Magnetic properties of a spin-$\frac{1}{2}$ quadrumer chain

We study a $S=1/2\mathrm{}$ cluster chain Hamiltonian that has recently been proposed in the context of the charge-ordered low-temperature phase of ${\alpha }^{\prime }-{\mathrm{NaV}}_2{\mathrm{O}}_5.$ We perform a detailed investigation of this model within a large range of parameters using perturbation theory and Lanczos diagonalization. Using model-specific local conservation laws and parameter-dependent mappings to various effective low-energy Hamiltonians, we uncover a rich phase diagram and several regimes of gapful spin-excitations. We find that the overall features of recent neutron scattering data on ${\alpha }^{\prime }-{\mathrm{NaV}}_2{\mathrm{O}}_5$ can be fitted within this model, however, using a set of parameters that seems unlikely.