Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.
The issue of ergodicity is often underestimated. The presence of zero-frequency excitations in bosonic Green's functions determine the appearance of zero-frequency momentum-dependent quantities in correlation functions. The implicit dependence of matrix elements make such quantities also relevant in the computation of susceptibilities. Consequently, the correct determination of these quantities is of great relevance and the well-established practice of fixing them by assuming the ergodicity of the dynamics is quite questionable as it is not justifiable a priori by no means. In this manuscript, we have investigated the ergodicity of the dynamics of the
z component of the spin in the one-dimensional Heisenberg model with anisotropic nearest-neighbor and isotropic next-nearest-neighbor interactions. We have obtained the zero-temperature phase diagram in the thermodynamic limit by extrapolating Exact and Lanczos diagonalization results computed on chains with sizes
L=6−−26. Two distinct nonergodic regions have been found: one for ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.3$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}<1$ and another for ${J}^{\ensuremath{'}}∕{J}_{z}\ensuremath{\lesssim}0.25$ and $\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}∕{J}_{z}=1$. On the contrary, finite-size scaling of $T\ensuremath{\ne}0$ results, obtained by means of Exact diagonalization on chains with sizes
L=4−−18, indicates an ergodic behavior of dynamics in the whole range of parameters.
The search for departures from standard hydrodynamics in many-body systems has yielded a number of promising leads, especially in low dimensions. Here, we study one of the simplest classical interacting lattice models, the nearest-neighbor Heisenberg chain, with temperature as the tuning parameter. Our numerics expose strikingly different spin dynamics between the antiferromagnet, where it is largely diffusive, and the ferromagnet, where we observe strong evidence either of spin superdiffusion or an extremely slow crossover to diffusion. This difference also governs the equilibration after a quench, and, remarkably, is apparent even at very high temperatures.
We study the low temperature dynamics of the classical Heisenberg antiferromagnet with nearest neighbor interaction on the frustrated pyrochlore lattice. We present extensive results for the wave vector and frequency dependence of the dynamical structure factor, obtained from simulations of the precessional dynamics. We also construct a solvable stochastic model for dynamics with conserved magnetization, which accurately reproduces most features of the precessional results. Spin correlations relax at a rate independent of the wave vector and proportional to the temperature.
We have performed large-scale molecular dynamics simulations of a dynamical variant of the classical Heisenberg model in which the spins are replaced by interacting "linear molecules," each of which has two rotational degrees of freedom. The Hamiltonian consists of rotational kinetic-energy terms as well as nearest-neighbor Heisenberg-like interactions $\ensuremath{-}J{\mathrm{s}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{s}}_{j}$, where
si is a unit vector specifying the orientation of the molecule at lattice site
i. Systems of size up to
643 spins on the simple cubic lattice have been studied, and thermostatting is used to maintain strictly constant temperature. We determined the dynamic structure function from the time- and space-displaced correlation functions, and found good agreement with the predictions of the dynamic scaling theory with a dynamic critical exponent of
z=1.5. Results are compared and contrasted with data from spin-dynamics calculations on classical Heisenberg ferromagnets and antiferromagnets.
The nearest-neighbor random-exchange Heisenberg antiferromagnetic Heisenberg chain (
s=21) is studied at low temperatures via an approximate renormalization-group method. This entails renormalization of the random-exchange coupling constant
J and of the probability law for
J.
P0(J). After
n iterations we find that the renormalized probability function
Pn(J(n)), a function of the renormalized coupling
J(n), develops singular behavior for small
J(n), independent of the initial form of
P0(J). This happens both for
P0(J) that diverge or go to zero as $J\ensuremath{\rightarrow}0$. The singular form of
Pn(J(n)) is such that it leads to a specific heat and susceptibility that behave like $C\ensuremath{\sim}{T}^{1\ensuremath{-}{\ensuremath{\alpha}}_{C}(T)}$ and $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)}$. ${\ensuremath{\alpha}}_{C}(T)$ and ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)$ are exponents weakly dependent on temperature (
T) that go to one as $T\ensuremath{\rightarrow}0$ and satisfy ${\ensuremath{\alpha}}_{\ensuremath{\chi}}(T)>{\ensuremath{\alpha}}_{C}(T)$, for arbitrary initial probability laws. The classical
n-vector models, and in particular the classical Heisenberg model are also studied using a renormalization-group approach and it is found that their behavior is different than that of the quantum model: singular behavior in the thermodynamic properties is only obtained if the starting
P0(J) is singular. An explanation of these results in terms of the scaling theory of localization is suggested. The relevance of our results to the experimental findings on the magnetic properties of tetracyanoquinodimethanide complexes is also discussed.