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

Motivated by the recent inelastic neutron scattering (INS) measurements in the iron pnictides which show a strong anisotropy of spin excitations in directions perpendicular and parallel to the ordering wave-vector even above the magnetic transition temperature $T_N$, we study the frustrated Heisenberg model with a biquadratic spin-spin exchange interaction. Using the Dyson-Maleev (DM) representation, which proves appropriate for all temperature regimes, we find that the spin-spin dynamical structure factors are in excellent agreement with experiment, exhibiting breaking of the $C_4$ symmetry even into the paramagnetic region $T_N<T<T_σ$ which we refer to as the Ising-nematic phase. In addition to the Heisenberg spin interaction, we include the biquadratic coupling $K (\mathbf{S}_i\cdot \mathbf{S}_j)^2$ and study its effect on the dynamical temperature range $T_σ-T_N$ of the Ising-nematic phase. We find that this range reduces dramatically when even small values of the interlayer exchange $J_c$ and biquadratic coupling $K$ are included. To supplement our analysis, we benchmark the results obtained using the DM method against those from different non-linear spin-wave theories, including the recently developed generalized spin-wave theory (GSWT), and find good qualitative agreement among the different theoretical approaches as well as experiment for both the spin-wave dispersions and the dynamical structure factors.