# 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 2018/9

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 2020/1 | WS 2019/20 | WS 2018/9 | WS 2017/8 |

### Basic Information

PH2256 is a semester module in German or English language at 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 2018/9 was Michael Knap.

### Content, Learning Outcome and Preconditions

#### Content

This module provides a modern introduction to many-body physics. It covers basic theoretical methods and their application to various problems of condensed matter theory, such as the interacting electron gas, phonons in solids, quantum magnetism, and superconductivity. Throughout the class relations between experiments and theory will be emphasized. This module will provide students the basic knowledge to follow state-of-the-art research in condensed matter physics and to be able to start their independent research project in that field.

Outline:

Introduction

(1) Landau Theory

(2) Quantum phases of matter

(3) Second Quantization

(4) Transverse Field Ising Model

Functional Field Integrals

(5) Feynman's Path Integral in Single-particle QM

(6) Bosonic and Fermionic Coherent States

(7) Functional Field Integrals for the Partition Function

Weakly Interacting Bose Gas

(8) Non-interacting bosons

(9) Weakly interacting bosons

(10) Consequences of a broken continuous symmetry

(11) Superfluidity

(12) Thermal disorder and BKT transition

Linear Response Theory

(13) Response functions

(14) Fluctuation-dissipation relations

(15) Analytic Properties of Correlation Functions

(16) Sum rules

(17) Structure Factor of a Superfluid

Fermi-Liquid Theory

(18) The non-interacting Fermi gas

(19) The main results of Fermi-Liquid Theory

(20) Quasi-particle excitations

(21) Interacting fermion Greens functions and self energy

(22) Momentum distribution function

(23) Landau's phenomenological approach

The interacting electron gas

(24) Hatree-Fock Approximation

(25) Coulomb interactions

(26) Screening and random phase approximation

(27) Collective modes

A general framework for studying broken symmetries and collective behavior

(28) Hubbard-Stratonovic transformation

(29) Functional integral perspective on the interacting electron gas

(30) Superconductivity

(31) Fluctuations and Ginzburg-Landau Theory

(32) Anderson-Higgs Mechanism

(33) Flux quantization and vortices in superconductors

(34) Spin-Liquids

The practical classes support the lectures with tutorials and problem sets. The tutorials provide complementary perspectives and the problem sets will help to understand and deepen the physical concepts presented in the lecture.

#### Learning Outcome

Students who have successfully participated in this module are able to:

- understand the concept of quantum phase transitions
- use technical tools such as second quantiyation, coherent states, functional integrals, and Hubbard-Stratonovic transformations
- explain the weakly interacting Bose gas and the consequences of spontaneous symmetry breaking
- apply linear response theory
- analyze the interacting electron gas and explain the basic concepts of Fermi liquid theory
- use functional integrals as a general technique to understtand symmetry broken phases

#### 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 | Pollmann, F. |
Mon, 10:00–12:00, virtuell Wed, 10:00–12:00, virtuell |
eLearning |

UE | 2 | Exercise to Quantum Many-Body Physics |
Responsible/Coordination: Pollmann, F. |
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

Blackboard presentation in combination with computer presentations to discuss experimental results.

#### Literature

- A. Altland & B.D. Simons:
*Condensed Matter Field Theory*, Cambridge University Press, (2010) - A.L. Fetter & J.D. Walecka:
*Quantum Theory of Many-Particle Systems*, Dover Publications, (2003)

### Module Exam

#### Description of exams and course work

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 of the results and a subsequent discussion.

For example an assignment in the exam might be:

- Describe Goldstone and Higgs modes in condensed matter.
- Explain the principles of unconventional superconductivity.
- Which role do vortices play in trapped condensates?
- Which collective excitations exist in superfluids? Describe them.

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

#### Exam Repetition

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