Critical spin dynamics of Heisenberg ferromagnets revisited

Abstract

We calculate the dynamic structure factor $S(\mathbf{k},\omega )$ 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 $\mathbf{k}$ and frequencies $\omega$ the dynamic structure factor assumes the scaling form $S(\mathbf{k},\omega )=(\tau TG(\mathbf{k})/\pi )\Phi (k\xi ,\omega \tau )$, where $G(\mathbf{k})$ is the static spin-spin correlation function, $\xi$ is the correlation length, and the characteristic time-scale $\tau$ is proportional to ${\xi }^{5/2}$. We explicitly calculate the dynamic scaling function $\Phi (x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\xi =\infty$ our result for the dynamic structure factor can be written as $S(\mathbf{k},\omega )=(\pi {\omega }_k{)}^{-1}{T}_{c}G(\mathbf{k}){\Psi }_c(\omega /{\omega }_k)$, where ${\omega }_k\propto k^{5/2}$. We find that ${\Psi }_c(\nu )$ vanishes as ${\nu }^{-13/5}$ for large $\nu$, and as ${\nu }^{3/5}$ for small $\nu$. While the large-frequency behavior of ${\Psi }_c(\nu )$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\epsilon =6-d$, 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 ${\Psi }_c(\nu )$. We also calculate the low-temperature behavior of $S(\mathbf{k},\omega )$ in one- and two dimensional ferromagnets and find that it satisfies dynamic scaling with exponent $z=2$ and exhibits a pseudogap for small frequencies.

arXiv comment19 pages, 18 figures

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