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Critical spin dynamics of Heisenberg ferromagnets revisited

We calculate the dynamic structure factor $S (\boldsymbol{k},ω)$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures $T$ close to the critical temperature $T_c$ using our recently developed functional renormalization group approach to quantum spin systems. In $d=3$ dimensions we find that for small momenta $\boldsymbol{k}$ and frequencies $ω$ the dynamic structure factor assumes the scaling form $S(\boldsymbol{k},ω) = (τT G (\boldsymbol{k})/π)Φ(kξ, ωτ)$, where $ G (\boldsymbol{k})$ is the static spin-spin correlation function, $ξ$ is the correlation length, and the characteristic time-scale $τ$ is proportional to $ξ^{5/2}$. We explicitly calculate the dynamic scaling function $Φ(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $ξ= \infty$ our result for the dynamic structure factor can be written as $S (\boldsymbol{k},ω) = (πω_k)^{-1} T_c G (\boldsymbol{k}) Ψ_c (ω/ω_k)$, where $ω_k \propto k^{5/2}$. We find that $Ψ_c(ν)$ vanishes as $ν^{-13/5}$ for large $ν$, and as $ν^{3/5}$ for small $ν$. While the large-frequency behavior of $Ψ_c (ν)$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $ε= 6-d$, our result for small frequencies disagrees with previous calculations. We argue that up until now neither experiments nor numerical simulations are sufficiently accurate to determine the low-frequency behavior of $Ψ_c (ν)$. We also calculate the low-temperature behavior of $S ( \boldsymbol{k},ω)$ in one- and two dimensional ferromagnets and find that it satisfies dynamic scaling with exponent $z=2$ and exhibits a pseudogap for small frequencies.