From the square lattice to the checkerboard lattice: Spin-wave and large-nlimit analysis

Within a spin wave analysis and a fermionic large-$n$ limit, it is shown that the antiferromagnetic Heisenberg model on the checkerboard lattice may have different ground states, depending on the spin size S. Through an additional exchange interaction that corresponds to an intertetrahedra coupling, the stability of the Néel state has been explored for all cases from the square lattice to the isotropic checkerboard lattice. Away from the isotropic limit and within the linear spin-wave approximation, it is shown that there exists a critical coupling for which the local magnetization of the Néel state vanishes for any value of the spin S. One the other hand, using the Dyson-Maleev approximation, this result is valid only in the case $S=\frac{1}{2}$ and the limit between a stable and an unstable Néel state is at $S=1.\mathrm{}$ For $S=\frac{1}{2},$ the fermionic large-$n$ limit suggests that the ground state is a valence-bond solid build with disconnected four-spin singlets. This analysis indicates that for low spin and in the isotropic limit, the checkerboard antiferromagnet may be close to an instability between an ordered $S=0\mathrm{}$ ground state and a magnetized ground state.