The course provides theoretical tools for understanding the properties of aggregate quantum systems with huge number of particles. In particular the crystalline solid, characterized by the lattice and the electronic degrees of freedom, is described at microscopic level, starting from the Schroedinger equation. Its low temperature ordered phases, the response to external perturbation, and its transport and optical properties are investigated, with reference to more recent applications, such as high Tc superconductivity, quantum Hall effect, nano and low dimensional physics, ultra cold gases of atom and molecules. An introduction to reference numerical techinques appropriate for their investigation is also provided.

Knowledge of the microscopic mechanisms, models and tools for describing the behavior of interacting ions, electrons and excitations in solids. Ability to apply the knowledges to the theoretical and numerical study of reference systems in condensed matter physics, from solids to ultra cold atomic gases.

Basic knowledges of quantum and statistical physics

1. From condensed matter to solid state physics. The fundamental Hamiltonian of a solid in first quantization; the Born Oppenheimer approximation. The crystalline solid; Bravais and reciprocal lattice. (5 hs) 2. Review of basic concepts of quantum statistical physics. Second quantization, density matrix, grand canonical ensamble, chemical potential. (6 hs) 3. Free Fermi and Bose gases. Bose-Einstein condensation; superfluids. (6 hs) 4. Single electron approximation. The Sommerfeld model; Bloch theorem; bands and Fermi surface; weak potential and tight binding approximations; graphene bands. (9 hs) 5. Lattice dynamics. The dynamical matrix; phonons; optical and acoustic modes; the Debye model and specific heat. (9 hs) 6. Electron-phonon interaction. The electron-phonon Froelich Hamiltonian; polarons; the Holstein model; second order processes and effective electronic Hamiltonian. (6 hs) 7. Transport properties: Drude conductivity, thermal conductivity and Wiedemann- Franz law. Classical and Quantum Hall effect. (6 hs) 8. Optical properties. Macroscopic formulation of electrodynamics in dispersive media: complex refraction index, absorption coefficient and dissipated power. Microscopic formulation: interaction of electrons with electromagnetic radiation; (5 hs) 9. Electron-electron interaction in mean field. The Hartree-Fock approximation; exchange interaction, the jellium model, ferromagnetism. Screening and Thomas Fermi treatment. . Introduction to Fermi and Luttinger liquid theories. (10 hs) 10. Conventional superconductors. The Cooper instability. BCS microscopic theory. (9 hs) 11. Models for electron-electron interaction. Mott insulator and the Hubbard model. Ferromagnetism and the Heisenberg Hamiltonian. Quantum phase transitions. Entanglement. (9 hs). 12. High Tc superconductors. Role of electron-electron interaction and modelization. (6 hs) 13. Numerical simulations: the quantum Monte Carlo technique. The Lanczos algorithm. (9 hs) 14. The concept of Nanostructures. K-dot-p theory, envelope function, quantum wells, wires and dots. (5 hs)

The course consists of frontal lectures accompanied by exercises and numerical simulations in assisted working subgroups.

N.W. Ashcroft, N.D. Mermin, Solid State Physics, Hartcourt Courtrige Pubiher, 1976 H. Bruus, and K. Flensberg, Introduction to many body quantum theory in condensed matter physics, 2002 C. Di Castro, R. Raimondi, Statistical mechanics and applications in condensed matter, Cambridge University Press, 2015 R.P. Feynman, Statistical mechanics: a set of lectures, Benjamin Cummings Publishing Company, 1972 U. Roessler, Solid state theory: an introduction, Physica Verlag, 2009 J. Solyom, Fundamentals of the physics of solids, vols 1,2,3, Springer, 2007-2010 A. Montorsi, Notes of the course, 2018

Modalità di esame: Prova orale obbligatoria; Elaborato scritto prodotto in gruppo; Prova scritta tramite PC con l'utilizzo della piattaforma di ateneo;

Both numerical and exercise homework will be assigned along the course, to be prepared in groups. Their evaluation will concur to the final grade. A written test has to be passed in order to be admitted to the oral exam. The oral exam will amount to questions on different aspects of the program.

Modalità di esame: Prova scritta (in aula); Prova orale obbligatoria; Elaborato scritto prodotto in gruppo;

Both numerical and exercise homework will be assigned along the course, to be prepared in groups. Their evaluation will concur to the final grade. A written test has to be passed in order to be admitted to the oral exam. The mandatory oral exam will amount to questions on different aspects of the program.