Condensed Matter Quantum Many-Body Systems and Field Theory 2

This module develops advanced methods to study interacting quantum many-particle systems in and out of equilibrium. The module starts with a short recapitulation of field theoretical fundamentals, in particular functional integral techniques. The next part of the module introduces the renormalization group as a central tool for understanding effective low-energy properties of interacting quantum many-particle systems. Topics include scaling, perturbative renormalization, RG flows and fixed points, as well as the Kondo effect and the superfluid-Mott insulator transition as examples. The module then covers fundamentals of low-dimensional systems (Luttinger liquids, bosonization) and the Keldysh technique to study many-particle systems out of equilibrium. Final topics include instantons and non-perturbative techniques as well as optional topics, such as the quantum Hall effect (integer and fractional), Chern-Simons theory, disorder in many-particle systems, high-Tc superconductivity and quantum phase transitions.

After completing the Module the student is able to:

1. Explain the basic ideas of a renormalization group transformation.

2. Understand the concept of RG flows and RG fixed points.

3. Perform perturbative renormalization group computations.

4. Explain what the Kondo effect is.

5. Use bosonization to understand properties of low-dimensional quantum systems.

6. Explain how the Keldysh formalism is used to study non-equilibrium phenomena.

7. Explain what an instanton is.

Quantum Many-Body Systems and Field Theory I, or equivalent knowledge of second quantization, functional integrals, Green’s functions.

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 One Note presentations, blackboard.

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

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

• „Scaling and Renormalization in Statistical Physics“, J. Cardy, Cambridge University Press

• "Field Theories of Condensed Matter Physics“, E. Fradkin, Cambridge University Press

• "Field Theory of Non-Equilibrium Systems“, A. Kamenev, Cambridge University Press

• „Quantum Physics in One Dimension“, T. Giamarchi, Clarendon Press

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:

• How do you perform a renormalization group transformation?
• What is the Wilson-Fisher fixed point?
• What are critical exponents and how do they relate to RG eigenvalues?
• How does the RG flow for the Kondo effect look like?
• Define the Keldysh contour.
• How are the different components of the Keldysh Green’s function related?
• Explain what the Luttinger parameter is.
• How does the bosonized electron density operator look like?
• Explain the phenomenon of spin-charge separation.