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

Abstract

For the isotropic Heisenberg Hamiltonian with ferromagnetic nearest-, antiferromagnetic next-nearest-neighbor interactions, and an external magnetic field: $H={J}_{1}\ensuremath{\Sigma}{n,\mathrm{NN}}^{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n+{\ensuremath{\delta}}_{1}}+{J}_{2}\ensuremath{\Sigma}{n,\mathrm{NNN}}^{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{n+{\ensuremath{\delta}}_{2}}\ensuremath{-}h\ensuremath{\Sigma}{n}^{}{S}_{n}^{z}$, ${J}_{1}l0$, ${J}_{2}g0$ 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 $h\ensuremath{\ne}0$. 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 $\frac{|{J}_{1}|}{{J}_{2}}l1$ and $\frac{|{J}_{1}|}{{J}_{2}}g1$, 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.