Exact spectra of periodic samples are computed up to N = 36. Evidence of an extensive set of low-lying levels, lower than the softest magnons, is exhibited. These low-lying quantum states are degenerated in the thermodynamic limit; their symmetries and dynamics as well as their finite-size scaling are strong arguments in favor of Neel order. It is shown that the Neel order parameter agrees with first-order spin-wave calculations. A simple explanation of the low-energy dynamics is given as well as the numerical determinations of the energies, order parameter, and spin susceptibilities of the studied samples. It is shown how suitable boundary conditions, which do not frustrate Neel order, allow the study of samples with N = 3@+1 spins. A thorough study of these situations is done in parallel with the more conventional case N = 3p.

A group symmetry analysis of the low lying levels of the spin-1/2 kagomé Heisenberg antiferromagnet is performed for small samples up to N = 27.This new approach allows to follow the effect of quantum fluctuations when the sample size increases. The results contradict the scenario of "order by disorder" which has been advanced on the basis of large S calculations. A large enough second neighbor ferromagnetic exchange coupling is needed to stabilize the √ 3 × √ 3 pattern: the finite size analysis indicates a quantum critical transition at a non zero coupling.