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Zeitschrift für Naturforschung A

The Ground-State Energy of Anisotropie Spin-Spin Interaction in One-Dimensional Chain

A. D. Jannussis

Published May 1, 1969• 0 citations
Publisher

Abstract

Abstract In the present work the integral equation of Yang and Yang is studied by the method of moments. In general the solution of the integral equation is reducible to a linear algebraic system which can be solved only approximately. From the solution of the system the ground-state energy and the magnetization of the anisotropic spin-spin interaction in a one-dimensional chain is de­termined.

DOI
10.1515/zna-1969-0508

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Abstract In the present work the integral equation of Yang and Yang is studied by the method of moments. In general the solution of the integral equation is reducible to a linear algebraic system which can be solved only approximately. From the solution of the system the ground-state energy and the magnetization of the anisotropic spin-spin interaction in a one-dimensional chain is de­termined.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.
The one-dimensional model which consists of two isotropic XY chains with spins S=1/2 coupled by three-spin interactions is considered with the help of the Bethe ansatz. It is shown that the diagonalization of the Hamiltonian can be reduced to solving a set of coupled nonlinear equations. The exact solution of these equations corresponding to the ground state of system is obtained. The model considered exhibits a ground state of Anderson type with the finite magnetization along the quantization axis.
The integral equations for the calculation of the ground-state energy of the repulsive delta-function interaction problem in one dimension for NNN particles are reduced to a system of algebraic equations by using the method of moments. We obtain explicit results in a limiting case.
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.
The ground-state energy 2f2f2f per lattice site for an infinite system is studied as a function of $\ensuremath{\Delta}$ and of the magnetization yyy. Analyticity properties of $f(\ensuremath{\Delta}, y)$ are proved. The behavior of $f(\ensuremath{\Delta}, y)$ at and near y=0y=0y=0 and y=1y=1y=1 are investigated.