Ground-State Energy of a Heisenberg-Ising Lattice

The Heisenberg-Ising Hamiltonian $H=\ensuremath{-}\frac{1}{2}{\ensuremath{\alpha}({\ensuremath{\sigma}}_{x}\ensuremath{\sigma}_{x}^{}{}_{}{}^{\ensuremath{'}}+{\ensuremath{\sigma}}_{y}\ensuremath{\sigma}_{y}^{}{}_{}{}^{\ensuremath{'}})+\ensuremath{\epsilon}({\ensuremath{\sigma}}_{z}\ensuremath{\sigma}_{z}^{}{}_{}{}^{\ensuremath{'}})}$ for rectangular one-, two-, or three-dimensional lattices are considered. The sum is over nearest neighbors and $\ensuremath{\Delta}=\frac{\ensuremath{\epsilon}}{\ensuremath{\alpha}}$ measures the anisotropy of the coupling. Upper and lower bounds for the ground-state energy are established and these bounds apply equally well to lattices of one, two, or three dimensions. Furthermore, it is shown that the ground-state energy per nearest-neighbor pair is nondecreasing as the dimension of the lattice (one, two or three) increases.

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