An introduction to the Hubbard model

In these notes we review some of the basic features of the 2D Hubbard model, thought of as the appropriate model for the description of the Cu - O planes in the cuprate superconductors. We discuss briefly the weak-coupling regime of the model and, in the opposite limit, the mapping of the one band Hubbard model onto an AFM Heisenberg model at half filling and onto the t - J model below half filling. We discuss next Emery's three band model and its mapping onto the so-called "spin-fermion" model. Its continuum limit is discussed by making use of an adiabatic followed by a gradient expansion. We review briefly how the model maps onto a nonlinear sigma model and some of the features of the latter.