Dissipative spin dynamics in hot quantum paramagnets

Abstract

We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B $99$, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbf{k},\omega )$ of quantum Heisenberg magnets at infinite temperature. For small wavevectors $\mathbf{k}$ and frequencies $\omega$ we find that $G(\mathbf{k},\omega )$ assumes in dimensions $d>2$ 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 $30\%$ 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 $\mathcal{D}(\omega )\propto {\omega }^{-1/3}$ in $d=1$. Our result in one dimension implies scaling with dynamical exponent $z=3/2$, 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 $S(\mathbf{k},\omega )$ for all wavevectors. We show how the short-wavelength behavior of $S(\mathbf{k},\omega )$ at high temperatures reflects the relative sign and strength of competing exchange interactions.

arXiv comment30 pages, 11 figures

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