  A description of different phases of two dimensional magnetic insulators is
given. The first chapters are devoted to the understanding of the symmetry
breaking mechanism in the semi-classical Neel phases. Order by disorder
selection is illustrated. All these phases break SU(2) symmetry and are gapless
phases with magnon excitations. Different gapful quantum phases exist in two
dimensions: the Valence Bond Crystal phases (VBC) which have long range order
in local S=0 objects (either dimers in the usual Valence Bond acception or
quadrumers..), but also Resonating Valence Bond Spin Liquids (RVBSL), which
have no long range order in any local order parameter and an absence of
susceptibility to any local probe. VBC have gapful S = 0, or 1 excitations,
RVBSL on the contrary have deconfined spin-1/2 excitations. Examples of these
two kinds of quantum phases are given in chapters 4 and 5. A special class of
magnets (on the kagome or pyrochlore lattices) has an infinite local degeneracy
in the classical limit: they give birth in the quantum limit to different
behaviors which are illustrated and questionned in the last lecture.
