one presented in Fig. 3.7, upper left panel. There, we simulated how the avoided crossing of two pure Fock states would look like in terms of D˜ 2. For this reason we know that at this peak, there are two avoided crossings of two nearly pure Fock states, which is confirmed by looking at the energy plot. With the observation that we are indeed dealing with the two widest avoided crossings in the system (this can be seen by comparing the different widths of the peaks in the D˜ 2 plot), we can identify the participating Fock states as 0,3,0 and 0,2,1 (avoided crossing at E/U 3), as well as 3,0,0 and 2,1,0 | 〉 | 〉 ≈ | 〉 | 〉 (avoided crossing at E/U 3), where only one boson needs to tunnel and where ≈ − all bosons are highly localized in real space.

24 Chapter 3. Features of the System for M N 3 = =

6

4

2 E U 0

-2

10-1 ˜ D2

10-2

10-3 0.5 1.0 1.5 2.0 F/U

Figure 3.8: D˜ (bottom) and energy E/U (top) versus F /U for J/U 2 = 0.1. The evolution of 4 specific energy levels and their corres- ponding eigenstate structure is highlighted in dark blue, green, red and black.

If we now have a look at the behaviour of D˜ around F /U 0.4-0.5, we dis- 2 ≈ cover three seemingly sharp jumps between different well defined D˜ 2 trajectories that correlate with three avoided crossings in the energy spectrum. They appear when following the evolution of the energy level that corresponds to the Fock state 3,0,0 in the J 0 case. After each of the first three avoided crossings the colour | 〉 = changes, from blue to green to red to black. When following the same colour code

25 Chapter 3. Features of the System for M N 3 = =

in the D˜ 2 plot, one observes a continuous line with very low values of D˜ 2. This indicates us that even for non-zero coupling J, the eigenstate still consists mostly of 3,0,0 , up to a certain value of F /U. Because of the very sharp jumps in D˜ , | 〉 2 we know that the three avoided crossings are quite narrow and that the coupling between the Fock states involved is very weak.

26 CHAPTER 4

LOCALIZATION AND ROBUST EIGENSTATE STRUCTURESIN FOCK SPACE

In this chapter we consider larger systems in which we discover certain eigen- states with a special robustness and localization in Fock space. Since these state were studied before, e.g., in Refs. [24, 25], we give a short overview about earlier descriptions of the phenomenology. Afterwards, we discuss our own approach to these states in a M N 4 system. Then, we enlarge the lattice size to values = = M 8 and M 12 and see which conclusions can be drawn. = =

4.1 “Solitonic States” in Tilted Lattices

A special feature of the tilted Bose-Hubbard Hamiltonian was studied in Ref. [25]. In Fig. 4.1 the energy spectrum E/J versus F /J for U 1 and J 0.5 of a system = = with N 3, M 11 is illustrated. One observes firstly, as shown in Fig. 4.1(b), = = states that seem to traverse the spectrum mixing very weakly with the bulk of eigenstates. When zooming out [see Fig. 4.1(a)], M such states are identified. We will shortly summarize the major aspects of these states as portrayed in Ref. [25]. There, they were called “solitonic states”, and we will keep this nomenclature in this section. One defining feature of the solitonic states is the constant slope in the E versus F plot. Using the Hellman-Feynman theorem [34] one finds that ¿ ˆ À * M + ∂E ∂H X ˜ lnˆl , (4.1) ∂F = ∂F = l 1 = where represents the expectation value with respect to a single eigenstate. This 〈·〉 means that the centre of mass of the bosonic density in a solitonic state stays

27 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

Figure 4.1: Spectrum E/J versus F /J, with U 1, J 1/2 for N 3 = = = bosons and M 11 lattice sites. The M eigenstates with the = largest IPR are plotted in red. (a) A set of (almost) straight-line energy levels traverse the chaotic background, without changing the slope. (b) A zoom into the vicinity of the two lowest solitonic levels reveals small avoided crossings. Taken from Ref. [25].

constant over a large range of F values. However, as one can also see in Fig. 4.1, from a certain F , the structure of the state changes, and the former bulk-crossing line is no more recognizable as such. In Ref. [25] the solitonic states were seen to have a high inverse participation ratio in the Fock basis, i.e., to exhibit localization in Fock space. It was also shown that they exhibit localization of the bosonic density in real space, while states in the bulk background are rather extended both in Fock and real space. The solitonic states were associated with M Fock states of the form 0,...,0,N,0,...,0 . | 〉 Their stability against dynamical changes in F was also demonstrated. As it can be seen in Fig. 4.1, the solitonic states do not exist for any tilt strength. They rather exhibit a threshold value of F for their existence that was estimated to be

F U(N 1) 2JpN. (4.2) t ≈ − − The determination of this value was based on the study of the first order avoided crossing involving the Fock state mentioned above. For a detailed description and analysis of the features of such solitonic states we refer the reader to Refs. [24,25].

4.2 Parameter Dependence of Eigenstate Structure

In this section we try to deepen into the eigenstate structure and investigate the existence of robust states in Fock space. We first consider a system with M N 4 = = and study it in different parameter areas. Afterwards, we enlarge the number of sites while keeping the boson number constant and see how that affects Fock space localization.

28 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

4.2.1 System with M N 4 = = From now on, we scale the energy by the interaction strength U. The range of tilt F is selected by the interval in which avoided crossings appear, i.e., F /U [0,4].1 The ∈ properties of the eigenstates will be analysed for different values of the hopping strength J.

Features of the eigenstates for weak hopping J/U

In Fig. 4.2(a) we show the energy spectrum versus F /U for J/U 0.079 in terms = of the energy density E Emin ² − . (4.3) ≡ E E max − min Here, E is the energy of an individual eigenstate and Emin/Emax are the minim- al/maximal eigenenergies in the spectrum for a certain F . This representation allows us to better resolve the spectrum for small values of F , for which the spectral lines are still very close to each other. For weak tilt one can identify five different manifolds of eigenenergies. These manifolds stem from the degeneracies present in the spectrum for J/U 0. In the absence of hopping there are 4 degenerate = manifolds of eigenstates located at energies ² 1,1/2,1/3,1/6, corresponding to = all Fock states resulting from the permutations of 4,0,0,0 , 3,1,0,0 , 2,2,0,0 | 〉 | 〉 | 〉 and 2,1,1,0 , respectively. Hence, these manifolds contain correspondingly 4, 12, | 〉 6 and 12 Fock states. Additionally, we identify the ground state 1,1,1,1 at ² 0 | 〉 = as the fifth manifold. Let us now describe the localization properties (in Fock space) of the different manifolds. For this purpose, we build the scatter plot D˜ versus ² for J/U 0.079 2 = and F /U 0.06 shown in Fig. 4.2(b). For the small chosen value of F /U, the = manifolds are still well separated in energy and the scatter plot provides a clear picture of the localization properties of the eigenstates within each manifold. There are different average values of D˜ 2 for the manifolds observable. To make clearer how the mixing of Fock states within the eigenstates occurs for small J/U, one can look at Fig. 4.3. There, all eigenstate coefficients in the Fock basis are presented. One can confirm the general localization properties observed in the scatter plot, e.g., states in the first manifold are localized on mostly one Fock state, whereas states from the second manifold look more extended. For small J/U the maximum of the coefficients in the eigenstates of each manifold correspond to the Fock states of the respective unperturbed manifold, which are marked in red in Fig. 4.3. We will therefore refer to the eigenstates as, e.g., Ψ that indicates that the | 0400〉 state is strongly localized on the specified Fock state, in this case 0,4,0,0 . The | 〉 eigenstates of the first manifold, labelled by i, ii, iii, iv in Fig. 4.2(b), correspond to Ψ , Ψ , Ψ and Ψ , respectively. Since for small J, Fock states | 4000〉 | 0400〉 | 0040〉 | 0004〉 mixing is dominated by first order transitions (i.e., transitions where one boson

1For small J/U, the last avoided crossing is expected to be at F U(N 1), see Sec. 2.1. = −

29 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

1 1.0 1 ˜ J/U=0.079 D2 0.8 ϵ 4 10-1 2 0.6 2 GS 3 0.4 10-2 3 1 ii iii 0.2 4 J/U=0.079 i F/U=0.060 iv 10-3 0.0 0 1 2 3 4 0. 0.2 0.4 0.6 0.8 1. (a) F/U (b) ϵ

Figure 4.2: Energy and D˜ spectra of a tilted BHH with M N 4. (a) 2 = = Energy density ² versus F /U. For small F /U, different manifolds of states labelled 1 to 4 can be identified at ² 1, 1/2, 1/3, 1/6, ≈ respectively. Numbers and a square bracket highlight the separ- ation of the different manifolds. (b) Scatter plot D˜ 2 versus ² for J 0.079, F 0.06. In the first manifold, the states are marked = = from i to iv.

tunnels from one site to its next neighbouring.), one can explain the higher D˜ 2 value of states ii and iii: States 0,4,0,0 and 0,0,4,0 have two possibilities of | 〉 | 〉 first order transitions to other Fock states, whereas 4,0,0,0 and 0,0,0,4 have | 〉 | 〉 only one due to HWBC. In order to study how the localization properties for small J/U depend on the tilt, we show in Fig. 4.4 a sequence of scatter plots for J/U 0.1 and increasing = value of F /U. Eigenstates of the form Ψ and permutations (first manifold) | 4000〉 and Ψ and permutations (second manifold) are marked in the scatter plots | 3100〉 by red and green points, respectively. One can observe how manifolds 2 till 5 spread out in energy and start to overlap. The localization properties of the states emerging from the second manifold are strongly affected by the tilt. In fact, these eigenstates are the first to exhibit a strong tilt-induced localization. In contrast, the eigenstates of the first manifold, although they spread in energy, retain their localization properties, i.e., D˜ 2 stays constant over a large interval of F . Hence, at least for small J/U, one can find states localized in Fock space and in real space which are robust against changes in F . In order to have a global picture of the influence of the tilt, we show in Fig. 4.5 the plot of D˜ 2 versus F /U correlated with the evolution of the energy spectrum. In Fig. 4.5 the trajectory of each eigenstate emerging from the first manifold is highlighted: Ψ (green), Ψ (orange), Ψ (black) and Ψ (red). | 4000〉 | 0400〉 | 0040〉 | 0004〉

30 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

i ii iii iv 2 1 1 Ψn 10 -1

¤ - 10 2

10 -3

1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400031003010 300122002110 21012020 2011200213001210 12011120 111111021030 10211012100304000310 03010220 021102020130 0121011201030040 0031002200130004

È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È 2 1 1 HΨnL 10 -1

¤ - 10 2

10 -3

2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400031003010 300122002110 21012020 2011200213001210 12011120 111111021030 10211012100304000310 03010220 021102020130 0121011201030040 0031002200130004

È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È 2 1 1 HΨnL 10 -1

¤ - 10 2

10 -3

3 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400031003010 300122002110 21012020 2011200213001210 12011120 111111021030 10211012100304000310 03010220 021102020130 0121011201030040 0031002200130004

È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È 2 1 1 HΨnL 10 -1

¤ - 10 2

10 -3

4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400031003010 300122002110 21012020 2011200213001210 12011120 111111021030 10211012100304000310 03010220 021102020130 0121011201030040 0031002200130004

È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È 2 1 1 HΨnL 10 -1

¤ - 10 2

10 -3

GS \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 400031003010 300122002110 21012020 2011200213001210 12011120 111111021030 10211012100304000310 03010220 021102020130 0121011201030040 0031002200130004

È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È È H L 2 Figure 4.3: Wavefunctions coefficients ψn versus Fock states for the | | tilted BHH with M N 4, U 1, F 0.06 and J 0.079. The = = = = = Fock states of the unperturbed manifolds (i.e., for J/U 0) are = highlighted in red. (1) First manifold, 4 states [with labelling i to iv as in Fig. 4.2(b)]. (2) second manifold, 12 states. (3) Third manifold, 6 states. (4) Fourth manifold, 12 states. (GS) Ground state. 31 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

Ž Ž Ž D2 D2 D2

10-1 10-1 10-1

10-2 10-2 10-2

F U=0.020 F U=0.160 F U=0.300 10-3 10-3 10-3 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1.  Ε Ž Ε Ž Ε Ž D2 D2 D2

10-1 10-1 10-1

10-2 10-2 10-2

F U=0.440 F U=0.580 F U=0.720 10-3 10-3 10-3 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1.  Ε Ž Ε Ž Ε Ž D2 D2 D2

10-1 10-1 10-1

10-2 10-2 10-2

F U=0.860 F U=0.980 F U=1.880 10-3 10-3 10-3 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1.  Ε  Ε  Ε

Figure 4.4: Evolution of scatter plots D˜ 2 versus ² of the tilted BHH with M N 4 for increasing values of F /U (from left to right and = = top to bottom) with coupling fixed to J/U 0.1. States marked = in red have 90% or more of their norm on the subspace spanned by the permutations of 4,0,0,0 . Green points highlight states | 〉 with 80% or more of their norm on the subspace spanned by the permutations of 3,1,0,0 . | 〉

For F /U between 0 and 1, states Ψ and Ψ (green and orange) go through | 4000〉 | 0400〉 several tiny avoided crossings that do not change their localization properties noticeably. The first avoided crossing for these states with noticeable width happens at F /U 1. With help of the energy plot, we can identify the actors = taking part. State Ψ crosses with Ψ , Ψ crosses with Ψ and | 4000〉 | 1300〉 | 0400〉 | 0130〉 Ψ crosses with Ψ . Here, three bosons tunnel at the same time, which | 0040〉 | 0013〉

32 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space means that the first manifold undergoes a crossing of third order. At F /U 3/2, a = second avoided crossing can be observed. Comparing the energy spectrum, one discovers that Ψ crosses with Ψ and Ψ crosses with Ψ . This | 4000〉 | 3010〉 | 0400〉 | 0301〉 second order transition is not possible for state Ψ , so it does not show a peak | 0040〉 in D˜ . At F /U 2, another wider avoided crossing takes place. Comparing again 2 = with the energy spectrum, one sees that state Ψ crosses with state Ψ | 4000〉 | 2200〉 and so on. This moving of two bosons corresponds to a mixing of second order. The widest avoided crossing for Ψ , Ψ and Ψ occurs at F /U 3, | 4000〉 | 0400〉 | 0040〉 = where Ψ crosses with Ψ . Here, we observe a crossing of first order which | 4000〉 | 3100〉 distorts the localization of the eigenstates considerably. The state with the highest energy, Ψ (red) does not undergo any avoided crossing and conserves its | 0004〉 localization properties over the whole range of F /U. Additionally, since we know that the widest avoided crossings in the energy plot are due to transitions of first order and that they exhibit the widest peaks in D˜ , we can identify all wide peaks at F /U 0,1,2 also with crossings of first order 2 = involving states emerging from other manifolds.

Stability of localization as a function of J/U

Up to here, we only considered small values of the hopping strength J/U. How- ever, we know that J is the parameter that controls the Fock space mixing. The localization and D˜ 2 in particular should therefore be strongly dependent on the hopping strength. We will now study the behaviour of the localization of the first manifold when increasing J/U. In Fig. 4.6 one can see the energy spectrum E/U combined with the D˜ 2 versus F /U plot for six different values of J/U. In the plots, the eigenstates highlighted are those with 90% or more of their norm localized on the space spanned by 4,0,0,0 and its permutations (i.e., the eigenstates stemming from the first man- | 〉 ifold that retain their localization properties). The choice of such threshold for the localization norm of the states induces an upper bound for the values of 2 D˜ 2. In the energy spectrum one can then see where these states are still well localized and where not. Generally, one can observe a shifting upwards af the average value of D˜ 2 for increasing J/U. This is expected, since larger J implies stronger mixing (wider avoided crossings), and hence delocalization in general. The widening of the avoided crossings for increasing J/U can be clearly observed, e.g., at F /U 1,2,3. = Therefore, the breaking of localization in the first manifold starts developing from the centre of the wider avoided crossings, e.g., the first order crossing at F /U 3. We must however emphasize that for sufficiently large J/U (cf. the case = J/U 0.25 in Fig. 4.6) even the third order crossing at F /U 1 breaks the 90% = = localization condition of the first manifold. Hence, the larger J/U, the shorter the range of F /U within which localization remains.

2 The maximal value of D˜ 2 can be estimated considering the eigenstate to be 90% localized on one Fock state and the remaining 10% of its norm to be evenly spread over the rest of Fock space.

33 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

1

10-1 Ž D2 10-2

10-3

10-4

1.

0.8

0.6 Ε 0.4

0.2

0. 0 1 2 3 4 F U

 Figure 4.5: D˜ 2 (upper panel) and energy density ² (lower panel) versus F /U, at fixed J/U 0.0501. Colour symbols highlight those ei- = genstates that have more that 90% of their norm concentrated on a single Fock state, as follows: Ψ (green), Ψ (or- | 4000〉 | 0400〉 ange), Ψ (black) and Ψ (red). Vertical dashed lines | 0040〉 | 0004〉 mark the position of the major avoided crossings. The higher density of points between 0 and 1 is due to a shorter step in F /U. This was done to have a better resolution of the tiny avoided crossings in D˜ 2.

34 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

Figure 4.6: D˜ 2 and energy E/U versus F /U for increasing values of J/U (from left to right and top to bottom) for the tilted BHH with M N 4. States marked in orange have 90% or more = = of their norm on the subspace spanned by the permutations of 4,0,0,0 . Note that the range of the vertical axis changes for | 〉 different rows. The higher density of points between 0 and 1 is due to a shorter step in F /U. This was done to have a better resolution of the tiny avoided crossings in D˜ 2. 35 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

Another interesting observation is that for any J/U above F /U 1 certain ≈ eigenstates exhibit lower D˜ 2 values than the states localized on the first manifold. Such behaviour (tilt-induced localization) we already encountered in Fig. 4.4.

4.2.2 Localization in Fock space for larger M We study now the robustness of the eigenstates of the first manifold when keeping the number of bosons constant at N 4 and enlarging the number of lattice = sites to 8 and 12 so that one reaches Hilbert space sizes of N 330 and 1365, = respectively. The system with N 4, M 12, is close to the one shown in Fig. 4.1 = = and studied in Ref. [25].

Localized manifolds for comparable values of J/U and F /U

We have seen that for small values of J/U, it is possible to identify localized eigenstates whose structure survives over a certain range of the tilt strength. This may be expected, since for weak hopping strengths, Fock space mixing is weak and thus delocalization is per se stronger. The question is whether it is possible to have robust localized eigenstate structures in a range where interaction (U), hopping (J) and tilt (F ) are comparable, and no individual term in the Hamiltonian dominates over the others. In Fig. 4.7, we show the evolution of the scatter plots D˜ versus ² for J/U 2 = 0.3981 and increasing values of the tilt, F /U {0.10,0.35,0.60}. Panels on the ∈ left (right) correspond to M 8 (M 12) with N 4. Red points highlight those = = = eigenstates for which at least 75% of their norm populate one Fock state with all bosons on one site.3 As one can observe, the first manifold can still be clearly separated from the other states in the system for both lattice sizes 8 and 12. We refer to all states that are not part of the first manifold as the bulk. The D˜ 2 values of the bulk and the first manifold differ by an order of magnitude, even for values of F /U comparable to J/U. Each state of the first manifold can still be associated with a Fock state with all bosons on one site. Furthermore, it is apparent that all bulk states exhibit similar localization properties in Fock space which give rise to a well defined cluster of D˜ 2 values that gets more recognisable the larger M. As F /U increases, the first manifold states spread out in energy and eventually fully overlap with the bulk along the ² axis. In this process, the states of the first manifold go through a large number of narrow avoided crossings that do not affect their eigenstate structure, i.e., D˜ 2 does not change noticeably. The localization manifold exists and is clearly decoupled from the bulk over a range of F /U values comparable or even larger then J/U, irrespective of the lattice size. In fact, one may argue that the effect is stronger for larger M, since the number of states in the localized manifold is precisely M.

3In Fig. 4.4, the threshold was taken to be higher. Note that the hopping strength J/U there was much smaller than here and thus D˜ 2 was globally lower.

36 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

˜ ˜ D2 D2

- - 10 1 10 1

F/U=0.100 F/U=0.100 - - 10 2 10 2 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. ϵ ϵ ˜ ˜ D2 D2

- - 10 1 10 1

F/U=0.350 F/U=0.350 - - 10 2 10 2 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. ϵ ϵ ˜ ˜ D2 D2

- - 10 1 10 1

F/U=0.600 F/U=0.600 - - 10 2 10 2 0. 0.2 0.4 0.6 0.8 1. 0. 0.2 0.4 0.6 0.8 1. ϵ ϵ

Figure 4.7: Scatter plots D˜ versus ² for J/U 0.398 and three differ- 2 = ent values of F /U (increasing from top to bottom) for the tilted BHH with 4 bosons and 8 sites (left panels) and 12 sites (right panels). The states marked in red have more than 75% of their norm localized on the subspace of Fock states with all bosons on one site. Note that in the bottom right panel, one state of the first manifold is undergoing an avoided crossing and thus breaks the 75% localization condition.

37 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

In Fig. 4.8, we provide a different perspective by showing the evolution of D˜ 2 versus F /U correlated with the level dynamics, for two values of J/U. The orange, red, and blue symbols highlight the eigenstates with more than 90%, between 90% and 80%, and between 80% and 70%, respectively, of their norm localized on Fock states with all bosons on one site. For J/U 0.0501 (top panels in Fig. 4.8), we can = see that the D˜ 2 trajectories of localized states versus F /U look the same for any M (cf. also with the results for M 4 shown in Fig. 4.6). Only three trajectories of D˜ = 2 are highlighted, corresponding to the eigenstates Ψ , Ψ , and the group | 40...0〉 | 0...04〉 of M 2 states of the form Ψ . As we discussed in Sec. 4.2.1, these three − | 0...040...0〉 groups have slightly different localization properties due to the use of HWBC. As F /U is increased, the distortion of localization around the major crossings at F /U 1,3/2,2,3 can be observed. The more relevant case, J/U 0.3981, is shown = = in the bottom panels of Fig. 4.8. By looking at the energy spectrum, we clearly see the linear trajectories associated to localized states traversing the whole bulk of the spectrum, as observed in Ref. [25] and shown in Fig. 4.1. In the evolution of D˜ 2, the localized manifold exhibits essentially three trajectories, as for the weak case, localization is however strongly distorted in this case. One may estimate that the localized structure of the manifold persists up to F /U 0.6-0.7, independently of ≈ the lattice size.

Robust states and threshold value for the tilt strength

It is now worth revisiting the results of Ref. [25], where “solitonic states” were reported, and comparing those against our findings. We have seen that in the tilted BHH, there exists a special manifold of states which exhibit robust localization properties well separated from those of the rest of the spectrum. Such states are strongly linked to Fock states with all bosons on one site, and hence it seems appropriate to single out this manifold. In Ref. [25], it was suggested that one may also speak of a second set of states with special localization properties that are associated with Fock states of the form 0,...,0,N 1,1,0,...,0 . They called these “solitonic states of second order”. | − 〉 Here, we have provided evidence that does not support such distinction. The localization properties of the set of states emerging from the mentioned Fock states are essentially those of the bulk of the spectrum. In order to monitor the evolution of states of the first manifold across the spec- trum, we suggest to look at the localization properties on the relevant subspace of Fock states, rather than searching for lowest D˜ 2 or highest IPR values (as done in Ref. [25]). From the results shown in Figs. 4.6 and 4.8, we know that the latter criterion may not be appropriate for small J/U values. Regarding the range of F /U within which the states of the robust manifold exist, an estimation for the upper value Ft was given in Ref. [25], and given in Eq. (4.2) on page 28. This value for Ft follows from the estimation of the width ¯ ® of the first order avoided crossing between states of the form ¯Ψ0...0,N,0...0 and ¯ ® ¯Ψ0...0,N 1,1,0...0 . For a system with N 4 and J/U 0.4, the estimation from − = ≈ 38 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

J/U=0.0501 J/U=0.0501 20 20

15 15

10 10

5 5

E/U 0 E/U 0

-5 -5

-10 -10

-1 -1 ˜ 10 ˜ 10 D2 D2 10-2 10-2

10-3 10-3

10-4 10-4 0 1 2 3 4 0 1 2 3 4 F/U F/U J/U=0.3981 J/U=0.3981 15 15

10 10

5 5 E/U E/U 0 0

-5 ˜ ˜ D2 D2

-1 10-1 10

10-2 10-2 0. 0.2 0.4 0.6 0.8 1. 1.2 0. 0.2 0.4 0.6 0.8 1. 1.2 F/U F/U

Figure 4.8: Energy E/U and D˜ 2 versus F /U for the tilted BHH with N 4 for two different coupling J/U 0.0501 (top panels), = = J/U 0.3981 (low panels). The left (right) panels correspond to = M 8 (M 12). The orange, red, and blue symbols highlight = = the eigenstates with more than 90%, between 90% and 80%, and between 80% and 70%, respectively, of their norm localized on Fock states with all bosons on one site. Note the different F /U values for different J/U.

39 Chapter 4. Localization and Robust Eigenstate Structures in Fock Space

Eq. (4.2) yields F /U 1.44, which, in view of the results shown in Fig. 4.8, is t ≈ clearly an overestimation. Certainly, one can argue that the norm threshold criterion that we use to monitor the states of the first manifold in Fig. 4.8 is debatable, but in any case, we have also shown in Figs. 4.6 and 4.8 that for values of the hopping strength J/U & 0.2 already the third order avoided crossing taking place at F /U 1 disturbs strongly the localization of the robust manifold. Hence, = for not small J/U the estimation of Ft via Eq. (4.2) does not seem to be justified. In this regard, further analysis of systems with a larger number of bosons seems to be necessary.

40 CHAPTER 5

CONCLUSIONAND OUTLOOK

In the present thesis we discussed the Fock space localization properties of many- body eigenstates and the spectrum of the tilted Bose-Hubbard model. In order to investigate the eigenstate structure in Hilbert space, we made use of an original approach based on generalized fractal dimensions, that allowed us to quantify the localization properties of the states in finite Hilbert spaces. We gave a detailed discussion of avoided crossings appearing in the energy spectrum and their relation to the structure of eigenstates. We constructed a toy model with which we could simulate avoided crossings involving a few Fock states. By considering the finite-size fractal dimensions D˜ q , we were able to make statements about the localization properties of states undergoing an avoided crossing, and more prominently, to identify avoided crossings in the parametric evolution of D˜ q . We could confirm the existence of one and only one manifold of states with special localization properties, both in Fock space and in real space. We saw that this manifold can be directly correlated with Fock states which have all bosons localized on one site, by showing that the eigenstates have a high percentage of their norm located on the mentioned Fock states. We could also make statements about the ranges of hopping strength J, interaction strength U and tilt strength F in which this manifold exists. Most importantly, we demonstrated, that the manifold survives in regimes where J, U and F have the same order of magnitude, i.e., when no individual term in the Hamiltonian dominates. Moreover, we saw, by considering systems with fixed J/U, variable F /U and different numbers of lattice sites, that the stability of the first manifold is independent of the number of lattice sites when fixing the number of bosons. In order to monitor the parametric evolution of the localized manifold, e.g., across the energy spectrum, we suggested to apply a criterion based on the eigen- state’s norm on a specific subspace of Fock states, namely on all states with all bosons concentrated on one site. This method seemed to be more effective than

41 Chapter 5. Conclusion and Outlook

considering the highest values of the inverse participation ratio of the states, as it was done in earlier studies. We also showed that the estimation of the upper value Ft of the tilt’s range in which the robust manifold prevails needs to be further investigated. Specific- ally, we concluded that in order to understand the stability of the manifold, it is important to consider higher order avoided crossings, in contrast to what was suggested in earlier studies. For the continuation of this work, it would be desirable to have a better cri- terion to quantify the degree of localization of the manifold. This would lead to clearer identification of the threshold value Ft for the tilt. One could think, 2 for example, of monitoring the evolution of the intensities ψn on the relevant | | Fock subspace as a function of the tilt strength. If such evolution were to reveal a sharp decrease beyond a certain value of F , the breaking of localization in Fock space could be consistently identified. Additionally, it would be interesting to investigate how the stability of localization and the eigenstate structure evolve when increasing the number of bosons N for a constant lattice size as well as when increasing N while keeping the filling factor N/M constant.

42 BIBLIOGRAPHY

[1] F.Haake, Quantum Signatures of Chaos (Springer, Berlin, Heidelberg, 2004). [2] A. R. Kolovsky, and A. Buchleitner, ‘Multiparticle quantum chaos in tilted optical lattices’, J. Mod. Opt. 51-6, 999–1003 (2004). [3] A. R. Kolovsky, and A. Buchleitner, ‘Quantum chaos in the Bose-Hubbard model’, Europhys. Lett. 68, 632–638 (2004). [4] C. Kollath, G. Roux, G. Biroli, and A. M. Läuchli, ‘Statistical properties of the spectrum of the extended Bose–Hubbard model’, J. Stat. Mech. Theory Exp. 2010, P08011 (2010). [5] M. Hiller, T. Kottos, and T. Geisel, ‘Complexity in parametric Bose-Hubbard Hamiltonians and structural analysis of eigenstates’, Phys. Rev. A 73, 061604 (2006). [6] B. B. Mandelbrot, ‘Multifractal Power Law Distributions: Negative and Crit- ical Dimensions and Other “Anomalies,” Explained by a Simple Example’,J. Stat. Phys. 110, 739–774 (2003). [7] G. Paladin, and A. Vulpiani, ‘Anomalous scaling laws in multifractal objects’, Phys. Rep. 156, 147–225 (1987). [8] F.Evers, and A. D. Mirlin, ‘Anderson transitions’, Rev. Mod. Phys. 80, 1355– 1417 (2008). [9] A. D. Mirlin, Y. V. Fyodorov, F. M. Dittes, J. Quezada, and T. H. Seligman, ‘Transition from localized to extended eigenstates in the ensemble of power- law random banded matrices’, Phys. Rev. E 54, 3221–3230 (1996). [10] Y. V. Fyodorov, A. Ossipov, and A. Rodriguez, ‘The Anderson localization transition and eigenfunction multifractality in an ensemble of ultrametric random matrices’, J. Stat. Mech. Theory Exp. 2009, L12001 (2009). [11] E. Bogomolny, and C. Schmit, ‘Spectral Statistics of a Quantum Interval- Exchange Map’, Phys. Rev. Lett. 93, 254102 (2004).

43 Bibliography

[12] J. Martin, I. García-Mata, O. Giraud, and B. Georgeot, ‘Multifractal wave functions of simple quantum maps’, Phys. Rev. E 82, 046206 (2010). [13] Y. Y. Atas, and E. Bogomolny, ‘Multifractality of eigenfunctions in spin chains’, Phys. Rev. E 86, 021104 (2012). [14] Y. Y. Atas, and E. Bogomolny, ‘Calculation of multi-fractal dimensions in spin chains’, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 372, 20120520 (2014). [15] E. Bogomolny, Multifractality in simple systems (presentation at the confer- ence “Complex patterns in wave functions: drums, graphs, and disorder" at the Kavli Royal Society Centre, UK, 2012). [16] J. Lindinger, ‘Multifractal properties of the ground state of the Bose-Hubbard model’, MSc thesis (Albert-Ludwigs-Universität Freiburg, 2017). [17] J. Lindinger, A. Buchleitner, and A. Rodríguez, ‘Many-Body Multifractality throughout Bosonic Superfluid and Mott Insulator Phases’, Phys. Rev. Lett. 122, 106603 (2019). [18] H. A. Gersch, and G. C. Knollman, ‘Quantum cell model for bosons’, Phys. Rev. 129, 959–967 (1963). [19] M. P.A. Fisher, P.B. Weichman, G. Grinstein, and D. S. Fisher, ‘Boson loc- alization and the superfluid-insulator transition’, Phys. Rev. B 40, 546–570 (1989). [20] I. Bloch, J. Dalibard, and S. Nascimbène, ‘Quantum simulations with ul- tracold quantum gases’, Nat. Phys. 8, 267–276 (2012). [21] I. Bloch, J. Dalibard, and W. Zwerger, ‘Many-body physics with ultracold gases’, Rev. Mod. Phys. 80, 885–964 (2008). [22] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, and U. Sen, ‘Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond’, Adv. Phys. 56, 243–379 (2007). [23] K. V. Krutitsky, ‘Ultracold bosons with short-range interaction in regular optical lattices’, Phys. Rep. 607, 1–101 (2016). [24] H. Venzl, ‘Ultracold bosons in tilted optical lattices – impact of spectral stat- istics on simulability, stability, and dynamics’, PhD Thesis (Albert-Ludwigs- Universität Freiburg, 2011). [25] M. Hiller, H. Venzl, T. Zech, B. Ole´s, F.Mintert, and A. Buchleitner, ‘Robust states of ultracold bosons in tilted optical lattices’, J. Phys. B At. Mol. Opt. Phys. 45, 095301 (2012). [26] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Folling, L. Pollet, and M. Greiner, ‘Probing the Superfluid-to-Mott Insulator Transition at the Single-Atom Level’, Science 329, 547–550 (2010).

44 Bibliography

[27] M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, ‘Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms’, Nature 415, 39–44 (2002). [28] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, ‘One di- mensional bosons: From condensed matter systems to ultracold gases’, Rev. Mod. Phys. 83, 1405–1466 (2011). [29] A. R. Kolovsky, and H. J. Korsch, ‘Bloch oscillations of cold atoms in optical lattices’, Int. J. Mod. Phys. B 18, 1235–1260 (2004). [30] T. Hartmann, F.Keck, H. J. Korsch, and S. Mossmann, ‘Dynamics of Bloch oscillations’, New J. Phys. 6, 2 (2004). [31] T. Brünner, ‘Signatures of partial distinguishability in the dynamics of inter- acting bosons’, PhD Thesis (Albert-Ludwigs-Universität Freiburg, 2018). [32] D. Stauffer, and A. Aharony, Introduction to percolation theory, 2nd rev. (Taylor & Francis, 2003). [33] H. Hentschel, and I. Procaccia, ‘The infinite number of generalized dimen- sions of fractals and strange attractors’, Phys. D Nonlinear Phenom. 8, 435– 444 (1983). [34] R. P.Feynman, ‘Forces on Molecules’, Phys. Rev. 56, 340–343 (1939).

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