Dynamic Ruderman-Kittel-Kasuya-Yosida indirect interaction

Abstract

We consider the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two spins, when they are affected by a frequency-dependent magnetic field. It is shown that the RKKY indirect interaction, i.e., value of the coupling, its spatial dependence, and the periods of spatial oscillations, in the three-dimensional free-electron gas for transversal component of the spins depends on the frequency. For the high-frequency range h\ensuremath{\omega}\ensuremath{\leqslant}${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{F}}$ three main uncommensurable periods of oscillations ${\mathrm{k}}_{\mathrm{F}}$, (1+$\sqrt{2}$)${\mathrm{k}}_{\mathrm{F}}$, ($\sqrt{2}$-1)${\mathrm{k}}_{\mathrm{F}}$ are found. The spatial dependence of the coupling for large ${\mathrm{k}}_{\mathrm{F}}$ r contains terms proportional to ${\mathrm{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathrm{r}}^{\mathrm{\ensuremath{-}}2}$. The value of the coupling is of the order (${\mathrm{k}}_{\mathrm{F}}$ r${)}^{2}$ as much as in the static case. For the frequency range h\ensuremath{\omega}\ensuremath{\ll}${\mathrm{\ensuremath{\epsilon}}}_{\mathrm{F}}$ the dynamic RKKY interaction contains the terms whose values and spatial periods are determined by the frequency of the magnetic field. It is shown that there is a frequency-dependent anisotropy of the dynamic RKKY interaction.