# Quantum theory of magnetism

**Introduction**

Lecture 0. About the lecture course [slides]

Lecture 1. Introduction, Magnetic susceptibility; Dia-, para-, ferro- and antiferromagnetism [slides]

Lecture 2. Dia-, para-, ferro- and antiferromagnetism, dipole-dipole interaction, Bohr-van Leeuwen theorem [slides]

**Magnetism of free ions**

Lecture 3. Spin, orbital, and total moments. Hund's rules, spin-orbit coupling, atomic diamagnetism [slides]

Lecture 4. Atomic diamagnetism, Pascal constants, Van Vleck paramagnetism [slides]

**Magnetism of ions in a crystal**

Lecture 4. Spherical and cubic harmonics, crystal-field splitting in a nut shell [slides]

Lecture 5. Group theory in a nutshell, its application for crystal-fields (not for exam) [slides]

Lecture 6. The Jahn-Teller effect, txE and txT problems [slides]

Lecture 7. Intra-atomic exchange interaction, spin-state transitions, orbital moment quenching [slides]

Lecture 8. Curie law, Brillouin function, imprtance of orbital moment quenching [slides]

**Correlation effects**

Lecture 9. Second quantization. Tight-binding approximation [slides]

Lecture 10. Mott insulators. Hubbard model. Metal-insulator transitions. Spectral function in Hubbard model. Phase diagram of non-degenerate Hubbard model on the square lattice. [slides]

Lecture 11. Different types of insulators (band, Mott, Slater, charge-transfer). [slides]

**Heisenberg-like models**

Lecture 11. Derivation of the Heisenberg model, symmtric and antisymmetric exchanges, Dzyaloshinskii-Moriya vector. Heisenberg's misperception. [slides]

Lecture 12. Classical Heisenberg, Ising, XY models. Connection between Hubbard and Heisenberg models. [slides]

Lecture 13. Exchange due to tunneling of electrons. Goodenough - Kanamori - Anderson rules. Kugel-Khomskii model [slides]

Lecture 14. Superexchange interaction. Double exchange. [slides]

**Different approaches to Heisenberg model**

Lecture 15. Magnetic spirals, Luttinger-Tizsa method for classical Heisenberg model. [slides]

Lecture 16. Mean-field approximation to Heisenberg model. [slides]

Lecture 17. Curie-Weiss law. Critical indexes. Spin-wave theory for ferromagnets. [slides]

Lecture 18. Mermin-Wagner theorem and why our world is 3D. Spin-wave theory for antiferromagnets. [slides]

**Low-dimensional magnetism **

Lecture 19. Dimers, spin gaps, Schottky anomaly. AFM chains in Ising model. AFM chain in Heisenberg model [slides]

Lecture 20. Spinons, Haldane chains. 2D Ising. Frustrations, spin liquids. Berezinskii-Kosterlitz-Thauless transiton. Frustration [slides]

Lecture 21. Kitaev model. Single-ion anisotropy. Spin ice. Magnetic monopoles. Order-by-disorder [slides]

**Magnetism of itinerant (metallic) electrons**

Lecture 22. Fermi gas. Pauli susceptebility. Screening of Coloumb interavtion in metals. Fermi-liquid [slides]

Lecture 23. Linear response theory, Lindhard's formula [slides]

Lecture 24. RKKY exchange. Nesting of the Fermi surface. Peierls transition, spin and charge density waves (SDW and CDW) [slides]

Lecture 25. Susceptibility of interacting electronic "gas". Generalized Stoner criterium. Mechanism of spin density waves (SDW) formation [slides]