MANY BODY PHYSICS

Course objectives

GENERAL OBJECTIVES: The aim of the course is to teach the main paradigms in many-body systems, particularly of fermionic systems, like electrons in metals, and to give an introduction to the methods of field theory in conndensed matter. At the end of the course the student should have acquired both technical competences (second quantization, Green function and Feynman diagrams at T=0 and T>0, response functions) and the physical understanding of the simplest approximations used to describe the many-body effects. In general the student should be able to understand both the language and the issues of modern research in correlated systems. SPECIFIC OBJECTIVES: A - Knowledge and understanding OF 1) Basic concepts of Landau Fermi liquid theory, the fundamental paradigm of the metallic state OF 2) Properties of the Green functions and their physical meaning OF 3) Interaction representation. S matrix. Wick's theorem and Feynman diagrams. OF 4) Self-energy and Dyson equation. Hartree-Fock approximation, RPA approximation OF 5) Linear response theory; response function. Analytic properties. Reactive and absorptive part. OF 6) Kramers-Kronig relations. Kubo formula Fluctuation and dissipation theorem B - Application skills OF 7) The second objective is to prepare the students to actively solve problems in physics where MB theory concepts are required. This will happen at first with problems structured within a conceptual scheme similar to the ones discussed and applied during the course. However, as their preparation progresses, students are also expected to use MB concepts for solving new problems in different applications. C - Autonomy of judgment OF 8) The third and more ambitious objective is to teach the students to think using concepts and methods from MB theory as a powerful problem solving tool in physics. D - Communication skills OF 9) Besides having a clear understanding of the new acquired concepts in MB theory, the students should correspondingly acquire the ability to communicate and transmit these concepts in a clear and direct way. E - Ability to learn OF 10) The students should become able to read and understand scientific books and articles where MB concepts are involved and should be able to deepen autonomously their knowledge in this field.

Channel 1
SERGIO CAPRARA Lecturers' profile

Program - Frequency - Exams

Course program
Second quantization. Fock space. Creation and annihilation operators and field operators. Operators in second quantization. Landau theory of normal Fermi liquids. Introduction to the concept of quasi-particle. Energy as a functional of the quasi-particle distribution function. Equilibrium properties of quasi-particles: effective mass, specific heat, compressibility, spin suceptibility. Stability of the ground state. Currents associated to the quasi-particles. Kinetic equations: collective modes and zero-sound. Green function and perturbation techniques. Single-particle Green functions at T=0. Spectral representation and poles. Equation of motion. Interaction representation. S matrix. Wick's theorem and Feynman diagrams. Diagrammatic rules for different interction types. Self-energy and Dyson equation. Hartree-Fock approximation, RPA. Perturbation theory at T>0. Matsubara Green function. Perturbation techniques and thermodynamic potential. Identification of the phenomenological parameters of Landau's theory with microscopic quantities. Quasi-particle lifetime. Linear response theory; response function. Analytic properties. Reactive and absorptive part. Kramers-Kronig relations. Kubo formula Fluctuation and dissipation theorem. Spectral representation and sum rules. Conductivity. Continuity equation and gauge invariance. Esplicit calculation of response functions in perturbation theory (RPA) in simple cases of physical interest. The Hubbard model: mean-field approach to the antiferromagnetic Néel state; magnetic instability and Stoner criterion. Electron-phonon interaction.
Prerequisites
The course requires the fundamental notion s acquired in the courses of quantum mechanics and statistical mechanics at the bachelor level, as well as in the course of condensed matter physics at the master level. In. particular, the student must know the description of a quantum state and its time evolution in quantum mechanics, perturbation theory, the definition of the main statistical ensembles (microcanonical/canonical/grandcanonical), the thermodynamic and statistical properties of Fermi and Bose gases, Bloch's theorem and the various schemes of description of one-electron states in crystals (nearly free electron, effective mass approximation, tight binding).
Books
L. D. Landau and E. M. Lifshits: Statistical Physics: Theory of the Condensed State (Pt 2) - Course of Theoretical Physics, volume 9 (Pergamon 1980) A.A. Abrikosov, L.P. Gorkov, and I.Y.Dzyaloshinskii: Methods of quantum field theory in statistical physics (Dover 1963)
Frequency
Attendance is not compulsory, but the students are encouraged to take the classes, because during the lectures the most subtle aspects of many b body theory will be discussed in details, and exercises will be proposed, aimed at a deeper understanding of the theory.
Exam mode
The exam consist in an oral colloquium lasting 30/45 minutes, during which the students will be asked two questions about the main topics of the course. The expositive clarity, the knowledge of the fundamental hypotheses and of the main results, the methodological rigour, and problem-solving skills (method and results) will be evaluated.
Lesson mode
Lectures and examples discussed on the board, with the opportunity for students to ask questions about the lesson content or the example being discussed. The lectures will serve to introduce and explain various concepts, both basic and auxiliary. The examples serve the twofold purpose of clarifying and illustrating the concepts covered in class and enabling students to develop their own physical representations of the concepts learned. Questions from students, during and at the end of the lesson, will serve to clarify less intuitive aspects of the concepts learned.
SERGIO CAPRARA Lecturers' profile

Program - Frequency - Exams

Course program
Second quantization. Fock space. Creation and annihilation operators and field operators. Operators in second quantization. Landau theory of normal Fermi liquids. Introduction to the concept of quasi-particle. Energy as a functional of the quasi-particle distribution function. Equilibrium properties of quasi-particles: effective mass, specific heat, compressibility, spin suceptibility. Stability of the ground state. Currents associated to the quasi-particles. Kinetic equations: collective modes and zero-sound. Green function and perturbation techniques. Single-particle Green functions at T=0. Spectral representation and poles. Equation of motion. Interaction representation. S matrix. Wick's theorem and Feynman diagrams. Diagrammatic rules for different interction types. Self-energy and Dyson equation. Hartree-Fock approximation, RPA. Perturbation theory at T>0. Matsubara Green function. Perturbation techniques and thermodynamic potential. Identification of the phenomenological parameters of Landau's theory with microscopic quantities. Quasi-particle lifetime. Linear response theory; response function. Analytic properties. Reactive and absorptive part. Kramers-Kronig relations. Kubo formula Fluctuation and dissipation theorem. Spectral representation and sum rules. Conductivity. Continuity equation and gauge invariance. Esplicit calculation of response functions in perturbation theory (RPA) in simple cases of physical interest. The Hubbard model: mean-field approach to the antiferromagnetic Néel state; magnetic instability and Stoner criterion. Electron-phonon interaction.
Prerequisites
The course requires the fundamental notion s acquired in the courses of quantum mechanics and statistical mechanics at the bachelor level, as well as in the course of condensed matter physics at the master level. In. particular, the student must know the description of a quantum state and its time evolution in quantum mechanics, perturbation theory, the definition of the main statistical ensembles (microcanonical/canonical/grandcanonical), the thermodynamic and statistical properties of Fermi and Bose gases, Bloch's theorem and the various schemes of description of one-electron states in crystals (nearly free electron, effective mass approximation, tight binding).
Books
L. D. Landau and E. M. Lifshits: Statistical Physics: Theory of the Condensed State (Pt 2) - Course of Theoretical Physics, volume 9 (Pergamon 1980) A.A. Abrikosov, L.P. Gorkov, and I.Y.Dzyaloshinskii: Methods of quantum field theory in statistical physics (Dover 1963)
Frequency
Attendance is not compulsory, but the students are encouraged to take the classes, because during the lectures the most subtle aspects of many b body theory will be discussed in details, and exercises will be proposed, aimed at a deeper understanding of the theory.
Exam mode
The exam consist in an oral colloquium lasting 30/45 minutes, during which the students will be asked two questions about the main topics of the course. The expositive clarity, the knowledge of the fundamental hypotheses and of the main results, the methodological rigour, and problem-solving skills (method and results) will be evaluated.
Lesson mode
Lectures and examples discussed on the board, with the opportunity for students to ask questions about the lesson content or the example being discussed. The lectures will serve to introduce and explain various concepts, both basic and auxiliary. The examples serve the twofold purpose of clarifying and illustrating the concepts covered in class and enabling students to develop their own physical representations of the concepts learned. Questions from students, during and at the end of the lesson, will serve to clarify less intuitive aspects of the concepts learned.
  • Lesson code10592567
  • Academic year2025/2026
  • CoursePhysics
  • CurriculumBiosistemi
  • Year2nd year
  • Semester1st semester
  • SSDFIS/03
  • CFU6