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 (Coulomb) interaction.The Hamiltonian of the Hubbard model consists of two parts: H hop which describes quantum mechanical hopping of electrons, and H int which describes non-linear repulsive interaction.Either H hop or H int alone is easy to analyse, and does not favour any specific order.But their sum H = H hop + H int is believed to exhibit various non-trivial phenomena including metalinsulator transition, antiferromagnetism, ferrimagnetism, ferromagnetism, Tomonaga-Luttinger liquid, and superconductivity.It is believed that we can find various interesting 'universality classes' of strongly interacting electron systems by studying the idealized Hubbard model.In the present article we review some mathematically rigorous results relating to the Hubbard model which shed light on the 'physics' of this fascinating model.We mainly concentrate on the magnetic properties of the model in its ground states.We discuss the Lieb-Mattis theorem on the absence of ferromagnetism in one dimension, Koma-Tasaki bounds on the decay of correlations at finite temperatures in two dimensions, the Yamanaka-Oshikawa-Affleck theorem on low-lying excitations in one dimension, Lieb's important theorem for the half-filled model on a bipartite lattice, Kubo-Kishi bounds on the charge and superconducting susceptibilities of halffilled models at finite temperatures, and three rigorous examples 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. Contents1. H Tasaki 5.1.Perturbation for U t 5.2.Lieb's theorem 5.3.Lieb's ferrimagnetism 5.4.Kubo-Kishi bounds on susceptibilities at finite temperatures 6. Ferromagnetism 6.1.Instability of ferromagnetism 6.2.The toy model with two electrons 6.3.Nagaoka's ferromagnetism 6.4.Mielke's ferromagnetism and flat-band ferromagnetism 6.5.Ferromagnetism in a non-singular Hubbard model