Magnon damping, spin stiffness, and dynamic scaling in the two-dimensional quantum Heisenberg ferromagnet at low temperatures.

Abstract

We combine Dyson-Maleev formalism with Schwinger-boson mean-field theory to study magnon damping ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{k}}$ in the two-dimensional quantum Heisenberg ferromagnet at low temperatures T and for wavelengths ${\mathit{k}}^{\mathrm{\ensuremath{-}}1}$ large compared with the thermal de Broglie wavelength ${\ensuremath{\lambda}}_{\mathrm{th}}$. For k${\ensuremath{\lambda}}_{\mathrm{th}}$\ensuremath{\ll}1 we obtain perturbatively \ensuremath{\Elzxh}${\mathrm{\ensuremath{\Gamma}}}_{\mathbf{k}}$/${\mathit{E}}_{\mathbf{k}}$=2\ensuremath{\pi}(T/2\ensuremath{\pi}${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}^{0}$${)}^{2}$, where ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}^{0}$ is the zero-temperature limit of the spin stiffness. To incorporate nonperturbative corrections due to the finiteness of the correlation length \ensuremath{\xi}, we replace ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}^{0}$ by the momentum-dependent spin stiffness ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$(k,T) at finite temperature. Defining the momentum- and frequency-dependent spin stiffness ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$(k,\ensuremath{\omega},T) via the change of the free energy of the system due to a slowly varying twist in the direction of the local magnetization of wavelength ${\mathit{k}}^{\mathrm{\ensuremath{-}}1}$ and frequency \ensuremath{\omega}, we find that ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$(k,\ensuremath{\omega},T) is given by the sum of a constant diamagnetic term and a paramagnetic term. The latter can be written in terms of a retarded correlation function of a spin current. We evaluate ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$(k,\ensuremath{\omega}=0,T) approximately using Schwinger-boson mean-field theory, and explicitly obtain \ensuremath{\Elzxh}${\mathrm{\ensuremath{\Gamma}}}_{\mathbf{k}}$/${\mathit{E}}_{\mathbf{k}}$ as a function of k\ensuremath{\xi} alone, in agreement with the dynamic-scaling hypothesis.