Quantum matter
The goal of this course is to introduce somewhat "advanced" topics in quantum matter, tackle truly quantum-entangled, strongly interacting, phases of matter and materials, and present how quantum matter is a particularly rich field, with many open theoretical problems.
We will briefly review introductory topics in solid state/condensed matter physics for those who have not had a first class on the subject, and introduce / review second quantization. We will emphasize the role and consequences of symmetries, including time-reversal symmetry, and symmetry-based theorems. We will see how insulators can appear from adding interactions to free electrons in metals and how spin-spin interactions emerge, and we will also introduce enough quantum chemistry to understand how to build models for materials. In order to illustrate how new phenomena emerge from many interacting particles, we will study several spin models where long-range quantum entanglement emerges, with its host of fractional particles. We will introduce the theoretical tools to understand several experimental techniques, as well as a few modern numerical techniques.
- Motivation and introduction
- What is quantum matter?
- Scales of quantum matter
- The Hamiltonian
- Basics
- Second quantization
- The translation operator
- Bloch's theorem et al.
- Tight-binding
- Band conductors and insulators
- Some interacting models
- The Hubbard model
- Anderson exchange
- The Haldane chain model
- Microscopics et al.
- Symmetries, phase transitions
- Wigner's theorem
- Time reversal symmetry and Kramers theorem
- Phases and phase transitions
- Goldstone's theorem
- Spin dynamics of product states
- The Heisenberg ferromagnet: an exact solution
- Spin-waves
- Quantum spin liquids
- General features
- Fractionalized particles
- The Kitaev honeycomb model
- Experimental probes
- Numerical techniques
Written exam
Required:
- Quantum mechanics
- Statistical physics
A plus, but not required:
- A first course in condensed matter/solid state physics
- Advanced quantum mechanics
- Advanced statistical physics
- Introduction to field theory
- Introduction to atomic and molecular physics
