Quantum Many-Body Physics

This module provides an introduction into quantum many-body physics. We will cover basic quantum field-theoretical methods and their application to various many-body problems of condensed matter theory, such as Fermi and Luttinger liquids, superfluids and superconductors and quantum Hall fluids. This module will provide students with a basic knowledge for starting independent research in quantum condensed matter physics. The exercise classes will supplement the lectures with regular instructions and problem sets. The problem sets will help to understand and deepen the physical concepts presented in the lecture.

Outline:

• Introduction into quantum many-body problem:
  emergence and collective behavior, quantum fields, second quantization
  
  
• Path integral formulation of quantum field theory: 
  single-particle quantum mechanics from the path integral, coherent states and functional integrals, partition function as a functional integral
    
• Linear response theory
  response functions, fluctuation-dissipation theorem, conductivity of Fermi gas
  
  
• Fermi liquid theory:
  Fermi liquid ground state, quasiparticles and their stability, collective modes, Landau damping, non-Fermi liquids
  
  
• Luttinger liquids:
  pecularities of physics in one dimension, Luttinger model, basic of bosonization, correlation functions, relation to conformal field theories and two-dimensional classical XY model
  
  
• Superfluids and superconductors:
  physical properties of superfluids and superconductors, BCS theory, phase stiffness, vortices, rotating superfluids, boson-vortex duality in two dimensions, Berezinskii-Kosterlitz-Thouless transition, chiral superfluids and superconductors 
  
  
• Quantum Hall fluids:
  basics of quantum Hall effect, flux attachment and Chern-Simons theory, topological order and anyons in fractional quantum Hall fluids, abelian Chern-Simons theory and the hierarchy of quanum Hall states, edge of quantum Hall fluids, non-abelian quantum Hall states, Dirac fermion duality

After successful completion of this module students will be able to

• apply field theory techniques in condensed matter physics
• use second quantization, coherent states, path integrals, linear resonse theory to solve many-body problems
• understand theoretical paradigms that are central in modern condensed matter physics
• have a working knowledge of the physics of Fermi liquids, one-dimensional Luttinger liquids, superfluids and superconductors, quantum Hall liquids

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

The course consists of a lecture and exercise classes. 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 exercises by the students themselves under the supervision of a tutor in order to develop the skills to explain a physics problem logically.

In classroom lectures the content is presented on blackboard. Questions from students are welcome. The handwritten script will appear on the web-page of the lecture, where students can also find the relevant literature for self-study.

In tutorials we will discuss the solutions of home-work assignments which would provide students with a practical knowledge of the material discussed in the lecture and will prepare them for doing research in condensed matter physics.

• P. Coleman, Introduction to Many-Body Physics
• A. Altland & B. Simons, Condensed Matter Field Theory 
• T. Giamachi, Quantum Physics in One Dimension
• E. Fradkin, Field Theories of Condensed Matter Physics
• X.-G. Wen, Quantum Field Theory of Many-Body Systems

There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Using perturbation theory, calculate the Landau parameters for fermions with a weak short-range potential
• What is the main difference between Luttinger and Fermi liquids
• Analyze the fermionic energy spectrum and derive the Chern number of a chiral superconductor
• What are anyons and why they emerge only in two-dimensional world?

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