Condensed Matter Quantum Many-Body Systems and Field Theory 1

The aim of this module is to learn basic methods of modern quantum many-body theory and to apply them to various problems in condensed matter physics. The module starts with an introduction to second quantization and its application to paradigmatic models of interacting electrons, such as the Hubbard- and Heisenberg models, the Bogoliubov theory of weakly interacting bosons, Hartree-Fock mean-field theory and the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. The subsequent, main part of this module develops functional integral techniques for bosons and fermions in the finite-temperature Matsubara formalism, discusses Green’s functions and their analytic properties, and introduces perturbation theory using Feynman diagrams and elementary non-perturbative methods such as the Hubbard-Stratonovich transformation. These methods are then used to study properties of interacting electron systems (random-phase approximation, screening and plasmon excitations) and to discuss Fermi liquid theory. The next chapter covers the linear response formalism (Kubo formula) as the central tool to establish a connection between theoretically computable correlation functions and experimental observables. The final core topic is an extended discussion of the BCS theory of superconductivity, starting from the functional integral representation.

After completing the Module the student is able to:

1. Understand and apply the formalism of second quantization to study interacting quantum many-particle systems.

2. Explain the main ideas behind common approximation schemes, in particular mean-field theory and the Bogoliubov transformation.

3. Understand the functional integral representation of partition functions, manipulate functional integrals, and apply a Hubbard-Stratonovich decoupling.

4. Explain the properties of Green’s functions and their use in diagrammatic perturbation theory.

5. Understand and use the linear response formalism to compute experimental observables of interacting many-particle systems.

6. Understand the theory of BCS superconductivity.

7. Follow current research topics and use the toolbox of many-body methods to start independent research.

Quantum mechanics, statistical physics, solid state physics, at the level of elementary modules from a Bachelor’s degree in physics.

The module consists of a lecture series (4 SWS) and exercise classes (2 SWS), comprising two lecture sessions and one exercise session per week. 

The main teaching material is presented on the blackboard or by beamer. Lectures are supplemented by weekly problem sets, deepening the understanding of core concepts through concrete calculations. Solutions to the problem sets are discussed in the exercise sessions.

Participation in the exercise classes is strongly recommended, since the exercises are aids for acquiring a deeper understanding of the core tools of condensed matter many-body physics and field theory and for practicing to solve typical exam problems.

Power point and Keynote presentations, blackboard.

Standard textbooks on many-body theory, e.g.:

• „Condensed Matter Field Theory“, A. Altland, B. Simons, Cambridge University Press

• „Introduction to Many-Body Physics“, P. Coleman, Cambridge University Press

• "Many-Body Quantum Theory in Condensed Matter Physics: An Introduction“, H. Bruus, K. Flensberg, Oxford University Press

• „Quantum Many-Particle Systems“, J.W. Negele, H. Orland, Perseus Books

• „Many-particle physics“, G.D. Mahan, Springer

• „Interacting Electrons and Quantum Magnetism“, A. Auerbach, Springer

There will be a written exam of 180 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using conceptual questions and computational tasks.

For example an assignment in the exam might be:

• What is Fock space?
• How does the mean-field approximation work?
• Write down the functional integral representation of the partition function for electrons with pairwise interactions.
• What is the difference between the retarded and the advanced Green’s function?
• What are the Kramers-Kronig relations?
• Explain the Dyson-equation and its relation to the self-energy operator.
• What is the random phase approximation?
• How is the electrical conductivity related to the current-current correlation function?
• What is a plasmon and how does its dispersion look like?