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Journal of Physics C Solid State Physics

On the eigenstates of a Heisenberg chain with complex wavenumbers not forming strings

Ferenc Woynarovich

Published Nov 10, 1982• 41 citations
Publisher

Abstract

The nature of those S z =0 excitations of an anisotropic Heisenberg chain that have been found to be typical in the low-energy excited states is considered. A simple estimation is used to take into account the effect of complex phase shifts on the long-distance decay properties of the wavefunction. It is established that pairs of complex wavenumbers that do not form two-strings correspond to pairs of free spins, while quartets described two independent pairs of bound spins.

DOI
10.1088/0022-3719/15/31/015

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The nature of those S z =0 excitations of an anisotropic Heisenberg chain that have been found to be typical in the low-energy excited states is considered. A simple estimation is used to take into account the effect of complex phase shifts on the long-distance decay properties of the wavefunction. It is established that pairs of complex wavenumbers that do not form two-strings correspond to pairs of free spins, while quartets described two independent pairs of bound spins.
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.
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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.
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.
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