01VIYPF

A.A. 2022/23

Course Language

Inglese

Course degree

Master of science-level of the Bologna process in Physics Of Complex Systems (Fisica Dei Sistemi Complessi) - Torino/Trieste/Parigi

Course structure

Teaching | Hours |
---|---|

Lezioni | 76 |

Esercitazioni in aula | 15 |

Esercitazioni in laboratorio | 9 |

Teachers

Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
---|---|---|---|---|---|---|---|

Tocchio Luca Fausto | Professore Associato | FIS/03 | 60 | 30 | 0 | 0 | 2 |

Teaching assistant

Context

SSD | CFU | Activities | Area context |
---|---|---|---|

FIS/03 | 10 | B - Caratterizzanti | Discipline matematiche, fisiche e informatiche |

2022/23

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.

The course provides theoretical tools for understanding the properties of aggregate quantum systems with a huge number of particles. First, we introduce the crystalline solid and we describe, at a microscopic level, the electronic properties in the potential given by the ions and the lattice vibrations, starting from the Schroedinger equation. Then, we model the low temperature ordered phases that emerge when electron-electron interaction and the interaction of electrons with lattice vibrations are taken into account; we investigate also the response to external perturbations, the transport and the optical properties. All these phenomena are presented with reference to recent applications, such as high-Tc superconductivity, quantum Hall effect, nano and low dimensional physics, and atomic physics. Exercise sessions along the course allow the students to better understand and apply the theoretical models that are introduced along the course. A short introduction to numerical techniques appropriate for model 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.

Knowledge of the microscopic mechanisms, models and tools for describing the behavior of interacting ions, electrons and excitations in solids.
Ability to apply the knowledge 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.

Basic knowledge of quantum mechanics 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.

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. Short 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. Electron-electron interaction in momentum space. The Hartree-Fock approximation; direct and exchange interaction. The jellium model and ferromagnetism. Screening and Thomas Fermi semi-classical theory.
8. Introduction to Fermi liquids.
9. Conventional superconductors. The Cooper instability; BCS microscopic theory; the gap equation and the specific heat.
10. Electron-electron interaction in Wannier basis.; ferromagnetism and the Heisenberg Hamiltonian.; Mott insulators and the Hubbard model; quantum phase transitions and the mean-field phase diagram; origin of antiferromagnetism. The Bose-Hubbard and the SSH models.
11. Phenomenology and modelization of high-Tc superconductors.
12. Numerical simulations: the Density Matrix Renormalization Group (DMRG) and the Lanczos methods.
13. The concept of Nanostructures. K-dot-p theory, envelope function, quantum wells, wires and dots. Optical properties of materials.

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

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

Lecture notes available for this course:
A. Montorsi and L.F. Tocchio, Notes of the course, 2022
General books about solid state physics:
N.W. Ashcroft, N.D. Mermin, Solid State Physics, Hartcourt Courtrige Pubiher, 1976
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
G. Grosso and G. Pastori Parravicini, Solid State Physics, Academic Press, 2000
Further reading:
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
P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific, 1999

...
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.

The final mark will be obtained from the three following compulsory steps, that will verify the expected learning outcomes:
-) 2-hour written exam with exercises and questions on the topics of the course. No material (books, notes,...) can be used during the written exam, while the calculator is not needed. Examples of previous written exams will be available. The written exam will be evaluated up to 20 points.
-) Oral exam with a discussion of the written exam and the presentation of a topic chosen by the student, within the course program. The oral exam will be evaluated up to 8 points.
-) Numerical project, to be prepared in groups as a homework. The numerical project will consist in the application of the presented numerical methods to the study of some models and it will be evaluated up to 4 points.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.

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Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY

Corso Duca degli Abruzzi, 24 - 10129 Torino, ITALY