**Description:**

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.

Contents:

**Preface **

**Acknowledgements **

**Author biography**

**Symbols to topological insulators**

**Chapter 1. Symmetries and effective Hamiltonians**** • Crash course on symmetry transformations • Unitary
symmetry transformations • Action of symmetry transformations on operators •
Antiunitary symmetry transformations: time reversal • Symmetry groups •
Translations, Bloch’s theorem and space groups • Effective Hamiltonians for
bulk III—V semiconductors • Effective Hamiltonian about the ?-point: plain
vanilla model • Cubic crystalline effects and double covering groups • Bulk
inversion asymmetry • Confinement and structural inversion asymmetry •
Hands-on: symmetry analysis of a triple quantum dot • References **

**Chapter 2. Electron-coupling to external fields and
transport theory**** •
Electromagnetic potentials, fields and currents • Minimal coupling and electric
charge conservation law • Charge current in lattice systems • Linear response
and current—current correlation functions • Matsubara technique and thermal
Green functions • Matsubara formulation of linear response • Charge
conductivity of an electron gas • Thermoelectric and thermal transport • Energy
conservation and heat current • Luttinger’s gravitational field approach •
Nature of the gravitational field • Hands-on: magnetoconductivity tensor of a
triangular triple quantum dot • Hands-on: Boltzmann transport equation •
References **

**Chapter 3. Jackiw—Rebbi model and Goldstone—Wilezek
formula**** • Helical electrons in nanowires:
emergent Jackiw–Rebbi model • Zero-energy solutions in the Jackiw–Rebbi model •
The Jackiw–Rebbi model in condensed matter physics • Polyacetylene and the
Su–Schrieffer–Heeger model • One-dimensional conductors and sliding charge
density waves • Goldstone–Wilczek formula and dissipationless current •
Connection to Dirac physics and chiral anomaly • Fractional electric charge at
solitons and electric charge pumping
• Hands-on: derivation of the
Goldstone–Wilczek formula for a sliding • charge density wave conductor •
References **

**Chapter 4. Topological insulators in 1+1 dimensions**** • Prototypical topological-insulator model in 1+1
dimensions • Hamiltonian and zero-energy edge states • Topological invariant •
Homotopy mapping and winding number • Topological invariance • Generalised
winding number • Lattice topological-insulator model and higher winding numbers
• Adiabatic transport: Thouless pump and Berry curvature • Continuum model •
Relation between Chern and winding numbers • Lattice model and electric
polarisation • Berry phase • Hands-on: winding number in a 3+1d model •
Hands-on: current and electric polarisation formula • Hands-on: violation of
chiral symmetry and electric polarisation • References **

**Chapter 5. Chern insulators—fundamentals**** • Jackiw–Rebbi model and Dirac physics in 2 + 1d •
Electric charge and current responses of the chiral edge modes • Chiral edge
modes in the quantum Hall effect: Laughlin’s argument • Connection to Dirac
physics and parity anomaly • Maxwell-Chem-Simons action and topological
Meissner effect • Chern insulator in 2 + 1d • Continuum model • Lattice model •
Quantised Hall conductance and Chern number—bulk approach • Bulk eigenstates •
Adiabatic Hall transport and Berry curvature • Homotopy mapping and Chern number
• Chern insulators in higher dimensions • Chern-insulator model in 4 + 1d •
Second Chern number and non-Abelian Berry gauge potentials • 4 + 1d
Chern-Simons action and four-dimensional quantum Hall effect • Generalisation to arbitrary
dimensions • Dimensional reduction: chiral anomaly • Hands-on: Chern-Simons
action • Hands-on: Chern number for interacting systems • Hands-on: second
Chern number • References**

**Chapter 6. Chern insulators—applications**** • Dynamical anomalous Hall response and polar Kerr effect •
Dynamical anomalous Hall conductivity
• Polar Kerr effect • Dielectric
tensor and circular-polarisation birefringence • Kerr-angle formula • Polar
Kerr effect in a 2 + 1d Chern insulator • Chern insulators in an external
magnetic field • High-field limit and the formation of Landau levels • Theory
of orbital magnetisation—a Green-function method • Anomalous thermoelectric and
thermal Hall transport • Thermoelectric conductivity tensor • Thermal
conductivity tensor • Diathermal contributions to the conductivities and
transport • current • Hands-on: magnetic-field-induced Chern
systems • Hands-on: thermoelectric transport in the Haldane model • References **

**Chapter 7. Z**_{2} topological insulators** • ****Z**_{2}** topological insulators in 2 + 1 dimensions • Bottom-up construction
based on Chern insulators: BHZ model • Violation of chiral symmetry and Z**_{2
}topological invariant • Z_{2} topological insulators in 3 + 1
dimensions • Crystal structure and model Hamiltonian • Surface states for
negligible warping • Consequences of warping and ? Berry phase •
Magnetoelectric polarisation and **Z**_{2
}**topological invariants in
3 + 1d • Dimensional reduction from a 4 + 1d Chern insulator and
magnetoelectric coupling • Dimensional
reduction • Magnetoelectric polarisation domain wall and quantum anomalous Hall
effect • Hands-on: quasiparticle interference on the topological surface •
Hands-on: topological Kondo insulator • References **

**Chapter 8. Topological classification of insulators and
beyond**** • Generalised
antinunitary symmetries and symmetry classes • The art of topological
classification • Complex symmetry classes • Real symmetry classes •**** Z**_{2}** classification and relative Chem and winding numbers •
Weak topological invariants and flat bands • Topological classification with
unitary symmetries • Crystalline topological insulators • Topological
classification of gapless systems • 2 + l d semimetals—graphene • Weyl
semimetals • Topological classification of insulators and defects • Topological
superconductors and Majorana fermions • Further topics and outlook • Hands-on:
Berry magnetic monopoles in hole-like semiconductors • Hands-on: Floquet
topological insulator • References **

**Index **

About the Author:

**Dr Panagiotis Kotetes** recently embarked on
his five-year faculty appointment at the Institute of Theoretical Physics of
the Chinese Academy of Sciences in Beijing. During 2015-2018 he was a
postdoctoral researcher at the Niels Bohr Institute of the University of
Copenhagen, where this book was mainly written. His first postdoctoral
appointment was at the Karlsruhe Institute of Technology, where he worked for
five years. Panagiotis carried out his Diploma, Masters and PhD studies at the
National Technical University of Athens in Greece. His research interests and
activity cover the topics of topological systems, unconventional
superconductivity, exotic magnetism, and quantum computing.

Target Audience:

This
book is particularly recommended for graduate students or researchers who are
new to the field of topological matter. Useful for people interested in
materials science and physics.