## Non-relativistic Quantum Mechanics of Many Body Systems

- Fock space and operators of second quantization for fermions and bosons.
- Physical vacuum and quasiparticles. Particle-hole transformation and changes in the electronic structure; transformation of interaction in the Hubbard model.
- Field operators and their relation to the Slater determinant.
- One- and Two-particle (interaction) operators in the second quantization formalism (with derivation).
- Para Cooper pair as a bound state. Change in the electronic spectrum and the gap.
- BCS model: Hamiltonian and the BCS wave-function. Solution of the variational problem at
*T*= 0 and the parameter (0). - Hamiltonian BCS Hamiltonian in mean field approach. Diagonalization and Bogoliubov transformation for fermions. BCS state as a vacuum state for quasiparticles.
- BCS model at
*T*> 0: one-particle excitations and a Cooper pair in the excited state. Equations for (T) and*T*._{c} - Universal ratio (0)/
*k*in the BCS theory and the superconductivity as a weak coupling problem. Modifcation of the density of states at_{B}T*T < T*and the gap 2._{c} - Instability of electron gas with long-range Coulomb interaction.
- Hartree-Fock approximation (with the derivation of HF equation).
- Wick's theorem and its application to calculate averages of fermion operators.
- Interpretation of Hartree and Fock terms (qualitative and using Feynman diagrams). Correlation energy.
- Hubbard model — electron localization in the Hartree-Fock approximation, conclusions.
- Gutzwiller wave function and Gutzwiller approximation — electron localization at half lling for large
*U >> t*. - Metal-insulator transition in the Gutzwiller approximation. Reduction of charge fluctuations in a Mott insulator and its physical consequences.
- Method of canonical transformation for the Hubbard model and the effective
*t-J*model. - Possible ground states of a Mott insulator with an antiferromagnetic interaction: dimension dependence.
- Zubarev Green's functions. Equations of motion and the fluctuation-dissipation theorem.
- 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.
- Hubbard I approximation: calculation method. Spectral function and Hubbard subbands.
- Coulomb interactions for 3
*d*atomic states. Degenerate Hubbard model with*U*and*J*. Eigen-states and eigenenergies for two electrons in d_{H}^{2}(or two holes in*d*^{8}). - 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.
- Antiferromagnetic state for the 2D Heisenberg model on square lattice. Derivation of spin waves with the Bogoliubov transformation for bosons.
- Quantum uctuations in an antiferromagnet for the Heisenberg model. Quantum corrections to energy and order parameter in 1D and 2D models, conclusions.
- 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. - CuO
_{2}planes: Charge transfer insulator and the microscopic derivation of the AF exchange interaction. - 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.
- Hole in the antiferromagnetic state in the Heisenberg model. Quasiparticle: Explain why a single hole may move in this state in a coherent way.
- Self-consistent Born approximation. Equation on self-energy. Qualitative character of the spectral function.
- 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). - Singlet states on bonds <
*ij>*in the*t-J*model. Hartree-Fock approximation and superconductivity in the*t-J*model. - Phase diagram of high-Tc superconductors. Instability of the normal state of doped CuO
_{2}planes and stripe phases. Why the lines of nonmagnetic atoms occur as domain walls and not as defects within a single domain (solitons)? - 1D model for two degenerate orbitals in the limit of vanishing Hund's exchange,
*J*→ 0. SU(4) model and entanglement._{H} - Complementarity of spin and orbital correlations in models: SU(4) and Kugel-Khomskii. Goodenough-Kanamori rules in classical and quantum case.
- Realistic model
*d*for KCuF^{9}_{3}: Kugel-Khomski model. Projection operators for spin and orbital states; the structure of spin-orbital model. - Phase diagram of the Kugel-Khomskii model in mean eld and three possible phases with long-range order. Quantum critical point.
- Frustration of interactions in the Kugel-Khomskii model. Explain why Hund's exchange stabilizes A-AF phase and why in this cubic crystal (KCuF
_{3}) AF and FM interactions coexist. - Energies of charge excitations in models:
*d*— similarities and differences.^{9}, d^{4}i d^{1} - 2D compass model and Kitaev model. Nematic order in the compass model and spin liquid in the Kitev model (hexagonal lattice).
- Double exchange: Colossal magneto-resistance and the mechanism of phase transition to a metallic ferromagnetic phase in doped manganites.
- 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.
- 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?
- Matrix Product State representation of the wave function of the spin chain. How it is related with the Schmidt decomposition?