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Physical Review B

Spectral properties of alternating antiferromagnetic-ferromagnetic spin-half chains and the emergence of composite spin-one behavior

Weihong Zheng, Chris J. Hamer, Rajiv R. P. Singh

Published Nov 13, 2006• 8 citations
Publisher

Abstract

We study the spectral properties of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings $J$ and $\ensuremath{-}\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}J$. The crossover from the gapped dimerized spectra at small $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$ to the spin-one chain behavior at large $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$ is signaled by the merging of the one-particle state with the bottom of the continuum, and its vanishing spectral weight, near $k=0$. At large $\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}$, the model represents a class of spin-one chains with varying degrees of spatial and temporal compositeness and provides an excellent quantitative description of the neutron scattering spectra for such materials as ${(\mathrm{C}{\mathrm{D}}_{3})}_{2}\mathrm{N}{\mathrm{D}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{3}$ and $\mathrm{IPA}\text{\ensuremath{-}}\mathrm{Cu}{\mathrm{Cl}}_{3}$.

DOI
10.1103/physrevb.74.172407

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We study the spectral properties of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings JJJ and \ensuremath−\ensuremath∣\ensuremathλ\ensuremath∣J\ensuremath{-}\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}J\ensuremath−\ensuremath∣\ensuremathλ\ensuremath∣J. The crossover from the gapped dimerized spectra at small \ensuremath∣\ensuremathλ\ensuremath∣\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}\ensuremath∣\ensuremathλ\ensuremath∣ to the spin-one chain behavior at large \ensuremath∣\ensuremathλ\ensuremath∣\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}\ensuremath∣\ensuremathλ\ensuremath∣ is signaled by the merging of the one-particle state with the bottom of the continuum, and its vanishing spectral weight, near k=0k=0k=0. At large \ensuremath∣\ensuremathλ\ensuremath∣\ensuremath{\mid}\ensuremath{\lambda}\ensuremath{\mid}\ensuremath∣\ensuremathλ\ensuremath∣, the model represents a class of spin-one chains with varying degrees of spatial and temporal compositeness and provides an excellent quantitative description of the neutron scattering spectra for such materials as (CD3)2ND2CuCl3{(\mathrm{C}{\mathrm{D}}_{3})}_{2}\mathrm{N}{\mathrm{D}}_{2}\mathrm{Cu}{\mathrm{Cl}}_{3}(CD3​)2​ND2​CuCl3​ and IPA\ensuremath-CuCl3\mathrm{IPA}\text{\ensuremath{-}}\mathrm{Cu}{\mathrm{Cl}}_{3}IPA\ensuremath-CuCl3​.
We calculate the excitation spectrum and spectral weights of the alternating antiferromagnetic-ferromagnetic spin-half Heisenberg chain with exchange couplings JJJ and −∣λ∣J-|λ|J−∣λ∣J as a power series in λλλ. For small ∣λ∣|λ|∣λ∣, the gapped one-particle spectrum has a maximum at k=0k=0k=0 and there is a rich structure of bound (and anti-bound) states below (and above) the 2-particle continuum. As ∣λ∣|λ|∣λ∣ is increased past unity the spectrum crosses over to the Haldane regime, where the peak shifts away from k=0k=0k=0, the one particle states merge with the bottom of the continuum near k=0k=0k=0, and the spectral weights associated with the one-particle states become very small. Extrapolation of the spectrum to large ∣λ∣|λ|∣λ∣ confirms that the ground state energy and excitation gap map onto those of the spin-one chain.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
We study the one-dimensional Heisenberg spin-1∕21∕21∕2 alternating bond chain in which the nearest neighbor exchange couplings are ferromagnetic (FM) and antiferromagnetic (AF) alternatively. By using exact diagonalization and the density matrix renormalization group method, we discuss how the system approaches the AF uniform spin-1 chain under certain conditions. When the ratio of AF to FM coupling strengths, $\ensuremath{\alpha}$ $(\ensuremath{\alpha}={J}_{AF}∕{J}_{F})$, is very small, physical properties of the alternating bond chain such as spin-spin correlation, the string correlation function, and the spin density coincide with those of the AF uniform spin-1 chain. The edge state problem is discussed in the present model with the small $\ensuremath{\alpha}$ limit. In addition, the Haldane gap of the AF uniform spin-1 chain is four times the gap of the system considered.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum kkk by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small kkk, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.
The natural explanation of the excitation spectrum of the spin-1 antiferromagnetic Heisenberg chain is given from the viewpoint of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain. The energy spectrum of the latter is calculated with fixed momentum k by numerical diagonalization of finite size systems. It consists of a branch of propagating triplet pair (triplet wave) and the continuum of multiple triplet waves for weak ferromagnetic coupling. As the ferromagnetic coupling increases, the triplet wave branch is absorbed in the continuum for small k, reproducing the characteristics of the spin-1 antiferromagnetic Heisenberg chain.