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Signatures of topology in a quasicrystal: a case study of the non-interacting and superconducting Fibonacci chain

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Signatures of topology in a quasicrystal: a case study of the non-interacting and superconducting Fibonacci chain

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Content Signatures of topology in a quasicrystal: A case study of the non-interacting and superconducting Fibonacci chain by Gautam Rai A dissertation presented to the Faculty of the USC graduate school University of Southern California in partial fulllment of the requirements for the degree Doctor of Philosophy Physics December 2021 Copyright 2021 Gautam Rai In memory of Hritik Sampat ii Acknowledgements It takes a village. The rst and most important person that I need to thank for their part in this thesis is my advisor Stephan Haas. I am forever grateful to Stephan for the opportunity he gave me to learn and do physics at the highest level. He has an uncanny ability to see from the perspective of his students, to recognize what they need to succeed, and he has the wherewithal to provide it. I had a happy and fullled time in graduate school, and much of the credit goes to him. Anuradha Jagannathan eectively played the role of a second advisor to me. A little over three years ago, Anu introduced me to quasicrystals in what was to be my rst project in graduate school. Soon after, she invited me to spend a couple of weeks in Paris to get the collaboration going. The rest of the story is in this thesis. Much of this work would not have been possible if Anu hadn't been so patient and persistent with me. I was surrounded by a set of excellent students at USC, and I had the distinct pleasure of working with some of them on my projects. Henning Schl omer has the enviable knack of nding a way to make a meaningful contribution to every discussion. The enterprising Chris Matsumura was barely a junior when he started working with us and has a bright future in front of him. Ying Wang's output has been prodigious in the short time since she began, and I can rest assured that the quasicrystal project at USC is in good hands. Finally, Malte R osner helped me through my rst steps in graduate school. He walked me through my rst mini-project on inhomogeneous superconductors. iii Signatures of topology in a quasicrystal I thoroughly enjoyed every elective course I took at USC. Hubert Saleur and Paolo Zanardi consistently found ways to eectively communicate even the most forbidding topics. Satish Kumar Thittamaranahalli taught his computer science course in a way that was both accessible to physicists and profound. Thanks is also due to my dissertation committee: Paolo Zanardi, Rosa Di Felice, Eli Levenson-Falk, and Susumu Takahashi. As easy as it is to forget sometimes, there is more to life than physics. Ahis, John, Christina, Tiany, Namit, Johannes, Felipe, Nicolas, Zhihao, Arash, Gerogios, and George were a joy to be around. Robert Walker gets a special mention as my personal dispenser of sage advice and an all-round stand-up fellow. The biggest presence of all was Jack Lashner, my roommate for four years. By the third time we met, I knew we were going to be friends forever. The physicist's journey begins long before graduate school. I was fortunate to have two extraordinary teachers in high school. Parag Mehta and Hritik Sampat are the reasons I work in the natural sciences at all. Parag can incite wonder and curiosity for mathematics in anybody. Hritik might have the world's highest conversion ratio of high school students to PhDs in physics. Two of my professors at my alma mater Jacobs University Bremen had particular impact on me. Peter Schupp is one of the best lecturers around and the reason I ended up in graduate school. Stefan Kettemann taught me my rst course in solid state physics and continues to be a great collaborator. Any discussion of my time at Jacobs would be incomplete without mentioning my friends and fellow physics majors, Max Schallwig and Sven Wasmus. I was constantly pushed to be a better student with them raising the bar so consistently high. The nal year of work for my PhD was done during the Covid pandemic out of the Breseman house. A proper thanks is due to Rick and Eileen for having me around, and to Kelsey, Andre, Ryan, and Margaret for providing the diversions. Zeke, Piper, and Cora helped keep my spirits up, but Nimbu is the best cat of them all and most responsible for iv 0. my career success. Mummy and Papa get the nal credit for everything I do. I didn't make it easy, but they always knew the right thing to do. My brother put up with me at my worst behavior and still talks to me. If I am nicer now, it's because I learn from him. At long last, Dana, my partner, my best friend, and my love, what should we do next? v Contents Dedication ii Acknowledgements iii List of Tables viii List of Figures ix Abstract xi Preface xiii 1 Introduction 1 1.1 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Quasicrystals 5 2.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Physical properties of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 A new era of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 The Fibonacci chain 12 3.1 The Geometry of the Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Approximants of the Fibonacci chain . . . . . . . . . . . . . . . . . . 14 3.2 The cut and project method . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 The phason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2.2 Connection to the Aubry-Andr e-Harper model . . . . . . . . . . . . . 16 3.3 Perpendicular space of the Fibonacci chain . . . . . . . . . . . . . . . . . . . 17 3.3.1 Conumbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3.2 Connectivity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 The Fibonacci hopping model . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.4.1 The gap labelling theorem . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 The topology of the Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . 21 3.5.1 Relationship to the Hofstadter model . . . . . . . . . . . . . . . . . . 21 3.5.2 The topology of the Hofstadter model . . . . . . . . . . . . . . . . . . 23 3.5.3 Gap label$ Chern number . . . . . . . . . . . . . . . . . . . . . . . 24 vi 0. CONTENTS 3.5.4 Edge states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 The weak modulation limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.7 The strong modulation limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Superconductivity 35 4.1 Bardeen-Cooper-Schrieer formalism . . . . . . . . . . . . . . . . . . . . . . 36 4.1.1 The instability of the normal state to attractive interactions . . . . . 36 4.1.2 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.1.3 Solution by canonical transformation . . . . . . . . . . . . . . . . . . 39 4.2 The real space Bogoliubov-de Gennes approach . . . . . . . . . . . . . . . . 40 4.3 The proximity eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Superconductivity in quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . 46 5 Bulk signatures of topology in the Fibonacci chain 50 5.1 Charge density in perpendicular space . . . . . . . . . . . . . . . . . . . . . 50 5.2 In the weak modulation limit . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.3 In the strong modulation limit . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3.1 Generalization to arbitrary q . . . . . . . . . . . . . . . . . . . . . . . 56 5.4 Fourier transform of the charge density . . . . . . . . . . . . . . . . . . . . . 57 5.5 Robustness of the oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.6 Entanglement entropy in perpendicular space . . . . . . . . . . . . . . . . . 59 6 The proximity eect in the Fibonacci chain 62 6.1 Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.2 Order parameter prole in the Fibonacci chain . . . . . . . . . . . . . . . . . 66 6.3 Order parameter penetration in the Fibonacci chain . . . . . . . . . . . . . . 68 6.4 Ensemble statistics of the order parameter in the Fibonacci chain . . . . . . 69 6.5 Disordered chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.5.1 Weak Disorder Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.5.2 Strong Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.6 Topological labels in proximity induced SC . . . . . . . . . . . . . . . . . . . 78 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7 Outlook 84 7.1 Intrinsic superconductivity in the Fibonacci chain . . . . . . . . . . . . . . . 84 7.2 Signatures of topology in other systems . . . . . . . . . . . . . . . . . . . . . 89 References 92 A Bogoliubov-de Gennes implementation 98 vii List of Tables 3.1 Approximants of the Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The eigensystem of the unperturbed 1D chain . . . . . . . . . . . . . . . . . 30 6.1 The proximity eect in extended, critical, and localized systems . . . . . . . 82 viii List of Figures 2.1 Diraction pattern of a quasicrystal . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Characteristic structure factor of three types of solids . . . . . . . . . . . . . 7 2.3 Quasicrystals in metamaterials and 2D heterostructures . . . . . . . . . . . . 11 3.1 The rst 18 sites of the Fibonacci chain. . . . . . . . . . . . . . . . . . . . . 13 3.2 The cut and project method for the Fibonacci chain . . . . . . . . . . . . . . 15 3.3 AAH model$ Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Integrated density of state of the Fibonacci hopping model and related systems 20 3.5 Topological equivalence of the AAH and Fibonacci models . . . . . . . . . . 24 3.6 Phason evolution of the eigenvalues and edge states of the Fibonacci chain . 26 3.7 Measured spectrum and edge state energies of the Fibonacci chain in a po- laritonic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.8 Gap openings of the Fibonacci chain . . . . . . . . . . . . . . . . . . . . . . 31 3.9 Molecular and atomic RG in the strong modulation limit . . . . . . . . . . . 33 4.1 Superconducting order parameter prole in the Penrose tiling . . . . . . . . 48 5.1 Charge density in Perpendicular space . . . . . . . . . . . . . . . . . . . . . 52 5.2 Example RG-based analysis in the strong modulation limit . . . . . . . . . . 54 5.3 Fourier components of the perpendicular space charge density . . . . . . . . 57 5.4 Perpendicular space charge density in disordered and interacting systems . . 58 5.5 Oscillations in the entanglement entropy . . . . . . . . . . . . . . . . . . . . 60 6.1 Sketch of the hybrid ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Induced order parameter in metallic systems . . . . . . . . . . . . . . . . . . 64 6.3 Order parameter prole in the Fibonacci chain . . . . . . . . . . . . . . . . . 66 6.4 Self-similarity and symmetric structures in order parameter prole . . . . . . 67 6.5 Ensemble statistics in the Fibonacci chain . . . . . . . . . . . . . . . . . . . 69 6.6 Order parameter prole in disordered systems . . . . . . . . . . . . . . . . . 72 6.7 Order parameter penetration in disordered systems . . . . . . . . . . . . . . 73 6.8 Order parameter ensemble statistics for weak disorder . . . . . . . . . . . . . 74 6.9 Order parameter ensemble statistics for strong disorder . . . . . . . . . . . . 77 6.10 Induced order parameter away from half-lling . . . . . . . . . . . . . . . . . 79 6.11 Phason evolution of the induced order parameter . . . . . . . . . . . . . . . 80 6.12 Phason evolution of the open Fibonacci chain . . . . . . . . . . . . . . . . . 81 ix Signatures of topology in a quasicrystal 7.1 DOS in superconducting FC . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.2 Local DOS and order parameter in FC . . . . . . . . . . . . . . . . . . . . . 87 7.3 Critical temperature of the superconducting FC . . . . . . . . . . . . . . . . 87 7.4 Phase diagram of FC with attractive point interaction . . . . . . . . . . . . . 88 7.5 The generalizes Rauzy tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 x Abstract Orientational order and periodicity were long believed to be equivalent concepts. In the 1980s, the discovery of quasicrystals upended this thinking. Quasicrystals are aperiodic arrangements of atoms that have perfect long-range orientational order. The inclusion of this new class of solids has expanded the repertoire of ordered materials that we can design: i) non-crystallographic symmetries are now possible, and ii) topological phases that would otherwise be disallowed in a given space dimension can now be realized. A simple prototype of a one-dimensional quasicrystal is the Fibonacci chain. The Fi- bonacci chain is topologically equivalent to the two-dimensional Hofstadter model. The Chern numbers of the Hofstadter model have a physical interpretation in the form of the Hall current carried by a given band. This leads to the question: what is the physical inter- pretation of the Chern numbers of the Fibonacci chain? Previous studies have shown that the Chern number can be physically measured by tracking the localization, or the number of gap crossings, of an edge state as the Fibonacci chain Hamiltonian is transformed along a closed orbit by a sequence of phason ips. In my work at USC, I have identied a physical interpretation that does not invoke phason ips. We describe a way to measure the Chern numbers from a single sample of the Fibonacci chain. When mapped to appropriate coordinates, position-dependent quantities such as the real space charge density prole, oscillate with a frequency related to the Chern number of the gap that the system is tuned to. This is a direct demonstration of the bulk-boundary correspondence in the Fibonacci chain. xi Signatures of topology in a quasicrystal By explicit calculations in the presence of s-wave superconducting pairing, my co-authors and I show that the same Chern numbers survive upon the introduction of interactions to the Hamiltonian. If pairing is introduced by the proximity eect, then we can also measure the Chern numbers by tracking how the induced order parameter changes over a closed sequence of phason ips. We also investigate the general characteristics of the proximity eect in a superconductor- quasicrystal hybrid ring. We nd that the proximity eect is strong, with the superconduct- ing order parameter decaying only as a power law at zero temperature. This is comparable to order parameter penetration in clean metals. The ensemble statistics of the induced or- der parameter follow a log-normal distribution function. This is comparable to the strongly disordered dirty case. This dichotomy of behavior of the induced order parameter is a con- sequence of the criticality of the wavefunctions in the Fibonacci chain; they are neither extended nor localized. xii Preface This dissertation is a compilation of my research on quasicrystalline systems over the course of my PhD at University of Southern California in Stephan Haas' group. The work presented here includes contributions from my co-authors. Notably, Chris Matsumura ran some of the numerical simulations presented in Chapter 7, and Henning Schl omer is responsible for some of the calculations presented in Chapter 5. My doctoral advisor Stephan Haas, and co- advisor in spirit Anuradha Jagannathan, are responsible for substantial contributions in the interpretation and analysis of the data, the writing of the manuscripts, and most importantly, the setting of the research agenda. xiii Signatures of topology in a quasicrystal xiv Chapter 1 Introduction Ordinary matter is made up of only three elementary particles: the electron, the proton, and the neutron. Yet, the number and diversity of materials we encounter on an everyday basis is immense. This is because the properties of a material depend not on the constituents that make it up, but on the way that these constituents are arranged. This is the principle of emergence, and it is at the foundation of condensed matter physics. A particular organization of atoms with its set of material properties is called a phase, and every phase is associated with a certain order. In the 1950s, Landau formulated a general theory of phase transitions based on the idea of symmetry-breaking. In this picture, symmetry and order are equivalent concepts. Dierent materials are in dierent phases if the symmetry of the way its constituents are organized is dierent. A phase transition involves the breaking of a symmetry. The symmetry-breaking picture was enormously successful, and it was thought to be complete. But cracks began to emerge in this picture with the discovery of the fractional quantum Hall eect and rapid development in the eld of highT c superconductors. These systems exhibited distinct quan- tum phases separated by phase transitions even though they have the same symmetry group. These phases were described by a new kind of order beyond the symmetry description. This new kind of order is called topological order. 1 Signatures of topology in a quasicrystal In parallel, the eld of crystallography was experiencing its own revolution. Solids can be broadly categorized into crystalline or amorphous solids. Crystalline solids, or crystals, con- sist of neat periodic arrangements of their constituents; they are considered orientationally ordered. The kind of orientational order is related to the symmetry group of the underlying lattice. Amorphous solids, on the other hand, have their constituents haphazardly placed with only limited correlation between the positions of two atoms. These solids are considered disordered. For a long time, it was thought that periodicity was a prerequisite for order. In 1984, the rst quasicrystal was discovered. Quasicrystals are materials where the arrange- ment of the constituents is not periodic, yet there is a precise sense in which the arrangement is ordered. The positions of two atoms are correlated over long distances. An open problem at the intersection of these developments is the full classication of the kind of topological orders present in quasicrystals. This question is particularly interesting in the context of symmetry protected topological (SPT) phases because quasicrystals can circumvent the dimensional constraints on the kind of SPT phases that are possible [1]. This dissertation focuses on my work on the Fibonacci chain, a one-dimensional model of a quasicrystal. The Fibonacci chain is a topological insulator characterized by the integer topological invariants that can be assigned to the gaps in its energy spectrum. The com- mon theme in my work is the question of how these integer labels manifest themselves in experimentally measurable quantities. 1.1 Summary of key results This dissertation compiles the results of the following papers o which I am a co-author: 1. Bulk topological signatures of a quasicrystal [2]: It was previously known that the integer label of a gap of the Fibonacci chain can be obtained by measuring the energy of its edge states. This requires synthesizing multiple versions of the Fibonacci chain that are locally isomorphic to each other and tracking how the edge state energies 2 1. INTRODUCTION evolve over a set path through the dierent realizations. In this paper, we nd that there is a way to obtain these gap labels by making only bulk measurements of a single realization of the Fibonacci chain. This is done by mapping position-dependent quantities, such as the charge density or entanglement entropy, to perpendicular space. It is customary to use the 2D extensions of the Fibonacci chain to assign Chern numbers to individual realizations. Our nding justies this assertion and directly demonstrates the bulk-boundary correspondence in the Fibonacci chain. 2. Superconducting proximity eect and order parameter uctuations in disordered and quasiperiodic systems [3]: Around a normal-superconductor (N-SC) junction, the su- perconducting proximity eect can be a sensitive probe of the electronic properties of the N material. This paper studies how the unconventional properties of a quasicrys- tal aect the strength and real-space prole of the induced superconductivity when it is placed next to a superconductor. We focus on the dierences and similarity with the proximity eect in disordered systems. We nd that the proximity eect at 0T is strong in quasicrystals, comparable to metals, with the induced order parameter decaying slowly as a power law. The ensemble statistics of the induced order param- eter in quasicrystals follow a log-normal relationship, as do those of strongly localized systems. This dichotomy of behavior is a consequence of the critical wave functions of the quasicrystal, which are neither localized nor extended. 3. Proximity eect in a superconductor-quasicrystal hybrid ring [4]: In this paper, we study the superconductor-Fibonacci chain junction and examine the eect of its topo- logical edge states on the induced superconductivity. We nd that the topological edge states have a measurable in uence on the induced superconducting order param- eter. We track the phason evolution of the induced order parameter and nd that the strongest frequencies are the integer labels of the most prominent gaps close to the Fermi level. 3 Signatures of topology in a quasicrystal 1.2 Overview This dissertation is organized into three parts: I) Background, II) Original work, and III) Outlook. Part I, which consists of Chapters 2{4, is concerned with the essential background re- quired for a graduate-level physicist to follow the key reasoning provided in the aforemen- tioned papers. Chapter 2 introduces quasicrystals with an emphasis on their mathematical formulation as cut and project sets. Chapter 3 introduces the Fibonacci chain, our predom- inant subject of research. There is a vast body of literature on this model system (see the references in [5]). I decided to focus primarily on what the topological labels of the Fibonacci chain are and where they come from. Chapter 4 gives a brief introduction to superconduc- tivity. The treatment here focuses on the Bogoliubov-de Gennes formalism and how it can be applied to the problem of the superconducting proximity eect. In part II, which consists of Chapters 5 & 6, I describe our original results in detail. Chapter 5 is associated with the article Bulk topological signatures of a quasicrystal [2]. Chapter 6 discusses the other two articles [3, 6], which are both about the superconducting proximity eect in the Fibonacci chain. Part III consists of Chapter 8. I discuss the avenues of research that directly follow the results described in this dissertation. In Chapter 5, I demonstrated the usefulness o the conumber map to measure topological properties of the Fibonacci chain. I expect that this technique has a fairly general domain of applicability. I describe a higher-dimensional model system, the Generalized Rauzy tiling, where this mapping may have similar utility. A second future direction is the question of intrinsic superconductivity in the Fibonacci chain. I present some preliminary results on this problem. 4 Chapter 2 Quasicrystals In 1984, Dan Shechtman found that one of his Al-Mn samples had a perplexing X-ray diraction pattern|it had sharp Bragg peaks, but the pattern was 5-fold symmetric, an impossible symmetry for a periodic structure [7]. He concluded that the atoms in his sample had formed a long-range ordered, aperiodic arrangement. His assertion was controversial at the time, as many of his contemporaries 1 were unconvinced that elementary particles could self-organize into an ordered and aperiodic conguration. Over the next few years, more alloys with quasicrystalline phases were discovered under tightly controlled conditions, the more mundane explanations 2 of these observations were ruled out, and it became clear that Shechtman had discovered a new paradigm of ordered solids [8]. In 2011, Shechtman was awarded Nobel prize for Chemistry for his discovery. The prevailing view at the time was that point-like diraction patterns and lattice sym- metries (translational symmetry in d independent directions) are equivalent concepts. To see this, let the positions of all the atoms in a material be given by the point set X. The 1 (In)famously including the two-time Nobel prize-winning Linus Pauling 2 It was brie y posited that non-crystallographic diraction patterns could be the result of twinning| intergrowth of rotated copies of a crystal. 5 Signatures of topology in a quasicrystal Figure 2.1: Selected area diraction pattern from a single grain of Shechtman's Icosahedral Al-Mn (Taken from [7]) atomic density is given by: = X x2X x (2.1) The diraction pattern of a structure is related to its Fourier transform ~ 3 . IfX is a lattice, then ~ is the atomic density of another lattice ~ X called the reciprocal lattice. ~ = X x2 ~ X x (2.2) The bright spots in the diraction pattern directly corrrespond to ~ X and the primary task in crystallography is to use the diraction pattern to determine the lattice symmetries and composition of the unit cell. Quasicrystals are materials without lattice symmetries where the Fourier transform of the atomic density is composed of a countable set of weighted functions [9]. ~ = X x2 ~ X c x x (2.3) 3 It is the Fourier transform of its autocorrelation, but for our purpose, it is sucient to consider the Fourier transform of itself as the two quantities are very closely related 6 2. QUASICRYSTALS Structure factor, S(q) q Lattice sketch x Crystal Random Quasicrystal Figure 2.2: The structure factor of three types of solids: Crystals have a discrete diraction pattern on account of their translational symmetry. The diraction peaks themselves form a lattice. Disordered systems have smooth diraction patterns. The quasicrystal also has a discrete diraction pattern, but the pattern does not form a lattice. It is made up of a countable set of weighted Dirac deltas. The number of points visible in the diraction pattern depend on a cuto parameter based on the resolution of the experiment: X x2 ~ X:cxj> jc x j 2 x (2.4) At the time, one of the best known examples of a point set that fullls this criteria was the set of the vertices of the Penrose tiling. Levine and Steinhardt [8] used the Penrose tiling as the starting point to describe an innite set of 2D and 3D structures that have a discrete diraction pattern but are not periodic. They realized that the translational order of these materials was best characterized using the theory of quasiperiodic functions and coined the term quasicrystals. 7 Signatures of topology in a quasicrystal 2.1 Mathematical formulation The current mathematical formulation of quasicrystals is based on the theory of Meyer sets 4 [10]. A subset R k is a Meyer set if and only if: 1. is a Delone set|there is a nite distance r such that there is no more than one 2 in any r-ball, and a nite distance R such that there is at least one 2 in any R-ball. 2. There is a nite set F such that F . Note that, in the case of a periodic lattice, we have F =f0g. Meyer's formalism has the advantage of being succinct and elegant, but for our purposes a more suitable language is provided by the cut and project formalism [11], which is geometric rather than algebraic. The main idea behind the cut and project scheme is to obtain the structure by projecting some sites from a point set in a higher dimension onto the physical space. The two formulations are essentially equivalent|Meyer showed that all Meyer sets are subsets of cut and project sets [10]. The following denition of a cut and project set can be generalized in many ways. The denition I will use here is from [9]. It is more constrained than the standard denition of cut and project sets or model sets but it includes all physical quasicrystals. We start with a lattice ~ R d+m . We dene two orthogonal projections, a parallel and a perpendicular projection: p k :R d+m !R d ; p ? :R d+m !R m (2.5) The parallel projection is one-to-one and the perpendicular projection is dense inR m . Given a non-empty, compact subset WR m , a cut and project set R d is given by (W ) =fp k (x)jx2 ~ ;p ? 2Wg (2.6) 4 Incidentally, Meyer's work predates the discovery of quasicrystals by fteen years! 8 2. QUASICRYSTALS The motivation for studying these came from de Bruijn's 1981 result that the vertices of the Penrose tiling form a cut and project set. In this case, d =m = 2. Most three-dimensional quasicrystals found in various metal alloys are one of a small set of structures with icosahedral symmetry with d = 3;m = 3. The simplest cut and project set and the primary subject of my research is the Fibonacci chain. It is formed by projecting points from the 2D square lattice onto a line with slope equal to the Golden ratio = 1+ p 5 2 (d =m = 1). In 1995, Hof showed that all cut and project sets are pure point diractive [12]. It is useful to dene the map () by composition of the two projections: () :=p ? p 1 k :R d !R m (2.7) In the case of the quasicrystal, () is a one-to-one mapping. If is a periodic structure, the kernel of () is its translation group. The conumber transform of the Fibonacci chain (see Sec. 3.3) is a variation on this map and is the crucial ingredient of the results in Chapter 5. 2.2 Physical properties of quasicrystals As a consequence of the cut and project scheme, the Fourier transform of the lattice, ~ X, can be shown to be generated by linear combinations of d +m basis vectors ~ X = d+m X i=1 n i ~ e i (2.8) In the case of a crystal m = 0, the ~ e i are linearly independent, and ~ X is the reciprocal lattice. In the case of a quasicrystal, m > 0, the ~ e i are incommensurate in the sense that ~ X lls space densely. Dierences in basic transport properties can be inferred directly from these observations about the structure of ~ X. Following [13], periodic structures have plane waves for all wave vectors k that do not satisfy the diraction condition, 2 ~ Xkj ~ Xj 2 = 0. These states can freely propagate in the structure and contribute to the high electrical and thermal conductivities in metals. Since ~ X is dense for a quasicrystal, all k vectors satisfy 9 Signatures of topology in a quasicrystal the diraction condition. By itself, this suggests that there are no propagation phenomena in quasicrystals. But the real picture is a bit more complex. The mechanism of non-propagation at the diraction vectors is encapsulated by the opening of a gap !, which for crystals is of the order 10 12 Hz. The lifetime of a particle at this energy is given by 1 ! which corresponds to a mean free path of < 10 A for typical materials. In quasicrystals, the gaps are dense, but relatively smaller, 10 8 Hz, which corresponds to mean free paths of several micrometers. This analysis is a better indicator of the experimental results which show reduced propagation [14]. Quasicrystals combine low thermal and electrical conductivity with high hardness, re- duced wetting, low friction coecient, and increased brittleness. This unique combination of properties has made them particularly interesting for certain industrial application such as in lubricants and non-stick frying pan coatings [13, 14] 2.3 A new era of quasicrystals Originally, quasicrystals were primarily found in various metal alloys. The recent resurgence of research on quasicrystals is due to technological advances that allow quasicrystals to be physically realized in two new contexts|metamaterials and 2D stacked heterostructures. Metamaterials are heterostructures made of at least two materials with diering dielectric (or elastic) properties [15]. By arranging the materials in periodic, quasiperiodic, or ran- dom patterns, a metamaterial can mimic a crystal, a quasicrystal, or a disordered material. The eective potential on electromagnetic (or acoustic) waves can be tuned to resemble the Schr odinger equation for electrons in a given structure. The lattice constant in these struc- ture is much larger than for a real crystal| m for photonic crystals. For quasicrystals, metamaterials have been particularly useful to experimentally study the 1D quasicrystalline chains such as the Fibonacci chain [16{18], and 2D tilings such as the Penrose tiling [19]. Modern fabrication techniques not only allow the designing of otherwise impossible struc- 10 2. QUASICRYSTALS Figure 2.3: A new era of quasicrystals: 30 twisted bilayer graphene forms the 12-fold symmetric dodecagonal tiling (Taken from [20]). tures, but aord unprecedented control over the parameters of the eective potential. The discovery of superconductivity in magic-angle twisted bilayer graphene (TBLG) [21] set o a storm of activity studying the properties of Moir e materials. A Moir e material consists of two 2D crystals stacked on top of each other such that the unit cell o the combined lattice, the Moir e supercell, is larger than the unit cell of either structure alone. Twisted bilayer graphene is made of two stacked layers of graphene where the layers are rotated by a twist angle with respect to each other. Its electronic properties are extremely sensitive to the twist angle and the carrier density [22]. Most twist angles are incommensurate|the corresponding Moir e supercell is innite. Some result in a quasicrystal. 30 TBLG notably reproduces the dodecagonal tiling [23]. The world of quasicrystals formed by twisted 2D heterostructures has only just begun to be scratched. In addition to these, there are two more modern contexts in which quasicrystals can be designed that haven't received been fully explored yet. Cold atoms have been used to simulate the Fibonacci chain [24]. Recent developments in the world of superconducting qubits have shown that they can be connected to each other to simulate lattices in what is now dubbed circuit quantum electrodynamics [25, 26]. 11 Chapter 3 The Fibonacci chain The Fibonacci chain is the most widely studied quasicrystal. This is largely because it is one of the simplest possible quasicrystals, it is one-dimensional allowing for the use of large-scale numerical techniques, and it exhibits nearly every exotic feature that make quasicrystals so interesting. Partly, this is because it has a close relationship with the Golden ratio and the Fibonacci number sequence which imparts to it a certain elegance. There is a large body of work on the various physical models that can be built on top of the Fibonacci structure, a large portion of which can be found in the reference list of [5]. In this introduction, my main objective is to describe the topological origins of the gap labels of the Fibonacci chain. Sec. 3.1-3.3 describe the construction of the Fibonacci chain and dene important terms. Sec. 3.4 describes the basic properties of the Fibonacci hopping model. Finally, in Sec. 3.5, I discuss the connection between the gap labels and topology. Sec. 3.6 and Sec. 3.7 introduce approximate techniques that are applicable to the Fibonacci in the two opposite limits of very weak and very strong Fibonacci potential respectively. These techniques will prove to be very useful in interpreting the results of Chapter. 5. 12 3. THE FIBONACCI CHAIN Table 3.1: The rst few approximants of the Fibonacci chain. The number of letters in an approximant is always equal to a Fibonacci number (shown in the third column). n S n F n 0 B 1 1 A 1 2 AB 2 3 ABA 3 4 ABAAB 5 5 ABAABAAB 8 A B A B A A B A A B A B A A B A A B Figure 3.1: The rst 18 sites of the Fibonacci chain. 3.1 The Geometry of the Fibonacci chain The Fibonacci chain is a 1-dimensional quasicrystal. Identical atoms are arranged in a row such that the distances between successive atoms take one of two values: d A andd B . Whether the distance between a particular pair of neighboring atoms isd A ord B is determined by the Fibonacci word, denoted by S 1 . It is a semi-innite string of the letters, A and B, dened by the recursion relation: S n = Concatenate(S n1 ;S n2 ) with the initial valuesS 0 = [B] and S 1 = [A]. This method of building the Fibonacci chain is called the concatenation rule. The rst few terms generated in this way are shown in Table 3.1. A chain of atoms arranged according toS 1 has the dening property of quasicrystals|a discrete diraction pattern despite having no translation symmetry. This can be seen by calculating the structure factor of the Fibonacci chain, shown in Fig. 2.2. The same sequence of strings can be equivalently generated using the substitution rule. The map acts on a string of the letters A and B by replacing every occurrence of A with B and every occurrence ofB withAB. TheS n are obtained by repeatedly applying to the stringS 0 = [B], i.e. S n =(::: | {z } n (S 0 )). S 1 is invariant upon application of . This is at the heart of the self-similarity property of the Fibonacci chain. The map provides a natural 13 Signatures of topology in a quasicrystal way to apply renormalization group techniques to the Fibonacci chain. 3.1.1 Approximants of the Fibonacci chain An approximant of a quasicrystal is a periodic structure with an arbitrarily large supercell where the supercell retains the quasiperiodic character of the quasicrystal. The supercell of a given approximant is always a piece of the innite quasicrsytal, but not all pieces of the innite quasicrystal are the supercell of an approximant. The string S n generated after a nite number of applications of the concatenation (or substitution) rule is the supercell of thenth order approximant of the Fibonacci chain. I will use the symbolS n to represent the supercell as well as the approximant structure generated by this supercell. The length of the supercell, S n , is always a Fibonacci number F n 1 . The representative supercell for an approximant of ordern is theS n given in Table 3.1. For each ordern, there areF n distinct appproximant supercells and they can be obtained by cyclically permutingS n . It is worth noting that there areF n + 1 distinct subsequences of lengthF n in S 1 , i.e. there is one F n -length subsequence of the S 1 that is not an approximant. 3.2 The cut and project method While the substitution and concatenation rules are the simplest way to generate the Fi- bonacci chain, there is a more general method called the cut and project method. I included a brief abstract treatment of the cut and project method in Sec. 2.1. Here, I will focus on what the scheme looks like for the Fibonacci chain. For the higher-dimensional lattice ~ , we choose the regular 2D square latticeZ 2 R 2 . The direct and internal spaces are the orthogonal axes L;L ? rotated with respect to the standard basis of ~ such that the slope is equal to the Golden ratio, = 1+ p 5 2 . The window W is any interval of length p 3 onL ? . Fig. 3.2 shows an illustration of the procedure. All 1 We use the convention F 0 =F 1 = 1 14 3. THE FIBONACCI CHAIN (a) Real space 1 2 3 4 5 6 7 8 9 10 (b) Perpendicular space 1 2 3 4 5 6 7 8 9 10 A A B A A B (c) Figure 3.2: a) Cut-and-project method to obtain the Fibonacci chain: points in a regular 2D grid are projected onto the physical line of slope 1+ p 5 2 . The projections (solid lines) represent sites of the quasicrystalline lattice. The projections onto a plane perpendicular to the physical line (dashed lines) dene the conumber of each site. (b) The sites corresponding to (a) in real space. (c) The reordering of the sites in (b) according to their projection in perpendicular space: also known as conumbering. accepted points lie on a strip of width given by W parallel to L. The projections p k of the accepted point are the sites of the Fibonacci chain 3.2.1 The phason In the cut and project framework, the nth order approximant, S n , is obtained by replacing the golden ratio by its (n 1)th order rational convergent, n1 = 1 1 + 1 1 +::: | {z } (n1) = F n1 F n2 (3.1) Since the slope is rational, the resulting structure has a repeating sequence of distances d A and d B . The period is F n and the supercell is the S n obtained previously up to a cyclic permutation. Since the length of the supercell isF n , there areF n distinct choices for how to represent the supercell by a string. We can transform from one choice to the other by cyclic permutations, but also by performing a number of phason ips. To perform a phason ip, 15 Signatures of topology in a quasicrystal Figure 3.3: Modulating the potential between the Fibonacci and AAH limit (Taken from Kraus and Zilberberg [27]). V S n (k;) is the same as (3.3) with (V S ! ~ ;n!i;k!) shift the window W along L ? (along the red line in Fig. 3.2). At certain values of the shift, a point in that used to be in the acceptance window is replaced by a dierent point such that a consecutive pair of bonds BA somewhere in the chain is replaced by the pair AB (or vice versa depending on the direction of the shift). This is called a phason ip. There will beF n such phason ips in the supercellS n generating the entire family of possible supercells of the approximant of this size. The shift is parametrized by so that the supercells can be labelled S n (). As varies from 0! 2, S n () denes a closed orbit in the space of approximant supercells such that two consecutive supercells dier only by a single phason ip. 3.2.2 Connection to the Aubry-Andr e-Harper model The cut and project scheme for the Fibonacci chain can be condensed into a closed form expression for the approximant S n . The i-th letter i of the Fibonacci approximant S n , wherei2 0; 1;:::;F n 1, has the following closed form expression, where we identify(+) with A(B): i = sign cos 2i n + cos n1 (3.2) This formulation lays bare the connection between the Aubry-Andr e-Harper (AAH) [28] model and the Fibonacci chain. The two models can be obtained from the following general 16 3. THE FIBONACCI CHAIN expression by taking opposite limits of the control parameter . ~ i = tanh [ (cos(2i 1 +) cos( 1 ))] tanh () (3.3) The !1 limit recovers (3.2) while the ! 0 limit recovers the AAH potential. The topological equivalence of the two models is shown in Sec. 3.5. 3.3 Perpendicular space of the Fibonacci chain A perpendicular space[11] (or internal space) analysis of a cut and project structure involves considering the dual structure = fx jx 2 g, the point set formed by the accepted points of the superlattice projected onto L ? . for the Fibonacci chain consists of densely distributed points over an interval equal to the window W . This is shown by the green dashed lines in Fig. 3.2(c). Under fairly loose conditions, is uniformly distributed in W . This maps discrete averages on to integration over the interval W . For this reason, the mapping to perpendicular space is often an extremely useful tool. 3.3.1 Conumbering For a Fibonacci approximant, there is a closely related concept called conumbering [29]. The conumber index of a site x2 is found by numbering its image x from left to right along the window W . This arrangement places sites with similar local environments clos to each other. For example, if the F n sites of an nth order approximant are arranged according to their conumber, then the leftmost (rightmost) F n2 sites will have anA(B) bond to the left and a B(A) bond to the right, and the central cluster of F n3 sites have an A bond to both sides. Inside each of these three clusters, the sites are separated according to the hopping values in a larger neighborhood around the site. For example, in the righmost cluster ofF n2 sites, the central F n5 site have a local hopping pattern of AABAAB. The left and right subclusters have the local hopping patterns AABABA and BABAAB respectively. By 17 Signatures of topology in a quasicrystal numbering all the projections according to their positions in perpendicular space from left to right, we can dene the co-number, j, of the ith site of a given Fibonacci approximant: C :Z!Z. The larger the size of the local neighborhood where the hopping patterns around two sites agree, the closer their conumbers will be. 3.3.2 Connectivity matrix The conumber map is an extremely useful tool in the analysis of the Fibonacci chain because of its eect on the weighted adjacency matrix,T . T has elementsT i 1 i 2 such thatT i 1 i 2 =A(B) if sites i 1 and i 2 are adjacent to each other and separated by a distance d A (d B ). T is tridiagonal in real space. For example, for the approximantS 5 =ABAABAAB,T takes the following form: 0 B B B B B B B B B B @ 0 A 0 0 0 0 0 B A 0 B 0 0 0 0 0 0 B 0 A 0 0 0 0 0 0 A 0 A 0 0 0 0 0 0 A 0 B 0 0 0 0 0 0 B 0 A 0 0 0 0 0 0 A 0 A B 0 0 0 0 0 A 0 1 C C C C C C C C C C A ; (3.4) where I have used periodic boundary conditions. The representation of T in perpendicular space|indexing by the conumber of a site instead of its position|results in a matrix with a simpler structure. The weighted adjacency matrix ofS n is anF n F n matrix withAs along theF n2 th diagonal andBs along theF n1 th diagonal. For example, the perpendicular space representation of the example matrix above is 18 3. THE FIBONACCI CHAIN 0 B B B B B B B B B B @ 0 0 0 A 0 B 0 0 0 0 0 0 A 0 B 0 0 0 0 0 0 A 0 B A 0 0 0 0 0 A 0 0 A 0 0 0 0 0 A B 0 A 0 0 0 0 0 0 B 0 A 0 0 0 0 0 0 B 0 A 0 0 0 1 C C C C C C C C C C A : (3.5) In perpendicular space, T is a symmetric Toeplitz matrix with the only non-zero entries in theF n2 th and theF n1 th diagonal. 3.4 The Fibonacci hopping model The Fibonacci hopping model is the simplest physical model built out of the Fibonacci chain. It has been used as a prototype system to study the general properties of the Fibonacci modulation as well as quasicrystals in general to great success. It is a spinless tight-binding model with nearest neighbor hopping only: ^ H = X i c y i c i X i t i c y i c i+1 +h:c:; (3.6) c i ;c y i are the Fermionic annihilation and creation operators at site i, t i can take one of two values t A or t B depending on the Fibonacci word and is a constant that determines the lling. Fibonacci chain, a one-dimensional quasicrystal, is the best understood with many known results for the single electron spectrum and states. The spectrum is singular continuous for an innitesimally small quasiperiodic modulation [30]. In fact, it is a Cantor set [30, 31]. Singular continuous spectra have a one-to-one correspondence with critical states. Critical states are neither extended (associated with continuous spectra), nor localized (associated with point-like spectra) [32]. 19 Signatures of topology in a quasicrystal 2 0 2 0.0 0.2 0.4 0.6 0.8 1.0 Integrated density of states Fibonacci hopping model 2 0 2 Energy/t Fibonacci on-site model 2 0 2 Diagonal AAH model q=1 q=2 q=3 q=-2 q=-3 q=-1 Figure 3.4: The integrated density of states for the Fibonacci hopping model, the FIbonacci on-site model, and the diagonal AAH model. The six largest gaps are marked with dashed lines and their gap labels. The lling factor that corresponds to each gap is the same for all three models. 3.4.1 The gap labelling theorem There are an innite number of gaps, and each gap can be labelled by a unique integer q. According to the gap-labelling theorem [33], the number of states below a gap with label q is given by the integrated density of states (IDOS), IDOS() 2q-gap =q 1 mod 1: (3.7) A modied version of the gap-labelling theorem applies to the approximants, IDOS() 2q-gap =q F n1 F n mod 1; (3.8) where q belongs to a nite subset of the integers depending on the size of the approximant. The same gap labels apply to a family of models characterized by the golden ratio. This includes the Fibonacci on-site model|where the diagonal terms in the Hamitonian V i c y i c i take one of two values V A (V B ) according to the Fibonacci chain, the mixed model|where the hopping and the on-site potential are both modulated, the AAH model (diagonal, o- diagonal, or mixed) as long as the modulation frequency is set to the golden ratio, as well as 20 3. THE FIBONACCI CHAIN models that interpolate between the two via the control parameter as in (3.3). A member of this family has the following tight-binding representation ^ H = X i (V i )c y i c i X i t i c y i c i+1 +h:c:; (3.9) t i = t +t tanh [ t (cos(2i 1 + t ) cos( 1 ))] tanh ( t ) (3.10) V i = V +V tanh [ V (cos(2i 1 + V ) cos( 1 ))] tanh ( V ) (3.11) where one of t or V must be non-zero, as must one of t or t. is the golden ratio. As we will see, all of these models are topologically non-trivial, adiabatically connected to each other, and belong to the same symmetry protected topological equivalence class [27]. 3.5 The topology of the Fibonacci chain 3.5.1 Relationship to the Hofstadter model The gap indices of the Fibonacci chain (and its related models) have topological origins. They are related to the Chern numbers of a 2D Model that is obtained by treating the phason as a second physical dimension. I will refer to the extended 2D model as the ancestor Hamiltonian. The ancestor Hamiltonian of the Fibonacci chain describes electrons hopping on a 2D rectangular lattice pierced by a magnetic eld. It is adiabatically connected to the Hofstadter model with 1 ux quanta per unit cell, which is the ancestor Hamiltonian of the AAH model (as long as the modulation frequency is the Golden ratio). In this section, I will describe how the Hofstadter model and the AAH model are connected. The analogous calculation for the Fibonacci chain and the proof of the topological equivalence between the ancestor Hamiltonians of the two models is quite techincal. The interested reader may refer to the original calculation by Kraus and Zilberberg in [27]. 21 Signatures of topology in a quasicrystal The Hamiltonian for the AAH model is X i t(c y i c i+1 +c y i+1 c i ) +V cos(2 1 i +): (3.12) is the frequency of the potential modulation. When this is equal to the golden ratio, this model is connected to the Fibonacci chain. The Hofstadter problem [34] describes electrons hopping on a 2D rectangular lattice pierced by a uniform magnetic eld, b. X i;j t 0 (c y i;j c i+1;j +c y i+1;j c i;j ) +t(e i2bi c y i;j+1 c i;j +e i2bi c y i;j c i;j+1 ) (3.13) Here t 0 (t) are the nearest neighbour hopping amplitudes in the i(j) direction. The eect of the magnetic eld in Landau gauge is the appearance of the Peierl's phase in front of the transverse hopping terms. We set the lattice constant a = 1, and the spatial extent of the system to be niteLL. Now, we can apply periodic boundary conditions and express the Hamiltonian in momentum space. In the absence of a magnetic eld, the lattice momenta ~ k lie on a torus T 2 , called the Brillouin zone. The torus is parametrized byk x ;k y < and the momenta take quantized values in units of 2 L . With the introduction of the ux b, it is convenient to introduce the concept of the magnetic Brillouin zone. Let us assume for the moment that b is rational. In order to later connect with the Fibonacci approximants, we choose b = 1 = F n1 Fn . Notice that with our choice of gauge, the Hamiltonian is periodic in the x direction with a period of F n sites. If we treat these F n sites as a unit cell, we can recover translational invariance in the Hamiltionian 2 . We accommodate this by introducing the following Fourier transform: c r (k) = X i;j e i(kx+2 F n1 Fn r)i e ikyj c i;j (3.14) where the momentum k = (k x ;k y ) runs over the the magnetic Brillouin zone, which is F n times smaller in the x direction: Fn k x < Fn , k y < . At each point in 2 This statement can be made more rigorous in the language of the magnetic translation group as in [35, 36] 22 3. THE FIBONACCI CHAIN the magnetic Brillouin zone, there are F n subbands labelled by r = 0;:::;F n 1. The Hofstadter Hamiltonian in the magnetic Brillouin zone is decoupled in momentum, so we can right ^ H Hofstadter = R dk ^ H(k) with: ^ H(k) = X r h te iky c y r+1 (k)c r (k) +h:c: i + 2t 0 cos 2 F n1 F n r +k x c y r (k)c r (k) (3.15) Comparing (3.12) and (3.15), we see that the AAH model for a given phase shift describes the physics of the k = (; 0) momentum mode of the Hofstadter model with magnetic ux b = 1 and transverse hopping t 0 = V 2 . A magnetic ux 2 0 threaded through the AAH ring acts by adding the phase factors e to the hopping terms in (3.12). With this additional phase twist, the AAH model fully maps onto the Harper equation for the Hofstadter model with ( 1 !b;V! 2t 0 ;!k x ;!k y ). 3.5.2 The topology of the Hofstadter model The Hofstadter model is a classic example of a topological insulator. The Hall current h = e 2 h is quantized in units of e 2 h = X r;Er<E f C r ; (3.16) where C r is the Chern number associated with the rth band. The integer is a topological invariant called the TKNN invariant after a seminal paper by Thouless and colleagues [34]. They showed that calculating was equivalent to solving the Diophantine equation r =F n1 +sF n (3.17) where is constrained to lie betweenF n =2 and F n =2. Arriving at this expression requires a tedious calculation. TKNN used the Kubo formula to calculate the Hall conductance and 23 Signatures of topology in a quasicrystal Hofstadter model b = 1 AAH model Ancestor model of the Fibonacci chain Fibonacci chain 2D 1D Diophantine equation Projection to momentum modes Figure 3.5: The Hofstadter model and ancestor model of the Fibonacci chain are topo- logically equivalent because the Diophantine equation that describes their Chern numbers are the same. By projection to individual momentum modes, the two ancestor models are topologically equivalent to the Fibonacci chain and the AAH model respectively. found the now familiar integral over the Berry connection, or the Chern number h = ie 2 2~ X E<E f I d 2 kd 2 x @u @k 1 @u @k 2 @u @k 2 @u @k 1 (3.18) The integrand above is necessarily an integer. Calculating this by carefully applying per- turbation theory to (3.15) yields the Diophantine equation. Dana et al. showed that this equation is a more general result stemming from the magnetic translation symmetry of the problem [37]. A detailed calculation can be found in the book by Fradkin [36]. 3.5.3 Gap label$ Chern number The Hall current carried by the rth band of the Hofstadter model is given by the Chern number associated with that band, which is an integral over a closed two-dimensional sur- face of the Berry connection. In the case of the Hofstadter model, the two-dimensional surface is the magnetic Brillouin zone, which is a two-dimensional torus, T 2 . In general, the surface over which the Chern number is dened could be given by any two parameters of the Hamiltonian. 24 3. THE FIBONACCI CHAIN The Hamiltonian of the Hofstadter model maps to the AAH model by identifying the two moment with the phase twist and the phase shift respectively, (k x ! ;k y ! ) in (3.15). We can also assign the same Chern numbers to the AAH model, except these are calculated over a torus T 2 dened by the two phases (;) that are parameters in the AAH Hamiltonian. Comparing the Diophantine equation for the Hall current with the gap labelling theorem (3.8), we nd that the the gap labels are directly related to the Chern numbers by q = , where gives the Hall current when the system is tuned to the gap with label q. Following the same argument that relates the AAH model to the Hofstadter model, the Fibonacci chain can be related to a complicated 2D ancestor model, H ancestor , of electrons hopping on a two-dimensional lattice. Kraus and Zilberberg [27] have dened the 2D ancestor model of the Fibonacci chain and argue following [37], that the Diophantine equation for the Hall conductivity of this model is the same as that for the Hofstadter model. Additionally, the same equation holds true for a continuum of models that lie between the Hofstadter model and H ancestor . Stringing the various relationships together, they show that the Fibonacci chain is topologically equivalent to H ancestor , which itself is topologically equivalent to the Hofstadter model which is topologically equivalent to the AAH model. The interpretation of the gap labels as Chern numbers can be made by mapping momenta in the 2D model to phase parameters in the Fibonacci chain in an entirely analogous way as was done for the AAH model. 3.5.4 Edge states We have seen that the gap labels have topological origins, but the question of how the topological invariants manifest themselves physically remains. In the Hofstadter model, the Chern number of a gap corresponds to the hall conductivity of the system when the Fermi level is tuned to that gap. Transverse charge transport is facilitated by chiral edge state and the number of these edge states is controlled by the Chern number. 25 Signatures of topology in a quasicrystal 0 /2 3 /2 2.0 1.5 1.0 0.5 0.0 Eigenvalues (a) 0.0 0.5 | i | 2 , q = 1 (b) 0 50 100 150 Site number, i 0.0 0.5 0 /2 3 /2 0 50 100 Site number, i (c) | i | 2 , q = 3 (d) Figure 3.6: (a) The evolution of the eigenvalues of an open Fibonacci chain as a function of . The q =1 edge state is marked for two values of around a level crossing. (b) The edge state switches its localization around a level crossing. (c) A color plot showing the q =3 edge state switching its localization three times as the completes an orbit. (d) An adiabatic pump built into a photonic crystal based on (b) (subgure (d) taken from Verbin et al. [18]). The Fibonacci chain also presents symmetry protected edge states, and by observing these edge states, we can recover the Chern number corresponding to a given gap. To demonstrate this, I apply open boundary conditions to the Fibonacci hopping model. The corresponding Hamiltonian for the nth order approximant is ^ H() = Fn X i=0 c y i c i Fn1 X i=0 t i ()c y i c i+1 +h:c:; (3.19) This tight binding Hamiltonian represents a linear chain with F n + 1 sites where hopping term between the site at i = 0 and the site at i = F n is 0 and t i () follows the Fibonacci approximantsS n (). Since we have open boundary conditions, theF n members of the family of supercells S n () represent dierent physical systems. Fig. 3.6 (a) shows how the eigenvalue spectrum of the open Fibonacci chain evolves as a function of . The rst thing to note is that familiar self-similar structure of a hierarchy of gaps labelled by integer indices. Varying leaves the bulk unchanged except for small 26 3. THE FIBONACCI CHAIN corrections. Turning our attention to the eigenstatse (Fig. 3.6 (b)-(c)), there are a few interesting observations: 1. Every gap has a pair of states whose energy is well within the gap for certain intervals of . When this is the case, these gap states are localized at the edges. 2. The energy of a gap state with label q crosses the gapjqj times. 3. Every gap crossing corresponds to the state abruptly switching its localization from one edge to the other. The sectrum and eigenstates of the Fibonacci chain have been measured and manipulated in experiments where the Fibonacci chain has been realized as one or the other articial crystal 3 . In a polaritonic crystal, the fractal energy spectrum was observed and the self smilarity conrmed [16] and he gap state energies and their localization as a function of the phason were measured by photoemission spectroscopy [17] (see Fig. 3.7). Using the idea that the the edge states can switch localization with a small number of phason ips and that the Fibonacci chain is topologically equivalent to the AAH model, an adiabatic pump was built using a photonic crystal [18, 38] (see Fig. 3.6). Position space and phason space were simultaneously probed using a diraction experiment in [39]. In all of the studies mentioned above, the variation of the phason, which is an external tuning parameter, plays a crucial role in observing topological features. To build the adia- batic pump, one needs to modulate the Fibonacci potential from one value of to another across the gap crossing, In order to measure the number of gap crossings in a gap, one needs to synthesize allF n member of the family ofnth order approximants. But our assertion that that the Chern number can be associated with an individual Fibonacci chain suggests that it should be measurable without invoking the phason at all. In [2], we report on a way to do precisely that. This is the subject of Chapter 5. 3 also referred to as metamaterials 27 Signatures of topology in a quasicrystal Figure 3.7: The Fibonacci chain realized as a polaritonic crystal: (a) The fractal density of state measured via photoemission spectroscopy (Taken from Tanese et al. [16]), (b) Edge state crossing the gap as a function of measured by photoemission spectroscopy (Taken from Baboux et al. [17]) 3.6 The weak modulation limit When the modulation strength is weak, t A =t B 1, we can treat the Fibonacci modulation as a perturbation to the homogeneous chain [29, 40]. In real space, the Fibonacci hopping Hamiltonian splits into two parts, ^ H = Fn X i=1 t A c y i c i+1 +h:c: | {z } ^ H 0 Fn X i=1 (t i t A )c y i c i+1 +h:c: | {z } ^ Hw (3.20) ^ H 0 is the Hamiltonian of a periodic chain with nearest neighbor hopping t A , and ^ H w intro- duces the Fibonacci modulation. We use periodic boundary conditions, so the index i (and j in the following) are understood to wrap around moduloF n . The matrix representation of the Fibonacci hopping Hamiltonian in perpendicular space, which I will denote with ^ H has a particularly pleasing form. The only non-zero hopping terms lie on four diagonals following the connectivity matrix in (3.5). Thet A s go in theF n2 th diagonal and thet B s go in theF n1 th diagonal. The decomposition of ^ H as a sum of a term that represents the homogeneous chain ^ H 0 28 3. THE FIBONACCI CHAIN and the Fibonacci perturbation ^ H w takes the following form: ^ H 0 = Fn X j=1 t A c y j c j+F n2 +t A c y j c j+F n1 +h:c: (3.21) ^ H w = F n2 X j=1 2wc y j c j+F n1 +h:c: (3.22) where we have used 2w = t A t B . ^ H w connects each of the leftmost F n2 sites to the site F n1 places to its right. ^ H 0 is a circulant matrix and ^ H w is a Toeplitz matrix with only the F n1 th diagonals lled. Before applying the Fibonacci modulation, consider the eigenstates of the unperturbed Hamiltonian. Recall that the eigenvalues of ^ H 0 are given byE m =2t A cos 2m Fn with the integer m constrained by 0 mbF n =2c. Almost all of these are doubly degenerate. If F n is odd, the m = 0 eigenvalue is non-degenerate. If F n is even, then addditionally, the m =bF n =2c eigenvalue is also non-degenerate. The corresponding eigenstates take the form of plane waves with wavenumber 2 Fn m. Remarkably, the unperturbed Hamiltonian represented in perpendicular space is also a circulant matrix. This means that it is also diagonalized by a Fourier transformation and the eigenstates, which are plane waves in real space, are also plane waves in perpendicular space. But the wavenumbers are shued or the wavenumber that corresponds to a plane wave with a given energy is dierent. In perpendicular space, the eigenstates are plane waves labelled by the integer = 0; 1;:::bF n =2c. As in real space, the eigenstates are almost all doubly degenerate, and the eigenstate with label has the wavenumber 2 Fn . However, the energy of this plane wave isE =2t A cos 2F n1 Fn . It is straightforward to check that theE labelled by and theE m labelled bym represent the same set of energies. They are simply ordered dierently. The two cases are compared in Table 3.2. The perpendicular space and real space wavenumbers can be related to each other (modulo F n ) by equating E m andE mF n1 (mod F n ) (3.23) 29 Signatures of topology in a quasicrystal Table 3.2: The eigenvalues and eigenstates of the periodic nearest neighbor hopping Hamil- tonian in real and perpendicular space Real space, i Perpendicular space, j ^ H 0 = 0 B B B B @ 0 t A t A t A & & & t A 1 C C C C A H 0 = 0 B B B B B B B B B B @ 0 t A t A & & & t A & t A & 1 C C C C C C C C C C A E m = 2t A cos 2m Fn ; E = 2t A cos 2F n1 Fn ; m = 0; 1;:::; Fn 2 = 0; 1;::: Fn 2 hijm;i = e 2i mi Fn hjj;i = e 2i j Fn Bearing in mind that we will shortly introduce a perturbation with an F n -site supercell, we describe the band structure of ^ H 0 as consisting of F n bands in a Brillouin zone that spans the interval k2 h Fn ; Fn h . For the unperturbed problem, all the bands touch each other, re ecting the fact that we have done nothing other than downfold the usual cos 2k Fn dispersion onto a smaller Brillouin zone. Introducing the perturbation ^ H w splits the doubly degenerate states of ^ H 0 , and opens F n 1 gaps. The gap opening can be seen using rst- order perturbation theory. The edges of gaps with an even (odd) number of states below them are at k = 0 Fn . I will show the calculation for the even case in detail. The odd case is entirely analogous. The rank 2 degenerate subspace labelled by has two statesj;i with energy2t A cos 2F n1 Fn . The rst order corrections are given by the eigensystem of a 2 2 matrix ^ V whose entries 30 3. THE FIBONACCI CHAIN 0.0 0.1 0.2 w = t B t A 2 2 1 0 1 2 Energy bands Exact result 0.0 0.1 0.2 w = t B t A 2 2 1 0 1 2 Energy bands Perturbation theory q=1 q=-1 q=2 q=-2 q=3 q=-3 q=±4 Figure 3.8: The opening of gaps with increasing modulation strength in the band structure of the F n = 8 approximant: a comparison of rst order perturbation theory with the exact result. The solid lines are the eigenvalues at the center of the Brillouin zone and the dashed lines are the eigenvalues at the edges of the Brillouin zone. The shaded regions are the bandsd. are V =h;j ^ H w j;i with ; =. V ++ = 1 e 2i Fn ::: e 2i (Fn1) Fn ^ H w 0 B B B B B B B @ 1 e 2i Fn . . . e 2i (Fn1) Fn 1 C C C C C C C A = 2w F n2 e 2i F n1 Fn + e 2i F n1 Fn = 4w F n2 cos 2 F n1 F n (3.24) The other terms are similarly computed resulting in V = 4w F n 0 B @ F n2 cos 2 F n1 Fn sin(2 Fn F n2) sin(2 Fn ) sin(2 Fn F n2) sin(2 Fn ) F n2 cos 2 F n1 Fn 1 C A (3.25) The eigenvalues of V are = 4w F n 2 4 F n2 cos 2 F n1 F n sin 2 Fn F n2 sin 2 Fn 3 5 , and the 31 Signatures of topology in a quasicrystal corresponding eigenvectors are 1 p 2 ; 1 p 2 . So, we have that the plane waves with the wavenumber 2 Fn split into the statesj;i with the energiesE E = 2t A + 4w F n2 F n cos 2F n1 F n 4w F n sin 2 F n2 Fn sin 2 Fn (3.26) hjj; +i = r 2 F n cos 2j F n ;hjj;i = r 2 F n sin 2j F n (3.27) Fig. 3.8 shows the level splitting leading to the gap opening as the modulation w is turned on. By combining the gap labelling theorem and (3.23), we can compute the perpendicular space wavenumber of the slightly perturbed plane waves around each gap. E m is monotonic when m2 [0;bF n =2c]. Therefore, for gaps with an even number of states below them 2mqF n1 2F n1 (mod F n ) =) = cF n +q 2 (3.28) where c is some integer. For gaps with an odd number of states below them, an analogous calculation at the edge of the Brillouin zone results in 4 : 2m + 1qF n1 (2 + 1)F n1 (mod F n ) =) = cF n +q 1 2 (3.29) 3.7 The strong modulation limit In the strong modulation limit of the Fibonacci hopping model, one of the hoppings is much larger than the other, = t A t B ! 0(1) [41, 42]. The work presented in this thesis is restricted 4 Using the following convention for the eigenstates at the edges of the Brillouin zone: E = 2t A cos 2 Fn1 Fn ( + 1 2 ) ;hjj;i = e 2i ( + 1 2 ) j Fn 32 3. THE FIBONACCI CHAIN Figure 3.9: An illustration of the (a) atomic RG, (b) molecular RG. (Taken from Niu & Nori [41] ). to the case t A < t B , so we will only consider the ! 0 limit. When t A = 0, the system consists of a disconnected set of atom and molecule sites. An atom site is one that has an A bond to both sides, while a molecule site has a B bond to one of its neighbors. There are only three energy levels in the system, the molecule bonding states with energyt B , the atom states with energy 0 and the molecule anti-bonding states with energy +t B . The way thatA andB bounds are distributed ensures that in a Fibonacci approximant of length F n , there are F n2 molecule sites and F n3 atom sites. Turning on the Fibonacci modulation with a small positive will split each level into three clusters, each of which in turn splits into three subclusters recursively. To rst order in perturbation theory, once separated levels never mix 5 . The recursive trifurcation is best seen using renormalization group (RG) techniques. Consider the F n2 molecular bonding states with energyt B localized at the molecule sites. The molecular RG consists of treating each molecule as a site in a rescaled chain. The chain formed by connecting each molecule is itself a Fibonacci approximant of length F n2 . This is shown in Fig. 3.9 (b). Degenerate state perturbation theory shows that the new eective Hamiltonian is also a Fibonacci hopping Hamiltonian up to a constant shift with new hoppingt 0 A = 2 t A ;t 0 B = 2 t B . The bonding cluster therefore splits into three subclusters separated in energy byt 0 B ; 0. The anti-bonding cluster splits in an identical way. The middle cluster is made up of atom states. The chain formed by the F n3 atom 5 In fact, it can be shown that the strong and weak modulation limits are adaiabatically connected, which means that all gaps remain open all the way between the ! 0 and ! 1 limits. 33 Signatures of topology in a quasicrystal sites is an F n3 -length Fibonacci approximant. This is shown in Fig. 3.9 (a) The eective Hamiltonian is again the Fibonacci hopping Hamiltonian. The renormalized hoppings are t 0 A = 2 t A ;t 0 B = 2 t B . As before, the new eective Hamiltonian splits the cluster into three subclusters with energy shifts oft 0 B ; 0. Each eigenstate of a Fibonacci chain (or approximant) can be assigned a unique label based on its renormaliztion path. The label consists of a string off0; +;g based on which cluster the state belongs to at each renormalization step. For example, a state in the middle subcluster of the upper subscluster of the middle cluster would have a label that goes as 0 + 0::: . 34 Chapter 4 Superconductivity Superconductivity was discovered by J. Kamerlingh Onnes in 1911, when he observed that the electical resistance of metals like mercury, lead, and tin vanished if the material was brought to temperatures below a material-dependent critical temperature T c 10 K. The most sensitive experiments show that persistent currents set up in rings made out of super- conducting material ow without any measurable decay 1 . Perfect conductivity is the most striking feature of superconductors. There is another more subtle hallmark of superconductivity| perfect diamagnetism. In 1933, Meissner and Ochsenfeld discovered that if a material placed in a magnetic eld H is cooled to below its critical temperature, and the eld is not too strong, H < H c , then below the superconducting transition, the material will expel the magnetic eld. The eld of superconductivity has undergone two major revolutions. In the 1950s, Bardeen Cooper and Shrieer (BCS) proposed a microscopic theory that has been enor- mously successful at explaining the physics of low-T c conventional superconductors. The second revolution came in 1986, the rst high T c superconductor was found with a super- conducting transition around 30 K [44]. The highest T c for a superconductor at ambient pressure has been reported to be 130 K [45]. The microscopic mechanism responsible for high T c superconductors is an open question at the time of writing. 1 The characteristic time scale of any decay has been bounded below by 10 5 years [43] 35 Signatures of topology in a quasicrystal The question of superconducting quasicrystals is relatively new. The rst superconduct- ing quasicrystal was discovered in 2018 [46] and it is believed to be of the conventional BCS kind. In our work, we have applied standard BCS theory in a modied form, the real space Bogoliubov-de Gennes approach, to tease out the basic properties of the quasicrystalline superconducting state. With that in mind, I have chosen to restrict my introduction to the conventional microscopic model. Sec. 4.1 introduces BCS theory. Sec. 4.2 introduces the Bogoliubov-de Gennes approach. Sec. 4.3 discusses the concept of the superconducting proximity eect. Sec. 4.4 summarizes the existing literature on superconducting quasicrystals. 4.1 Bardeen-Cooper-Schrieer formalism The introduction provided here is adapted from the textbooks by Tinkham [43]x3 and de Gennes [47]x4. 4.1.1 The instability of the normal state to attractive interactions In 1957, Cooper showed that the Fermi sea is unstable to a innitesimal attractive el-el interaction. This can be understood by considering ground state of two particular electrons at positions r 1 and r 2 while all other electrons are treated as a free electron gas. The two electrons attract each other while they interact with the other electrons only by the Pauli exclusion principle. The lowest energy state of the two electrons is one where the center of mass is at rest. Therefore, we can expand the combined wavefunction (r 1 ;r 2 ) in plane waves: (r 1 ;r 2 ) = X k g(k)e ik(r 1 r 2 ) (4.1) g(k) is the probability of nding one electron in the plane wave state with momentum~k and the other one with momentum~k. Anticipating that the attractive interaction will 36 4. SUPERCONDUCTIVITY bring the electrons closer together, we will assume the spin state of the pair to be a singlet state. The Pauli exclusion principle implies that: g(k) = 0 when k<k F (4.2) The Schr odinger equation for the two electrons is: ~ 2 2m (r 2 1 +r 2 2 ) (r 1 ;r 2 ) +V (r 1 ;r 2 ) = E + ~ 2 k 2 F 2m ; (4.3) whereV (r 1 ;r 2 ) is negative and the energyE is measured with respect to the top of the Fermi sea. We nd an equation for g(k) by inserting (4.1) into (4.3): ~ 2 k 2 m g(k) + X k 0 g(k 0 )V k;k 0 = (E + 2E F )g(k) (4.4) V k;k 0 = 1 L 3 Z drV (r)e i(kk 0 )r (4.5) If there are solutions to this equation with energyE < 0, then the electrons can form bound states that lower the energy of the Fermi sea and the Fermi sea is unstable. This can be explicitly shown to be the case using the simplied interaction: V k;k 0 = 8 > > < > > : V L 3 E f + ~ 2 k 2 2m <~! D 0 otherwise (4.6) The interaction is attractive and constant in a band below the cuto~! D . Solving (4.4) in the physically relevant regime N(0)V 1 where N(0) is the density of states at the Fermi level, we nd the solution: E =2~! D e 2=N(0)V (4.7) The energy is negative, and therefore a bound state exists. The pair of bound electrons is called a Cooper pair. Even a very weak attraction between the electrons in a free electron gas will immediately break the Fermi sea. The electrons will instead prefer to form Cooper pairs. In conventional superconductors, the attractive interaction is known to originate from 37 Signatures of topology in a quasicrystal the eective interaction mediated by phonons. In this introduction, we will simply assume that there is an attractive interaction that can be modelled by (4.6) and not worry about where it comes from. 4.1.2 The BCS ground state Now that we know that the Fermi sea is unstable, the question of what the ground state looks like arises. We know that it is made up of Cooper pairs. In the language of second quantization, a Cooper pair is written as j i = X k>k F g k c y k" c y k# jFi (4.8) wherejFi is the Fermi sea. The most general wavefunction for N electrons with Cooper pairing built in looks like: j N i = X g(k 1 ;k 2 ;:::;k N=2 )c y k 1 " c y k 1 # :::c y k N=2 " c y k N=2 # j0i (4.9) wherej0i is the vacuum with no particles, and g(k 1 ;:::;k N=2 ) is the weight with which the product of that particular set of N=2 momenta appears in the sum. This wavefunction is too general and has too many unknown parameters. To make this problem tractable, BCS chose to use a form that doesn't x N, but instead xes the expectation value N. This is the same idea as working in the grand canonical ensemble. The approximate form of the ground state is j G i = Y k (u k +v k c y k" c y k# )j0i (4.10) The coecients obeyju k j 2 +jv k j 2 = 1. The probability of the pair (k";k#) being occupied is given by v 2 k . 38 4. SUPERCONDUCTIVITY 4.1.3 Solution by canonical transformation To specify the ground state, we need to determine all the coecientsu k ;v k . This is ordinarily done by a variational method where the functional to be minimized is the expectation value of the energy h G jH X k c y k c k j G i (4.11) whereH is the reduced pairing Hamiltonian|all terms which have a zero expectation in the BCS ground state are neglected H = X k k c y k c k + X kl V kl c y k" c y k# c l# c l" (4.12) The eect of the chemical potential term is to control the expectation value of the number of electrons. However, since we want to build up to the reals pace Bogoliubov-de Gennes formalism, I will show how the coecients are found using an alternate approach|the solution by canonical transformation. We stat with the observation that the BCS ground state is a phase coherent superposition of a many-body state with pairs of opposite momenta states occupied. We expect the expec- tation value of operators such asc k" c k# to be non-zero. Since we are in the thermodynamic limit, we expect the uctuations away from the expectation value to be small c k" c k# = (c k" c k# b k ) +b k (4.13) b k =hc k" c k# i (4.14) Neglecting uctuations, we construct the mean-eld Hamiltonian from (4.12) H MF =H = X k ( k )c y k c k X k k c y k" c y k# + k c k" c k# k b k (4.15) 39 Signatures of topology in a quasicrystal where k is dened as: k = X l V kl b l (4.16) The Hamiltonian (4.15) is bilinear and therefore diagonalizable by dening new Fermionic operators k related to the original operators through the Bogoliubov transformation c k" =u k k0 +v k y k1 (4.17) c k# =v k k0 +u k y k1 (4.18) The numerical coecientsu k ;v K satisfy the constraintju k j 2 +jv k j 2 = 1. The task then is to choose u k and v k such that (4.15) written in terms of the k is diagonal. A bit of algebra shows that the solution to this is 2( k k )u k v k + k v 2 k k u 2 k = 0 (4.19) Solving for the coecients and using k = k k , jv k j 2 = 1ju k j 2 = 1 2 1 k p 2 k + 2 k ! (4.20) The calculation must be self-consistent, so the wavefunctions calculated in this way must reproduce the k used to calculate them. k must satisfy k = X kl l 2 p 2 l + 2 l tanh E l 2 (4.21) k is often referred to as the superconducting order parameter. It is closely related to the probability of the electrons with momentum k being in the superconducting condensate b k . 4.2 The real space Bogoliubov-de Gennes approach To treat the problem of superconductivity in quasicrystals, we chose to use the real space Bogoliubov-de Gennes formalism. This approach was used by de Gennes in the 1960s to 40 4. SUPERCONDUCTIVITY treat the boundary problem [48]. The advantage of using a real-space approach is that it allows us to incorporate the inhomogeneous potential exactly, in the form of an appropriate modulation of the tight-binding parameters in real space. This derivation followsx5 of the book by de Gennes [47], except I have adapted all expressions to a 1D tight-binding model. The starting point of the formalism is the attractive Hubbard model [47], ^ H = X i ( X ( 0 i i )c y i c i + X t i c y i c i+1 +h:c: X V i 2 c y i c y i c i c i ) : (4.22) Here c i is the electron annihilation operator at site i (i = 1;:::;L tot , where L tot is the total number of sites) and with spin . 0 i and i are the on-site potential and the chemical potential respectively at site i, t i is the hopping amplitude between sites i and i + 1 and V i > 0 is the strength of the attractive Hubbard interaction between spin-up and spin-down electrons at site i. In the Bogoliubov-de Gennes method, we introduce the mean elds HF i and i to decouple the quartic interaction term. The resulting mean eld Hamiltonian is given by ^ H mf = X i ( X i z }| { ( 0 i + HF i i )c y i c i + X t i c y i c i+1 +h:c: +V i i c y i" c y i# +h:c: ) : (4.23) ^ H mf can be diagonalized by introducing the Bogoliubov transformation: 41 Signatures of topology in a quasicrystal c i" = X n n" u in y n# v in ; (4.24) c i# = X n n# u in + y n" v in : (4.25) Choosing the correct coecients u in ;v in to diagonalize the mean eld Hamiltonian (4.23) is equivalent to solving the Bogoliubov-de Gennes equations for theL tot -component vectorsu n and v n , where u n (resp. v n ) are the coecients of the Bogoliubov transformation u in (resp. v in ) with i = 1;::;L tot . The matrix form of these equations reads 0 B @ ^ K ^ ^ ^ K 1 C A 0 B @ u n v n 1 C A =E n 0 B @ u n v n 1 C A ; (4.26) where E n is the associated eigenvalue. The components of the matrices ^ K and ^ are given in terms of the Kronecker delta ij by ij = X i V i i ij ; K ij = X i i ij +t i i+1j +t i1 i1j : (4.27) The matrix operator in (4.26) is Hermitian, and it is convenient to think of it as a pseudo- Hamiltonian, with eigenenergies E n and eigenvectors u n ;v n . Due to the redundancy of this system of equations, it suces to keep only the positive energy solutions E n > 0. The eigenvectors satisfy normalization conditions P n ju in j 2 +jv in j 2 = 1 at each site i. The mean-eld Hamiltonian (4.23) contains two L tot -component local mean-elds. The superconducting order parameters i and local eective onsite energies i = 0 i + HF i i . In order the minimize the free energy of the system, these quantities must satisfy the self- 42 4. SUPERCONDUCTIVITY consistency equations, HF i =V X n ju in j 2 f(E n ;T ) +jv in j 2 (1f(E n ;T )); i = X n v in u in (1 2f(E n ;T )); (4.28) wheref(E n ;T ) is the Fermi-Dirac distribution function at temperatureT . In our numerical calculations, we begin with a starting ansatz for the 2L tot mean-eld parameters. We then iteratively solve the Bogoliubov-de Gennes equations Eq. (4.26), and compute the new values of these parameters using Eq. (4.28). This procedure is repeated until convergence has been achieved to the required accuracy. An algorithm to perform this self-consistent calculation is described in Appendix A It is worth noting here that in the translationally invariant case, the label n is replaced by the momentum k, and the resulting set of equations resemble the BCS expressions. i is the analog to the gap paramter k in BCS theory. It is a measure of the proportion of charge carriers in the condensate or the number of superconducting electrons. In BCS theory, the gap was directly proportional to . In the inhomogeneous case and especially when the attraction strength V varies in space, while the gap width is still correlated with , the relationship is no longer as simple [49]. Mean-eld methods similar to this have been previously applied to the problem of inho- mogeneous superconductors in [4, 49{56]. Quantum uctuations are generally too strong for mean-eld treatments to suciently describe low-dimensional systems. There are however two contexts in which a one-dimensional mean-eld description is meaningful: 1) In an ex- periment, a 1D quantum wire will be embedded in a 3D environment. Substrate and other surrounding media may conspire to subdue quantum uctuations while the eective descrip- tion for the quantum wire is 1D. 2) Quasi-1D systems such as nanotubes or mesoscopic wires often admit an eective 1D description, while the true dimensionality of the system is 3D and the mean-eld results apply. The Bogoliubov-de Gennes method has been very useful in studying superconductivity 43 Signatures of topology in a quasicrystal in inhomogeneous systems. Ghosal et al [51] obtained some early results for the distribution of order parameters for bulk disordered superconductors. Their theoretical prediction of superconducting islands in a non-superconducting sea have led to STM based experiments on 2D lms [57, 58]. Bulk superconductivity in 2D quasicrystals has been studied [50, 56]. Similar mean-eld calculations have been used to study the behavior of the superconducting singlet and triplet order parameter near edges and impurities in 1D and 2D systems [54]. 4.3 The proximity eect The superconducting proximity eect at a junction between a superconductor (S) and a non- superconducting material (N) is a phenomenon where the superconducting order parameter is induced in the N part by the leakage of Cooper pairs across the interface. The existence of such an eect was rst suggested by experiments on superconducting wires separated by a thin gold layer in 1960 by Meissner [59]. He found that range of the leakage was of the order of 10 3 A. Early on, de Gennes posited that the long range of this eect could lead to important applications in device where full control over the electronic properties of a material is important [60]. This has indeed been the case with|to name the most prominent applications|the far reaching applications of Josephson junction based devices in quantum computing [61], quantum simulation [25], and the proposal of a superconductor-topological insulator as a candidate system to realize Majorana bound states [62]. The literature on the topic is vast and I will restrict my discussion to the case of where N is ordinarily a normal conductor as this background will be the most relevant to our Superconductor-quasicrystal results. I would like to start by reviewing some well-known properties of the pair amplitude =hc # c " i in the vicinity of a Superconductor-metal junc- tion [60]. At nite temperatures, if the metal is clean (meaning the mean free path is large compared to the coherence length), the probability of nding a Cooper pair in a distance x 44 4. SUPERCONDUCTIVITY away from the interface is (x) =(x)e Kx (x large) (4.29) where (x) is a slowly varying function of x. The length scale K 1 =~v n =2kT , where v n is the Fermi velocity. At zero temperature, however goes as a power law. / 1 x +x 1 ; (4.30) When the N part is dirty, the the tunneling of the Cooper pairs is a diusion process, and the length scale of the penetration is the coherence length N = (~D=2kT ) 1=2 where D is the diusion constant, D = 1 3 v n l n . There are two important phenomena that occur in conjunction with the proximity eect in SNS junctions|a non-superconducting material sandwiched between two superconductors. Andreev bound states [63] can form in the N region due to multiple Andreev re ections. A phase dierence, , between the order parameter in the left and the right S layers results in a Josephson current [64]. Andreev re ection occurs at one of the NS interfaces when and electron (or hole) from the N side is incident on the interface with energy within the superconducting gapjEj<E g . Since there are no quasiparticles in the superconductor at this energy, the incident electron (or hole) gets re ected as a hole (or electron). As a result a charge of 2e gets transferred to the superconducting condensate in the form of an extra Cooper pair. If the N part of the SNS junction is short enough, the induced Cooper pair has a nite probability o impinging on the other NS interface, where the Cooper pair scatters with an electron (or hole) on the other side, re ecting back in the N region. Multiple Andreev re ectins of this form result in the formation of bound states with energies that are submultiples of the energy gap, E bound = 2 n ;n2Z. These are the Andreev bound states [63]. Josephson current is a zero-voltage current that ows across a thin insulating barrier be- tween two superconductors. The current magnitude depends on the phase dierence between 45 Signatures of topology in a quasicrystal the superconducting order parameter on the two sides, , and the critical supercurrent I c : I s =I c sin (4.31) This is called the DC Josephson eect. If a constant voltage dierence is maintained across the junction, the phase dierence between the two superconductors evolves as: d dt = 2eV ~ (4.32) The results in an oscillating supercurrent I s (t). The frequency of the oscillation is 2eV=~. This is called the AC Josephson eect. The Bogoliubov-de Gennes method is an eective tool to treat these proximity related problems. It has previously been used to study the decay away from the interface of the induced spectral gap, and the pairing correlations when the N part is a topological insulator a semimetal [49]. It has been applied to the problem of Andreev bound states and Josephson current in interacting one-dimensional liquids [65] and graphene [53]. In all cases, the spectral features of the material forming the N region was found to have an appreciable impact on the qualitative features on the ensuing proximity eect. We apply similar techniques to the problem where the N region comprises of a quasicrystal. I have reproduced our ndings [3, 4] in Chapter 6. Following [55, 66], we employed the geometry of an NS hybrid ring|a one-dimensional chain made up of a superconducting and a non-superconducting part with periodic boundary conditions. 4.4 Superconductivity in quasicrystals Superconductivity in a quasicrystal was rst reported by Kamiya et al. [46] in 2018. The authors found that icosahedral phase of an Al-Zn-Mg quasicrystal transitions to bulk su- perconductivity at a critical temperature of 0:05 K 2 . This discovery instigated several theoretical studies into the nature o the superconducting state in quasiperiodic systems [50, 2 Since then, a critical temperature of 1 K was reported in a Y-Au-Si approximant [67]. 46 4. SUPERCONDUCTIVITY 56, 68{70]. The lack of translational symmetry means that k-space methods are generally inapplicable which renders making progress by analytical means quite challenging. The re- search eort has therefore been primarily numerical in nature. As is typically the case with real-space based numerical studies, eciently simulating system sizes that are large enough is not trivial and various approximations have to be used. To the best of my knowledge calculations on 3D quasicrystalline superconductors have not been performed. The two- dimensioanal Penrose and Ammann-Beenker tilings have remained the favored systems to study. The numerical studies on the 2D structures have yielded a handful of insights into the nature of the quasicrystalline superconducting state. The typical starting point is the attractive-Hubbard model on the Penrose (or Ammann-Beenker) vertex model. H =t X hiji c y i c j; X i n i c i U X i n i" n i# (4.33) c y i creates an electron at a site placed at a vertex of the Penrose (or Ammann-Beenker) tiling and n i is the number operator c y i c i . A real-space dynamical mean eld theory (RDMFT) treatment of (4.33) was used to study the spatial pattern of the superconducting order parameter. The authors recognize three regimes in the parameter space of the attraction strength U and the lling n. 1. small U and arbitrary n: extended non-standard Cooper pairs. Pairing is between time-reversed states as suggested by Anderson for disordered materials. 2. largeU and small n: the order parameter is strongly correlated with the charge density. The charge density and the order parameter primarily depend on the local geometry, characterized by the coordination number. 3. large U and large n: The charge density and order parameter modulations are weak. The relationship between the order parameter and the charge density depends on the coordination number of the site. 47 Signatures of topology in a quasicrystal Figure 4.1: The real space order parameter prole (top) and charge density (bottom) in the three regimes: (a) small U and arbitrary n, (b) large U and small n, (c) large U and large n (Taken from [68]). In the latter two cases, the Fourier transformed pair amplitudehc k" c k 0 # i is strongly peaked on the k =k 0 line. These regimes can be understood to form a Bose-Einstein condensate| which is in a sense, the weak coupling limit of BCS theory. In the rst case, the pair amplitude has considerable weight away from the k =k 0 line, suggesting that the nature of the pairing state departs from the BCS picture. The applicability of BCS theory was further tested by a Bogoliubov-de Gennes approach to the Hubbard model on the Amman- Beenker tiling [56]. The authors observed that there is no evidence of the formation of superconducting islands in the quasicrystal. This is in stark contrast to the disordered case [51, 52], where the appearance of superconducting islands in a non-superconducting bulk dominate transport in the strong disorder limit. The authors further conclude by a scaling analysis that superconductivity in quasicrystals is governed primarily by the energy spectrum, and only weakly by the form of the wave functions, suggesting that mean-eld BCS results should generally hold. In another study emplying the Bogoliubov-de Gennes approach [69], the typical physical observables such as the specic heat and the superconducting gap width were computed around the transition temperature for the superconducting Penrose tiling. The numerical constants were found to be close to the BCS expectation in accordance with the previous studies. Other related works have studied the superconducting state in the 48 4. SUPERCONDUCTIVITY presence of a magnetic eld [50], and the competition between various electron instabilities in quasicrystals [70]. 49 Chapter 5 Bulk signatures of topology in the Fibonacci chain I co-authored a paper [2] that showed that the gap indices of the Fibonacci chain can be experimentally observed by measuring the real space charge density. The oscillations that appear when the charge density is mapped to appropriate coordinates (perpendicular space) directly indicate the Chern number of the gap in which the Fermi energy lies. 1 5.1 Charge density in perpendicular space The charge density, n i , of the Fibonacci hopping model (3.6) is inhomogeneous and is given by n i =hc y i c i i = X k c y i c i f( k f ;T ); (5.1) where i 2 f0; 1; 2;:::F n 1g labels each site according to its position along the chain, f(;T ) is the Fermi distribution function at temperature T , and k are the eigenvalues of the Hamiltonian. The charge density at a given site i depends primarily on the local neighborhood|the values that the hopping integralst j take forj close toi. Since the hopping 1 I consider this work to be the foremost achievement of my graduate school career. 50 5. BULK SIGNATURES OF TOPOLOGY sequence is quasiperiodic,n i is naturally also a quasiperiodic curve. Whether particular local neighborhoods are favored or not depends on the Fermi level. In the following, we always place the Fermi level in a spectral gap. To extract the Chern number, we map the charge density to perpendicular space, j = n C 1 (j) . This corresponds to reshuing the sites such that they are arranged according to their positions in perpendicular space, or their conumber index. Fig. 5.1 shows the results of a tight-binding calculation performed on a 233-site approxi- mant with periodic boundary conditions 2 . Subgure (a) shows the chage density in real and perpendicular space for three dierent values of the chemical potential|the Fermi level is tuned to lie in theq =1; 3, and 4 gaps from top to bottom. The main result is that the gap label corresponding to the Fermi level is given directly by the frequency of the oscillations in perpendicular space|the charge density oscillatesjqj times when the Fermi level is in a gap with label q. There are two ways in which this observation advances thestatus quo: 1. Our method needs only bulk measurements, and works just as a well without edges (torus geometry). This is the rst indication of the bulk-boundary correspondence shown for the Fibonacci chain. 2. Our method does not need to vary|there are no phason ips involved. It is custom- ary to assign Chern numbers to the Fibonacci chain by appropriately extending to a 2D ancestor Hamiltonian by treating the phason as a second dimension [4, 17, 18, 39]. However, all previous demonstrations of the topological nature of the Fibonacci chain involved varying the phason. We have shown that the Chern number is physically relevant property of a single realization of the Fibonacci chain. These oscillations persist as long as the gap labelling theorem is valid. Panel (b) shows the oscillations in the charge density for a wide range of values of the modulation strength 2 I used the Python package Kwant [71] to set up the Hamiltonian matrix 51 Signatures of topology in a quasicrystal 0.3 0.4 0.12 0.15 0.18 0 100 200 Site number, i 0.46 0.48 0 100 200 Conumber, i < c i c i > (a) q = 1 q = 3 q = 4 0.23 0.24 0.998 q = 2 (b) 0.20 0.25 0.8 0 50 100 150 Conumber, i 0.0 0.5 0.003 M 2 M 1 M 2 A M 2 M 1 M 2 Charge density, c i c i Weak modulation Strong modulation Figure 5.1: (a) The charge density,hc y i c i i, in real space (left) and perpendicular space (right), with the Fermi level tuned to theq =1;3, and 4 gaps. (b) The charge density in theq = 2 gap for three dierent values of representing strong, intermediate, and weak modulation. = t A t B . When the Fibonacci chain is close to the homogeneous chain, 1, the oscillations are weak and sinusoidal. We will see shortly how this behavior is reproduced by rst order perturbation theory. Approaching the opposite limit of a weakly coupled chain of monomers and dimers, 1, the curve takes the form of a sequence of plateaus. The number of oscillations, dened by the number of local minima/maxima, is preserved in this picture. This limit admits a simple intuitive explanation using a renormalization group (RG) decimation scheme. The intermediate case interpolates between the two limits. 5.2 In the weak modulation limit In the weak modulation regime, we follow the perturbative analysis of Sire and Mosseri [29], writing the Hamiltonian as a sum of two terms, ^ H = Fn X i=1 t A c y i c i+1 +hc | {z } ^ H 0 Fn X i=1 (t i t A )c y i c i+1 +hc | {z } ^ Hw (5.2) ^ H 0 is the Hamiltonian of a 1D homogeneous tight binding chain with only nearest neighbor hopping t A . ^ H w introduces the Fibonacci modulation. 52 5. BULK SIGNATURES OF TOPOLOGY Recall from Sec. 3.6 that the eigenvalues ofH 0 can be labelled by the integers = 0; 1;:::;bF n =2c and are given byE =2t A cos 2F n1 Fn . They are (almost all) doubly degenerate and the corresponding eigenvectors are given byv (j) = 1 p Fn exp 2i Fn j . The perturbation ^ H w immediately opens the gaps of the Fibonacci chain. The opening of the gap at energyE is associated with the splitting of the degenerate plane wave states. In Sec. 3.6, we showed that the slightly perturbed plane waves around the gaps with an even (odd) number of states below them have the wave number = cFn+q 2 ( = cFn+q1 2 ) whereq is the gap label andc is some integer. Therefore, if the Fermi level is tuned to a gap q, then only one of a pair of states with this wave number is occupied. The contribution of this state to the charge density q j is given by the squared amplitude of one of v (j): q j / 1 2 1 cos qj F n (5.3) All other occupied states appear in pairs, so their contribution to the charge density is constant. This is precisely the behavior seen in the top panel of Fig. 5.1(b). The oscillations are sinusoidal with q periods as the conumber j varies in the interval [0;F n 1]. They arise from the fact that the unperturbed perpendicular space eigenstates at the energy where the gap opens are plane waves with exactly the wave vector needed to obtain q oscillations. 5.3 In the strong modulation limit In the strong modulation limit, when = 0, a Fibonacci approximant of length F n breaks into F n3 atom sites (with an A bond on both sides) and 2F n2 molecule sites (with a B bond on either side). There are three kinds of electronic states available: F n2 (anti-)bonding molecular states with energyt B , and F n3 atomic states with zero energy. As described in Sec. 3.7, when is slowly turned on, the self-similar eigenvalue spectrum can be built by splitting each cluster of states into three subclusters recursively [42]. The charge density in the strong modulation limit can be recovered by considering the real space structure of all 53 Signatures of topology in a quasicrystal Molecular RG Occupy F n 2 states Reverse RG Figure 5.2: The renormalization group based algorithm to nd the real space charge distri- bution in the strong modulation limit|an example: to ll the q =2 gap in the 13-length chain, we perform the molecular RG decimation once, ll in the bonding and atomic states, then reverse the RG step. Compare with the bottom panel of Fig. 5.1(b). the states below a certain gap. I will show how this can be done by carefully applying Niu and Nori's renormalization group method [41]. First, let's assume that the Fermi level is in a gap whose label is a Fibonacci number q = F m . Then, the number of electrons that are present in the F n -site chain is also a Fibonacci numberF nm1 . In order to distribute the electron weight appropriately between the F n -site, we propose the following algorithm: If m is odd: 1. Use the molecular RG decimation scheme on the original chain m1 2 times. This will produce a chain with F nm+1 sites. 2. The lowest energy states in this chain are theF nm1 bonding states. Place one electron in each of the this. Every molecule site has the weight of half an electron. 3. All the electrons have now been placed. Reverse the RG steps. At every reverse RG step, redistribute any electron weight evenly among the ancestor sites of an occupied site. If m is even: 54 5. BULK SIGNATURES OF TOPOLOGY 1. Use the molecular RG decimation scheme on the original chain m 2 times. This will produce a chain with F nm sites. 2. The lowest energy states in this chain are theF nm2 bonding states. Place one electron in each of these. Every molecule site has the weight of half an electron. 3. There areF nm3 electrons remaining. The next lowest energy levels available are the F nm3 atom states. Place one electron in each of these. Every atom site has the weight of one electron 4. Reverse the RG steps. At every reverse RG step, redistribute any electron weight evenly among the ancestor sites of an occupied site. Fig. 5.2 shows an example of this procedure. A 13-site Fibonacci chain is tuned to the q = 2 gap. By the gap labelling theorem, this corresponds to three electrons. According to the procedure above, we rst perform one step of the molecular RG decimation scheme resulting in a Fibonacci chain with 4 molecule sites and 1 atom site. We place the rst two electrons in the two available bonding states. We place the last electron in the atom states. We reverses the decimation step, so that the ancestor sites of the half occupied molecule site gains 1=4 electron weight, while the ancestor site of the fully occupied atom site gains 1=2 electron weight. This corresponds precisely to the situation in the bottom panel of Fig. 5.1b. We nd a charge density of 0:5 in regions of perpendicular space populated by molecule sites in molecules with atom sites to both sides (labelled byM 1 ). These are precisely the molecule sites that become atom sites after one decimation step. Other molecule sites (M 2 ) have a charge density of 0:25 and atom sites (A) have a charge density of 0. This procedure demonstrates that in the strong modulation limit, the charge density at a given site depends only on a small local neighborhood around it. The size of the relevant local neighborhood for a given gap depends on the number of RG steps needed to ll in the electrons up to that gap, which is larger for small gaps with large q-labels. Distinguishing 55 Signatures of topology in a quasicrystal large local neighborhoods requires resolving smaller regions in perpendicular space, which results in faster oscillations in perpendicular space. 5.3.1 Generalization to arbitrary q This procedure generalizes to arbitrary q by applying Zeckendorf's theorem [72]. When an F n -site Fibonacci chain is tuned to a gap with an arbitrary label q, the number of electrons in the system,I q , may not be a Fibonacci number. However, Zeckendorf's theorem guarantees that we can uniquely express any positive integer as a sum of distinct Fibonacci numbers with the constraint that the sum does not include any two consecutive Fibonacci numbers. We write I q = F m 1 +F m 2 +::: +F m M with the constraintjm m j 2 and m 1 >m 2 >m 3 :::>m M . The key idea is to split the problem of lling I q electrons in the original Fibonacci chain toM independent problems of llingF m ( = 1; 2;:::;M) electrons in Fibonacci chains of dierent lengths. This is done recursively. In the rst step, we follow the procedure outlined in the main text to ll F m 1 electrons in the original Fibonacci chain but stop before the nal step of reversing the molecular RG. If m 1 was odd, we now have an F nm 1 +1 -site Fibonacci chain with all the molecular bonding states lled. Now, the question of how the remaining electrons are distributed among the remaining atom sites is an independent problem. This can be formalized by performing one decimation step of the atomic RG [41] which keeps all the atom sites. The new nearest neighbors are connected with a renormalized strong bond if there used to be one molecule between them, and with a renormalized weak bond if there used to be two molecules between them. Now, the problem has reduced to lling F m 2 +F m 3 +::: +F m M electrons in a F nm 1 2 -site Fibonacci. If m 1 was even we have an F nm 1 +1 -site Fibonacci chain with all the molecular bonding states and atom states lled. As before, the problem of distributing the remaining electron weight of F m 2 +F m 3 +:::F m M electrons among the anti-bonding states is an independent 56 5. BULK SIGNATURES OF TOPOLOGY IDOS (---) 0.0 2.5 5.0 7.5 (a) Zero T clean (b) kT = 0.1t clean 0.0 0.2 0.4 2.0 1.5 1.0 0.5 0.0 0.0 2.5 5.0 7.5 (c) Zero T weak disorder 2.0 1.5 1.0 0.5 0.0 (d) kT = 0.5t clean = 1 = 2 = 3 = 4 = 6 0.0 0.2 0.4 q=-1 q=2 q=-3 q=4 q=-6 Figure 5.3: The rst few Fourier components of the perpendicular space charge density are plotted as a function of the chemical potential, . Also shown is the integrated density of states (dashed line) at energy , with the ve largest gaps in the lower half of the spectrum marked with their q-labels. (a) Zero temperature, clean, (b) kT = 0:1t, clean, (c) zero temperature, t = 0:1, (d) kT = 0:5t, clean. problem which can be formalized by performing one molecular RG step. By performing this procedure M times and then reversing every RG step, taking care to distribute any electron weight in a de ated chain evenly among the ancestor states we recover the charge density prole in the original chain in the strong modulation limit. 5.4 Fourier transform of the charge density A fourier transform provides a useful bird's eye view of the charge density oscillations in perpendicular space is, ~ = Fn1 X j=0 e 2i j Fn j : (5.4) In Fig. 5.3(a), the th Fourier component, ~ , is plotted as a function of the chemical potential, for corresponding to the q-labels of the ve largest gaps in the spectrum of the 57 Signatures of topology in a quasicrystal hc † i c i i |q|=1 |q|=4 clean, free disordered, interacting,V/t=0.5 Conumber, C i t/t=0.5 Figure 5.4: The perpendicular space charge density for the clean, free (black), disordered (blue) and interacting (orange) Fibonacci chain. For the disordered case, t=t = 0:5, and the disorder average is taken over 1500 realizations. The light blue background indicates the statistical error to the mean. This gure is the work of my co-author Henning Schl omer. Fibonacci chain (jqj = 1; 2; 3; 4; 6). The same plot also shows the integrated density of states of the Fibonacci chain, corresponding to the energy. The ve largest gaps are marked with a shaded background. The magnitude of the th Fourier component, ~ , is largest precisely when the Fermi level is tuned to a gap withjqj =. The eect is particularly prominent for small q (large gaps), where the qth component dominates over all the other frequencies. In the smaller gaps, such as the q =6 gap, the qth component is not predominant, as its magnitude is less than that of the more fundamental frequencies (in this case = 1; 2). Here, the topological invariant can be identied by checking that the maximum of the curve ~ () is in thejqj = gap. 5.5 Robustness of the oscillations The observed oscillations are robust to moderate amounts of temperature, disorder, local interactions, as well as edge eects. Fig. 5.3 (b)&(d) show the eect of temperature on the Fourier components ~ (). Our calculations shows that the oscillations are robust as long as the temperature is a suciently small fraction of the bandwidth, kT / 0:1t. 58 5. BULK SIGNATURES OF TOPOLOGY The oscillations are strongly protected against disorder. We add o-diagonal noise to the Hamiltonian: t i !t i +t i withft i g a set of uniformly distributed random variables on the interval [t;t]. Fig. 5.4 depicts the disorder averaged result over 1500 realizations with disorder strength t=t = 0:5 and xed particle number, showing how the charge oscillations in perpendicular space prevail even in regimes where the disorder strength exceeds the gap size. Only in the strongly disordered limit t t, the averaged density loses its oscillating structure and uctuates around the value of the lling factor. The oscillations persist upon introducing local interactions as well. We study the eect of nearest neighbor repulsion of the formVc y i c y i+1 c i+1 c i using the density-matrix renormalization group (DMRG) [73, 74]. As shown in Fig. 5.4, the oscillation persists with a renormalized amplitude similar to the disordered case. Open boundary conditions introduce deviations to the charge density for a small number of sites close to the edges, but since sites close to the edge in real space are uniformly distributed in perpendicular space, these sites can be safely ignored without changing the Fourier amplitudes in a signicant way. 5.6 Entanglement entropy in perpendicular space It has been shown that the entanglement spectrum in the bulk of symmetry protected topo- logical (SPT) quantum chains can be used to characterize their topology [75]. In topologically non-trivial gaps of the 1D quantum dimer- (SSH) and trimer-chains, strong parity eects appear in the entanglement spectrum that persist deep in the bulk. This motivates the analysis of entanglement oscillations in higher order topological phases, as present in e.g. the Fibonacci chain. The von Neumann entanglement entropy (EE) of subsystem A ` = [1;:::;`] with B ` = [` + 1;:::;N] is given by S ` =S( A ` ) =Tr [ln( A ` ) A ` ]: (5.5) 59 Signatures of topology in a quasicrystal |q|=1 |q|=4 S(`) Subsystem size ` Subsystem conumber C ` S(C ` ) Figure 5.5: The entanglement entropy, S ` , in real (upper row) and co-number (lower row) space in gaps with labelsq =1 andq = 4. We only map data points from the bulk (marked by a shaded background) to perpendicular space. This gure is the work of Henning Schl omer The entanglement entropy (EE) of an approximant chain using open boundary conditions is shown in the upper row of Fig. 5.5 for two dierent llings. The bottom row show the same data for the EE plotted versus co-numbers. In the lower plots, only the data for sites in the interior of the chain (the grey zone of the upper plot) are shown, since cutting the chain introduces spurious edge eects. Just like the uctuations presented above for the charge density, the bulk entanglement spectrum oscillates between particular values when the chemical potential is tuned inside a gap. These uctuations map to full-period oscillations in perpendicular space, where the number of oscillations predicts precisely the topological label of the corresponding gap. Single period oscillations corresponding tojqj = 1 are already visible in the lowest order approximant of the Fibonacci chain, i.e. the SSH model. Here, perpendicular space only contains two in-equivalent points (corresponding to A and B sites), and the entanglement alternates between two values in the bulk [75], which can be interpreted by the formation of a valence bond at dimers in the chain in the large modulation limit. When analyzing larger approximants, new higher labelled gaps open in the single particle spectrum, leading to more complex entanglement structures which ultimately lead to the appearance of higher order oscillations {similar to the results and RG analysis presented previously for 60 5. BULK SIGNATURES OF TOPOLOGY the charge density. 61 Chapter 6 The proximity eect in the Fibonacci chain In 2018, Kamiya et al. found an Al-Zn-Mg quasicrystal with a superconducting transition at 0:05K [46]. Since then, there have been a number of theoretical studies attempting to explain the quasicrystalline superconducting state [50, 56, 68]. There are two important open questions in: 1) How does superconductivity set in a quasicrystal? 2) What does the superconducting state look like in a quasicrystal. I co-authored two papers [3] approaching this problem indirectly through the supercon- ducting proximity eect. We studied induced superconductivity in a quasicrystal when it is placed next to a conventional homogeneous superconductor. In this setup, we ignore the problem of how superconductivity sets in and focus on what the properties of the supercon- ducting state are. For numerical and analytical simplicity we focus on a 1D N-SC hybrid ring. As sketched in Fig. 6.1, it consists of a 1D tight-binding chain with periodic boundary conditions where one part (S) is homogeneous and superconducting and the other part (N) is a non-superconducting Fibonacci chain. The proximity eect is known to be a useful probe of electronic properties. In particular, our goal is study the eect of the critical multifractal wave functions and the topological edge states of the Fibonacci chain. Our rst question 62 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN SC Normal (a) Figure 6.1: Superconducting proximity eect in an N-SC hybrid ring. Part of the ring (right) is a one-dimensional superconductor (SC), the other part (left) is a non-superconducting chain (N). The N part will be chosen to be a clean metal, a highly disordered system, and a Fibonacci chain. is how the proximity eect is dierent in quasicrystals compared to disordered systems, the other archetype of inhomogeneous systems. To describe the disordered system, we will re- place the N part with the Anderson model [76]. The wavefunctions of disordered systems are localized, where the wave functions of the Fibonacci chain are critical|neither localized nor delocalized. We nd that the proximity eect in the Fibonacci chain is strong compared to the disordered case|the order parameter (x) decays as a power law away from the interface at zero temperature compared to exponential decay for the localized case. But since the wave functions are not completely delocalized either, we nd that the ensemble statistics of the induced superconductivity follow a log-normal distribution|reminiscent of systems with localized wave functions. This dichotomy is a signature of the criticality of the Fibonacci chain. The eect should be observable in a proximity eect experiment. These results were reported in [3] and are reproduced in Sec. 6.2-6.5. In [4], we reported on the eect of the topological edge states on the proximity eect. We study how the induced superconductivity evolves upon variation of the phason . We nd that the topological invariants|the gap labels|of the Fibonacci chain leave their imprint in the form of oscillations in the induced order parameter as a function of the phason . This phenomenon is the subject of Sec. 6.6. 63 Signatures of topology in a quasicrystal (a) (b) Figure 6.2: Superconducting proximity eect in a normal one-dimensional metal: (a) Spatial variation of the superconducting order parameter i along the chain for a clean N-SC hybrid ring|both parts are bulk-periodic. (b) Spatial decay of the superconducting order parameter in the normal part of a clean N-SC ring for zero and nite temperature ( = 1=kT ), extracted by nite-size scaling of mid for chains with varyingL. The best t parameters arex o = 10:0 and = 6:4. decays inversely with distance at zero temperature and exponentially with distance at nite temperature. 6.1 Description of the model The Bogoliubov-de Gennes mean eld Hamiltonian takes the following form for the 1D problem. ^ H mf = X i ( X i z }| { ( 0 i + HF i i )c y i c i + X t ii+1 c y i c i+1 +h:c: +V i i c y i" c y i# +h:c: ) : (6.1) The system is fully specied by xing three sets ofL tot parameters: ( i = i );t i;i+1 ;V i , where L tot is the number of sites in the ring. As our primary goal is to examine the eects of disorder on the proximity eect, we will focus on the ideal case where the interfaces are transparent. We will measure all energies in units of the strength of the nearest neighbor hopping in the superconductor, t. The hopping energy in the superconductor is then t =1. The hopping amplitudes in the N region will be chosen to be of comparable strength, i.e. of the order of unity. In each of the models, the band llings are xed at 1 2 , so that the Fermi level is in 64 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN the middle of the spectrum and particle-hole symmetry is maintained. The strength of the attractive Hubbard interaction in the SC is set to a xed value 1 for LiL tot 1.. The lengths of the SC region are chosen to be large enough that the OP relaxes to attain the expected bulk value well inside the SC region. The length of the N region is likewise chosen large enough, such that the bulk penetration laws can be properly determined by nite size scaling. This amounts to the following set of choices for the parameters: The normal region corresponds to the rstL sites,i = 0;::;L1. The superconducting region is of length L SC , corresponding to indices i =L;L + 1;:::;L tot 1 whereL tot = L+L SC is the total number of sites. L SC = 200 everywhere,L ranges from 90 to 1598. The hoppings at the two interfaces are taken to be unity, i.e. t L1 =t Ltot1 =1. There are no interactions in the normal region, i.e. V i = 0 for 0iL 1. Within the N region,t i values are either sampled from a random distribution function taken from the Fibonacci sequence. Within the (translationally invariant) superconductor , t i =1;V i =t = 1:5. Both regions are at half-lling, i.e. the Fermi level is in the middle of the spectrum. In the normal part, this means 0 i i = 0. In the superconducting part, we need to account for the Hartree-Fock shift which implies 0 i i = HF i = V 2 . Throughout this study, the central observable of interest is the strength of the supercon- ducting order parameter at the mid-point of the normal region of the ring, mid . This is given by L1 2 for odd chains and 1 2 L 2 + L 2 1 for even chains. We compute mid for an ensemble of rings with a given size and disorder/modulation strength. To obtain the spatial decay of the order parameter as a function of distance from the interface, we t values of 1 The BCS pairing attraction is chosen to be of the same order of magnitude as the nearest-neighbor hopping in order to resolve superconducting features in systems of this size. 65 Signatures of topology in a quasicrystal (b) Figure 6.3: (a) Superconducting proximity eect in a Fibonacci chain Real-space prole of the superconducting order parameter when the normal segment is a Fibonacci chain. Modulation strength: W=t = 0:1. (b) Finite-size scaling ofh mid i with the length of the normal region L shows power law decay of away from the interface in N{SC hybrid rings where N is a Fibonacci chain. Modulation strength: W/t = 0.005. Fit oset x 0 = 10:85. the ensemble averageh mid i for xed disorder strength and dierent chain lengths 2 . We use histograms of mid for xed system size and disorder strength to study the distributions of the induced order parameter (OP). Before moving on to inhomogeneous systems, it is useful to recall results for the periodic case, when the N chain hopping amplitudes are uniform, i.e. t i =1. The real space prole of the order parameter for the clean N-SC ring is shown in Fig. 6.2a. We nd inverse distance behavior at zero temperature and exponential decay at nite temperatures (see Fig. 6.2b). These results are in agreement with analytical calculations using Gor'kov's Green function method to compute [60, 77]. 6.2 Order parameter prole in the Fibonacci chain Fig. 6.3a shows the spatial behavior of the superconducting order parameter in a hybrid ring composed of a Fibonacci chain coupled to a BCS superconductor. This order parameter prole displays a self-similarity, which can be seen in Fig. 6.2 where we zoom into successively smaller sections in the center of the Fibonacci chain. The plots show a central region where the number of sites is reduced successively by a factor 3 . We will discuss shortly how the 2 We nd this method of tting to be more accurate and unambiguous than directly tting the curves in real space. 66 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN (a) i Position (c) t A t B i (b) 792 1791 Site number i Figure 6.4: (a) The local order parameter and its correlation with the symmetry of the local environment: the peaks in i occur at sites with high re ection symmetry represented by i . (b) Superconducting order parameter prole in the central 987 (top), 233 (middle), and 55 (bottom) segments of the Fibonacci chain in a hybrid Fibonacci-SC ring. The order- parameter proles are self-similar upon scaling by the renormalization parameter 3 . The data in panel (b) is taken from a hybrid ring with 2585 and 200 sites in the normal and superconducting regions respectively. (c) The hopping sequence corresponding to the sites shown in the last panel in (b) 3 scaling factor appears in an analysis based on the atomic RG described in Sec. 3.6, which is valid around half lling. The symmetry of the local environment plays an important role in the value of the OP on a given site. The lowest panel of Fig. 6.2 shows the hopping sequence for the region shown in Fig. 6.2, with t A and t B shown by yellow (resp. black) bands. Note the correlation between the heights of OP peaks and the symmetry of the local environment. This correlation is made quantitative by dening the local resonator size i as the distance up to which re ection symmetry is present around a given site i: specically i is the smallest whole number d such thatt i+1+d 6=t id . In of Fig. 6.2, notice that peaks in i coincide with higher values of i . This type of characterization of local environments was presented by R ontgen et al. [78] who they nd that edge states with certain energies localize in such high symmetry regions, which they call local resonators. This self-similarity and the local environment dependence of the OP have simple expla- nations in terms of the theoretical description of Fibonacci chain eigenstates in [79] and 67 Signatures of topology in a quasicrystal [80]. We assume that the sum over states can be replaced by the E = 0 contribution. The OP is thus determined by the structure of the E = 0 wavefunctions. In the limit of strong quasiperiodic modulation, these wavefunctions tend to have their support concentrated on the sites which are surrounded on either side by t A bonds (and were called \atom" sites in the renormalization group (RG) approach due to Niu and Nori and Kalugin et al [41, 42]). Under a renormalization transformation, the absolute value of the E = 0 wave function of a site in the nth generation chain is related to that of a site in the (n 3)th generation by the recursion formula j n E=0 (i)j = p j n3 E=0 (i 0 )j (6.2) where i and i 0 are the site indices of the old and the new (renormalized) chain, and is a wavefunction rescaling factor which can be computed as a function of the hopping parameters [80]. The superconducting OP is given by the product of two such wavefunctions. The highest amplitude is found for the sites which remain after the largest number of RG transformations. Under RG transformation, the number of such sites is reduced by a factor 3 . The distance out to which the site possesses re ection symmetry increases by the same factor. Thus the RG theory of the Fibonacci chain explains the numerical observations of a) self similarity of the OP, and b) the correspondence between the OP and the local environment. 6.3 Order parameter penetration in the Fibonacci chain In this section we consider the properties of the typical value of the OPh mid i g |after averaging over all values of 3 |and focus in particular on its spatial decay away from the N-SC interface. To study the spatial decay ofh mid i g as one moves away from the interface, we compute this quantity for chains of dierent lengthsL. Plotting it as a function of length L=2, as shown in Fig. 6.3b yields a power law withh mid (L)i g L , where the exponent 3 hi g refers to the geometric mean of 68 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN 12 11 10 9 8 7 6 5 log mid Number of realiztions (a) log (L/2) Var(log mid ) w = 0.25 w = 0.1 (b) Figure 6.5: Order parameter ensemble statistics in the Fibonacci chain: (a) The distribution of log mid in N-SC rings is symmetric when N is a Fibonacci chain. L = 988, L SC = 200. (b) The dependence of the mean and standard deviation of mid on the modulation strength W=t. Inset: The functional dependence of the width,(log mid ), on the ratio of the hopping parameters. depends on the modulation of hopping amplitudes. The observed power law decay is consistent with the presence of critical states: the eective exponent is non-universal and depends on the average values of the density of states and the states close to the Fermi level. An explicit calculation is outside the scope of the present discussion. We remark simply that when the ratio t A ! t B , the power ! 1, i.e. approaching the decay law for the simple non-modulated chain (see inset of Fig. 6.3b). As one might expect, increasing the strength W of the quasiperiodic modulation results in wave functions becoming less extended, leading to a faster spatial decay of the OP as one moves away from the interface. 6.4 Ensemble statistics of the order parameter in the Fibonacci chain It is well-known that all electronic states of the Fibonacci chain are multi-fractal and critical [81]. Such states are characterized by very large uctuations, and wave function amplitudes decay with dierent power laws, depending on the local environment. One way to study such uctuations is by looking at ensemble statistics. In the hybrid ring geometry, edge eects 69 Signatures of topology in a quasicrystal play a role and therefore the Fibonacci approximants of a given size separated by phason ips describe distinguishable systems. Treating the -parametrized members of the family S n (), we can consider the distribution of the order parameter, mid , as is often done for disordered systems. However, there is a signicant dierence between the two cases: whereas in the disordered system one can generate as many realizations of the chain as one wishes, for the Fibonacci chain there are only F n realizations of a chain of length F n . It is therefore necessary to go to very large systems to obtain a good t for the decay law ofh mid i and to t the distribution function to a smooth form. In this work, the biggest system that we have studied consists of N = 1; 597 bonds. The distribution of the induced order parameters are shown in Fig. 6.5a for a chain of length L = 988 for two dierent values of the modulation strength W , while the depen- dence of its mean and standard deviation on the disorder strength is shown in Fig. 6.5b. More precisely, Fig. 6.5a shows distributions of the logarithm of mid , which are symmetric, suggesting log-normal behavior. We can analytically motivate the log-normal distribution in this context. First, we approximate the sum in Eq. (4.28) for mid by keeping only theE = 0 term. Due to the bipartite character of the hopping Hamiltonian, the single particle states exactly atE = 0 are known to have a stretched exponential form [82, 83], meaning a spatial decay faster than a power law but slower than a pure exponential. This is most readily shown by using the tight-binding equations forE = 0 to relate the wave function amplitude (i) in the interior of the chain to the wave function on the boundary (0) and (1). For a site located at a distance 2m from the boundary, the local wave function amplitude is E=0 (2m)/ Y 1lm (1) m t 2l t 2l1 (6.3) Specializing to the Fibonacci chain, one can show that the wave function can be expressed in terms of a height function [79]. (6.3) reduces to (2m) =const(1) m exp(h(2m)); (6.4) 70 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN where = log(t A =t B ). The height function h, which depends solely on the geometry, can be computed for a given sequence of hopping amplitudes using the following relations for the height changes, which can take three values depending on the value of the hopping amplitudes between the two sites, h(2m) = 8 > > > > > > < > > > > > > : 0 if t 2m1 =t 2m =t A 1 if t 2m1 =t A ;t 2m =t B 1 if t 2m1 =t B ;t 2m =t A (6.5) where h(2m) = h(2m)h(2m 2). A similar structure holds for the state on the odd sublattice. To proceed, we use the renormalization transformation of Fibonacci chains, which relates a given chain to the next generation, to write a recursion relation for the height function. From this relation, we can deduce that for suciently long chains the distribution ofh-values must tend to a Gaussian of width proportional to lnL. The multi-fractal scaling properties of the E = 0 state can be deduced from the distribution of h. In particular, for this critical state all the generalized exponents describing its spatial characteristics have been computed exactly. The analysis shows that heights follow a gaussian distribution with the variance given by [79] hh 2 ihhi 2 = 1 p 5 log(L) log() (6.6) Returning to the proximity eect, the order parameter at the midpoint is determined by the wave functions u andv at the midpoint of the chain which both have the form given by (6.4), diering only in the values of the prefactor. The changes of values mid result from the phason ips that occur when the parameter is varied. From (6.4) and (6.6) the resulting distribution of mid must therefore be log-normal. This can be seen in Fig. 6.5a. According to (6.4) the width of this distribution should increase with the strength of quasiperiodic modulation as ln(t B =t A ). 71 Signatures of topology in a quasicrystal 0 100 200 300 400 Site number 0.00 0.02 0.04 0.06 0.08 i SC Normal mid Figure 6.6: Superconducting proximity eect in a disordered one-dimensional metal: spatial prole of the superconducting order parameter along the chain in a hybrid N-SC system with o-diagonal disorder in the normal region. Disorder strength: W/t = 0.08. 6.5 Disordered chains We want to compare these result withe the case when the metallic state disappears upon the addition of disorder. It is well known that adding arbitrarily small disorder in the one- dimensional periodic model leads to Anderson localization|the Lyapunov exponent (inverse localization length) is non-zero for all eigenstates. In a nite chain, however, we can identify a weak disorder regime, in which the localization length (E) of single particle eigenstates is much larger than the system size L, i.e. L. Upon increasing the disorder strength, there is a crossover to a strong disorder regime, where the localization length is smaller than L, i.e. <L. For T = 0, a third length scale, the inelastic (phase breaking) length scale, in this noninteracting model is innite and thus plays no role. In the following we present results corresponding to the two disorder regimes using the self-consistent theory outlined above. We show, rstly, that in the weak disorder case, the proximity induced superconducting order parameter (OP) decays as a power law of the distance from the interface. In contrast, the OP decays exponentially in the strong disorder regime. In addition, we obtain the full probability distributions of the OP and show how they dier in the two regimes. We will consider the o-diagonal disordered Anderson model, in which the on-site terms 72 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN 100 200 300 400 500 L/2 0.002 0.004 0.006 mid 1 L/2 + x 0 (a) 200 400 L/2 1 2 3 Var( mid ) ×10 9 0 200 400 600 800 1000 1200 1400 L/2 0.000 0.001 0.002 0.003 0.004 0.005 mid exp ( L 2 ) (b) L/2 10 20 30 Var(log mid ) Figure 6.7: Finite-size scaling ofh mid i with the length of the normal region L shows (a) inverse law decay of away from the interface in N{SC hybrid rings where N is weakly disordered. The variance of mid is also inversely proportional to the distance. Disorder strength: W/t = 0.005. Fit oset x 0 = 10:4. (b) When N is strongly disordered, we see stretched exponential decay of away from the interface in N{SC hybrid rings where N is strongly disordered. Disorder strength: W/t = 0.7. = 8:4 are uniform and set equal to zero, while the hopping amplitudes t i = t + i are random. The independent random variables i are drawn from a uniform (box) distribution P () = (+ W 2 )( W 2 ) W where W=t < 2 is the disorder strength, and () is the Heaviside step function. The width of the distribution is restricted to be less than 2, so that the hopping amplitudes t i are all strictly negative. Although we work with a box distribution, the exact form of the randomness is not expected to matter for our results, which should also hold for more general random distributions 4 . 6.5.1 Weak Disorder Regime In a weakly disordered nite chain, one can compute the (sample-dependent) corrections to the energies and wavefunctions of the clean system using perturbation theory. In this limit, the semi-classical viewpoint|in which the principal eect of the randomness is to randomize the phase of the wavefunctions|is useful. That wavefunctions in reality remain extended, can be readily seen from the fact that the average probability at each siteh 2 n (i)i tracks the values obtained in the clean system. We therefore use the term quasi-extended to denote this type of wave function. Fig. 6.6 shows a typical order parameter prole in the weak disorder 4 In preliminary calculations on the diagonal Anderson model, we found the same qualitative behaviour 73 Signatures of topology in a quasicrystal 4.4297 4.5283 4.6269 mid ×10 3 Number of realizations (a) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 w/t 4.50 4.51 4.52 4.53 mid ×10 3 (b) 0 2 4 6 ( mid ) ×10 4 Figure 6.8: Superconducting proximity eect in the weak disorder regime:(a) Normal distri- bution of mid in N-SC rings in the weak-disordered regime. 10000 realizations,W=t = 0:005, L = 90,L SC = 200. (b) The dependence of the mean and standard deviation of mid on the disorder strength W=t in the weak-disordered regime. regime. The order parameter decays as a power law away from the interface into the normal part of the ring (Fig. 6.7a). mid for a given system size and disorder strength is normally distributed (Fig. 6.8a). mid on average diers from the clean case by a term proportional to W 2 (Fig. 6.8b). These observations can be explained by means of perturbation theory in the variables i . In the clean limit, the eigenfunctions and energies of the hybrid chain system have been analytically studied in [55, 66]. Considering a Bogoliubov-de Gennes type model in which they xed the order parameter in the superconductor to a constant value, these authors showed that there is a nite, constant, density of states within the gap. These states should exist even after relaxing the constraint on the OP, as we do in our self-consistent approach. These eigenstates are of interest in the following perturbative argument for the induced OP within the N chain. We write the solutions of Eq. (4.26) in terms of the coecients for the clean system u n , v n and corrections to these, u n and v n . Within perturbation theory, the correction terms u n and v n up to second order in i are kept. At zero temperature, we have an expression for the order parameter at a given site (we suppress the index i) from 74 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN Eq. (4.28) 5 , = X n (v n +v n )(u n +u n ) = 0 + X n v n u n +v n u n +v n u n ; (6.7) where 0 is the OP in the clean case. The normalization condition P n ju n j 2 +jv n j 2 = 1 leads to X n u n u n +v n v n = 1 2 X n (ju n j 2 +jv n j 2 ): (6.8) Taken together, 0 = X n (v n u n )(u n v n ) 1 2 (ju n j 2 +jv n j 2 ) : (6.9) Averaging over the disorder then yields h 0 i 1 2 X n h(ju n j 2 +jv n j 2 );i (6.10) where we neglected the contribution of the rst term in (6.9) compared to that of the second term. This follows because the averages ofu n andv n are very small (the linear corrections in average to zero, and the second order corrections are small) compared to the average of ju n j 2 andjv n j 2 . Note that the correction term due to disorder in (6.10) is always negative and proportional to W 2 . For a chain of length L, Eq. (6.9) is a sum of L=2 random variables of variance propor- tional toW 2 . Therefore, we can invoke the central limit theorem to see that the distribution of mid must be a Gaussian, with a width proportional to W , centered at 0 c(W=t) 2 where c is a constant. The scaling of the width of the distribution with system size is given by the product of a factor of 1=L (normalization), and a factor p L (from the variance of the 5 We have exploited the fact the u, v, and can be chosen to be real in the absence of magnetic elds and that all n > 0 75 Signatures of topology in a quasicrystal sum over L random variables). These features are observed in the inset of Fig. 6.7a and in Fig. 6.8b. We now discuss the spatial decay of the OP, which we compute from the dependence of the average OP value at the midpoint of the chain,h mid i, for dierent lengthsL. As shown in Fig. 6.7a,h mid i has an inverse distance or 1=L dependence. This is expected in view of the weak localization physics we expect in this regime. According to theory, the averaged density-density correlations are known to obey a diusion equation [84]. These correlations therefore decay as the inverse power of distance, similar as in a pure metal. In our present context, this property implies that the spatial dependence of the proximity induced averaged pair correlation function will be similar to that of the clean system. In other words, it should fall o with the inverse of the distance from the interface, (x) 1=x [60]. These T = 0 properties should carry over at nite temperatures as long as the phase breaking length scale remains large compared to the system size. 6.5.2 Strong Disorder The strong disorder regime corresponds, in our model, to values ofW=t of order 0.1 or larger. In this regime, the wave functions have an exponentially decaying envelope function. The proximity eect is expected to be short-ranged, in contrast to the weak disorder regime. Indeed this is observed in Fig. 6.7b, where the value ofh mid i for a xed disorder strength W=t = 0:7 is plotted as a function of system sizeL. The characteristic decay length decreases as W is increased, in accordance with the theoretically predicted exponential behavior. Fig. 6.9a shows the distribution function ofh mid i in the strong disorder regime. This distribution is well-described by a log-normal form, i.e. the variable y = log( mid ) is dis- tributed according to a Gaussian, P (y) =Ce (yy 0 ) 2 =2(W;L) ; (6.11) where C is a normalization constant. The width of the Gaussian, (W;L), is found to grow 76 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN 1.0e-03 4.0e-03 9.9e-03 mid Number of realizations (a) 0.0 0.1 0.2 0.3 0.4 w/t 5.60 5.55 5.50 5.45 5.40 log mid (b) 0.0 0.2 0.4 0.6 0.8 (log mid ) Figure 6.9: Superconducting proximity eect in the strong disorder regime: (a) Log-normal distribution of mid in N-SC rings in the strong-disordered regime. 9000 realizations, W/t = 0.2, L = 90, L SC = 200. (b) The dependence of the mean and standard deviation of log( mid ) on the disorder strength W=t in the strong-disordered regime. linearly with W and with the system size L, as shown in Fig. 6.9b. For strongly disordered systems, we follow (6.12) to express the logarithm of the E = 0 wave function at the midpoint of the chain of length L = 2M as a sum, log E=0 = M X i=1 x i +const; (6.12) where the random numbers x i are related to the hopping amplitudes by x i = log(t 2i =t 2i1 ) [85]. The above equation shows that, according to the central limit theorem, ln is a Gaussian distributed random variable. Its variance increases with the number of x i , that is, is proportional to the chain length L. In the strong disorder regime, we assume that mid can be approximately written as mid u E=0 L=2 v E=0 L=2 (6.13) i.e. that the contributions from nite energy states in the sum (4.28) can be neglected, so the OP at the midpoint is determined principally by the wave functions u and v at E = 0. The OP at the midpoint is thus determined by the wave functions u and v at the midpoint of the chain which both have the form given by Eq.6.12, diering only in the values of the prefactor. The central limit theorem applied to the logarithm of the product uv tells us that log( mid ) log( ) must have a Gaussian distribution of width proportional to W , 77 Signatures of topology in a quasicrystal increasing with system size as p L. This is in agreement with the results shown in Fig. 6.9b. Note that for strong disorder the distribution width of the OP grows with the system size, in contrast with the distribution in the weak disorder limit where it decreases with L. The proximity induced OP is clearly a strongly uctuating quantity, analogous to the distribution of values of the resistivity in 1D systems [85, 86]. To conclude this section, we have shown that extended states lead to a power law decay of the OP and a Gaussian distribution in the weak disorder regime, whereas localized states lead to a stretched exponential decay and a log-normal distribution in the strong disorder regime. We have checked that these features in the induced order parameter in o-diagonal Anderson model carry over to the the diagonal Anderson model except that for high values of W , the OP decay is t better by an exponential rather than a stretched exponential. Although both distributions are log-normal, there is a signicant dierence between theL dependence of the widths in the Fibonacci chain (FC) as compared to the strongly disordered chain. For the FC, the width of the distribution grows only logarithmically, much much more slowly than in the random case. This is indeed seen numerically as shown in the inset of Fig. 6.5a. In this regard as in many others, the properties of the quasicrystal are intermediate between those of the weakly disordered and strongly disordered chains. 6.6 Topological labels in proximity induced SC In [4], we studied the eect of the topological edge states of the Fibonacci chain on the induced superconductivity in the N-SC hybrid ring. The key idea was to compare the strength of the induced order parameter in approximants with dierent values of . We used a slightly dierent set of parameters for this project. This time, in (6.1), we xed t B = 1 and varied t A < t B . We set ( 0 i i ) = 0 in the superconductor as well as in the non-superconducting region. With this choice, the overall diagonal term i = 0 i + HF i 0 is non-zero because of the Hartree-Fock shift, and the superconductor is not at half-lling. 78 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN Figure 6.10: Self consistently computed order parameter in real space for various t A =t B . The 90-site quasicrystal is on the left, and the 90-site superconductor is on the right. The blue(green) curve has been shifted up by 0.025(0.05) for clarity. This has no qualitative eect on the observable that we are interested in|the strength of the induced superconductivity as measured by mid . As an aside, we note that other observables do see an eect|there are uctuations in the order parameter prole near the interface on the S side (Compare Fig. 6.10 with Fig. 6.3a). We consider the penetration strength for the Fibonacci approximants for dierent values. Directly tting the OP curves turns out to be dicult and gives unreliable results due to the strong oscillations. We therefore proceed in two steps: rst tting i to a power law given by the expression (6.14), and then evaluating the t function at the center of the chain to obtain mid . fit i =c +b(i + (L + 1i) ) (6.14) L represents the eective FC chain length, as the location of the interfaces are slightly displaced with respect to the interfaces in the uncoupled problem. The best ts, with the smallest mean square deviation, were found for L =N n + 2. We have assumed a symmetric decay with respect to left (i = 1) and right (i =L) edges (theis are appropriately relabelled 79 Signatures of topology in a quasicrystal Figure 6.11: a) & b) The penetration strength mid of the order parameter into the Fibonacci chain as a function of the used to generate the chain. In (a), the FC is 145-site long and the dierent colors represent dierent modulation strengths. In (b), the modulation is constant (t A =t B = 0:9), but the length of the quasicrystalN is varied. c) Power spectrum of the penetration strength (blue curve in Fig 6.11a)jF[ mid ()]j 2 with the prominent peaks highlighted. to account for the shifted interfaces). Across the systems we studied, the t parameter , the decay power, lay in the range 0:3 1:0 depending on the system parameters and the details of the t. The value of for a given set of system parameters varies signicantly depending on how the t is taken. For instance the parameter set (N n = 90, N s = 35, t A =0:9) leads to the t parameters ( = 0:7330;b = 0:09285;c =0:003255) if L is taken to be N n + 2, and the t parameters ( = 0:4258;b = 0:06979;c =0:0181) if L is taken to be N n . For this reason, we have chosen to use the t-independent, appropriately smoothed order parameter amplitude at a given point (the center of the chain) as our measure for the strength of the proximity eect. In Fig 6.11a, we show the results thus obtained for mid as a function of for the set of FC of length N n = 145. The dierent colors represent dierent modulation strengths. These curves show that for xed system size, mid uctuates with increasing amplitude as the quasiperiodic modulation increases. In Fig 6.11b, we show this quantity as a function of for FCs of length 35 and 145 with modulation t A =t B = 0:9. A few remarks are in order. Firstly, for particular values of, mid is higher in the quasicrystal than it is in the periodic chain, i.e. for these values of the superconducting order parameter travels further into 80 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN 0 1 2 3 4 5 6 / 2.00 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 Eigenvalues q = 4 q = -9 q = -17 Figure 6.12: The lower half of the spectrum of the open Fibonacci chain of length 145 as a function of with t A =t B = 0:5. The gaps closest to the Fermi level are highlighted. the quasicrystal than it does into the periodic chain. Secondly, there appear to be some oscillations which conserve the same period as one goes from smaller to longer chains. To understand the mechanisms which lie behind these observations, we look at the Fourier components of the curves in Fig 6.11b. Fig 6.11c shows the power spectrum of the blue curve in Fig 6.11a, (t A =t B = 0:9;N n = 145). One sees that there are strong peaks corresponding to periods of 4, 9 and 17 along with smaller peaks not labeled in the plot. The plot in Fourier space suggests a relation between the delta exponent and the edge states in the main gaps which are labeled by winding numbersq as the numbers 4, 9 and 17 are special for the Fibonacci chain. Consider Fig 6.12, which shows how the energy eigenvalues of the 145-site open Fibonacci chain vary with 6 . The x-axis represents the values of , which vary for dierent members of the family, while the y-axis corresponds to the set of energy eigenvalues of the chains. One sees clearly how the energy levels of certain states|corresponding to edge states|wind within the gaps and it is easy to check that the number of gap crossings for a given gap is nothing but the index of the gap. The prominent gaps closest to half-lling are expected to contribute the most to the OP { these, as can be seen in the gure have the labels q =17; 4 and9, precisely the periodicities we saw to be present in the power 6 We have checked, by turning on the interface coupling slowly and directly looking at the wave functions and energy eigenstates of the coupled problem, that the Fibonacci gaps along with theirq-labels are preserved in the coupled problem. 81 Signatures of topology in a quasicrystal Table 6.1: A summary of the characteristics of the distribution function for mid when the electronic states in the normal side are i) (quasi)-extended, as in periodic and weakly disordered systems, ii) critical, as in quasicrystals, iii) localized, as in strongly disordered systems. and t refer to the average (arithmetic mean) and typical (geometric mean) values of the distribution. Electronic states h mid i P ( mid ) Variance Extended/Quasi-extended (Periodic/Weak Anderson model) 1=L e ( ) 2 2 2 ( mid )/ 1=L Critical (Fibonacci chain) 1=L 1 e (ln ln t ) 2 2 2 (ln mid )/ lnL Localized (Strong Anderson model) e p L= 1 e (ln ln t ) 2 2 2 (ln mid )/L spectrum of the OP. 6.7 Conclusion We have examined the proximity eect in inhomogeneous normal wires coupled to a super- conductor at T = 0, focusing on three important situations : I) when the N component is a weakly disordered crystal, in which wave functions are extended and perturbative calcu- lations can be performed, II) when the disorder is strong enough such that the localization lengths are smaller than the sample size, and III) when states are critical, as in the o- diagonal Fibonacci tight-binding model. Our main results are summarized in table 6.1. We nd, rstly, that the typical value of the OP has a power law decay as one moves away from the interface when states are extended or critical. On the other hand, for strongly disordered systems, where states are localized, the typical OP decays faster, in the present case as a stretched exponential. For arbitrary positions of the Fermi energy, away from the special point E = 0 we expect that a regular exponential decay should be observed. The OP uctuations in such systems are large. We have computed the distribution function of OP values and shown that they have gaussian or log-normal shapes. We have presented arguments to explain the forms of the distributions and their scaling as a function 82 6. THE PROXIMITY EFFECT IN THE FIBONACCI CHAIN of sample size and disorder strength for each of the cases considered. In the one-dimensional models we considered there are no mobility edges. However, one can speculate that our results are more generally applicable in other models where there are mobility edges. In that case, the typical values and the uctuations of the OP would depend on the spatial characteristics of the states close to the Fermi level. The penetration strength for an individual realization, as measured by the order param- eter amplitude at the center of the chain, mid , depends on the phase associated with the chain. We nd that for some values of the order parameter will travel further into the quasicrystal than into the periodic chain. Moreover the dependence of the penetration strength on has a fractal structure with a hierarchy of smaller peaks surrounding the larger ones. The positions of the peaks correspond directly with the positions of level crossings in the gaps in the spectrum of the corresponding open Fibonacci chain, and the periodicities of mid () are exactly theq labels of the gaps closest to the Fermi level. Since theq label is an inherited property of the edge state living in the gap, this indicates a direct correspondence between the penetration strength and the edge state and suggests the possibility of using the proximity eect as a probe for the topological edge states of the Fibonacci chain in partic- ular, and higher dimensional quasicrystals as well as other systems that exhibit topological edge states. In conclusion, we suggest that it will be interesting to study the proximity eect and associated topological signatures experimentally. One possibility for experiment consists of inducing superconductivity in quasiperiodic akes. Monolayers of quasiperiodically organised lead atoms have in fact been successfully deposited on specially chosen surfaces, and a number of other promising systems are possible as described in [87]. If such mesoscopic akes could be put into contact with larger superconducting lead islands, the resulting superconducting gaps could then be investigated via STM. Although we expect that qualitative features of our results will persist in two dimensions, we plan to extend our approach so as to obtain results specic to 2D systems in future work. 83 Chapter 7 Outlook In this chapter, I discuss two research direction that immediately follow from the results of the previous two chapters that. The rst is the problem of intrinsic superconductivity in the Fibonacci chain. I have included some preliminary results on the topic in Sec. 7.1. In Sec. 7.2, I discuss the generalization of the method described in Chapter. 5 to measure the topological invariants of higher dimensional quasicrystals. 7.1 Intrinsic superconductivity in the Fibonacci chain In Chapter 6, we studied the properties of a superconducting quasicrystalline state by in- ducing superconductivity into the Fibonacci chain by exploiting the proximity eect. We can also use the Bogoliubov-de Gennes formalism to treat the problem of an intrinsically superconducting quasicrystal. In this chapter I will describe some preliminary results from a mean-eld calculation on the Fibonacci chain with an attractive Hubbard interaction. The primary objective is to answer the following questions. To what extent can the onset of superconductivity in quasicrystals be described by BCS theory? What is the nature of the bound pairs? Can they be explained by Anderson's pairing of exact eigenstates idea? What kinds of persistent current can be induced in this model? In the case of the Fibonacci chain, what is the eect of the topological gap labels? In Chapter 6, we established that proximity 84 7. OUTLOOK induced superconductivity are aected by the topological edge states as these states are localized at the interface between the S and N parts. We also saw, in Chapter 5, that the topological labels have a physically observable bulk eect on the non-interacting Fibonacci chain. Is there a way in which the topological labels appear directly aect the physics of the interacting Fibonacci chain? Here, I will show some preliminary calculations I performed with Chris Matsumura, an undergraduate student at the University of Southern California. We use the Bogoliubov-de Gennes approach to study the attractive Hubbard model as described in Sec. 4.2. We take the Fibonacci hopping model and introduce an on-site attractive Hubbard interaction V . ^ H = X i ( X c y i c i + X t i c y i c i+1 +h:c: V 2 X c y i c y i c i c i ) : (7.1) t i =t A or t B according the Fibonacci approximantS n . We choose periodic boundary condi- tions, so the indexi is understood to wrap around moduloF n for thenth order approximant S n . The chemical potential controls the number of electrons in the system and V is the strength of the on-site attraction. Using the Hartree-Fock approximation, we get the eective Hamiltonian ^ H mf = X i ( X i z }| { ( HF i )c y i c i + X t i c y i c i+1 +h:c: +V i c y i" c y i# +h:c: ) : (7.2) that depends on the mean elds HF i and i . The mean elds are self-consistently computed according to (4.28). 85 Signatures of topology in a quasicrystal 2 0 2 E/t 0.00 0.25 0.50 0.75 1.00 DOS (a) v=0 2 0 2 E/t (b) v=1.3t Figure 7.1: The density of states with (b) and without (a) an attractive interaction V . A gap opens up around the Fermi level. The single-particle states weight gets shifted to the coherence peaks around the gap. We choose t A (t B ) such that the bandwidth remains xed. This is done by xing the average hoppingt = F n1 t A +F n2 t B Fn = 1. The strength of the modulation is parameterized by w = t A t B 2 . Our primary observables are the real space order parameter i and the global (DOS) and local density of states (LDOS). In the Bogoliubov-de Gennes framework, the local density of states at a given site, i, is given by LDOS i () = X n;i ju in j 2 ( n ) +jv in j 2 ( + n ) (7.3) The global density of states is the average of LDOS i over the entire sample. Introducing the BCS attraction results in a non-zero order parameter i and opens a gap in the Fibonacci chain density of states. The We can obtain the critical temperature by introducing nite temperature and checking when the self-consistent solution for the average order parameter, , is no longer non- zero. We nd that (T ) follows the BCS expression around the critical temperature, (T )/ p TT c . This relationship is shown for dierent values of the modulation strength w in Fig. 7.1. Remarkably, we nd that the order parameter becomes non-zero throughout the sample at the same critical temperature regardless of the strength of the modulation. This is consistent with precious calculations on 2D quasiperiodic tilings [56] and marks a serious departure from the disordered case. In a suciently disordered system, some parts 86 7. OUTLOOK 0 100 200 Site Number 0.08 0.10 0.12 (a) 0 100 200 Conumber (b) 2 0 2 E/t 0 2 4 6 LDOS ×10 3 (c) 0.075 0.100 0.125 1 2 CPH ×10 3 Figure 7.2: The order parameter i in (a) real and (b) perpendicular space. The order parameter in real space has strong self-similar uctuations. In perpendicular space, is relatively smooth and exhibits plateau-like behavior. (c) The local density of states at four representative sites with various values of the order parameter. The gap width remains constant throughout the sample, but the height of the coherence peak is strongly correlated with the magnitude of the order parameter (see inset). Figure 7.3: Temperature dependence of the average order parameter for dierent values of the modulation strength w. of the system becomes superconducting and others don't, resulting in the formation of su- perconducting islands in a non-superconducting sea [51, 52]. We also nd that the critical temperature increases as the modulation strength is increased. Fig. 7.3 shows that T c w 2 . This is a consequence of the fact that the size of the gaps of the Fibonacci chain grow mono- tonically with w. But, since the bandwidth remains constant, the states in the bands are forced to be closer to each other resulting in a higher density of states wherever there is no gap. In BCS theory, the critical temperature depends on the strength of the attraction and the density of states around the Fermi level asT c /e 1 N( F )V . Since,N( F ) is monotonically 87 Signatures of topology in a quasicrystal Figure 7.4: (a) a color map of the order parameter in perpendicular space as the chemical potential is varied along the y-axis (V = 0.9). (b) A color map of the average order parameter in V - space. increasing with w, the critical temperature must also increase with w. We know from the results of Sec. 5.5, that when the Fibonacci chain is tuned to a large enough gap (when compared to theV ), even in the presence of an attractive point interaction, the system is insulating with the charge density modulating such that when mapped to perpendicular space, it oscillates a number of times related to the label of the gap. When the Fermi level is tuned to a band, we get a superconductor instead. Because of the complex structure of gaps in the spectrum of the Fibonacci chain, we nd a phase diagram that alternates between superconducting and insulating many times as the Fermi level is varied over the total bandwidth. This is shown in Fig. 7.4a, which tracks the perpendicular space prole of the superconducting order parameter as a function of the chemical potential. The number of insulating regions depends on the number of gaps with a gap width that is large compared to the potential gap width. This results in a competition between the modulation strength w and the attraction strength V . This is clearly visible in the -V phase diagram shown in Fig. 7.4b. The color scale represents the strength of the average order parameter. The insulating regions appear as dark blue lobes on the left side of the phase diagram. The strength of the attraction increases as we move to the right and the superconducting state reappears. The fractal density of states of the Fibonacci chain facilitates a unique phase diagram in the presence of a Hubbard interaction. A superconductor-insulator transition can occur a 88 7. OUTLOOK Figure 7.5: A piece of the generalized Rauzy tiling (Taken from [88]) number of times by varying the chemical potential as well as the strength of the attraction V . 7.2 Signatures of topology in other systems In Chapter 5, we saw how the conumber transform could be used to measure the topological invariants of the Fibonacci chain and related models directly from bulk measurements on the system. Can this method be applied to other systems? What are the characteristics of a system where this works? In order to dene the conumber, we need perpendicular space to be one-dimensional; the quasicrystal is constructed by the projection p k :R d+1 !R d . The perpendicular space projection of the accepted points form a dense points set on a one-dimensional interval. By using a rational convergent of the irrational slope for the projection, we obtain an approxi- mant. In this case, we can number the perpendicular projections from left to right and this denes the conumber. There is a two dimensional system that has this property and also has topological invari- ants characterized by a Diophantine equation. This is the generalized Rauzy tilings in the presence of a magnetic eld [88, 89]. This tiling has a lot of similarities with the Fibonacci chain. The cut and project scheme projects sites from the three-dimensional cubic lattice 89 Signatures of topology in a quasicrystal to a plane. The irrational root (analogous to the Golden ratio) related to this tiling is the Pisot root of x 3 =x 2 +x + 1; 1:839 (7.4) The corresponding number sequence is the Tribonacci sequence, which is dened by T n+1 =T n +T n1 +T n2 (7.5) T 1 = 0; T 0 = 1; T 1 = 1: (7.6) The ordern approximant is made up ofT n+1 sites arranged in a tiling made up of isometric rhombi. The connectivity matrix of this tiling in perpendicular space is a Toeplitz matrix, just like the Fibonacci chain, except this time with 6 non-zero diagonals. If a nite ux f per plaquette is applied, the spectrum of the Rauzy tiling is similar to the Hofstadter butter y; the gaps can be labelled by the TKNN invariant. For certain values of the ux, the tiling presents an alternate set of gaps that are due to the quasiperiodicity analogous to the gaps of the Fibonacci chain. This gap labelling theorem can be expressed in terms of the integrated density of states in a gap IDOS = (T 1 +U 2 ) mod 1 (7.7) This time, there are two integers compared to the one integer label for the Fibonacci chain. The various analogies to the Fibonacci chain suggest that the same kind of analysis may be possible for this systems as well as in higher-dimensional generalized Rauzy tilings as they all have co-dimension m = 1, and therefore a conumber can be dened. The nite ux f is a new ingredient that needs to be taken into account as the Rauzy tiling is gapless when f = 0. As a nal speculative note, in systems with m > 1, we cannot dene a conumber as 90 7. 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Mosseri, \Generalized quasiperiodic Rauzy tilings", en, Journal of Physics A: Mathematical and General 34, 3927{3938 (2001). 97 Appendix A Bogoliubov-de Gennes implementation The python program that I wrote to perform the Bogoliubov-de Gennes calculations on the superconductor-quasicrystal hybrid ring is hosted on GitHub 1 . Here, I have included a minimal version of the program. Here are the critical packages and denitions for the minimal program. For all linear algebra routines, I used the standard implementations from the NumPy and SciPy libraries. For setting up and manipulating tight-binding Hamiltonians, I made heavy used of the kwant[71] software package. 1 import numpy as np 2 import kwant 3 import tinyarray 4 import scipy.linalg as lin 5 6 # Pauli matrices defined as tinyarrays for convenience and efficiency 7 tau_x = tinyarray.array([[0, 1], [1, 0]]) 8 tau_y = tinyarray.array([[0, -1j], [1j, 0]]) 1 https://github.com/gautamra/Proximity-Effect-Fibonacci 98 A. BOGOLIUBOV-DE GENNES IMPLEMENTATION 9 tau_z = tinyarray.array([[1, 0], [0, -1]]) 10 tau_0 = tinyarray.array([[1, 0], [0, 1]]) 11 12 # Lorentzians around every eiganvalue are used to define the density of states 13 def Lorentzian(eex, ee, gam): 14 return (gam/np.pi)*(1/((eex-ee)**2 + gam**2)) 15 16 # The Fermi function. beta large corresponds to zero temperature 17 def Fermi(eps, beta = 20000): 18 return 1/(1+np.exp(beta*eps)) A given 1D tight-binding Bogoilubov-de Gennes Hamiltonian is implemented by the TBmodel class. The Bogoliubov-de Gennes equations in (4.26) is treated like the action of a pseudo-Hamiltonian on a monoatomic two-orbital square lattice. The coecients u in ;v in are given by the eigenvectors of this pseudo-Hamiltonian. 1 class TBmodel: 2 # A given system is defined by the number of sites LL, and three lists of length LL: ts, us, vs representing respectively the hoppings t_i, the on site terms epsilon_i - mu_i, and the pairing strength v_i. 3 def __init__(self, LL, ts, us, vs): 4 self.LL = LL 5 self.a = 1 6 self.ts, self.us, self.vs = ts, us, vs 7 self.Delta = np.array([1]*self.LL) 8 self.Pot = np.zeros(self.LL) 9 10 # The onsite and hopping terms depend on two parameters: Pot[ x], the Hartree-Fock term, and Delta[x], the local order parameter 11 def onsite(self, site, Delta, Pot): 99 Signatures of topology in a quasicrystal 12 (x,y) = site.tag 13 return (self.us[x]+Pot[x])*tau_z - self.vs[x]*Delta[x]* tau_x 14 def hopping(self,site1,site2): 15 (x2,y2) = site2.tag 16 (x1,y1) = site1.tag 17 return self.ts[x1]*tau_z 18 19 # Using kwant's functionality to define the Hamiltonian 20 def make_syst(self): 21 self.syst = kwant.Builder() 22 self.lat = kwant.lattice.square(self.a, norbs = 2) 23 self.syst[(self.lat(x,0) for x in range(self.LL))] = self .onsite 24 self.syst[((self.lat(x+1,0),self.lat(x,0)) for x in range (self.LL-1))] = self.hopping 25 self.syst[((self.lat(0,0), self.lat(self.LL-1,0)))] = self.hopping 26 self.fsyst = self.syst.finalized() 27 return 28 29 # Keeps only the positive eigenvalues. The Eigenvectors of the pseudo-Hamiltonian are the Bogoliubov coefficients u and v 30 def solve(self,H): 31 (evals, evecs) = lin.eigh(H) 32 uvecs = evecs[::2] 33 vvecs = evecs[1::2] 34 return (evals[self.LL:],uvecs[:,self.LL:],vvecs[:,self.LL :]) 35 36 # Self-consistent loop for the order parameter and Hartree Fock term 37 def iterate(self): 100 A. BOGOLIUBOV-DE GENNES IMPLEMENTATION 38 39 # Calculates the coefficients u and v for a particular values of Pot and Delta. Calculates new mean fields Pot and Delta using the self-consistency equations. 40 def self_cons(H): 41 (evals, uvecs, vvecs) = self.solve(H) 42 self.evals, self.uvecs, self.vvecs = (evals, uvecs, vvecs) 43 Delta = np.zeros(self.LL, dtype = "complex128") 44 for ee, uvec, vvec in zip(evals, uvecs.T, vvecs.T): 45 Delta += (1-2*Fermi(ee, beta = self.beta))*uvec* vvec.conjugate() 46 occupancy = np.zeros(self.LL) 47 for ee, uvec, vvec in zip(evals, uvecs.T, vvecs.T): 48 occupancy += Fermi(ee, beta = self.beta)*2*np .abs(vvec)**2 + (1-Fermi(ee, beta = self.beta))*2*np.abs(uvec) **2 49 Pot = 1/2*self.vs*occupancy 50 Pot = Pot + 0.0001*np.ones(len(Pot)) 51 return (Delta, Pot) 52 53 # If error between new and old quantities is above a certain threshold, iterates. 54 err_Delta = np.ones(1) 55 cc = 0 56 while any([err/Del>0.001 and err>0.01*(np.max(self.Delta) +0.01) for err,Del in zip(err_Delta, self.Delta)]): 57 H = self.fsyst.hamiltonian_submatrix(params = dict( Delta = self.Delta, Pot = self.Pot)) 58 newDelta, newPot = self_cons(H) 59 err_Delta = np.abs(newDelta - self.Delta) 60 cc += 1 61 self.Delta, self.Pot = newDelta, newPot 101 Signatures of topology in a quasicrystal 62 self.H = H 63 return self.Delta, self.Pot 64 65 # Calculate the quasiparticle density of state for a given system 66 def get_DOS(self, gam = None, Num_es = 1000): 67 emax = np.max(np.abs(self.evals)) 68 if gam == None: 69 gam = 2*emax/self.LL 70 eex = np.linspace(-1.2*emax,1.2*emax, Num_es) 71 DOSu = np.zeros(eex.shape) 72 DOSv = np.zeros(eex.shape) 73 for ee, uvec, vvec in zip(self.evals, self.uvecs.T, self. vvecs.T): 74 if ee>0: 75 DOSu += np.linalg.norm(uvec)**2*Lorentzian(eex,ee ,gam) 76 DOSv += np.linalg.norm(vvec)**2*Lorentzian(eex,- ee,gam) 77 self.DOS = (DOSu + DOSv)/self.LL 78 return self.DOS , eex 102

Abstract (if available)

Abstract Orientational order and periodicity were long believed to be equivalent concepts. In the 1980s, the discovery of quasicrystals upended this thinking. Quasicrystals are aperiodic arrangements of atoms that have perfect long-range orientational order. The inclusion of this new class of solids has expanded the repertoire of ordered materials that we can design: i) non-crystallographic symmetries are now possible, and ii) topological phases that would otherwise be disallowed in a given space dimension can now be realized. ❧ A simple prototype of a one-dimensional quasicrystal is the Fibonacci chain. The Fibonacci chain is topologically equivalent to the two-dimensional Hofstadter model. The Chern numbers of the Hofstadter model have a physical interpretation in the form of the Hall current carried by a given band. This leads to the question: what is the physical interpretation of the Chern numbers of the Fibonacci chain? Previous studies have shown that the Chern number can be physically measured by tracking the localization, or the number of gap crossings, of an edge state as the Fibonacci chain Hamiltonian is transformed along a closed orbit by a sequence of phason flips. ❧ In my work at USC, I have identified a physical interpretation that does not invoke phason flips. We describe a way to measure the Chern numbers from a single sample of the Fibonacci chain. When mapped to appropriate coordinates, position-dependent quantities such as the real space charge density profile, oscillate with a frequency related to the Chern number of the gap that the system is tuned to. This is a direct demonstration of the bulk-boundary correspondence in the Fibonacci chain. ❧ By explicit calculations in the presence of s-wave superconducting pairing, my co-authors and I show that the same Chern numbers survive upon the introduction of interactions to the Hamiltonian. If pairing is introduced by the proximity effect, then we can also measure the Chern numbers by tracking how the induced order parameter changes over a closed sequence of phason flips. ❧ We also investigate the general characteristics of the proximity effect in a superconductor-quasicrystal hybrid ring. We find that the proximity effect is strong, with the superconducting order parameter decaying only as a power law at zero temperature. This is comparable to order parameter penetration in clean metals. The ensemble statistics of the induced order parameter follow a log-normal distribution function. This is comparable to the strongly disordered dirty case. This dichotomy of behavior of the induced order parameter is a consequence of the criticality of the wavefunctions in the Fibonacci chain; they are neither extended nor localized.

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Asset Metadata

Creator Rai, Gautam (author)

Core Title Signatures of topology in a quasicrystal: a case study of the non-interacting and superconducting Fibonacci chain

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 11/19/2021

Defense Date 09/24/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag AAH model,Aubry-André-Harper model,Chern number,conumber,Fibonacci chain,Hofstadter model,OAI-PMH Harvest,proximity effect,quasicrystal,SPT,superconductivity,symmetry-protected topological phases,topology,winding number

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan (committee chair), Di Felice, Rosa (committee member), Levenson-Falk, Eli (committee member), Takahashi, Susumu (committee member), Zanardi, Paolo (committee member)

Creator Email gautamr.qv@gmail.com,gautamra@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC17138477

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