Can a frustrated spin-cluster model describe the low-temperature physics of NaV₂O₅ ?

Recent experimental evidence suggest the existence of three distinct V-valence states (V^{+4}, V^{+4.5} and V^{+5}) in the low-temperature phase of NaV_2O_5 in apparent discrepancy with the observed spin-gap. We investigate a novel spin cluster model, consisting of weakly coupled, frustrated four-spin clusters aligned along the crystallographic b-axis that was recently proposed to reconcile these experimental observations. We have studied the phase diagram and the magnon dispersion relation of this model using DMRG, exact diagonalization and a novel cluster-operator theory. We find a spin-gap for all parameter values and two distinct phases, a cluster phase and a Haldane phase. We evaluate the size of the gap and the magnon dispersion and find no parameter regime which would reproduce the experimental results. We conclude that this model is inappropriate for the low-temperature regime of NaV_2O_5.