The course provides theoretical tools for understanding the properties of aggregate quantum systems with a huge number of particles. In particular, the crystalline solid, characterized by the ion lattice and the electronic degrees of freedom, is described at a microscopic level, starting from the Schroedinger equation. The low temperature ordered phases, the response to external perturbations, and the 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, and atomic physics. An introduction to 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 the crystalline solid. The fundamental Hamiltonian of a solid in first quantization; the Born Oppenheimer approximation. The crystalline solid; Bravais and reciprocal lattice. 2. Review of basic concepts of quantum statistical physics. Second quantization, density matrix, grand canonical ensamble, chemical potential. Free Fermi and Bose gases. 3. Single electron approximation. The Sommerfeld model; specific heat and effective mass. Bloch theorem; bands and Fermi surface; weak potential and tight binding approximations; graphene bands. 4. Lattice dynamics. The dynamical matrix; phonons; optical and acoustic modes; the Debye model and specific heat. 5. Electron-phonon interaction. The electron-phonon Froelich Hamiltonian; polarons; the Holstein model; second order processes and effective electronic Hamiltonian. 6. Transport properties: Drude conductivity, thermal conductivity and Wiedemann- Franz law. Classical and Quantum Hall effect. 7. 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; 8. Electron-electron interaction in momentum space. The Hartree-Fock approximation; direct and exchange interaction. The jellium model and ferromagneticm. Screening and Thomas Fermi semiclassical theory. Density Functional Theory. 9. Introduction to Fermi and Luttinger liquids. 10. Conventional superconductors. The Cooper instability. BCS microscopic theory. The gap equation. 11. Electron-electron interaction in Wannier basis. Ferromagnetism and the Heisenberg Hamiltonian. Mott insulator and the Hubbard model. Quantum phase transitions and mean field phase diagram. 12. Phenomenology and modelisation of high-Tc superconductors. 13. Numerical simulations: the Density Matrix Renormalization Group (DMRG) method 14. The concept of Nanostructures. K-dot-p theory, envelope function, quantum wells, wires and dots.

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

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 N.W. Ashcroft, N.D. Mermin, Solid State Physics, Hartcourt Courtrige Pubiher, 1976 G. Grosso and G. Pastori Parravicini, Solid State Physics, Academic Press, 2000 P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific, 1999 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, 2021

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

Exam: Written test; Compulsory oral exam; Group project;

... The final mark will be obtained from the three following compulsory steps: -) Written exam (exercises and questions on the topics of the course): up to 20 points; -) Oral exam (discussion of the written exam and presentation of a topic chosen by the student): up to 8 points; -) Numerical project in groups: up to 4 points.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.

Modalità di esame: Prova orale obbligatoria; Prova scritta su carta con videosorveglianza dei docenti; Elaborato progettuale in gruppo;

The final mark will be obtained from the three following compulsory steps: -) Written exam (exercises and questions on the topics of the course): up to 20 points; -) Oral exam (discussion of the written exam and presentation of a topic chosen by the student): up to 8 points; -) Numerical project in groups: up to 4 points.

Modalità di esame: Prova scritta (in aula); Prova orale obbligatoria; Prova scritta su carta con videosorveglianza dei docenti; Elaborato progettuale in gruppo;

The final mark will be obtained from the three following compulsory steps: -) Written exam (exercises and questions on the topics of the course): up to 20 points; -) Oral exam (discussion of the written exam and presentation of a topic chosen by the student): up to 8 points; -) Numerical project in groups: up to 4 points.