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1.3.2 Weak Topological Insulators (WTI)

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1.3.2 Weak Topological Insulators (WTI) Classification and characterization of topological insulators and superconductors by Roger Mong A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Joel E. Moore, Chair Professor Dung-Hai Lee Professor Michael Hutchings Spring 2012 The dissertation of Roger Mong, titled Classification and characterization of topological insulators and superconductors, is approved: Chair Date Date Date University of California, Berkeley Spring 2012 Classification and characterization of topological insulators and superconductors Copyright 2012 by Roger Mong 1 Abstract Classification and characterization of topological insulators and superconductors by Roger Mong Doctor of Philosophy in Physics University of California, Berkeley Professor Joel E. Moore, Chair Topological insulators (TIs) are a new class of materials which, until recently, have been overlooked despite decades of study in band insulators. Like semiconductors and ordinary insulators, TIs have a bulk gap, but feature robust surfaces excitations which are protected from disorder and interactions which do not close the bulk gap. TIs are distinguished from ordinary insulators not by the symmetries they possess (or break), but by topological in- variants characterizing their bulk band structures. These two pictures, the existence of gapless surface modes, and the nontrivial topology of the bulk states, yield two contrasting approaches to the study of TIs. At the heart of the subject, they are connected by the bulk-boundary correspondence, relating bulk and surface degrees of freedom. In this work, we study both aspects of topological insulators, at the same time providing an illumination to their mysterious connection. First, we present a systematic approach to the classification of bulk states of systems with inversion-like symmetries, deriving a complete set of topological invariants for such ensembles. We find that the topological invariants in all dimensions may be computed al- gebraically via exact sequences. In particular, systems with spatial inversion symmetries in one-, two-, and three-dimensions can be classified by, respectively, 2, 5, and 11 integer invariants. The values of these integers are related to physical observables such as polariza- tion, Hall conductivity, and magnetoelectric coupling. We also find that, for systems with \antiferromagnetic symmetry," there is a Z2 classification in three-dimensions, and hence a class of \antiferromagnetic topological insulators" (AFTIs) which are distinguished from ordinary antiferromagnets. From the perspective of the bulk, AFTI exhibits the quantized magnetoelectric effect, whereas on the surface, gapless one-dimensional chiral modes emerge at step-defects. Next, we study how the surface spectrum can be computed from bulk quantities. Specif- ically, we present an analytic prescription for computing the edge dispersion E(k) of a tight- binding Dirac Hamiltonian terminated at an abrupt crystalline edge, based on the bulk Hamiltonian. The result is presented as a geometric formula, relating the existence of sur- face states as well as their energy dispersion to properties of the bulk Hamiltonian. We 2 further prove the bulk-boundary correspondence for this specific class of systems, connect- ing the Chern number and the chiral edge modes for quantum Hall systems given in terms of Dirac Hamiltonians. In similar spirit, we examine the existence of Majorana zero modes in superconducting doped-TIs. We find that Majorana zero modes indeed appear but only if the doped Fermi energy is below a critical chemical potential. The critical doping is asso- ciated with a topological phase transition of vortex lines, which supports gapless excitations spanning their length. For weak pairing, the critical point is dependent on the non-abelian Berry phase of the bulk Fermi surface. Finally, we investigate the transport properties on the surfaces of TIs. While the surfaces of \strong topological insulators" { TIs with an odd number of Dirac cones in their surface spectrum { have been well studied in literature, studies of their counterpart \weak topolog- ical insulators" (WTIs) are meager, with conflicting claims. Because WTIs have an even number of Dirac cones in their surface spectrum, they are thought to be unstable to disor- der, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states, rather, presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form, reminiscent of the quantum Hall phase transition. i To my parents, ii Contents Acknowledgments vi List of Acronyms viii Preface ix I Introduction 1 1 Brief introduction to topological insulators 2 1.1 Quantum Hall effect . .4 1.1.1 Landau levels and band theory . .5 1.1.2 Berry phase, Chern integer, and topology . .6 1.1.3 Gapless chiral edge modes . .9 1.2 Quantum spin Hall effect . 10 1.2.1 Z2 classification and edge states . 11 1.3 Topological insulators in 3D . 13 1.3.1 Strong topological insulators (STI) . 15 1.3.2 Weak topological insulators (WTI) . 17 1.4 Topological superconductors . 18 1.4.1 Symmetries, Altland-Zirnbauer classification . 18 1.4.2 Realizing topological superconductors . 23 1.5 Applications and outlook . 24 2 Overview of thesis 26 II Classification of Band Insulators 28 Notation and conventions 29 Contents iii 3 Topology of Hamiltonian spaces 31 3.1 Topological classification of band structures . 33 3.2 Classification with Brillouin zone symmetries . 40 4 Inversion symmetric insulators 47 4.1 Summary of results . 50 4.2 Classifying inversion symmetric insulators . 57 4.2.1 Outline of the argument . 59 4.2.2 Topology of the space of Hamiltonians . 61 4.2.3 Hamiltonians, classifying spaces and homotopy groups . 62 4.2.4 Relative homotopy groups and exact sequences . 63 4.2.5 One-dimension . 65 4.2.6 Two-dimensions . 67 4.2.7 Going to higher dimensions . 71 4.3 Physical properties and the parities . 74 4.3.1 A coarser classification: grouping phases with identical responses . 75 4.3.2 Constraint on parities in gapped materials . 78 4.3.3 Polarization and Hall conductivity . 81 4.3.4 Magnetoelectric response . 84 4.4 Parities and the entanglement spectrum . 85 4.5 Conclusions . 92 5 Antiferromagnetic topological insulators 94 5.1 Z2 topological invariant . 95 5.1.1 Relation to the time-reversal invariant topological insulator . 97 5.1.2 Magnetoelectric effect and the Chern-Simons integral . 98 5.1.3 Invariance to choice of unit cell . 99 5.2 Construction of effective Hamiltonian models . 101 5.2.1 Construction from strong topological insulators . 101 5.2.2 Construction from magnetically induced spin-orbit coupling . 102 5.3 Surface band structure . 104 5.4 Surfaces with half-integer quantum Hall effect . 107 5.5 Conclusions and possible physical realizations . 110 III Edge Spectrum and the Bulk-Boundary Correspondence 112 6 Computing the edge spectrum in Dirac Hamiltonians 113 6.1 Characterization of the nearest-layer Hamiltonian . 114 6.2 Edge state theorem . 115 6.2.1 Edge state energy . 116 6.2.2 Discussion . 118 Contents iv 6.3 Proof by Green's functions . 118 6.3.1 Bulk Green's function . 119 6.3.2 Green's function of the open system . 121 6.3.3 Existence and spectrum of edge modes . 122 2 6.3.4 Constraints on hk and E ........................ 122 6.3.5 Sign of the energy . 125 6.4 Proof by transfer matrices . 125 6.4.1 Relation between left-right boundaries . 126 6.4.2 Algebraic relation between λa, λb and E ................ 127 6.4.3 Introducing functions L, L¯ ........................ 128 6.4.4 Edge state energy . 128 6.4.5 Existence of edge states . 129 6.4.6 Sign of edge state energy . 130 6.4.7 Effective surface Hamiltonian . 131 6.5 Bulk Chern number and chiral edge correspondence . 131 6.5.1 Discussion . 134 6.6 Applications of Theorem 1 . 134 6.6.1 Example: graphene . 134 6.6.2 Example: p + ip superconductor (lattice model) . 135 6.6.3 Example: 3D topological insulator . 136 6.7 Continuum Hamiltonians quadratic in momentum . 138 6.7.1 Discussion . 139 6.7.2 Example: p + ip superconductor (continuum model) . 139 6.7.3 Proof for continuum Hamiltonians . 140 6.8 Outlook . 142 7 Majorana fermions at the ends of superconductor vortices in doped topo- logical insulators 143 7.1 A vortex in superconducting topological insulators . 144 7.2 Vortex phase transition in models and numerical results . 146 7.2.1 Lattice model . 146 7.2.2 Continuum model . 148 7.3 General Fermi surface Berry phase condition . 149 7.3.1 Summary of the argument and results . 149 7.3.2 Bogoliubov-de Gennes Hamiltonian . 150 7.3.3 Pairing potential . 151 7.3.4 Projecting to the low energy states . 153 7.3.5 Explicit solution with rotational symmetry . 154 7.3.6 General solution without rotational symmetry . 156 7.3.7 Vortex bound states . 159 7.4 Candidate materials . 160 7.4.1 CuxBi2Se3 with s-wave pairing . 160 Contents v 7.4.2 p-doped TlBiTe2, p-doped Bi2Te3 and PdxBi2Te3 ........... 162 7.4.3 Majorana zero modes from trivial insulators . 162 IV Surface Transport 164 8 Quantum transport on the surface of weak topological insulators 165 8.1 Hamiltonian and disorder structure . 166 8.1.1 Chirality of the Dirac cones . 168 8.1.2 Constructing the mass for an arbitrary even number of Dirac cones .
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