Order versus disorder in the quantum Heisenberg antiferromagnet on the kagomé lattice using 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 one 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 $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ pattern: the finite-size analysis indicates a quantum critical transition at a nonzero coupling.