COLLECTIVE PROPERTIES OF CONDENSED MATTER SYSTEMS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
To fully appreciate the course, the student should have some aquaintince with Solid State Physics, Statistical Physics and Many body Physics.
The final exam consists in an oral test centered on a few subjects presented in the course. The student should be able to derive the results obtained in class. Moreover, the student will be also asked to use the notions learned in the course to identify the physical mechanisms at play in some practical example posed by the teacher. To pass the exam the student should correctly answer the initial, rather basic, questions. The successful student will face more demanding problems during the discussion in order to establish the final mark.
In the last forty years, new conceptual paradigms in condensed matter physics emerged, beyond the standard single particle picture usually adopted in solid state physics. The use of field theory methods, developed in the framework of high energy physics, was instrumental to uncover novel descriptions of interacting systems in condensed matter. New symmetries, able to explain puzzling experimental facts, emerged at long wave-lengths, paving the way to new predictions. The topological characterization of either single particle and collective states proved extremely useful in classical and quantum magnets, in the correct interpretation of the quantum Hall effect and, more recently, in topological insulators and semi-metals. The aim of this course is to provide an introduction to the use of field theory methods and topological considerations in condensed matter systems. The new concepts will be introduced starting from fully workable examples and then generalized to more complex cases.
At the end of the course the student will be able to:
1) critically read and understand the recent literature on these subjects;
2) recognize the underlying general features of an interacting system on the basis of the paradigms presented in the course: global symmetries and topological properties;
3) use the appropriate field theory to describe the long wave-length behavior of the model.
The course is divided into two main sections and a topical additional part.
A) Classical systems.
1) Classical fluids. The origin of inter-particle interactions. X-ray and neutron scattering techniques. The description of correlations.
2) Density functional theory. The freezing transition. Discrete and continuous symmetries. Symmetry breaking. Long wave-length fluctuations and effective field theory. The critical point. Mermin-Wagner theorem.
3) Lattice models. The two dimensional XY model and its realizations. Low and high temperature expansion. Topological defects and effective interactions. Mapping to a two dimensional Coulomb gas.
B) Quantum magnets.
1) Spontaneous symmetry breaking in quantum mechanics. Goldstone modes.
2) Coherent states. Low energy limit and mapping to a classical system. Berry phase, topological term and Haldane conjecture.
3) Solution of the two dimensional Ising model by use of Majorana fermions.
4) The Heisenberg antiferromagnet in one dimension. Luttinger liquids.
C) Topical part.
Considering the interests of the audience, we will examine one of the following topics:
1) Topological insulators and the Chern-Simons term.
2) Integer quantum Hall effect and its topological origin.
3) Fractional quantum Hall effect and the statistics of quasi-particles.
The reference book for this course is "Modern Condensed Matter Theory" by S.M. Girvin and K.Yang, Cambridge University Press (2019).
A useful reference is: "Principles of Condensed Matter Physics" by P.M. Chaikin and T.C. Lubensky, Cambridge University Press (2000).
Another reference is the review paper by J.B. Kogut "An introduction to lattice gauge theory and spin systems", Rev. Mod. Phys. 51, 659, (1979).
The course is essentially based on lectures where the teacher presents each topic in full detail, including mathematical derivations followed by a discussion with the students on the physical implications of the results.
The teacher is available for questions by appointment. His e-mail is:
alberto.parola@uninsubria.it