Advanced Methods in Quantum Many-Body Theory

• Introduction to semi-classical methods, focusing on many-body dynamics. Beyond mean-field dynamics: truncated Wigner approximation (TWA).

• TWA in classical phase space: heuristic arguments, and derivation from non-equilibrium Keldysh path integral. Example: dynamics of the sine-Gordon model.

• TWA for out-of-equilibrium spin systems. Construction of discrete phase space, and discrete TWA.

• Introduction to exactly solvable models in classical and quantum statistical mechanics.

• Algebraic Bethe Ansatz.

• Introduction to the thermodynamics and out-of-equilibrium properties of 1d quantum integrable models.

• Quantum quenches, lack of thermalization and relazation to the Generalized Gibbs Ensemble: the example of the 1d Bose gas.

• Emergent hydrodynamics in integrable models.

• Introduction to group theory, lattice gauge theory, and projective symmetry group (PSG) theory

• Projective construction of quantum spin liquids using PSG. Example: Kitaev honeycomb model with (1) four-Majorana exact solution (2) projective construction solution

• Beyond mean-field theory: Guztwiller projection.

• Gorkov equation and anomalous Green’s function

• Phenomenological theory of multicomponent-superconductivity

• Topological aspects of superconductivity

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.
• Explain why we can use emergent hydrodynamics to study integrable models
• Argue how to solve the algebraic PSG for square lattice antiferromagnetic Heisenberg model
• Sketch how to solve the Kitaev honeycomb model using projective construction
• Explain the energetical stabilization of specific pairing

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 70% of exercise points