Showing papers by "P. C. Hohenberg published in 1971"

Dynamics of an Antiferromagnet at Low Temperatures: Spin-Wave Damping and Hydrodynamics

Abstract: The Dyson-Maleev boson formulation is used to investigate the dynamical properties of Heisenberg antiferromagnets at long wavelengths and low temperatures. Various regimes for the decay rate of spin waves are found, depending on the relation between the wave vector $k$, the temperature $T$, and the anisotropy energy $\ensuremath{\hbar}{\ensuremath{\omega}}_{A}$, and in all cases the decay rate is much smaller than the spin-wave frequency. This result implies that spin waves are well-defined elementary excitations, which interact weakly at low temperatures and long wavelengths, in contrast to results obtained by previous authors, but in close analogy with the ferromagnetic case. When the long-wavelength limit is taken at fixed temperature, the decay rate ${\ensuremath{\Gamma}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$ is proportional to the square of the frequency ${\ensuremath{\omega}}_{E}{\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}$, where ${\ensuremath{\omega}}_{E}$ is the exchange frequency. In the quantum-mechanical low-temperature limit ($\mathrm{ST}\ensuremath{\ll}{T}_{N}$), we find ${\ensuremath{\Gamma}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}=2{\ensuremath{\omega}}_{E}{S}^{\ensuremath{-}2}{\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{{}^{2}}{\ensuremath{\tau}}^{3}{(2\ensuremath{\pi})}^{\ensuremath{-}3}(a|\mathrm{ln}\ensuremath{\tau}|+{a}^{\ensuremath{'}})$ for ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}\ensuremath{\ll}{\ensuremath{\tau}}^{3}\ensuremath{\ll}1$, where $\ensuremath{\tau}=\frac{2{k}_{B}T}{\ensuremath{\hbar}{\ensuremath{\omega}}_{E}}$, and $S$ is the spin quantum number. In the classical low-temperature limit ($\frac{{T}_{N}}{S}\ensuremath{\ll}T\ensuremath{\ll}{T}_{N}$), we find ${\ensuremath{\Gamma}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}=(\frac{4\ensuremath{\eta}}{3\ensuremath{\pi}}){\ensuremath{\omega}}_{E}{(\frac{T}{{T}_{N}})}^{2}{\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{2}$ for ${\ensuremath{\epsilon}}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}\ensuremath{\ll}1$. For small uniaxial single-ion anisotropy [${\ensuremath{\epsilon}}_{0}\ensuremath{\sim}{(\frac{2{\ensuremath{\omega}}_{A}}{{\ensuremath{\omega}}_{E}})}^{\frac{1}{2}}\ensuremath{\ll}1$], we find ${\ensuremath{\Gamma}}_{0}=\frac{3}{2}{\ensuremath{\omega}}_{E}{S}^{\ensuremath{-}2}{\ensuremath{\epsilon}}_{0}^{2}{\ensuremath{\tau}}^{3}{(2\ensuremath{\pi})}^{\ensuremath{-}3}(a|\mathrm{ln}\ensuremath{\tau}|+{a}^{\ensuremath{'}\ensuremath{'}})$ for ${\ensuremath{\epsilon}}_{0}\ensuremath{\ll}{\ensuremath{\tau}}^{3}\ensuremath{\ll}1$. (In these expressions, $a$, ${a}^{\ensuremath{'}}$, $\ensuremath{\eta}$, and ${a}^{\ensuremath{'}\ensuremath{'}}$ are all constants of order unity.) Results are also obtained for other regimes, and for the damping of a spin wave driven off resonance. In each case, the nature and self-consistency of the perturbation expansion are examined in detail. For the isotropic system, the full frequency-dependent transverse spin-correlation functions are calculated in the long-wavelength limit, and are found to agree with the forms previously obtained by hydrodynamic arguments. By a comparison of the two forms, the transport coefficients are determined at low temperatures. Several of the calculations have been performed using the Holstein-Primakoff as well as the Dyson-Maleev representations. The results for observable quantities agree in the two formalisms, except at the longest wavelengths, where the Holstein-Primakoff expressions are not self-consistent in lowest order. Finally, the possibility of experimental verification of the present calculations is briefly discussed.