Order versus Disorder in the Quantum Heisenberg Antiferromagnet on the Kagom{\'e} lattice: an approach through exact spectra analysis

A group symmetry analysis of the low lying levels of the spin-1/2 kagom\'e Heisenberg antiferromagnet is performed for small samples up to N=27. This 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 $\sqrt 3 \times \sqrt 3$ pattern: the finite size analysis indicates a quantum critical transition at a non zero coupling.