Quantum Many-Body Physics in Non-Equilibrium

This module provides an introduction to the non-equilibrium many-body theory of quantum systems, required for the description of transport phenomena in quantum systems and their response to external perturbations, and of modern non-equilibrium measurements in condensed matter systems. The fundamental theoretical concepts to be taught are non-equilibrium Green’s functions and the Keldysh formalism. The module starts with a short repetition of second quantization and important models of condensed matter physics. It proceeds to formulating the non-equilibrium evolution of quantum systems and the evaluation of time-dependent averages and correlations based on contour integration (Keldysh formalism). After introducing the hierarchy of many-particle Green’s functions, the module elucidates the physical content of (non-equilibrium) single-particle and two-particle Green’s functions. In order to actually evaluate Green’s functions, the module will then move through several of the key approximations made, mean-field approximations, Kadanoff-Baym conserving approximation, and many-body perturbation theory. The formalisms will then be applied to important problems of non-equilibrium many body theory, namely linear-response theory and transport phenomena. The use of the formalisms will be illustrated by their use for the understanding of actual experiments.

After completing the Module the student is able to:

1. Explain and use the main models of interacting condensed-matter systems

2. Understand the physical meaning and use of single-particle and two-particle Green’s functions out of equilibrium

3. Set up calculations of physical observables and correlations in non-equilibrium quantum systems using the Keldysh contour formalism 

4. Understand and use the main approximations required to actually evaluate the formal expressions of 2 and 3.

5. Derive and use key results from quantum linear response and transport theory.

In addition to the requirements for the Master’s program in Quantum Science and Technology, the completion of a module in (equilibrium) quantum many-body physics is advisable.

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 will be presented on the blackboard. This will be supplemented by power point / keynote presentations to summarize / illustrate state-of-the-art applications of the methods taught in the module. Weekly problem sets are offered to obtain a better comprehension of the lecture content and to improve their familiarity with them. The solutions to the problem sets are discussed in weekly exercise classes.

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

Power point /keynote and One Note presentation, blackboard.

Standard textbooks of quantum many-body theory that also treat non-equilibrium, e.g.:

• Gianluca Stefanucci, Robert van Leeuwen, Non-equilibrium Many-Body Theory of Quantum Systems, Cambridge University Press 2013: Comprehensive textbook on non-equilibrium theory in a unified presentation  

• Jorgen Rammer, Quantum Transport Theory, Perseus Books, 1998: Non-equilibrium quantum theory with special interest in transport phenomena

• Henrik Bruus, Karsten Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press, 2004: Popular introduction to equilibrium quantum many-body theory with an introduction to non-equilibrium phenomena, laying the ground for more advanced non-equilibrium problems

• Leonid Keldysh, Progress in Non-equilibrium Green’s functions II, ed. M. Bonitz and D. Semkat, World Scientific 2003: Perspective on the field by its leading inventor

There will be a written exam of 120 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:

• Derive the formal solution of a non-equilibrium quantum problem in the Keldysh contour formalism.
• Discuss the physical content of a non-equilibrium Green’s function.
• Explain the Luttinger-Ward theorem.
• What is the linked cluster theorem?
• Calculate time-dependent screening in an electron gas.
• Applications of the Landauer-Büttiker formula