Advanced Methods in Quantum Many-Body Theory

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

•The Bose gas and its non-interacting points: the strongly repulsive limit and its fermionization.

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

• Emergent hydrodynamics in integrable models.

2. Quantum Spin Liquids:

• 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.

• Beyond mean-field theory: Guztwiller projection.

3. Phenomenological theory of topological superconductivity:

• Short review about BCS theory and pairing order parameters

• Group theory

• Ginzburg-Landau theory

• Gorkov equation and path integral

• Topological aspects of superconductivity

4. Quantum criticality in itinerant magnetic systems:

• Short review about critical phenomena and Wilsonian renormalization group

• Stoner theory of itinerant magnetism

• Wilson-Fisher fixed point of the \phi^4 theory

• Hertz-Millis-Moriya theory and its breakdown

• Learning exact techniques to study integrable models out of equilibrium, with emphasis on transport and hydrodynamics.

• Becoming acquainted with the 1d interacting Bose gas, a paradigmatic model for cold atom experiments.

• Learning parton mean-field descriptions of quantum spin liquids using PSG and understand experimental consequences of spin fractionalization. <\p>

• Understanding the classification and topological properties of multicomponent-superconductivity. <\p>

• Understanding the critical properties of itinerant magnets and learning RG techniques with interacting fermions. <\p>

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

Knowledge of Quantum Many-Body Theory and Statistical Mechanics 2 is helpful.

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

• An introduction to integrable techniques for one-dimensional quantum systems, F. Franchini https://arxiv.org/abs/1609.02100

• Lecture notes on Generalized Hydrodynamics, B. Doyon SciPost Phys. Lect. Notes 18 (2020)

• Quantum field theory of many-body systems. Xiao-Gang Wen

• Anyons in an exactly solved model and beyond. Alexei Kitaev

• Quantum orders and symmetric spin liquids. Xiao-Gang Wen

• Group Theory and Quantum Mechanics. Michael Tinkham

• Physics of projection wavefunctions. Claudius Gros

• Introduction to Unconventional Superconductivity, V.P. Mineev and K.V. Samokhin

• Introduction to Unconventional Superconductivity, M. Sigrist

• Superconducting classes in heavy-fermion systems, G. E. Volovik and L. P. Gorkov

• Aspects of Topological Superconductivity, M. Sigrist

• Lecture Notes on Electron Correlation and Magnetism, P. Fazekas

• Lecture Notes on Electron Correlation and Magnetism, P. Fazekas

• Quantum Field Theory in Strongly Correlated Electronic Systems, N. Nagaosa

• Condensed Matter Field Theory, A. Altland and B. Simons

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