  We study the kagome antiferromagnet for quantum spin-1/2 with first J1,
second J2 and third J3 neighbour exchanges, along the J2 = J3 = J line. We use
Schwinger-boson mean-field theory for the precise determination of the phase
diagram, and two different rewritings of the Hamiltonian to build an intuition
about the origin of the transitions. The spin liquid obtained at J = 0 remains
essentially stable over a large window, up to J = 1/3, because it is only
weakly frustrated by the J term. Then at J = 1/2, the intermediate Z2 spin
liquid condenses into a long-range chiral order because of the change of nature
of local magnetic fluctuations. As a side benefit, our Hamiltonian rewriting
offers an exact solution for the ground state of our model on a Husimi cactus.

Strongly Correlated Electrons;Statistical Mechanics