In this lecture, and the next, I'll be emphasizing the essential role of fluctuations, particularly magnetic fluctuations, in the two-dimensional case.
The lecture will begin with a description of these fluctuations using an effective low-energy theory: the non-linear sigma model. I will then show how the coupling of these fluctuations to fermionic degrees of freedom leads to a " pseudogap " in the spectrum of excitations in the weak coupling regime when the correlation length is greater than the thermal wavelength (Vilk and Tremblay criterion). I will also present some results of a recent study comparing a large number of numerical methods in weak/intermediate coupling.
