  In this paper we give a general introduction to quantum critical phenomena,
which we practically illustrate by a detailed study of the low energy
properties of the spin boson model (SBM), describing the dynamics of a spin 1/2
impurity (or more generically a two-level system) coupled to a bath of
independent harmonic oscillators. We show that the behavior of the model is
very sensitive to the bath spectrum, in particular how the properties of the
quantum critical point in the SBM are affected by the functional form of the
bath Density of States (DoS). To this effect, we review the renormalization
group (RG) treatment of the SBM for various bath DoS, based on an
unconventional Majorana representation of the spin 1/2 degree of freedom. We
also discuss the derivation of Shiba's relation for the sub-ohmic SBM, and
explicitely derive an effective action vindicating the quantum to classical
mapping.
