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Physical review. B, Condensed matter

Dynamical properties of antiferromagnetic Heisenberg spin chains

Stephan Haas, José A. Riera, Elbio R. Dagotto

Published Aug 1, 1993• 30 citations
Publisher

Abstract

The dynamical properties of spin-1 and spin-1/2 antiferromagnetic Heisenberg chains (AHC's) are studied by diagonalizing exactly clusters of up to 18 and 26 sites, respectively. It is shown that the spin-1 AHC has a quasi-single-mode spectrum for momentum k\ensuremath{\ge}0.3\ensuremath{\pi}, while the low-energy edge of the spin-1/2 AHC is dominated by a spin-wave continuum. The dispersion curve obtained for the spin-1 chain is in excellent agreement with recent experiments on Ni(${\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}$. The size dependence of the low-energy spectral weights is also analyzed.

DOI
10.1103/physrevb.48.3281

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The dynamical properties of spin-1 and spin-1/2 antiferromagnetic Heisenberg chains (AHC's) are studied by diagonalizing exactly clusters of up to 18 and 26 sites, respectively. It is shown that the spin-1 AHC has a quasi-single-mode spectrum for momentum k\ensuremath{\ge}0.3\ensuremath{\pi}, while the low-energy edge of the spin-1/2 AHC is dominated by a spin-wave continuum. The dispersion curve obtained for the spin-1 chain is in excellent agreement with recent experiments on Ni({\mathrm{C}}_{2}$${\mathrm{H}}_{8}$${\mathrm{N}}_{2}$${)}_{2}$${\mathrm{NO}}_{2}$${\mathrm{ClO}}_{4}. The size dependence of the low-energy spectral weights is also analyzed.
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.
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 spin pair-correlation function of finite bipartite antiferromagnetic Heisenberg quantum spin rings was studied numerically by means of exact diagonalization techniques. For large spins, the spectrum of the spin pair-correlation function is of a remarkably simple structure: It consists of few characteristic peaks at low, and a broad featureless signal at high temperatures. This arises as the energy spectrum exhibits a set of parallel rotational bands at low energies emerging into a quasi-continuum of states, with transitions from the lowest rotational band to the quasi-continuum being highly suppressed. The energies of the rotational bands can be accurately described by a generalized dispersion relation that depends on the total spin quantum number and, as usual, on the shift quantum number. These regularities are better fulfilled the larger the spin length but the smaller the ring size. It will be shown that all these features are associated with the underlying sublattice structure of the spin rings, and it is argued that they are valid for a more general class of finite Heisenberg systems than rings.
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.
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 authors study the dynamic spin correlation function of a spin-1 antiferromagnetic chain with easy-plane single-ion anisotropy. They use exact diagonalization by the Lanczos method for chains of length up to N=16 spins. They show that a single-mode approximation is an excellent description of the dynamical properties. A variational calculation allows them to clarify the nature of the excitations. The existence of a two-particle continuum near-zero wavevector is clearly seen, both in finite-size effects and in the dynamical structure factor. The recent neutron scattering experiments on the quasi-one-dimensional antiferromagnet NENP are fully explained by the results.