Dynamical properties of the inorganic spin-Peierls compound ${\ensuremath{\alpha}}^{\ensuremath{'}}\ensuremath{-}{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ are investigated using a one-dimensional dimerized Heisenberg model. By exact diagonalizations of chains with up to 28 sites, supplemented by a finite-size scaling analysis, the dimerization parameter \ensuremath{\delta} is determined by requiring that the model reproduces the experimentally observed spin gap \ensuremath{\Delta}. The dynamical and static spin structure factors are calculated. As for
CuGeO3, the existence of a low-energy magnon branch separated from the continuum is predicted. The present calculations also suggest that a large magnetic Raman scattering intensity should appear above an energy threshold of 1.9\ensuremath{\Delta}. The predicted photoemission spectrum is qualitatively similar to results for an undimerized chain due to the presence of sizable short-range antiferromagnetic correlations.
Spin dynamics as well as static properties of the one-dimensional J-J' model (S=1/2, J>0 and 0\le α=J'/J\le 0.5) are studied by the exact diagonalization and the recursion method of finite systems up to 26 sites. Especially, the dynamical structure factor S(q,ω) is investigated carefully for various values of α. As αincreases beyond the gapless-gapful critical value α_c=0.2411, there appear features definitely different from the Heisenberg model but the same with the Majumdar-Ghosh model. Some of these features depend only on the value of αand not on δ: a parameter introduced for the coupling alternation. By comparing these results with a recent inelastic neutron scattering spectrum of an inorganic spin-Peierls compound CuGeO_3 [M. Arai et al.: Phys. Rev. Lett. 77 (1996) 3649], it is found that the frustration by J' in CuGeO_3 is unexpectedly strong (α=0.4-0.45), and at least αmust be larger than α_c to some extent. The value of J is evaluated at \sim 180K consistent with other estimations. The coupling alternation is extremely small. This large frustration is a primary origin of the various anomalous properties CuGeO_3 possesses. For comparison we refer also to α'-NaV_2O_5.
The dynamical spin structure factor and the Raman response are calculated for structurally dimerized and spin-Peierls chains in a magnetic field, using exact diagonalization techniques. In both cases there is a spin liquid phase composed of interacting singlet dimers at small fields
h<hc1, an incommensurate regime
(hc1<h<hc2) in which the modulation of the triplet excitation spectra adapts to the applied field, and a fully spin-polarized phase above an upper critical field
hc2. For structurally dimerized chains, the spin gap closes in the incommensurate phase, whereas spin-Peierls chains remain gapped. In the spin liquid regimes, the dominant feature of the triplet spectra is a one-magnon bound state, separated from a continuum of states at higher energies. There are also indications of a singlet bound state above the one-magnon triplet.
Spin fluctuations in the Haldane ground state should exhibit a quantum spin gap observable with Raman scattering. The Raman spectra of single-crystalline
Y2BaNiO5, a linear spin-1 Heisenberg antiferromagnet, exhibit broad continuum scattering between \ensuremath{\sim}200 and \ensuremath{\sim}800 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, in the energy range expected for magnetic excitations (\ensuremath{\sim}J=224 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$). Numerous weak and broad features superimposed on this continuum scattering possibly result from quantization of the magnon continuum by local molecular fields from nearby chains. A strong and relatively narrow mode at \ensuremath{\sim}680 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, near 3J, unrelated to lattice vibrations, may also be of magnetic origin. All of these features are strongly resonant in the red (${\ensuremath{\lambda}}_{\mathit{L}}$=6328 \AA{}), disappearing for shorter excitation wavelengths. These spectra provide a view of the Haldane spin system excitations complementary to that given by inelastic neutron scattering. Modeling the excitations with Schwinger bosons or spin waves predicts a highly asymmetric scattering feature with a sharp onset near 2\ensuremath{\Delta}(\ensuremath{\sim}0.8J), gradually diminishing toward higher energies. The observed spectra disagree with this theoretical expectation, perhaps due to spin-phonon coupling which may provide an efficient decay channel for the magnon pairs.
The spin-Peierls transition at
TSP of spin-
21 chains with isotropic exchange interactions has previously been modeled as correlated for
T>TSP and mean field for
T<TSP. We use correlated states throughout in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model with antiferromagnetic exchange
J1 and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between first and second neighbors, respectively, and variable frustration $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}0.50$. The thermodynamic limit is reached at high
T by exact diagonalization of short chains and at low
T by density matrix renormalization group calculations of progressively longer chains. In contrast to mean field results, correlated states of 1D models with linear spin-phonon coupling and a harmonic adiabatic lattice provide an internally consistent description in which the parameter
TSP yields both the stiffness and the lattice dimerization $\ensuremath{\delta}(T)$. The relation between
TSP and $\mathrm{\ensuremath{\Delta}}(\ensuremath{\delta},\ensuremath{\alpha})$, the
T=0 gap induced by dimerization, depends strongly on $\ensuremath{\alpha}$ and deviates from the BCS gap relation that holds in uncorrelated spin chains. Correlated states account quantitatively for the magnetic susceptibility of TTF-
CuS4C4(CF3)4 crystals (
J1=79K, $\ensuremath{\alpha}=0$,
TSP=12K) and
CuGeO3 crystals (
J1=160K, $\ensuremath{\alpha}=0.35$,
TSP=14K). The same parameters describe the specific heat anomaly of
CuGeO3 and inelastic neutron scattering. Modeling the spin-Peierls transition with correlated states exploits the fact that $\ensuremath{\delta}(0)$ limits the range of spin correlations at
T=0 while
T>0 limits the range at $\ensuremath{\delta}=0$.
We show that to describe properly the low-energy excitations of
CuGeO3 one must include the effects of a transverse antiferromagnetic coupling, which is estimated to be ${J}_{\ensuremath{\perp}}=0.15J.$ Owing to this coupling the frustration in the chains is significantly lower than recent one-dimensional (1D) estimates based on purely one-dimensional arguments, we find
J2=0.2J. Furthermore, we have found a strong modulation of the nearest-neighbor coupling due to the static distortion $\ensuremath{\delta}=0.065,$ which is five times higher than that previously deduced from a 1D chain approach. Our set of parameters gives (i) a value of the distortion which agrees well with some recent estimate for a lower bound we are able to perfectly reproduce, (ii) the dispersions, and (iii) the experimental susceptibility at both high and low temperatures. By performing density matrix renormalization-group (DMRG) calculations for two coupled chains we have analyzed the effect of the transverse coupling on the ratio of singlet to triplet gaps. The ratio is very sensitive to the parameters and the universality reported in the strict one-dimensional case is lost. As an additional point, we provide a simple picture to explain the interesting feature observed in recent inelastic neutron-scattering experiments: the existence of a second branch of excitations.