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Geometric Phases in Graphene and Topological Insulators CORE Metadata, citation and similar papers at core.ac.uk Provided by SISSA Digital Library Scuola Internazionale Superiore di Studi Avanzati - Trieste Area of Mathematics Ph.D. in Mathematical Physics Thesis Geometric phases in graphene and topological insulators Candidate Supervisor Domenico Monaco prof. Gianluca Panati (“La Sapienza” Università di Roma) Internal co-Supervisor prof. Ludwik D ˛abrowski Thesis submitted in partial fulfillment of the requirements for the degree of Philosophiæ Doctor academic year 2014-15 SISSA - Via Bonomea 265 - 34136 TRIESTE - ITALY Thesis defended in front of a Board of Examiners composed by prof. Ludwik D ˛abrowski, dr. Alessandro Michelangeli, and prof. Cesare Reina as in- ternal members, and prof. Gianluca Panati (“La Sapienza” Università di Roma), prof. Stefan Teufel (Eber- hard Karls Universität Tübingen) as external members on September 15th, 2015 Abstract This thesis collects three of the publications that the candidate produced during his Ph.D. studies. They all focus on geometric phases in solid state physics. We first study topological phases of 2-dimensional periodic quantum systems, in absence of a spectral gap, like e. g. (multilayer) graphene. A topological invariant nv Z, baptized eigenspace vorticity, is attached to any intersection of the energy bands,2 and characterizes the local topology of the eigenprojectors around that intersection. With the help of explicit models, each associated to a value of nv Z, we are able to extract the decay at infinity of the single-band Wannier function 2w in mono- and 2 bilayer graphene, obtaining w(x) const x ¡ as x . Next, we investigate gappedj periodicj · quantum¢ j j j systems,j ! 1 in presence of time- reversal symmetry. When the time-reversal operator £ is of bosonic type, i. e. it sat- isfies £2 1l, we provide an explicit algorithm to construct a frame of smooth, pe- riodic andÆ time-reversal symmetric (quasi-)Bloch functions, or equivalently a frame of almost-exponentially localized, real-valued (composite) Wannier functions, in di- mension d 3. In the case instead of a fermionic time-reversal operator, satisfying £2 1l,we· show that the existence of such a Bloch frame is in general topologically Æ¡ obstructed in dimension d 2 and d 3. This obstruction is encoded in Z2-valued topological invariants, whichÆ agree withÆ the ones proposed in the solid state litera- ture by Fu, Kane and Mele. iii Acknowledgements First and foremost, I want to thank my supervisor, Gianluca Panati, who lured me into Quantum Mechanics and Solid State Physics when I was already hooked on Topology and Differential Geometry, and then showed me that there isn’t really that much of a difference if you know where to look. During the years of my Ph.D. studies, I have had the occasion to discuss with several experts in Mathematics and Physics, who guided my train of thoughts into new and fascinating directions, and to whom I am definitely obliged. Big thanks to Jean Bellissard, Raffaello Bianco, Horia Cornean, Ludwik D ˛abrowski, Giuseppe De Nittis, Gianfausto dell’Antonio, Domenico Fiorenza, Gian Michele Graf, Peter Kuch- ment, Alessandro Michelangeli, Adriano Pisante, Marcello Porta, Emil Prodan, Raf- faele Resta, Shinsei Ryu, Hermann Schulz-Baldes, Clément Tauber, Stefan Teufel, and Andrea Trombettoni. I am also extremely grateful to Riccardo Adami, Claudio Cacciapuoti, Raffaele Carlone, Michele Correggi, Pavel Exner, Luca Fanelli, Rodolfo Figari, Alessandro Giuliani, Ennio Gozzi, Gianni Landi, Mathieu Lewin, Gherardo Piacitelli, Alexander Sobolev, Jan Philip Solovej, Alessandro Teta, and Martin Zirnbauer, for inviting me to the events they organized and granting me the opportunity to meet many amusing and impressive people. v Contents Introduction .............................................................. ix 1 Geometric phases of quantum matter . ix 1.1 Quantum Hall effect . ix 1.2 Quantum spin Hall effect and topological insulators. xii 2 Analysis, geometry and physics of periodic Schrödinger operators . xv 2.1 Analysis: Bloch-Floquet-Zak transform . xv 2.2 Geometry: Bloch bundle . xxi 2.3 Physics: Wannier functions and their localization . xxvi 3 Structure of the thesis . xxvii References . xxix Part I Graphene Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene ..................................................... 3 Domenico Monaco and Gianluca Panati 1 Introduction . .3 2 Basic concepts . .5 2.1 Bloch Hamiltonians. .5 2.2 From insulators to semimetals . .7 2.3 Tight-binding Hamiltonians in graphene . .7 2.4 Singular families of projectors . .9 3 Topology of a singular 2-dimensional family of projectors . .9 3.1 A geometric Z-invariant: eigenspace vorticity . 10 3.2 The canonical models for an intersection of eigenvalues . 14 3.3 Comparison with the pseudospin winding number. 19 4 Universality of the canonical models . 26 5 Decrease of Wannier functions in graphene. 29 5.1 Reduction to a local problem around the intersection points . 31 5.2 Asymptotic decrease of the n-canonical Wannier function . 32 5.3 Asymptotic decrease of the true Wannier function. 37 A Distributional Berry curvature for eigenvalue intersections . 47 References . 49 Part II Topological Insulators vii viii Contents Construction of real-valued localized composite Wannier functions for insulators ................................................................. 55 Domenico Fiorenza, Domenico Monaco, and Gianluca Panati 1 Introduction . 55 2 From Schrödinger operators to covariant families of projectors . 58 3 Assumptions and main results . 62 4 Proof: Construction of a smooth symmetric Bloch frame. 65 4.1 The relevant group action . 66 4.2 Solving the vertex conditions . 67 4.3 Construction in the 1-dimensional case . 69 4.4 Construction in the 2-dimensional case . 69 4.5 Interlude: abstracting from the 1- and 2-dimensional case . 75 4.6 Construction in the 3-dimensional case . 77 4.7 A glimpse to the higher-dimensional cases . 82 5 A symmetry-preserving smoothing procedure . 83 References . 87 Z2 invariants of topological insulators as geometric obstructions . 89 Domenico Fiorenza, Domenico Monaco, and Gianluca Panati 1 Introduction . 89 2 Setting and main results . 92 2.1 Statement of the problem and main results . 92 2.2 Properties of the reshuffling matrix " ........................ 94 3 Construction of a symmetric Bloch frame in 2d ..................... 96 3.1 Effective unit cell, vertices and edges . 97 3.2 Solving the vertex conditions . 99 3.3 Extending to the edges . 101 3.4 Extending to the face: a Z2 obstruction . 102 3.5 Well-posedness of the definition of ± ........................ 106 3.6 Topological invariance of ± ................................. 107 4 Comparison with the Fu-Kane index . 109 5 A simpler formula for the Z2 invariant . 114 6 Construction of a symmetric Bloch frame in 3d ..................... 116 6.1 Vertex conditions and edge extension . 116 6.2 Extension to the faces: four Z2 obstructions. 117 6.3 Proof of Theorem 2.3 . 119 6.4 Comparison with the Fu-Kane-Mele indices . 122 A Smoothing procedure . 124 References . 126 Conclusions Open problems and perspectives ........................................... 131 1 Disordered topological insulators . 131 2 Topological invariants for other symmetry classes . 132 3 Magnetic Wannier functions . 132 References . 133 Introduction 1 Geometric phases of quantum matter In the last 30 years, the origin of many interesting phenomena which were discov- ered in quantum mechanical systems was established to lie in geometric phases [45]. The archetype of such phases is named after sir Michael V. Berry [8], and was early related by Barry Simon [46] to the holonomy of a certain U(1)-bundle over the Bril- louin torus (see Section 2.1). Berry’s phase is a dynamical phase factor that is ac- quired by a quantum state on top of the standard energy phase, when the evolution is driven through a loop in some parameter space. In applications to condensed mat- ter systems, this parameter space is usually the Brillouin zone. As an example, one of the most prominent incarnations of Berry’s phase effects in solid state physics can be found in the modern theory of polarization [47], in which changes of the elec- tronic terms in the polarization vector, ¢Pel, through an adiabatic cycle are indeed expressed as differences of geometric phases. This result, first obtained by R.D. King- Smith and David Vanderbilt [26], was later elaborated by Raffaele Resta [42]. In more recent years, the Altland-Zirnbauer classification of random Hamiltonians in pres- ence of discrete simmetries [1] sparked the interest in geometric labels attached to quantum phases of matter, and paved the way for the advent of topological insula- tors (see Section 1.2). After being confined to the realm of high-energy physics and gauge theories, topology and geometry made their entrance, through Berry’s phase, in the low- energy world of condensed matter systems. This Section is devoted to giving an in- formal overview on two instances of such geometric effects in solid state physics, namely the quantum Hall effect and the quantum spin Hall effect, which are of rele- vance also for the candidate’s works presented in this thesis in Parts I and II. 1.1 Quantum Hall effect One of the first and most striking occurrences of a topological index attached to a quantum phase of matter is that of the quantum Hall effect [50, 17]. The experimental setup for the Hall effect consists in putting a very thin slab of a crystal (making it effectively 2-dimensional) into a constant magnetic field, whose direction zˆ is orthogonal to the plane Ox y in which the sample lies. If an electric current is induced, say, in direction xˆ, the charge carriers will experience a Lorentz ix x Introduction force, which will make them accumulate along the edges in the transverse direction. If these edges are short-circuited, this will result in a transverse current flow j, called Hall current (see Figure 1). Hall effect ε F B L v j F x 0 0 z y x Fig.
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