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Introduces the idea of second quantized operators in the many-particle domain, Fock spaces, field operators, and vacuum states, and outlines how canonical transformations can be applied to solve many-body problems. Coherent states, as eigenstates of the annihilation operator, including the development of Grassmann’s algebra and calculus for fermions, are presented.
Delineates how the ideas of topological equivalence and adiabatic continuity lead to the emergence of distinct classes of insulator Hamiltonians, and how this, in turn, leads to bulk-boundary correspondence – the connection between bulk topological invariants and edge or surface states. Classification of topologically nontrivial and trivial phases, based on fundamental discrete symmetries and dimensionality, the “tenfold way," is explained. Mapping of d-dimensional Brillouin zones onto a d-dimensional Brillouin torus and Bloch Hamiltonians are defined. and construction of Bloch bundles on the torus base manifold is outlined. Time-reversal symmetry, Kramers’ band degeneracy, “time-reversal invariant momenta,” and the implied vanishing of Berry’s curvature are delineated. The integer quantum Hall effect and the modern theory of polarization are discussed in detail. Z2 topological invariant is derived using the sewing matrix, time-reversal polarization and the non-Abelian Berry connection.
Provides detailed analysis of mechanisms of exchange coupling: direct or potential exchange, kinetic exchange, superexchange, polarization exchange, Dzialoshinskii–Moriya, double exchange, and RKKY. The effect of crystal fields and the single-site anisotropy are also discussed.
Presents functional integral methods of quantum many-body theory. Starting with Feynman’s path integral, it develops functional integrals of partition functions in imaginary time and extends these techniques to many-body systems. It expands the formulation in the coherent-state basis, and describes the application of the Hubbard–Stratononvich transformation and the saddle-point approximation.
Deals with magnetism in itinerant systems. It starts with the Stoner mean-field theory, as derived from a simple Hubbard model, and Stoner excitations and spin-waves obtained with the aid of RPA. The concept of nesting and spin-density waves is then discussed. This is followed by a detailed presentation of Anderson’s magnetic impurity model and its relation to the Kondo model through the Schrieffer–Wolff transformation. Finally, a detailed account of the Kondo effect and the Kondo resonance is given.
Covers linear response from the one-electron viewpoint, including causality and the Kramers–Kronig relation. It develops the Kubo conductivity formula with special reference to the quantum Hall effect. The longitudinal and transverse dielectric functions are derived, and the ideas of intraband and interband, both direct and indirect, optical transitions are discussed.
Develops the one-particle formalism within Hartree–Fock and density functional frameworks, and examines validity bounds. The effects of exchange and correlation are also discussed, bringing out the idea of an exchange hole for fermions.
Explains the effect of dimensionality on electronic susceptibilities, including nesting effects. It describes the onset of instabilities, as manifest in the Peierls phenomenon, and delineates their emergent order parameters. It also introduces the idea of a Kohn anomaly, and derives the giant Kohn anomaly as a consequence of the one-dimensional Peierls instability.
Introduces the concept of Cooper pairing and develops a diagramatic approach to the Cooper instability. The BCS Hamiltonian is then constructed and solved with the aid of the Bogoliubov–Valatin transformation. Nambu–Gorkov formalism is introduced, and the Gor'kov anomalous Green function constructed. The Ginzburg–Landau formulation is derived from microscopic theory using the coherent-state partition function and the HS transformation. Detailed account of the Ginzburg–Landau perspective of superconductivity is given, ending with a derivation of the Meissner effect and an explanation of the Anderson–Higgs mechanism.
Reveals limitations of noninteracting fermion formulation. The chapter also introduces Landau’s Fermi liquid parameters and the conceptual basis of quasiparticles. Some suscptibilities are derived. Microscopic justification is explained.
Treats the Bose-Einstein condensation, and explains superfluidity from the Bogoliubov and Ginzburg-Landau perspectives. It also describes the concept of spontaneous symmetry breaking and Goldstone modes.
Presents relevant aspects of topology, such as homeomorphism, fiber and vector bundles, connection and curvature, parallel transport, and holonomy, and ends with establishing the relevance of topology to physics.