Non-relativistic Quantum Mechanics of Many Body Systems

  1. Fock space and operators of second quantization for fermions and bosons.
  2. Physical vacuum and quasiparticles. Particle-hole transformation and changes in the electronic structure; transformation of interaction in the Hubbard model.
  3. Field operators and their relation to the Slater determinant.
  4. One- and Two-particle (interaction) operators in the second quantization formalism (with derivation).
  5. Para Cooper pair as a bound state. Change in the electronic spectrum and the gap.
  6. BCS model: Hamiltonian and the BCS wave-function. Solution of the variational problem at T = 0 and the parameter (0).
  7. Hamiltonian BCS Hamiltonian in mean field approach. Diagonalization and Bogoliubov transformation for fermions. BCS state as a vacuum state for quasiparticles.
  8. BCS model at T > 0: one-particle excitations and a Cooper pair in the excited state. Equations for (T) and Tc.
  9. Universal ratio (0)/kBT in the BCS theory and the superconductivity as a weak coupling problem. Modifcation of the density of states at T < Tc and the gap 2.
  10. Instability of electron gas with long-range Coulomb interaction.
  11. Hartree-Fock approximation (with the derivation of HF equation).
  12. Wick's theorem and its application to calculate averages of fermion operators.
  13. Interpretation of Hartree and Fock terms (qualitative and using Feynman diagrams). Correlation energy.
  14. Hubbard model — electron localization in the Hartree-Fock approximation, conclusions.
  15. Gutzwiller wave function and Gutzwiller approximation — electron localization at half lling for large U >> t.
  16. Metal-insulator transition in the Gutzwiller approximation. Reduction of charge fluctuations in a Mott insulator and its physical consequences.
  17. Method of canonical transformation for the Hubbard model and the effective t-J model.
  18. Possible ground states of a Mott insulator with an antiferromagnetic interaction: dimension dependence.
  19. Zubarev Green's functions. Equations of motion and the fluctuation-dissipation theorem.
  20. Green's function and spectral density for the Hubbard model in the atomic limit. Interpretation. Compare this result with the spectral function obtained in the Hartree-Fock approximation.
  21. Hubbard I approximation: calculation method. Spectral function and Hubbard subbands.
  22. Coulomb interactions for 3d atomic states. Degenerate Hubbard model with U and JH. Eigen-states and eigenenergies for two electrons in d2 (or two holes in d8).
  23. Ferromagnetic state as an exact state of the Heisenberg model and spin-wave (magnon) excitations. Holstein-Primako transformation, and the diagonalization. Interpretation of the result.
  24. Antiferromagnetic state for the 2D Heisenberg model on square lattice. Derivation of spin waves with the Bogoliubov transformation for bosons.
  25. Quantum uctuations in an antiferromagnet for the Heisenberg model. Quantum corrections to energy and order parameter in 1D and 2D models, conclusions.
  26. Antiferromagnetism in the Hubbard model at weak coupling (perfect nesting). Relation between the gap and the value of U. Similarity to the instability of the Fermi surface in the BCS model.
  27. CuO2 planes: Charge transfer insulator and the microscopic derivation of the AF exchange interaction.
  28. Hole in the antiferromagnetic state in the Ising model. Explain charge con nement for a single hole and why two holes may move in the AF state.
  29. Hole in the antiferromagnetic state in the Heisenberg model. Quasiparticle: Explain why a single hole may move in this state in a coherent way.
  30. Self-consistent Born approximation. Equation on self-energy. Qualitative character of the spectral function.
  31. How is the disappearance of the quasiparticle seen in the Ising model? Self-energy and Green's function for J → 0. Spectral function for nite J (ladder spectrum).
  32. Singlet states on bonds <ij> in the t-J model. Hartree-Fock approximation and superconductivity in the t-J model.
  33. Phase diagram of high-Tc superconductors. Instability of the normal state of doped CuO2 planes and stripe phases. Why the lines of nonmagnetic atoms occur as domain walls and not as defects within a single domain (solitons)?
  34. 1D model for two degenerate orbitals in the limit of vanishing Hund's exchange, JH → 0. SU(4) model and entanglement.
  35. Complementarity of spin and orbital correlations in models: SU(4) and Kugel-Khomskii. Goodenough-Kanamori rules in classical and quantum case.
  36. Realistic model d9 for KCuF3: Kugel-Khomski model. Projection operators for spin and orbital states; the structure of spin-orbital model.
  37. Phase diagram of the Kugel-Khomskii model in mean eld and three possible phases with long-range order. Quantum critical point.
  38. Frustration of interactions in the Kugel-Khomskii model. Explain why Hund's exchange stabilizes A-AF phase and why in this cubic crystal (KCuF3) AF and FM interactions coexist.
  39. Energies of charge excitations in models: d9, d4 i d1 — similarities and differences.
  40. 2D compass model and Kitaev model. Nematic order in the compass model and spin liquid in the Kitev model (hexagonal lattice).
  41. Double exchange: Colossal magneto-resistance and the mechanism of phase transition to a metallic ferromagnetic phase in doped manganites.
  42. Explain why kinetic energy is responsible for FM interactions in doped manganites and for spin excitations. Magnons as a result of the expansion in Schwinger bosons.
  43. Entanglement entropy of bipartition and its connection with the Schmidt decomposition (for a pure state). What is an area law for entanglement entropy of bipartition?
  44. Matrix Product State representation of the wave function of the spin chain. How it is related with the Schmidt decomposition?

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