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Motivated by the recent inelastic neutron scattering (INS) measurements inthe iron pnictides which show a strong anisotropy of spin excitations indirections perpendicular and parallel to the ordering wave-vector even abovethe magnetic transition temperature $T_N$, we study the frustrated Heisenbergmodel with a biquadratic spin-spin exchange interaction. Using the Dyson-Maleev(DM) representation, which proves appropriate for all temperature regimes, wefind that the spin-spin dynamical structure factors are in excellent agreementwith experiment, exhibiting breaking of the $C_4$ symmetry even into theparamagnetic region $T_N<T<T_{\sigma}$ which we refer to as the Ising-nematicphase. In addition to the Heisenberg spin interaction, we include thebiquadratic coupling $K (\mathbf{S}_i\cdot \mathbf{S}_j)^2$ and study itseffect on the dynamical temperature range $T_{\sigma}-T_N$ of the Ising-nematicphase. We find that this range reduces dramatically when even small values ofthe interlayer exchange $J_c$ and biquadratic coupling $K$ are included. Tosupplement our analysis, we benchmark the results obtained using the DM methodagainst those from different non-linear spin-wave theories, including therecently developed generalized spin-wave theory (GSWT), and find goodqualitative agreement among the different theoretical approaches as well asexperiment for both the spin-wave dispersions and the dynamical structurefactors.

Ising-nematic order in the bilinear-biquadratic model for the iron pnictides