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Faculty of Physics, Lomonosov Moscow State University

Advanced Quantum Field Theory: Modern Applications in HEP, Astro & Cond-Mat
Instructor: assist. prof., Dr. Oleg Kharlanov

Term 2 Examination Syllabus (Fall 2014)
1. Tight-binding approximation and its origins. A particle in a periodic potential well, crystal momentum, hopping integrals, lattice wave function. Bloch functions and the Brillouin zone. Secondquantized version of the TB model; linear closed array of atoms, the effective mass. 2. Self-consistent field approximation: the Hartree­Fock and Hartree­Fock­Roothaan equations; a qualitative analysis of the approximations used. 3. Fundamentals of the Density functional theory: the energy functional, the Hohenberg­Kohn theorems, the universal functional F[n(x)] . Density functionals and correlation functions. The Thomas­ Fermi model. 4. A deeper look at the Density functional theory: the energy functional within the Kohn­Sham approach, the exchange-correlation potential and its approximations, the kinetic energy and the Kohn­ Sham equations. 5. Lattice field theory: `free fields'. Second quantization of the TB model. one-particle eigenstates, lattice `field' operator, the Fermi sea as a QFT vacuum, quasiparticles and anti-particles (holes). The 2point Green's function on a lattice. 6. Lattice field theory: symmetries. Symmetries of the TB model for a general periodic lattice and a bipartite lattice. Global U (1) and SU (2) symmetries; local spin and charge density operators; the Peierls substitution and the local U (1) gauge symmetry of the TB model. 7. Lattice field theory: `interacting fields'. The Coulomb interaction within the TB model and the role of selected Coulomb matrix elements. The Hubbard model, its symmetries and its weak-coupling regime, including the 2D rectangular lattice at half-filling. 8. The strong-coupling regime of the Hubbard model at half-filling: the Heisenberg model. The t­ J model. 9. Fermi surface instabilities: susceptibilities and nesting vectors, an example of the Hubbard model on a 2D rectangular lattice at half-filling. Mean-field theory for such a model around the ferromagnetic state and the corresponding critical on-site coupling. 10. Fermi surface instabilities: an example of an antiferromagnetic (NeÈl) order in the Hubbard model on a 2D rectangular lattice at half-filling. The mean-field theory, the `effective Dirac equation', and the gap equation. Estimation of the gap using the van Hove singularities of the DOS. 11. Graphene: the TB model for the -band electrons, 1-particle eigenstates; the Brillouin zone and the Fermi points; energy dispersion around the Fermi points. 12. Effective Dirac equation for the excitations about the Fermi points in graphene: the equation, the field operator, the charge and current density, the canonical commutation relations. 13. Graphene and external fields: slowly-varying strains (small variations of the hopping parameters) and electromagnetic fields. The interacting theory for graphene and its Lagrangian formulation.