The dynamical structure factor S(Q,\ensuremath{\omega}) of the S=1 antiferromagnetic Heisenberg chain with length 20 at zero temperature is calculated. The lowest-energy states have the \ensuremath{\delta}-function peak at the region \ensuremath{\pi}\ensuremath{\ge}\ensuremath{\Vert}Q\ensuremath{\Vert}>0.3\ensuremath{\pi}. At \ensuremath{\Vert}Q\ensuremath{\Vert}0.3\ensuremath{\pi} the lowest-energy states are the lower edge of the continuum of the scattering state, the strength of which decreases for large systems. This gives a reasonable explanation for the experimental fact that no clear peak is observed at the region Q0.3\ensuremath{\pi}. This situation is more apparent for the valence-bond solid state. On the contrary for S=1/2 antiferromagnetic Heisenberg chain the lowest-energy states are always the edge of the continuum.
The dynamical structure factor S(Q,\ensuremath{\omega}) of the S=1 antiferromagnetic Heisenberg chain with length 20 at zero temperature is calculated. The lowest-energy states have the \ensuremath{\delta}-function peak at the region \ensuremath{\pi}\ensuremath{\ge}\ensuremath{\Vert}Q\ensuremath{\Vert}>0.3\ensuremath{\pi}. At \ensuremath{\Vert}Q\ensuremath{\Vert}0.3\ensuremath{\pi} the lowest-energy states are the lower edge of the continuum of the scattering state, the strength of which decreases for large systems. This gives a reasonable explanation for the experimental fact that no clear peak is observed at the region Q0.3\ensuremath{\pi}. This situation is more apparent for the valence-bond solid state. On the contrary for S=1/2 antiferromagnetic Heisenberg chain the lowest-energy states are always the edge of the continuum.
We calculate the dynamical structural factor of the S=1 bond-alternating Heisenberg chain. In the Haldane phase, the lowest excited states form the lower edge of the multimagnon continuum in 0<q<qc and the one-magnon mode in ${q}_{c}<~q<~\ensuremath{\pi}.$ As the system approaches the gapless point, qc shifts towards $q=\ensuremath{\pi}$ and the largest integrated intensity of the one-magnon mode is decreased. In the singlet-dimer phase, the one-magnon mode appears in 0<q<qc. As the bond-alternation becomes strong, qc shifts towards $q=\ensuremath{\pi}.$ In the antiferromagnetic-ferromagnetic bond-alternation region with a strong ferromagnetic coupling, the lowest excited states form the lower edge of the multimagnon continuum in $0<~q<~0.2\ensuremath{\pi}$ and $0.8\ensuremath{\pi}<~q<~\ensuremath{\pi},$ and the one-magnon mode appears in $0.2\ensuremath{\pi}<q<0.8\ensuremath{\pi}.$ The largest integrated intensity of the one-magnon mode is 93 which is slightly smaller than that in the S=1 Haldane-gap system. We further discuss the dynamical structural factor in connection with the inelastic neutron-scattering experiments.
Dynamical structure factors of S =1/2 antiferromagnetic–ferromagnetic alternating chains are calculated by the numerical method based on the ground state obtained by exact diagonalization procedure and also by the pair dynamical correlated-effective-field approximation. A couple of characteristic natures of neutron scattering intensity due to magnetic excitations are predicted: the intensity of transition to the lowest triplet states takes the maximum value at q =π/2 in the whole range of the exchange ratio, and the continuum states above the second gap could be observable at large q .
Dynamical structure factors of the S =1 bond-alternating spin chains in the dimer state are calculated at finite temperature using the pair dynamical correlated-effective-field approximation. At T =0, the delta-function-type peak corresponding to the one-magnon mode appears. When temperature increases, such a sharp peak is broadened and the additional weak peak caused by the excitation from the triplet state to the quintet state emerges in the higher energy region. The results are discussed in comparison with those obtained by the exact diagonalization method.
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=1NSi\ensuremath{\cdot}Si+1+D tsum(Siz)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=\ifmmode\pm\else\textpm\fi{}1 and in the subspace of Sz=0 are different. The Sz=\ifmmode\pm\else\textpm\fi{}1 branch is lower than the 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(C2H8N2)2NO2ClO4.
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.
Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/T1 from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for Sr2CuO3 and CuGeO3. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)]3/2 and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)]1/2, expected from previous theoretical studies.