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Contributors; Foreword; Graphene and Relativistic Quantum Physics; 1. Introduction; 2. Early experiment; 3. Pseudospin chirality in graphene; 4. Berry phase in magneto-oscillations; 5. Pseudospin and Klein tunneling in graphene; 6. Conclusions; References; Dirac Fermions in Condensed Matter and Beyond; 1. Introduction; 2. Emergence of Dirac fermions in a generic two-band model; 3. Dirac fermions in tight-binding models and fermion doubling; 3.1. Rotation to a simplified model and spinorial form of the wave functions; 3.2. Berry phases and winding numbers
3.3. Basic properties of electrons in graphene4. Dirac fermions in a magnetic field; 4.1. Landau levels of Dirac fermions; 4.2. Degeneracy; 4.3. Semi-classical quantization rule; 5. Motion and merging of time-reversal-symmetric Dirac points; 6. Manipulation of Dirac points in artificial graphenes; 6.1. A lattice of cold atoms; 6.2. Propagation of microwaves; 7. More Dirac points; 7.1. Monolayer with third-neighbour coupling; 7.2. Manipulation of Dirac points in twisted bilayer: a second type of merging; 8. Conclusions; Acknowledgment; References
Quantum Transport in Graphene: Impurity Scattering as a Probe of the Dirac Spectrum1. Introduction; 2. Impurity scattering in graphene: determination of the transport and elastic scattering times; 3. Quantum transport: proximity induced superconductivity and specular Andreev reflection.; 4. Perspectives: inducing new functionalities in graphene by creating scattering centers; 4.1. Adatoms, agregates or molecules; 4.2. Engineering inhomogeneous strain in graphene; 5. Conclusion; Acknowledgment; References; Experimental Signatures of Topological Insulators; 1. Introduction
2. Bulked gap in strained mercury telluride3. ARPES spectra and surface mercury telluride; 4. Topological signatures in transport experiments; 5. Conclusions; References; Topology of Bands in Solids: From Insulators to Dirac Matter; 1. Introduction; 2. Bloch theory; 3. Geometrical phase and parallel transport; 3.1. Aharonov-Bohm effect; 3.2. Berry phase; 3.3. Berry phase and parallel transport on vector bundles; 3.4. Bloch vector bundle; 4. Topological properties of a valence band in an insulator; 4.1. Chern topological insulators
4.2. Kane-Mele topological index with time-reversal symmetry 4.2.1. Time-reversal symmetry in Bloch bands.4.3. Insulating materials displaying a topological valence band; 5. From insulators to semi-metals; 5.1. From topology in the bulk to demi-metallic surface states; 5.2. From topology in the bulk to critical semi-metals; 5.3. Topological properties of semi-metallic phases; 6. Conclusion and perspectives; Appendix: Two useful trivializations of the Bloch bundle; A.1. Trivialization by Fourier transform; A.2. Trivialization from periodic functions; Acknowledgment; References
3.3. Basic properties of electrons in graphene4. Dirac fermions in a magnetic field; 4.1. Landau levels of Dirac fermions; 4.2. Degeneracy; 4.3. Semi-classical quantization rule; 5. Motion and merging of time-reversal-symmetric Dirac points; 6. Manipulation of Dirac points in artificial graphenes; 6.1. A lattice of cold atoms; 6.2. Propagation of microwaves; 7. More Dirac points; 7.1. Monolayer with third-neighbour coupling; 7.2. Manipulation of Dirac points in twisted bilayer: a second type of merging; 8. Conclusions; Acknowledgment; References
Quantum Transport in Graphene: Impurity Scattering as a Probe of the Dirac Spectrum1. Introduction; 2. Impurity scattering in graphene: determination of the transport and elastic scattering times; 3. Quantum transport: proximity induced superconductivity and specular Andreev reflection.; 4. Perspectives: inducing new functionalities in graphene by creating scattering centers; 4.1. Adatoms, agregates or molecules; 4.2. Engineering inhomogeneous strain in graphene; 5. Conclusion; Acknowledgment; References; Experimental Signatures of Topological Insulators; 1. Introduction
2. Bulked gap in strained mercury telluride3. ARPES spectra and surface mercury telluride; 4. Topological signatures in transport experiments; 5. Conclusions; References; Topology of Bands in Solids: From Insulators to Dirac Matter; 1. Introduction; 2. Bloch theory; 3. Geometrical phase and parallel transport; 3.1. Aharonov-Bohm effect; 3.2. Berry phase; 3.3. Berry phase and parallel transport on vector bundles; 3.4. Bloch vector bundle; 4. Topological properties of a valence band in an insulator; 4.1. Chern topological insulators
4.2. Kane-Mele topological index with time-reversal symmetry 4.2.1. Time-reversal symmetry in Bloch bands.4.3. Insulating materials displaying a topological valence band; 5. From insulators to semi-metals; 5.1. From topology in the bulk to demi-metallic surface states; 5.2. From topology in the bulk to critical semi-metals; 5.3. Topological properties of semi-metallic phases; 6. Conclusion and perspectives; Appendix: Two useful trivializations of the Bloch bundle; A.1. Trivialization by Fourier transform; A.2. Trivialization from periodic functions; Acknowledgment; References