Quantum Many-Body Physics

This course 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 course 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.

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

1. follow the current literature of condensed matter theory
2. start their own research project in this field
3. understand the concept of quantum phase transitions
4. use technical tools such as second quantiyation, coherent states, functional integrals, and Hubbard-Stratonovic transformations
5. explain the weaklz interacting Bose gas and the consequences of spontaneous symmetry breaking
6. apply linear response theory
7. analyze the interacting electron gas and explain the basic concepts of Fermi liquid theory
8. use functional integrals as a general technique to understtand symmetry broken phases

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

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

A. Altland, B. Simons: "Condensed Matter Field Theory"
A. L. Fetter, J. D. Walecka: “Quantum theory of many-particle systems”

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: Goldstone and Higgs modes in condensed matter, Unconventional Superconductivity, Vortices in trapped condensates, Collective excitations in superfluids, etc

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