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Elucidating the nature of spin excitations is important to understanding themechanism of superconductivity in the iron pnictides. Motivated by recentinelastic neutron scattering measurements in the nearly 100% detwinnedBaFe$_{2}$As$_{2}$, we study the spin dynamics of an $S=1$ frustratedbilinear-biquadratic Heisenberg model in the antiferromagnetic phase withwavevector $(\pi,0)$. The biquadratic interactions are treated in a dynamicalway using a flavor-wave theory in an $SU(3)$ representation. Besides the magnon(dipolar) excitations, the biquadratic interactions give rise to quadrupolarexcitations at high energies. We find that the quadrupolar wave significantlyinfluences, in an energy dependent way, the anisotropy between the spinexcitation spectra along the $(\pi,0)$ and $(0,\pi)$ directions in the wavevector space. Our theoretical results capture the essential behavior of thespin dynamics measured in the antiferromagnetic phase of the detwinnedBaFe$_2$As$_2$. More generally, our results underscore the importance ofelectron correlation effects for the microscopic physics of the iron pnictides.

Anisotropic magnetic excitations of a frustrated bilinear-biquadratic spin model -- Implications for spin waves of detwinned iron pnictides