Luttinger's theorem is one of the most important results in the Fermi liquid theory. Luttinger's original proof of the theorem was based on perturbation theory which is rigorous but unintuitive. By using a flux insertion trick and large gauge transformation, Oshikawa gave a topological proof of the theorem [1]. In this talk, I will introduce these techniques to prove Luttinger's theorem. Furthermore, I will apply these techniques to Kondo lattice to show this model hosts a “large fermi surface”. Finally, I will discuss other applications of these techniques [2].
Reference:

Masaki Oshikawa, "Topological Approach to Luttinger's Theorem and the Fermi Surface of a Kondo Lattice", Phys. Rev. Lett. 84, 3370 (2000).

Masaki Oshikawa, "Commensurability, Excitation Gap, and Topology in Quantum ManyParticle Systems on a Periodic Lattice", Phys. Rev. Lett. 84, 1535 (2000).