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Journal of Physics A Mathematical and General

On the Sz=0 excited states of an anisotropic Heisenberg chain

Ferenc Woynarovich

Published Sep 1, 1982• 83 citations
Publisher

Abstract

The S z =0 excited states of the anisotropic antiferromagnetic Heisenberg Hamiltonian H= Sigma j=1 N (S j x S j+1 x +S j y S j+1 y + rho S j z S j+1 z ) are studied when 0<or= rho <or=1. The original set of secular equations is reduced to a simpler one, which contains the parameters of excitations only. The energy-momentum dispersion is also found. It is shown that the typical low-lying excitations are either two-strings or not of string form. The simplest excitations are described in more detail.

DOI
10.1088/0305-4470/15/9/044

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Quantum Reports•Oct 19, 2022•1 citation
The S z =0 excited states of the anisotropic antiferromagnetic Heisenberg Hamiltonian H= Sigma j=1 N (S j x S j+1 x +S j y S j+1 y + rho S j z S j+1 z ) are studied when 0<or= rho <or=1. The original set of secular equations is reduced to a simpler one, which contains the parameters of excitations only. The energy-momentum dispersion is also found. It is shown that the typical low-lying excitations are either two-strings or not of string form. The simplest excitations are described in more detail.
We calculate energies and wave functions of S=1 antiferromagnetic Heisenberg chains with single-ion anisotropy in a symmetry-breaking external magnetic field. From results for the string correlation function for low-lying excited states we identify these states as solitons with respect to the hidden string order. For finite anisotropic chains we investigate the transition to the fully magnetized state at high magnetic fields and conclude that Bose condensation in an external magnetic field occurs also for the anisotropic chain.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0{\mathit{e}}_{0}e0​\ensuremath{\simeq}-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length \ensuremath{\xi}\ensuremath{\simeq}6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(\ensuremath{\infty})\ensuremath{\simeq}-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=\ensuremath{\pi}, is the Haldane gap, which we find to be \ensuremath{\Delta}\ensuremath{\simeq}0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near \ensuremath{\pi}, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
The neutron-scattering intensity and energy eigenvalues of the elementary excitations of the S=1 antiferromagnetic Heisenberg chain with single-ion anisotropy [scrH=J tsumi=1N{\mathit{tsum}}_{\mathit{i}=1}^{\mathit{N}}tsumi=1N​ Si{\mathbf{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathit{i}+1}Si+1​+D tsum(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] are calculated with a numerical diagonalization method and a projector Monte Carlo method. We set D/J=0.2,0.0,-0.2 and N=14,16,18,32. For D\ensuremath{\ne}0, the elementary excitations in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of Sz{\mathit{S}}^{\mathit{z}}Sz=0 are different. The Sz{\mathit{S}}^{\mathit{z}}Sz=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the Sz{\mathit{S}}^{\mathit{z}}Sz=0 branch near Q=\ensuremath{\pi}; however, it becomes slightly higher in other regions. The crossing of levels occurs at Q\ensuremath{\simeq}\ifmmode\pm\else\textpm\fi{}0.7\ensuremath{\pi}. We find that the low-lying excited state has a weight of more than 90% in the dynamical structure factor, except for Q\ensuremath{\simeq}0. Our results explain recent experimental results on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​.
The ground state and low-lying excited states of a one-dimensional Heisenberg model with axial anisotropy are studied by extrapolation from finite lattices. The ground-state properties and the phase diagram are determined for quantum spins, especially for S=12,1,and32S=\frac{1}{2}, 1, \mathrm{and} \frac{3}{2}S=21​,1,and23​. Higher spin values do not give new features in the phase diagram. the limitations of finite-lattice extrapolation are discussed in connection with the possibility of a singlet phase for the S=1S=1S=1 isotropic Heisenberg antiferromagnet.
The T=0 excitation spectra of the antiferromagnetic (J>0) anisotropic Heisenberg chain of spins 1/2 are studied using the Bethe Ansatz equations for Δ=cos(π/n), n=3,4 and 5. The number of unknown functions is n−1 for Δ=cos(π/n) and can be solved numerically for a finite external field. The low-energy excitations form a Luttinger liquid parametrized by a conformal field theory with conformal charge of c=1. For higher energy excitations, the spectral functions display deviations from the Luttinger behavior arising from the curvature in the dispersion. Adding a corrective term of the form of a mobile impurity coupled to the Luttinger liquid modes corrects this difference. The “impurity” is an irrelevant operator, which if treated non-perturbatively, yields the threshold singularities in the one-spinwave particle and hole Green’s function correctly.