84536 - Interactions and Correlations in Condensed Matter

Learning outcomes

At the end of the course the student will acquire the basic notions about interacting electron systems, by discussing phenomena such as electron interactions in solids, Mott transition, superfluidity, electron-phonon interaction and elements of BCS superconductivity. He/she will also be acquainted with magnetic systems and their phase transitions.

Course contents

1) Second Quantization and Second Quantized Hamiltonian

Second quantization is a formalism to describe quantum many-body systems. This part of the course covers the changes from the first to the second quantization formalism, and highlights the advantages and applications of the latter approach. We will introduce the second quantized Hamiltonian for the homogeneous electron gas (see point below), and we will emphasize how this formalism simplifies the treatment of interactions and facilitates the study of quantum statistics.

2) The Homogeneous Interacting Electron Gas and Beyond

This section introduces the concepts of the homogeneous electron gas, a model that describes a system of electrons interacting with each other in a uniform and positive charged background. We will address its importance in condensed matter physics, investigate the effects of electron-electron interactions, and introduce the Hartree-Fock approximation used to study such systems.

3) Electrons and Phonons

The understanding of the interaction between electrons and phonons (quanta of lattice vibrations) is important for explaining many physical properties of solids. This section explores a specific model to describe lattice vibrations, namely the linear chain of atoms, as well as the electron-phonon coupling. We will introduce models used to describe these interactions, such as the Fröhlich Hamiltonian. The role of the electron-phonon coupling in phenomena such as superconductivity will also be introduced.

4) The Polaron Problem

The polaron problem addresses how an electron interacts and distorts the surrounding lattice, forming a quasiparticle known as a polaron. We will study the formation and behavior of polarons by using both perturbation theory and variational methods. This topic includes a discussion of different types of polarons, such as large and small polarons, and their implications for material properties.

5) One- and two-particle Green's Functions

Green's functions are essential tools for studying many-body interacting systems in condensed matter physics. This section introduces the idea of Green's functions, their mathematical definition and their physical interpretation. One and two-particle Green's functions will be introduced, as well as the concept of imaginary time. We will understand how Green's functions naturally describe the propagation of particles and interactions in correlated systems, opening the door to the description of complex phenomena in realistic quantum materials.

6) Feynman Diagrams: How to Solve the Many-Body Problem with Pictures

Feynman diagrams give a visual and intuitive pictorial method to represent and calculate interactions in many-body systems. This part of the course explains how to build and interpret Feynman diagrams, converting mathematical expressions, that we have addressed in the previous section, into simple graphical forms. We will apply these diagrams to various problems in condensed matter physics, such as the electron-electron and electron-phonon interactions.

7) Dielectric Function and Linear Response

The dielectric function characterizes the response of a material to external electric fields and is important for understanding many electronic properties. In this section, we will study the theory of linear response and Kubo's formula, which describe how a system responds to weak perturbations. We will derive the dielectric function within the framework of retarded response functions and two-particle Green's functions, and explore its applications, including screening, plasmons, and the optical properties of materials. Non perturbative resummation of Feynman diagrams such as the Random Phase Approximation (RPA) will also be explored.

8) Theory of Superconductivity

Superconductivity is a quantum phenomenon for which a material's electrical resistivity drops to zero below a critical temperature. This section covers the phenomenological principles and theoretical models of superconductivity. We will study the BCS theory, which accounts for the formation of Cooper pairs in the presence of electron-phonon interaction and the resulting superconducting state. If time permits, we will additionally discuss more advanced topics such as the Ginzburg-Landau theory, type-I and type-II superconductors, and recent developments in high-temperature superconductivity.

Readings/Bibliography

Necessary Readings

The only necessary bibliography consists in Lectures Notes and Slides that we will regularly upload on the Virtuale's webpage of the course.

Suggested Readings

There are several classical textbooks that students interested in deepening their knowledge about certain topics can consult:

1) "Many-Body Quantum Theory in Condensed Matter Physics: An Introduction" by Henrik Bruus and Karsten Flensberg.

2) "Quantum Field Theory for the gifted amateur" by T. Lancaster and S. J. Bundell

3) "A Guide to Feynman Diagrams in the Many-body Problem" by Richard D. Mattuck

4) "Introduction to Many-Body Physics" by Piers Coleman

5) "Many-Particle Physics" by G. D. Mahan

6) "A Course in Quantum Many-Body Theory" by Michele Fabrizio

Teaching methods

The teaching methods for this course are primarily based on front lectures, in order to give a high-quality delivery of the complex material we will encounter during the lectures. This approach makes sure that key concepts are clearly and systematically presented. The teaching methods include:

1) Front Lectures

Each topic in the syllabus will be introduced through front lectures. These lectures will give a clear and deep explanation of theoretical concepts and physical interpretations. Key points will be stressed out, and complex ideas will be addressed incrementally to facilitate the students' understanding.

2) Blackboard and Visual Aids

Lectures will be given with visual aids such as slides that are commented and annotated on real-time (first module given by Prof. Franchini) and at the blackboard (second module by Prof. Di Sante). These visual tools will turn to be useful to illustrate the main concepts of the course, making them more accessible. Interactive demonstrations from real-time numerical simulations will also be used to show examples of correlated physical phenomena.

4) Q&A and Discussions

Each lecture, primarily during the first 15 minutes, will include dedicated time for questions and answers about topics of the course that have been dealt previously, providing students with the opportunity to clarify their doubts and foster discussions.

5) Lecture Notes and Supplementary Materials

As already stated in "Readings/Bibliography" above, lecture notes and slides will be provided to accompany each lecture. These notes and slides will include all the detailed explanations, all the key equations, and all the important derivations. In addition, supplementary materials such as research papers and additional readings will also be recommended to deepen students' understanding.

6) Homework Assignments

Homework assignments will be given regularly on a weekly-based schedule to reinforce the material covered in lectures. Their solution is voluntary and at one's discretion. These assignments will include demonstrations of mathematical results that are not covered during the lecture for time constrains, theoretical questions, computational exercises, and practical problems. This will allow students to apply their knowledge and develop a deeper understanding of the content of the course.

Assessment methods

The final exam is oral and it is designed to comprehensively assess the student's understanding of the course material. It can be taken in one of two formats:

1) Comprehensive Questions

Students will answer three questions (15 minutes per question) that cover the entirety of the course content.

2) Project Presentation

Students will present a project and answer questions on a topic provided near the end of the course (30 minutes). This will be followed by an additional question (15 minutes) on a topic unrelated to the project.

As such, for both formats, the student will have 45 minutes to complete the exam. The maximum grade that can be achieved is 30 cum laude. To pass the oral exam, the student must achieve a minimum grade of 18/30.

Teaching tools

Blackboard, Slides, Computer applications

Office hours

See the website of Domenico Di Sante

See the website of Cesare Franchini