Ground-State Energy of a Heisenberg-Ising Lattice

Abstract

The Heisenberg-Ising Hamiltonian $H=-\frac{1}{2}\left\{\alpha \right({\sigma }_x\sigma _x^{}{}_{}{}^{\prime }+{\sigma }_y\sigma _y^{}{}_{}{}^{\prime })+\epsilon ({\sigma }_z\sigma _z^{}{}_{}{}^{\prime }\left)\right\}$ for rectangular one-, two-, or three-dimensional lattices are considered. The sum is over nearest neighbors and $\Delta =\frac{\epsilon }{\alpha }$ 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.