This book provides an introduction to topological matter with a focus on insulating bulk systems.
A number of prerequisite concepts and tools are first laid out, including the notion of symmetry transformations, the band theory of semiconductors and aspects of electronic transport. The main part of the book discusses realistic models for both time-reversal-preserving and -violating topological insulators, as well as their characteristic responses to external perturbations. Special emphasis is given to the study of the anomalous electric, thermal, and thermoelectric transport properties, the theory of orbital magnetisation, and the polar Kerr effect. The topological models studied throughout this book become unified and generalised by means of the tenfold topological-classification framework and the respective systematic construction of topological invariants. This approach is further extended to topological superconductors and topological semimetals. This book covers a wide range of topics and aims at the transparent presentation of the technical aspects involved. For this purpose, homework problems are also provided in dedicated Hands-on sections. Given its structure and the required background level of the reader, this book is particularly recommended for graduate students or researchers who are new to the field.
Author(s): Panagiotis Kotetes
Series: IOP Concise Physics
Publisher: IOP Publishing
Year: 2019
Language: English
Pages: 216
City: Bristol
PRELIMS.pdf
Preface
Acknowledgements
Author biography
Panagiotis Kotetes
Symbols to topological insulators
Outline placeholder
I. Abbreviations
II. Basic mathematical notation
III. Matrix and operator notation
IV. Symmetry operations
V. Symmetry groups and representations
VI. Symmetry classes
VII. Fundamental constants
VIII. Physical quantities – Latin symbols
IX. Physical quantities – Greek symbols
X. Relativistic notation
CH001.pdf
Chapter 1 Symmetries and effective Hamiltonians
1.1 Crash course on symmetry transformations
1.1.1 Unitary symmetry transformations
1.1.2 Action of symmetry transformations on operators
1.1.3 Antiunitary symmetry transformations: time reversal
1.1.4 Symmetry groups
1.1.5 Translations, Bloch's theorem and space groups
1.2 Effective Hamiltonians for bulk III–V semiconductors
1.2.1 Effective Hamiltonian about the Γ-point: plain vanilla model
1.2.2 Cubic crystalline effects and double covering groups
1.2.3 Bulk inversion asymmetry
1.2.4 Confinement and structural inversion asymmetry
1.3 Hands-on: symmetry analysis of a triple quantum dot
References
CH002.pdf
Chapter 2 Electron-coupling to external fields and transport theory
2.1 Electromagnetic potentials, fields and currents
2.2 Minimal coupling and electric charge conservation law
2.3 Charge current in lattice systems
2.4 Linear response and current–current correlation functions
2.5 Matsubara technique and thermal Green functions
2.6 Matsubara formulation of linear response
2.7 Charge conductivity of an electron gas
2.8 Thermoelectric and thermal transport
2.8.1 Energy conservation and heat current
2.8.2 Luttinger's gravitational field approach
2.8.3 Nature of the gravitational field
2.9 Hands-on: magnetoconductivity tensor of a triangular triple quantum dot
2.10 Hands-on: Boltzmann transport equation
References
CH003.pdf
Chapter 3 Jackiw–Rebbi model and Goldstone–Wilczek formula
3.1 Helical electrons in nanowires: emergent Jackiw–Rebbi model
3.2 Zero-energy solutions in the Jackiw–Rebbi model
3.3 The Jackiw–Rebbi model in condensed matter physics
3.3.1 Polyacetylene and the Su–Schrieffer–Heeger model
3.3.2 One-dimensional conductors and sliding charge density waves
3.4 Goldstone–Wilczek formula and dissipationless current
3.4.1 Connection to Dirac physics and chiral anomaly
3.4.2 Fractional electric charge at solitons and electric charge pumping
3.5 Hands-on: derivation of the Goldstone–Wilczek formula for a sliding charge density wave conductor
References
CH004.pdf
Chapter 4 Topological insulators in 1+1 dimensions
4.1 Prototypical topological-insulator model in 1+1 dimensions
4.1.1 Hamiltonian and zero-energy edge states
4.1.2 Topological invariant
4.1.3 Homotopy mapping and winding number
4.1.4 Topological invariance
4.1.5 Generalised winding number
4.2 Lattice topological-insulator model and higher winding numbers
4.3 Adiabatic transport: Thouless pump and Berry curvature
4.3.1 Continuum model
4.3.2 Relation between Chern and winding numbers
4.3.3 Lattice model and electric polarisation
4.4 Berry phase
4.5 Hands-on: winding number in a 3+1d model
4.6 Hands-on: current and electric polarisation formula
4.7 Hands-on: violation of chiral symmetry and electric polarisation
References
CH005.pdf
Chapter 5 Chern insulators—fundamentals
5.1 Jackiw–Rebbi model and Dirac physics in 2 + 1d
5.1.1 Electric charge and current responses of the chiral edge modes
5.1.2 Chiral edge modes in the quantum Hall effect: Laughlin’s argument
5.1.3 Connection to Dirac physics and parity anomaly
5.1.4 Maxwell–Chern–Simons action and topological Meissner effect
5.2 Chern insulator in 2 + 1d
5.2.1 Continuum model
5.2.2 Lattice model
5.3 Quantised Hall conductance and Chern number—bulk approach
5.3.1 Bulk eigenstates
5.3.2 Adiabatic Hall transport and Berry curvature
5.3.3 Homotopy mapping and Chern number
5.4 Chern insulators in higher dimensions
5.4.1 Chern-insulator model in 4 + 1d
5.4.2 Second Chern number and non-Abelian Berry gauge potentials
5.4.3 4 + 1d Chern–Simons action and four-dimensional quantum Hall effect
5.4.4 Generalisation to arbitrary dimensions
5.5 Dimensional reduction: chiral anomaly
5.6 Hands-on: Chern–Simons action
5.7 Hands-on: Chern number for interacting systems
5.8 Hands-on: second Chern number
References
CH006.pdf
Chapter 6 Chern insulators—applications
6.1 Dynamical anomalous Hall response and polar Kerr effect
6.1.1 Dynamical anomalous Hall conductivity
6.1.2 Polar Kerr effect
6.1.3 Dielectric tensor and circular-polarisation birefringence
6.1.4 Kerr-angle formula
6.1.5 Polar Kerr effect in a 2 + 1d Chern insulator
6.2 Chern insulators in an external magnetic field
6.2.1 High-field limit and the formation of Landau levels
6.2.2 Theory of orbital magnetisation—a Green-function method
6.3 Anomalous thermoelectric and thermal Hall transport
6.3.1 Thermoelectric conductivity tensor
6.3.2 Thermal conductivity tensor
6.3.3 Diathermal contributions to the conductivities and transport current
6.4 Hands-on: magnetic-field-induced Chern systems
6.5 Hands-on: thermoelectric transport in the Haldane model
References
CH007.pdf
Chapter 7 Z2 topological insulators
7.1 Z2 topological insulators in 2 + 1 dimensions
7.1.1 Bottom-up construction based on Chern insulators: BHZ model
7.1.2 Violation of chiral symmetry and Z2 topological invariant
7.2 Z2 topological insulators in 3 + 1 dimensions
7.2.1 Crystal structure and model Hamiltonian
7.2.2 Surface states for negligible warping
7.2.3 Consequences of warping and π Berry phase
7.2.4 Magnetoelectric polarisation and Z2 topological invariants in 3 + 1d
7.3 Dimensional reduction from a 4 + 1d Chern insulator and magnetoelectric coupling
7.3.1 Dimensional reduction
7.3.2 Magnetoelectric polarisation domain wall and quantum anomalous Hall effect
7.4 Hands-on: quasiparticle interference on the topological surface
7.5 Hands-on: topological Kondo insulator
References
CH008.pdf
Chapter 8 Topological classification of insulators and beyond
8.1 Generalised antinunitary symmetries and symmetry classes
8.2 The art of topological classification
8.2.1 Complex symmetry classes
8.2.2 Real symmetry classes
8.2.3 Z2 classification and relative Chern and winding numbers
Example
8.2.4 Weak topological invariants and flat bands
8.2.5 Topological classification with unitary symmetries
8.2.6 Crystalline topological insulators
8.3 Topological classification of gapless systems
8.3.1 2 + 1d semimetals—graphene
8.3.2 Weyl semimetals
8.4 Topological classification of insulators and defects
8.5 Topological superconductors and Majorana fermions
8.6 Further topics and outlook
8.7 Hands-on: Berry magnetic monopoles in hole-like semiconductors
8.8 Hands-on: Floquet topological insulator
References
INDEX.pdf
Index
