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Physical Review

Ground-State Energy of a Heisenberg-Ising Lattice

Chongming Yang, Chuan‐Zhang Yang

Published Jul 8, 1966• 166 citations

Abstract

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.

DOI: 10.1103/physrev.147.303

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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.

We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.

Antiferromagnetic ground-state energies for infinite Heisenberg linear chains with up to third-nearest-neighbor interactions have been determined by means of numerical extrapolations. On the other hand, new conditions to produce a ferromagnetic ground state have been established for finite chains. These conditions are expressed as lower bounds for the coupling constants. The particular case in which the coupling constant for third-nearest-neighbor interactions is zero is compared in two different ways with previous calculations by other authors, showing good correspondence.

The random Heisenberg model with a Gaussian distribution of nearest-neighbor interactions is studied for the pure spin-glass case where the average interaction vanishes. The distribution of domain-wall energies at zero temperature is calculated using a spin-quench algorithm to find the ground-state energy for finite lattices. A renormalization-group transformation is set up which preserves the domain-wall energy distribution when the lattice parameter is changed. In the strong-coupling regime (zero temperature) the model iterates toward weak coupling and therefore exhibits a ``phase transition at zero temperature'' in both two and three dimensions. The correlation-length exponent is \ensuremath{\nu}=0.714\ifmmode\pm\else\textpm\fi{}0.015 in two dimensions and \ensuremath{\nu}=1.54\ifmmode\pm\else\textpm\fi{}0.19 in three dimensions. The lower critical dimension is four.