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 acquaintance with Solid State Physics, Statistical Physics and Many body Physics.
The final exam consists in the discussion of the assignments. Each student has to present a written report with all the details of the solution to the assignment and then illustrate to the class the main physical concepts emerging in the assignment.
The student should make use of the notions learned in the course to identify the physical mechanisms at play in the assignment. To pass the exam the student should correctly solve the assignment. In order to establish the final mark, the successful student will face more demanding questions on the specific assignment and on its relations to other concepts discussed in the lectures.
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 which can be covered as a results of assignments for the students.
A) Classical systems.
1) Classical fluids. The concept of coarse-grain. The origin of inter-particle interactions. X-ray and neutron scattering techniques. The description of correlations. Freezing of hard spheres.
2) Discrete and continuous symmetries. Lattice models and symmetry breaking in one dimension: Peierls argument. Linear response theory. Long wave-length fluctuations and effective field theory. Classical "Goldstone theorem" and Mermin-Wagner theorem. Examples of symmetry breaking.
3) 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. Superfluid stiffness.
B) Quantum magnets.
1) Introduction to the second quantization formalism.
2) Spontaneous symmetry breaking in quantum mechanics. Coherent states and Golstone Theorem. Solution of the quantum XY model in one dimension.
3) Heisenberg antiferromagnets. Low energy limit and mapping to a classical system. Non linear sigma models. Berry phase, topological term and Haldane conjecture.
C) Topical part.
Considering the interests of the audience, the students will be given assignments on more advanced topics, like, for instance:
1) Topological insulators and the Chern number.
2) Integer quantum Hall effect and its topological origin.
3) Solution of the two dimensional Ising model by use of Majorana fermions.
4) Fractional quantum Hall effect and the statistics of quasi-particles.
5) Symmetry breaking in superfluidity and superconductivity.
The course covers a collection of topics in classical and quantum condensed matter physics.
Two reference books on these topics are:
"Modern Condensed Matter Theory" by S.M. Girvin and K.Yang, Cambridge University Press (2019).
"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 first two parts of the course are 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 topical part is instead covered by assignments performed by the students and then presented to the class.
The teacher is available for questions by appointment. His e-mail is:
alberto.parola@uninsubria.it