The dynamic-spin-correlation function S(q,\ensuremath{\omega}) is computed for the one-dimensional antiferromagnetic Heisenberg model for S=1/2 and S=1. Imaginary-time correlation functions are evaluated with the quantum Monte Carlo method and the maximum-entropy method is used to continue these results to real frequencies. This technique produces a value for the S=1 Haldane gap that agrees with previous work and also gives a good description of inelastic neutron scatter- ing data for the one-dimensional S=1/2 antiferromagnet
CuCl2\ensuremath{\cdot}2N(
C5D5).
A combination of world-line quantum Monte Carlo and maximum-entropy methods is used to calculate the dynamic structure factor ${\mathit{S}}_{\mathrm{\ensuremath{\alpha}}\mathrm{\ensuremath{\alpha}}}$(q,\ensuremath{\omega}) for one-dimensional antiferromagnets. A comparison to exact results demonstrates that qualitative features of spectra are reproduced, but are broadened. The broadening is frequency dependent with low-frequency features being reproduced most accurately. Peak positions are well described. Results for the Heisenberg model with S\ensuremath{\le}2 demonstrate the qualitative difference between the dynamics of half-integer and integer spins predicted by Haldane. From the positions of the low-temperature q=\ensuremath{\pi} peaks quantitative estimates are produced for the S=1 and 2 gaps. The q=\ensuremath{\pi} peak position for S=1 increases with temperature in agreement with experiment. When on-site anisotropy is included for S=1 a gap splitting results that resembles that found for NENP [Ni(
C2H8N2)2NO2ClO4].
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures
T close to the critical temperature
Tc using our recently developed functional renormalization group approach to quantum spin systems. In
d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent
z=2 and exhibits a pseudogap for small frequencies.
Well-defined spin-wave excitations were observed by using pulsed neutrons from CsVC
l3, an
S=3/2 Heisenberg antiferromagnetic chain. The observed spin-wave dispersion relation shows an enhancement of the excitation energy by a factor of 1.26, which must be compared with the quantum renormalization factor of $\ensuremath{\pi}/2$ for an
S=1/2 system. The linewidth of the spectrum at 40 K was broad and independent of
q. The
q-independent linewidth is the first experimental observation, which was predicted by rigorous theories for a classical chain at sufficiently low temperature.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.