Kinematics of Electrons near a Van Hove Singularity

A two-dimensional electronic system, where the Fermi surface is close to a Van Hove singularity, shows a variety of weak coupling instabilities. It is a convenient model to study the interplay between antiferromagnetism and anisotropic superconductivity. We present a detailed analysis of the kinematics of electron scattering in this model. The similarities and differences between a standard renormalization group approach and previous work based on parquet summations of ${log}^2$ divergences are analyzed, with emphasis on the underlying physical processes. General properties of the phase diagram are discussed.