Calculations are presented for the on-site dynamical spin pair correlation of the s=1/2 Heisenberg chain at T=∞. The result for the isotropic chain is consistent with short time analysis. The long time behavior may be interpreted as compatible with exponential decay as well as with power law decay interpretations. The chain with small Ising-like anisotropy also exhibits exponential-like long time decay as found in recent Mössbauer effect spectroscopy work. Preliminary calculations are also presented for the mathematically challenging case of extreme Ising-like anisotropy.
The Heisenberg equation of motion for the surface spin operator in a semi-infinite S=1/2 XY chain is exactly solved at infinite temperatures via the recurrence-relations method of Lee. It is found that the time evolution of the surface spin shows Bessel-like on-site relaxation that is drastically different from the corresponding Gaussian relaxation of a bulk spin. This difference is shown to be related to broken translational symmetry in the former case. The memory function and the random force for the surface spin is exactly determined. It is found that the surface spin dynamics is, in some sense, ``harmonic'' in nature.
We investigate full dynamical behaviors of the spin-Peierls transition in
S=1/2 Heisenberg antiferromagnetic chains by means of two-time Green's functions. The effect of temperature-dependent lattice displacements is incorporated self-consistently. Depending on temperature T and transfer wave number k, the imaginary part of the dynamical susceptibility ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ shows rich structures: At
T=0, it is nonvanishing in the frequency region ${\ensuremath{\Omega}}_{M}(k)<\ensuremath{\Omega} (\ensuremath{\equiv}\ensuremath{\Elzxh}\ensuremath{\omega}/J)<{\ensuremath{\Omega}}_{H}(k).$ ${\ensuremath{\Omega}}_{M}(k)$ denotes a lower edge of a triplet magnetic excitation continuum, and is finite at
k=0 due to lattice distortion. The upper edge ${\ensuremath{\Omega}}_{H}(k)$ increases as
k=0 is approached, which is completely different from the feature without lattice distortion. An asymmetric line shape of ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ is found with a peak at the frequency slightly higher than ${\ensuremath{\Omega}}_{M}(k).$ At
0<T<TSP (spin-Peierls transition point) there appears another structure due to thermal fluctuations in the lower-frequency region, $0<\ensuremath{\Omega}<{\ensuremath{\Omega}}_{L}(k).$ Since ${\ensuremath{\Omega}}_{L}(k)<{\ensuremath{\Omega}}_{M}(k),$ the forbidden band always exists at
T<TSP. Above
TSP, ${\ensuremath{\chi}}_{\mathrm{zz}}^{\ensuremath{''}}(k,\ensuremath{\omega})$ reduces to a smooth curve characterized by a single broad maximum. These features in dynamics are systematically explained in terms of different quasiparticle scattering processes proper to the spin-Peierls phase. The results are compared with recent experimental data of neutron scattering for the inorganic compound
CuGeO3.The Mori-Lee formalism for solving the Liouville (or Heisenberg) equation of motion for Hermitian systems demonstrates that the Laplace-transformed dynamical correlations in canonical ensembles can be written as (in)finite continued fractions. We show that a model-independent direct summation method allows accurate numerical evaluation of all known classes of these (in)finite continued fractions that arise in dynamics problems and thus provides a powerful technique to study the dynamics of many-body and few-body systems. Some studies on dynamical correlations in s=1/2 quantum spin chains are cited as applications of the method presented.
Exact coefficients of the short-time expansion of the spin-pair correlations are computed up to order t 30 in the one-dimensional spin-1/2 Heisenberg model with isotropic nearest-neighbour interaction at infinite temperature. The key points of the computational algorithm are sketched. The newly derived coefficients are used to obtain lower and upper bounds on the pair correlations which are graphically indistinguishable up to time nearly twice as large as those up to which previous bounds are reasonably close together. In view of the spin-diffusion hypothesis the spatial spread of the pair correlations is studied as a function of time, but no indication for a diffusive long-time behaviour is found in the extended time region, where the new bounds provide control.
We study the disordered Heisenberg spin chain, which exhibits many-body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}}$, whereas the conductivity exhibits a low-frequency power law $\ensuremath{\sigma}\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\alpha}}$. The exponents depict subdiffusive behavior $\ensuremath{\beta}<1/2,\phantom{\rule{0.28em}{0ex}}\ensuremath{\alpha}>0$ at all finite disorders and convergence to the scaling result $\ensuremath{\alpha}+2\ensuremath{\beta}=1$ at large disorders.