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Journal of Physics A Mathematical and General

The antiferromagnetic spin- 1/2 -XXZ model on rings with an odd number of sites

Michael Karbach, K -H Mutter

Published Aug 21, 1995• 22 citations
Publisher

Abstract

The ground-state energies of the antiferromagnetic XXZ model at a given spin are determined on chains with an odd number N of sites. Analytical and numerical solutions of the Bethe ansatz equations are compared for the N-even and N-odd case. The scaling properties of the ground-state energies enable the determination of the zero-temperature susceptibility. For the isotropic case, we analyse the logarithmic terms in the low-field limit.

DOI
10.1088/0305-4470/28/16/007

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The ground-state energies of the antiferromagnetic XXZ model at a given spin are determined on chains with an odd number N of sites. Analytical and numerical solutions of the Bethe ansatz equations are compared for the N-even and N-odd case. The scaling properties of the ground-state energies enable the determination of the zero-temperature susceptibility. For the isotropic case, we analyse the logarithmic terms in the low-field limit.
The thermodynamics of finite open antiferromagnetic XXZ chains is studied using field theory, Bethe ansatz and quantum Monte Carlo methods. For the susceptibility a parameter-free result as a function of the number of sites L and temperature T beyond the scaling limit is derived. The limiting cases ( J being the exchange constant), where the boundary correction shows a logarithmically suppressed Curie behaviour, and , where a crossover to the ground state behaviour takes place, are discussed in detail. On the basis of this analysis we present a simple formula for the averaged susceptibility of a spin-1/2 chain doped with non-magnetic impurities. We show that the effective Curie constant has a highly non-trivial temperature dependence and shows scaling in the low temperature limit. Finally, corrections due to intrachain and interchain couplings and implications for experiments on Sr 2 Cu 1− x Pd x O 3+δ and related compounds are discussed.
The scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is studied in the gapped antiferromagnetic phase. For a spin-chain ring having N_{x} sites, the universal Casimir scaling function, which characterizes the leading finite-size correction term in the large-N_{x} expansion of the ground-state energy, is calculated by numerical solution of the nonlinear integral equation of the convolution type. It is shown that the same scaling function describes the temperature dependence of the free energy of the infinite XXZ chain at low enough temperatures in the gapped scaling regime.
We investigate Bethe Ansatz equations for the one-dimensional spin-12\frac{1}{2}21​ Heisenberg XXX chain with a special interest in a finite system. Solutions for the two-particle sector are obtained. The ground state in antiferromagnetic case has been analytically studied through the logarithmic form of Bethe Ansatz equations.
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.