Frustrated quantum magnets

In these lectures we present a summary of the T = 0 properties of the ground state and First excitations of two-dimensional frustrated quantum magnets. Up to now four phases have been observed at T = 0: a Neel ordered phase with spontaneous symmetry breaking and soft modes (the antiferromagnetic magnons) and three phases with no SIT(2) symmetry breaking and a spin gap. The first four sections are devoted to the characterization of these four phases. We first explain the two main quantum effects which can take place in the semi-classical Neel phase: reduction of the order parameter by quantum fluctuations and in the case of a competition between various ordered phases: selection of the most symmetric one by quantum fluctuations. Amongst the three spin gapped phases, the Valence Bond Crystal has long range order in S = 0 valence bonds and all its excitations have integer spins. The other two, the Resonating Valence Bond Spin Liquids have no local order parameter and fractionalized excitations. After a characterization of these different phases we use information gathered from exact diagonalizations, toy models and large N expansions to emphasize criteria that seem important to find one or another of these spin gapped phases. The last section of this paper is concerned with the special behavior of frustrated magnets in high magnetic fields: metamagnetic behavior and quantized plateaus. Some insights are given on Chiral Spin Liquids and on a proposed connexion between Quantum Hall Effect and magnetization plateaus.