Self-consistent spin-wave theory for a frustrated Heisenberg model with biquadratic exchange in the columnar phase and its application to iron pnictides

Abstract

Recent neutron scattering studies revealed the three-dimensional character of the magnetism in the iron pnictides and a strong anisotropy between the exchange perpendicular and parallel to the spin stripes. We extend studies of the ${J}_{1}$-${J}_{2}$-${J}_{c}$ Heisenberg model with $S=1$ using self-consistent spin-wave theory. A discussion of two scenarios for the instability of the columnar phase is provided. The relevance of a biquadratic exchange term between in-plane nearest neighbors is discussed. We introduce mean-field decouplings for biquadratic terms using the Dyson-Maleev and the Schwinger boson representation. Remarkably their respective mean-field theories do not lead to the same results, even at zero temperature. They are gauged in the N\'eel phase in comparison to exact diagonalization and series expansion. The ${J}_{1}$-${J}_{2}$-${J}_{c}$ model is analyzed under the influence of the biquadratic exchange ${J}_{\text{bq}}$ and a detailed description of the staggered magnetization and of the magnetic excitations is given. The biquadratic exchange increases the renormalization of the in-plane exchange constants which enhances the anisotropy between the exchange parallel and perpendicular to the spin stripes. Applying the model to iron pnictides, it is possible to reproduce the spin-wave dispersion for CaFe${}_{2}$As${}_{2}$ in the direction perpendicular to the spin stripes and perpendicular to the planes. Discrepancies remain in the direction parallel to the spin stripes which can be resolved by passing from $S=1$ to $S=3/2\phantom{\rule{0.28em}{0ex}}\mathit{or}\phantom{\rule{0.28em}{0ex}}S=2$. In addition, results for the dynamical structure factor within the self-consistent spin-wave theory are provided.