Alternating spin-1/spin-$\frac{1}{2}$ sites in a diamond lattice: Ground state and excitations

We consider a diamond lattice with spin-1 and spin-$\frac{1}{2}$ sites in alternating positions. The system is shown to be mapped on to a spin-1 sawtooth lattice for a range of parameter values. With varying strengths of interactions between spin sites, we find a number of phases, including, gapped spin-1, dimerized, and gapless spin-1 phases. In the gapped phase, there exist two special lines, namely, disorder and the Lifshitz lines, which differentiate spin ordering from the Néel type to spiral type, in real and in the momentum space, respectively. The dimerized phase, which ends at disorder line, is gapless in this case and corresponds to nearest-neighbor spin-$\frac{1}{2}$ sites forming alternate singlet and triplet.