Frustrated Heisenberg antiferromagnet on the pyrochlore lattice

We investigate quantum phase transitions for the $s=\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice. By means of the series expansion method starting from isolated tetrahedra, the ground-state phase diagram is determined. When the ratio of the two competing exchange couplings is varied, a first-order (second-order) quantum phase transition occurs between two spin gap phases (the spin-gap and the antiferromagnetic phases). We also discuss some properties expected for the $s=1\mathrm{}$ pyrochlore spin system.