Modern Topics in Condensed Matter Physics

Catalog description

This course explores foundational and advanced topics in condensed?matter physics, emphasizing the role of topology in understanding the electronic properties of crystalline materials. We begin by introducing the very basic concepts of a Bravais lattice, quantum numbers, and Bloch's theorem. Next, the tight?binding method for a one?dimensional chain of atoms, along with the concepts of Bloch and Wannier functions, provides a framework for modeling electrons in crystals. We then consider tight?binding electrons in a crystal subjected to electric and magnetic fields, introducing the Wannier equation and the Peierls substitution. Then, essential concepts—such as the adiabatic theorem and Berry's set the stage for understanding topological phases of matter. As an example, we consider Thouless's charge pump. After introducing Second Quantization as the modern language of condensed-matter physics, we cover core models—such as the Su-Schrieffer-Heeger (SSH), Hubbard, Heisenberg and Ising models—which introduce concepts of quantum phase transitions, topological phases and states, including Majorana fermions. Furthermore, the course examines electron behavior in crystals under magnetic fields, covering topics such as the Integer Quantum Hall Effect, Hofstadter's model and butterfly, and representation of the Hall conductivity as a topological invariant. Finally, we explore simple models of Chern insulators and Haldane's model as a toy example of Quantum Anomalous Hall Effect. Prerequisite: 33-234