Basic diffusion propagator in small freq. & wave number limit Gaussian Propagators Paramagnon propagator spin-singlet spin-triplet heat diffusion Basic diffusion propagator in small freq. & wave number limit

Perturbation theory to all orders 3 points vertices The one-loop renormalization is a finite number (scale like (d-2))  n-loop skeleton diagrams scale like n(d-2) 4 points vertices vertex  k2 - GH renorm. of k dependent part -> finite number renorm. of  dependent part can give log singularities HOWEVER one-loop renorm of H from 4-point vertex is identical to that obtained by renormalizing the 2-point vertex

Perturbation theory to all orders 2 points vertices One-loop renormalization of the mixed 2 point vertex One-loop renormalization of the magnetic 2 point vertex One-loop renormalization of the fermionic 2 point vertex Renormalization to all orders fermionic vertex magnetization vertex D(in)=1/GH(i n) Ds(k, in)=u2 (k, in)/GKt D(0)=D0[g(ln 1/)]-1 Ds(k 0, 0)= Ds0(k)d-2[g(2ln kF/k)]-1