Advanced Methods in Quantum Many-Body Theory

Quantum many-body systems can exhbit extremely rich phenomena, ranging from novel phases of matter to exotic non-equilirbrium physics. Tackling such systems theretically is very challenging and requires advanced methods. This module serves as an introduction to several of these advanced analytical methods. The following topics are covered in this module:

• Bethe Ansatz techniques for zero and finite temperature thermodynamic properties of 2D classical statistical mechanical problems, 1D quantum spin chains, and 1D quantum field theories.
• The algebraic structure of topological quantum field theories (TQFTs).
• The string-net Hamiltonian construction for topologically ordered phases.
• Semiclassical methods to study out of equilibrium dynamics and transport in many-body systems, truncated Wigner approximation, quenches in the sine-Gordon model and in 1D spin chains.

After successful completion of the module, the students have an overview of several methods to analyze strongly interacting many-body systems, relevant in modern condensed matter physics. In particular, the students are able to

• Compute exactly thermodynamic properties of certain one-dimensional quantum spin models, certain 2D classical statistical mechanical models and certain interacting 1D quantum field theory models 
• Compute the value of TQFT planar diagrams corresponding to the braiding of abelian and non-abelian anyons.
• Study non-equilibrium dynamics within the framework of various semclassical approaches.

No preconditions in addition to the requirements for the Master’s program in Physics.

The lecture is designed for the presentation of the subject, usually by blackboard presentation. The focus resides on theoretical foundations of the field, presentation of methods and simple, illustrative examples. Command of basic methods is deepened and practised through homework problems, which cover important aspects of the field. The homework problems should develop the analytic skills of the students and their ability to perform calculations. The homework problems are discussed in the exercise classes by the students themselves under supervision in order to develop the skills to explain a physics problem logically.

Blackboard lectures, written notes for download, exercise sheets, course homepage

The achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using presentations independently prepared by the students. The exam of 25 minutes consists of the presentation and a subsequent discussion.

For example an assignment in the exam might be:

• Sketch the derivation of the (standard phase space) TWA from the Keldysh path integral.
• Describe the main steps of the discrete (or cluster) truncated Wigner approximation for a spin-1/2 chain.
• Sketch the derivation of the diagonalization of the Hamiltonian of the XXZ chain using Algebraic Bethe Ansatz.
• Calculate the ground state from the Bethe equations.
• What are the modular S and T matrices?
• Describe the string-net construction for the toric code.

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.

There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to the exam period directly following the lecture period (not to the exam repetition) and subject to the condition that the student passes the mid-term of active participation in the tutorials and at least 50% of exercise points