  A simple phenomenological real-space renormalization group method for quantum
Heisenberg spins with nearest and next nearest neighbour interactions on a
pyrochlore lattice is presented. Assuming a scaling law for the order parameter
of two clusters of different sizes a set of coupled equations that gives the
fixed points of the renormalization group transformation and, thus, the
critical temperatures and ordered phases of the system is found. The particular
case of spins $\frac{1}{2}$ is studied in detail. Furthermore, to simplify the
mathematical details, from all the possible phases arising from the
renormalization group transformation, only those phases in which the magnetic
lattice is commensurate with a subdivision of the crystal lattice into four
interlocked fcc sublattices are considered. These correspond to a quantum spin
liquid, ferromagnetic order, or non-collinear order in which the total magnetic
moment of a tetrahedral unit is zero. The corresponding phase diagram is
constructed and the differences with respect to the classical model are
analysed. It is found that this method reproduces fairly well the phase diagram
of the pyrochlore lattice under the aforementioned constrains.
