FIG. 1: The CAVO lattice. The nearest-neighbour bonds that connect two sites on a four-site plaquette are shown by the solid lines and have a bond strength given by J (=1). The nearest- neighbour bonds that connect two sites on different plaquettes (dimer bonds) are shown by the dotted lines and have a bond strength given by J $^{′}$. The unit cell of the lattice is shown by the square with the grey dashed lines. In the previous section, we saw that the exact dimerised state at the Majumdar-Ghosh point may be obtained for the N´ eel model state by appropriate choice of ket-state correla- tion coefficients. However, this is a case in which the translational symmetry of the lattice (the linear chain) is spontaneously broken. We now wish to a case in which dimerised and plaquette exact solutions follow the symmetry of the Hamiltonian and so do not form symmetry-broken states. We consider an antiferromagnetic Heisenberg model in which the basic geometric unit cell contains four neighbouring lattice sites on the underlying crystal- lographic lattice of the magnetic material CaV 4 O 9 (CAVO), shown in Fig. 1. There are two non-equivalent antiferromagnetic nearest-neighbour bonds J and J ′ belonging to dimers ( J $^{′}$) and to four-spin plaquettes ( J ) respectively. The ground state of the quantum model depends on the ratio J $^{′}$/J of the competing bonds. Using a unit cell as defined in Fig. 1, the plaquette bonds J are inside the four-site unit cell and the dimer bonds J ′ connect sites in different unit cells. We note that this model is not frustrated but the two non-equivalent B. The J – J ′ Heisenberg antiferromagnet on the CAVO lattice 6