Dissipative spin dynamics in hot quantum paramagnets

We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B $\mathbf{99}$, 060403® (2019)] to calculate the spin-spin correlation function $G(\mathit{k},\omega )$ of quantum Heisenberg magnets at infinite temperature. For small wavevectors $\mathit{k}$ and frequencies $\omega$ we find that $G ( \boldsymbol{k}, \omega )$assumesindimensions$d > 2$ the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient $\mathcal{D}$ is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about $30\mathrm{\%}$ larger than the measured high-temperature value of $\mathcal{D}$ in the Heisenberg ferromagnet Rb${}_2$CuBr${}_4\cdot$2H${}_2$O. In reduced dimensions $d\le 2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\omega )$ which diverges logarithmically in $d=2$, and as a power-law ${\cal{D}} ( \omega ) \propto \omega^{-1/3}$in$d=1$. Our result in one dimension implies scaling with dynamical exponent $z=3\mathrm{/}2$, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S ( \boldsymbol{k} , \omega )$ for all wavevectors. We show how the short-wavelength behavior of $S(\mathit{k},\omega )$ at high temperatures reflects the relative sign and strength of competing exchange interactions.