THE HUBBARD-MODEL - AN INTRODUCTION AND SELECTED RIGOROUS RESULTS

The Hubbard model is a 'highly oversimplified model' for electrons in a solid which interact with each other through extremely short-ranged repulsive (CouIomb) interaction. The Hamiltonian of the Hubbard model consists of two parts: H-hop which describes quantum mechanical hoppin g of electrons, and H-int which describes non-linear repulsive interac tion. Either H-hop or H-int alone is easy to analyse, and does not fav our any specific order. But their sam H = H-hop + H-int is believed to exhibit various non-trivial phenomena including metal-insulator trans ition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Lu ttinger liquid, and superconductivity. It is believed that we can find various interesting 'universality classes' of strongly interacting el ectron systems by studying the idealized Hubbard model. In the present article we review some mathematically rigorous results relating to th e Hubbard model which shed light on the 'physics' of this fascinating model. We mainly concentrate on the magnetic properties of the model i n its ground states. We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay o f correlations at finite temperatures in two dimensions, the Yamanaka- Oshikawa-Affleck theorem on lour-tying excitations in one dimension, L ieb's important theorem for the half-filled model on a bipartite latti ce, Kubo-Kishi bounds on the charge and superconducting susceptibiliti es of half-filled models at finite temperatures, and three rigorous ex amples of saturated ferromagnetism due to Nagaoka, Mielke, and Tasaki. We have tried to make the article accessible to non-experts by giving basic definitions and describing elementary materials in detail.