The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the ``hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the ``hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.
The critical behaviour of a magnetic system with O(N) spin symmetry, bounded by two (d-1)-dimensional hyperplanes meeting at an angle alpha , is studied within mean field theory and in d=4- epsilon dimensions. New exponents emerge for correlation functions, and magnetisation and susceptibilities, for spins close to the edge. They can be expressed in terms of known bulk and surface exponents, together with a single new edge exponent which depends, however, on the angle alpha . This exponent is computed to first order in epsilon .
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.
Employing Monte Carlo simulations, the equilibrium dynamic critical behavior or L×L×L simple cubic classical Heisenberg models with periodic boundary conditions are determined. From the long-time exponential decay of the time-displaced correlation function of the magnetization the linear (equilibrium) relaxation time τ is also determined. It is found that at the critical point Tc, τ grows like τ≂Lz with z=1.94±0.06. In the paramagnetic region a power-law divergence τ≂(T/Tc−1)−Δ was found with Δ=zν=1.38±0.04, which implies a dynamical critical exponent z=1.96±0.07. Both values are consistent with theoretical predictions and results of experimental studies concerning the dynamical critical behavior of Heisenberg magnets perturbed by order-parameter nonconserving forces.