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Le Journal de Physique Colloques

NEW SOLUTIONS FOR THE ANISOTROPIC ANTIFERROMAGNETIC CHAIN

Miguel Lagos, G. G. Cabrera

Published Dec 1, 1988• 0 citations
Publisher

Abstract

Coherent states are proposed as solutions for the antiferromagnetic Heisenberg Hamiltonian in the quasi-Ising asymptotic regime of high anisotropy. Since our solutions are given in closed analytical form, a deep insight concerning the structure of the ground state and its excitations is obtained. Our calculation is in excellent agreement with numerical simulations.

DOI
10.1051/jphyscol:19888632

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Coherent states are proposed as solutions for the antiferromagnetic Heisenberg Hamiltonian in the quasi-Ising asymptotic regime of high anisotropy. Since our solutions are given in closed analytical form, a deep insight concerning the structure of the ground state and its excitations is obtained. Our calculation is in excellent agreement with numerical simulations.
We propose a coherent-state solution for the one-dimensional antiferromagnetic Heisenberg Hamiltonian in the quasi-Ising asymptotic regime. We discuss the nature of the spectrum and the eigenvectors as a function of the anisotropy parameter. Results for the energy of the ground state and the short-range-order correlation agree with exact numerical calculations in a wide range of variation of anisotropy. We also present a canonical formulation of the theory which interpolates continuously from the Ising limit to cases of lower anisotropies, thus yielding a novel view of antiferromagnetism. Finally we discuss possible connections with the theory of superconductivity of high-Tc{T}_{c}Tc​ ceramics.
The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.
The exact solution is given for a linear chain of NNN atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.