# Quantum Many-Body Physics

## Module PH2256

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of WS 2021/2

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions | |||||
---|---|---|---|---|---|

WS 2022/3 | WS 2021/2 | WS 2020/1 | WS 2019/20 | WS 2018/9 | WS 2017/8 |

### Basic Information

PH2256 is a semester module in English language at Master’s level which is offered in winter semester.

This Module is included in the following catalogues within the study programs in physics.

- Theory courses for condensed matter physics
- Theory courses for Applied and Engineering Physics
- Focus Area Theoretical Quantum Science & Technology in M.Sc. Quantum Science & Technology
- Complementary catalogue of special courses for nuclear, particle, and astrophysics
- Complementary catalogue of special courses for Biophysics
- Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workload | Contact hours | Credits (ECTS) |
---|---|---|

300 h | 90 h | 10 CP |

Responsible coordinator of the module PH2256 in the version of WS 2021/2 was Frank Pollmann.

### Content, Learning Outcome and Preconditions

#### Content

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

- 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

#### Learning Outcome

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

#### Preconditions

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

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

Type | SWS | Title | Lecturer(s) | Dates | Links |
---|---|---|---|---|---|

VO | 4 | Quantum Many-Body Physics | Knap, M. |
Mon, 08:30–10:00, PH HS3 Wed, 16:00–18:00, PH HS2 |
eLearning |

UE | 2 | Exercise to Quantum Many-Body Physics |
Arzi, O.
Birnkammer, S.
Zechmann, P.
Zerba, C.
Responsible/Coordination: Knap, M. |
dates in groups | |

UE | 2 | Open Tutorial to Quantum Many-Body Physics |
Responsible/Coordination: Knap, M. |
dates in groups |

#### Learning and Teaching Methods

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.

#### Media

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.

#### Literature

- 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

### Module Exam

#### Description of exams and course work

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

#### Exam Repetition

The exam may be repeated at the end of the semester.