Conditions for a ferromagnetic ground state of Ising and Heisenberg models with an external magnetic field

For the isotropic Heisenberg Hamiltonian with ferromagnetic nearest-, antiferromagnetic next-nearest-neighbor interactions, and an external magnetic field: $H=J_1\Sigma {n,\mathrm{NN}}^{}{\overrightarrow{\mathrm{S}}}_n·{\overrightarrow{\mathrm{S}}}_{n+{\delta }_1}+J_2\Sigma {n,\mathrm{NNN}}^{}{\overrightarrow{\mathrm{S}}}_n·{\overrightarrow{\mathrm{S}}}_{n+{\delta }_2}-h\Sigma n^{}{S}_n^z$, $J_1<0\mathrm{}$, $J_2>0\mathrm{}$ necessary and sufficient conditions for a ferromagnetic ground state are obtained for some Bravais lattices with periodic boundaries and arbitrary spin $s$. For the square and cubic lattices the sufficient conditions possess a nontrivial $s$ dependence. These conditions are compared with the thresholds for the Ising and classical Heisenberg model. Threshold inequalities are generalized to the case $h\ne 0$. The zero-temperature magnetization and susceptibility are discussed for the classical and quantum case. For the square lattice with only 4 sites the magnetization as function of $h$ shows a qualitatively different behavior in the quantum case for $\frac{\left|J_1\right|}{J_2}<1\mathrm{}$ and $\frac{\left|J_1\right|}{J_2}>1\mathrm{}$, respectively. Sufficient, necessary, and threshold conditions are also derived for the nearest-neighbor antiferromagnet and the Heisenberg model with arbitrary coupling constants ($J_1,\dots ,J_r$) in an external field.