7 - Noninteracting Bands

Summary

1. Introduce the basic concepts of single-particle dispersion in a lattice.

2. Introduce maximally localized Wannier wave functions and their importance in constructing interacting models.

3. Introduce the basic idea of quantum simulation.

4. Introduce the Dirac point in the honeycomb lattice and the concept of a semimetal.

5. Illustrate the Dirac and the Weyl points as topological defects in the momentum space.

6. Introduce the concept of the symmetry protection for the topological stability of the Dirac point.

7. Introduce the Su-Schrieffer-Heeger model and the concept of symmetry-protected topology.

8. Introduce the Haldane model and the topological phase transition in the Haldane model.

9. Summarize the common mathematical structures between topological characters of a band theory and topological excitation in a Bose condensate.

10. Discuss the physical consequences of topological invariants for a band insulator in both near-equilibrium transport and the far-from-equilibrium quench dynamics.

11. Discuss the equivalence between neural atoms in a moving lattice and charged particles in an electric field.

12. Introduce the Floquet theory and the Floquet effective Hamiltonian from thehigh-frequency expansion.

13. Discuss various applications of periodical-driven optical lattices.

14. Introduce the Hamiltonian for atom-cavity system and highlight the role of the Langevin force term.

15. Introduce the superradiant quantum phase transition of a Bose condensate coupled to cavity light.