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

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 Néel 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 $S=3/2\mathit{or}S=2$. In addition, results for the dynamical structure factor within the self-consistent spin-wave theory are provided.

Figure 1

Three-dimensional collinear phase with antiferromagnetic interlayer exchange $J_c$ and biquadratic exchange $J_{\text{bq}}$ between in-plane nearest-neighbor sites.

Figure 2

Dependence of the ground-state energy on the relative angle $\varphi$ between the two sublattice magnetizations on the two intercalated Néel lattices, after Ref. 51.

Figure 3

Ratio $v_b/v_a$ as a function of $m_{\mathit{st}}$ for $S=1/2$ (upper panel) and $S=1$ (lower panel). The curves either end on the ordinate, i.e, for zero staggered magnetization, or on the abscissa, i.e., for zero spin-wave velocity $v_b<v_c<v_a$. Thus, two qualitatively different instabilities emerge naturally. The solid lines correspond to the magnetization-driven scenario (vanishing magnetization) while the dashed lines correspond to the velocity-driven scenario (vanishing spin-wave velocity). The dotted lines are extrapolations in the magnetization-driven scenario.

Figure 4

Influence of the biquadratic exchange on the excitation energies in the 2D Néel phase ($S=1$). The dashed lines are fits to the corresponding data points. Note the almost perfect straight lines obtained in exact diagonalization and in series expansion.

Figure 5

Staggered magnetization as a function of $x$ for $S=1$ and $\mu =0.25$. All curves are in the velocity-driven scenario which we presume to signal a strong first-order transition to the Néel phase. The magnetization remains finite at the end point where the columnar phase ceases to exist; for details see Sec. 2.

Figure 6

Renormalization of the effective exchange parameters by quantum fluctuations as function of $x$ for $S=1$ and $\mu =0.25$. All curves are in the velocity-driven scenario.

Figure 7

Ratios of the spin-wave velocities as a function of $x$ for $S=1$ and $\mu =0.25$. All curves are in the velocity-driven scenario; i.e., the ratio $v_b/v_a$ vanishes at the end point. This is at the limit of the resolution of the numerical evaluation of the three-dimensional integrals.

Figure 8

Relative anisotropy, cf. definitions (46a), as a function of the relative biquadratic exchange $\nu$ in the two-dimensional model for $\mu =0,x=1,\mathit{and}S=1$, $3/2$, and 2.

Figure 9

Spin-wave dispersion (40) resulting from the self-consistent mean-field theory of the model (35) for $S=1$ in the three-dimensional columnar phase with biquadratic exchange for various values of $\nu =J_{\mathit{bq}}/J_1$. The dispersions (lines) are plotted for the parameters given in Table . The red dots are experimental data extracted from INS for CaFe${}_2$As${}_2$ from Ref. 33. Wave vectors are given in units of $\pi$ over lattice constants assuming an orthorhombic crystal. Note that along the path in the BZ shown in the left panel all fits collapse in one curve. Significant differences occur only along the path shown in the right panel.

Figure 10

Same as in Fig. 9 but for $S=3/2$ and $S=2$; the parameters are given in Table .

Figure 11

Constant-energy cuts of the dynamical structure factor per spin (47) integrated over the given energy interval for a twinned sample for $S=1$ and the relative biquadratic exchange $\nu =0.5$. Other parameters are given in Table . The range of $k_{a,b}$ and the values of $k_c$ and $E$ have been chosen to match the rendering of the experimental data (Ref. 33). The reciprocal lattice vectors are given in units $\pi$/lattice constant. Note that the weights in the given energy intervals are dimensionless according to formula (47).

Figure 12

Same as Fig. 11, but for $S=2$. The parameters are given in Table . The plots for $S=3/2$ would look essentially the same.