Graduate

Assumes second quantization and the basics of many-body theory.

• Cuprates and the Mott Problem Mott Physics in Correlated Lattices Why high-temperature superconductivity forced condensed-matter physics to take strong electron correlation seriously — and why the story begins with an insulator that band theory says should be a metal.
• The Hubbard Model Mott Physics in Correlated Lattices The minimal model of reference for the cuprates — one hopping amplitude and one on-site repulsion — together with its particle–hole symmetry, Green's functions, and the two limits (atomic and non-interacting) that already reveal the Mott insulator.
• Weak Coupling: Mean-Field Theory and the Slater Insulator Mott Physics in Correlated Lattices Starting from the non-interacting limit, a mean-field decoupling of the Hubbard interaction with staggered magnetization gives an antiferromagnetic two-band insulator. Solving the self-consistent gap equation reveals a Slater gap that opens exponentially at weak coupling and crosses over to the Mott gap Δ → U/2 at strong coupling.
• Strong Coupling: Perturbation Theory and Superexchange Mott Physics in Correlated Lattices In the opposite limit U ≫ t, treating hopping as a perturbation and projecting out doubly occupied sites with a Schrieffer–Wolff transformation maps the half-filled Hubbard model onto the antiferromagnetic Heisenberg model, with superexchange J = 4t²/U.