8 - The Hubbard Model

Summary

1. Introduce quantum phase transition in the Bose-Hubbard model.

2. Emphasize vanishing energy scales and universality at the quantum critical point.

3. Introduce microscopic theories to describe the quantum phase transition, and explicitly show the vanishing of energy scales and the critical exponents.

4. Illustrate the emergent Lorentz symmetry and the Higgs mode at the quantum critical point of the Bose-Hubbard model with the particle-hole symmetry.

5. Discuss an experimental probe of the superfluid to the Mott insulator phase transition.

6. Show that the repulsive and the attractive Fermi-Hubbard models are related by the particle-hole symmetry.

7. Discuss the origin of antiferromagnetic order in the repulsive Fermi-Hubbard model at half-filling.

8. Introduce the enlarged $SO(4)$ symmetry of the Fermi-Hubbard model at half-filling and zero spin imbalance and its physical concequence.

9. Introduce important unsolved challenge issues in the Fermi-Hubbard model.

10. Introduce the concept of eigenstate thermalization hypothesis and many-body localization as opposite to thermalization.

11. Introduce a few metrics to characterize the many-body localization.

12. Emphasize the role of entanglement entropy in characterizing quantum thermalization, and discuss how to measure entanglement entropy in ultracold atom systems.