Collective Quantum Phenomena in Condensed Matter

Description

1. The course is based mainly on the book: “Introduction to Many-Body Physics” by Piers Coleman, 2015

2. Topics related to the RPA and Hartree-Fock approximations are partially based on the book: "Quantum Theory of Many-Particle Systems" (Dover Books on Physics) by A.L. Fetter, J.D. Walecka "

3. The topic related to Cooper instability in superconductors can be found in the book "Methods of Quantum Field Theory in Statistical Physics" by by A. A. Abrikosov , L. P. Gorkov , I. E. Dzyaloshinski

4. The topic related to Bose superfluidity is based on the book “Statistical physics” by R.K. Pathria, 1996

The basic strategy to complete the course is to solve Home work problems using the material explained during lectures. There are several sets of Home works on different topics of the course. During the exercise sessions we discuss these problems which students can hand-in afterwards. In principle attendance is not necessary but it will make it much faster to solve the HW assignments. The completion of the course (pass/fail) is then solving more than 1/2 of the HW problems. This would allow you to get practical experience with basic techniques o the course and at the same time should not be too time-consuming.

Course web page on Jyväskylä University site.

LEARNING OUTCOMES

At the end of this course, students will be able to

• Explain the role of interactions in many-body systems: electron in metals, atoms in quantum liquids and gases

• Explain the most common models of many-body systems such as the concepts of quasiparticles, Hartree-Fock approximation, Fermi liquid theory, Hubbard model

• Use basic theoretical tools such as the second quantization formalism, many-body Green's functions and Feynman diagram technique

• Explain superconductivity in metals and superfluidity in quantum liquids. Apply BCS model and Ginzburg-Landau theory to describe magnetic and thermodynamic properties of superconductors.

• Use Stoner and Hubbard models to describe magnetic phenomena in metals.

• Give a presentation of the scientific paper related to the topics of the course

CONTENT

• Second quantization, causal, retarded and advanced Green’s function of the many-body system. Free fermion and phonon propagators. Relation to observables. Connection between different types of the Green’s functions: retarded/advanced, real-time and imaginary time.

• Concept of quasiparticles

• Perturbation theory: Wick’s theorem, Feynman rules. Self-energy, Dyson’s equation, polarization operator. Example of Coulomb screening and plasma waves

• Hartree-Fock approximation, ground state energy of interacting system, stability of metals and Stoner criterion of magnetism

• Fermi liquid theory: susceptibilities, zero sound and spin waves

• Methods of the many-body theory in superconductivity. Cooper problem and pairing instability in particle-hole channel Green’s functions of a superconductor. Gor’kov equations, Bogolubov-de Gennes equations. Quasiparticles in superconductors.

• Ginzburg-Landau theory, Meissner effect, Abrikosov vortices and Anderson-Higgs mechanism.

• Magnetism in Hubbard model

• Antiferromanetism

• Bose systems: condensation, superfluidity in weakly interacting Bose gas. Gross-Pitaevskii equation.