Fundamentals of Quantum Materials

Course goals

The objective of this course is two-fold: First, it should provide students with the necessary background to follow more specialized courses on various types of quantum materials. Second: it should provide students with a first glimpse of the various types of quantum materials that are studied at Utrecht University, both experimentally and theoretically.

The learning objectives are:
1. The student is familiar with the second quantization formalism and can use this formalism to compute the band-structure of simple crystals.
2. The student has elementary knowledge of (crystalline) topological insulators, including topological invariants.
3. The student can apply the Hubbard model and understands Mott-Hubbard insulators.
4. The student is familiar with the classical Heisenberg model to describe ferromagnets and antiferromagnets.
5. The student understands the phenomenological description of superconductors based on London theory (Meissner effect)

Content

Place in the curriculum
The course “Fundamentals of Quantum Materials” is the first in a set of four courses in ‘Quantum Materials’. It covers the quantum mechanics needed for a basic understanding of these materials and provides a brief introduction to three types of quantum materials (topological insulators, superconductors and magnetic materials) that are currently being investigated experimentally and theoretically at Utrecht University. The content of this course is a prerequisite to follow the more specialized courses on topological insulators (SK-MQUTM), superconductors (SK-MQUSU) , and magnetic materials (SK-MMAMA)

Brief course content
This course covers the formalism of quantum mechanics, and how it can be applied to describe the electronic properties of crystalline solids. First, we will cover the required background in quantum mechanics, building up to the second quantization formalism. This method is then applied to crystals to compute band structures. Subsequently, we’ll see how topological states of matter can emerge. We then introduce the simplest model to describe electron-electron interactions, the Hubbard model and use it to understand Mott-insulators. Furthermore, using this model, we will find that for low energies only the spin-degree of freedom remains, leading to the Heisenberg model and a description of ferro- and anti-magnetism. Finally, a phenomenological description of superconductivity is given.