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 even above the magnetic transition temperature $T_N$, we study the spin dynamics within the frustrated Heisenberg model with biquadratic spin-spin exchange interactions. 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 $T_N<T<T_{\sigma }$, which we refer to as the Ising-nematic phase. In addition to the Heisenberg spin interaction, we include the biquadratic coupling $-K{({\mathbf{S}}_i·{\mathbf{S}}_j)}^2$ and study its effect on the dynamical temperature range $T_{\sigma }-T_N$ 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 full decoupling in the DM method against those from different nonlinear 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.

Figure 1

Evolution of the sublattice magnetization $m_s$ and the nematic order parameter $\left({\mathrm{\Gamma }}_y-{\mathrm{\Gamma }}_x\right)$ for the case of $S=1$ and $K=0.01J_2$, for different values of the interlayer coupling using the full decoupling of the DM bosons. As expected, $T_N\le T_{\sigma }$ for all cases although the nematic range of temperatures is negligible except for artificially small values of $J_c$ (corresponding effectively to a quasi-two-dimensional model). Note that the discontinuous first-order transition observed for the nematicity is just an artifice of the mean-field technique.

Figure 2

$T$ vs $J_c$ phase diagrams for the case of $S=1$ with a biquadratic coupling of (a) $K=0$ and (b) $K=0.01J_2$. The colored region corresponds to the Ising-nematic regime, which covers an appreciable range of temperatures only for unrealistically small values of $J_c$ (which turn this into an effective two-dimensional problem) and requires very precise fine-tuning for $K$.

Figure 3

(a) and (b) Calculated dynamical structure factor $S(\mathbf{q},\omega =0.5J_2)$ using the DM method for the case of $S=1, K=0$, and $J_c=0.1J_2$ at two temperatures: (a) $T=1.043T_N$ ($T_N<T<T_{\sigma }$), with its position indicated in the phase diagram of Figs. 2 and 2 $T_{\sigma }=1.085T_N$. We note that in (b), the figure was rotationally symmetrized due to the known limitation of the DM method [50], which gives an unphysical zero value of $S(\mathbf{q},\omega )$ above the temperature $T>T_{\sigma }$. (c) and (d) Experimental INS data on detwinned ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ above the Néel temperature $T_N=T_s\approx 138$ K, adopted from Lu et al. [16].

Figure 4

LSWT and NLSWT results for the spin-wave dispersions of the $S=1$ case for (a) $K=0$, (b) $0.4J_2$, and (c) $0.8J_2$. In agreement with the results in Ref. [49], quantum fluctuations suppress the appearance of a maximum at $(\pi ,\pi )$ for the fully decoupled NLSWT. In turn, the simpler decoupling scheme based on the Hubbard-Stratonovich transformation with ${\mathrm{\Gamma }}_{x,y}$ bond averages replicates the findings in Ref. [48], which underestimate the effect of quantum fluctuations.

Figure 5

LSWT and NLSWT results for the spin-wave dispersions of the $S=2$ case for (a) $K=0$, (b) $0.1J_2$, and (c) $0.2J_2$. The values of the biquadratic constant are reduced by a factor of $1/S^2$ with respect to those of the case of $S=1$ since the constant $K$ is of that order [48]. Note that in this case, the $(\pi ,\pi )$ point has become a maximum for all methods, due to the smaller relative importance of the fluctuations for a larger size of spin.

Figure 6

Discrepancies among the results from the different spin-wave theories. In (a), we plot the spin-wave dispersions of the magnetically ordered ground state of the $J_1-J_2-K$ model for the case of $S=1$, for a value of $K=0.8J_2$, using the results from all the methods we tried. The simplified decoupling of the NLSWT using the HS transformation is almost as ineffective as the LSWT at capturing the effect of the quantum fluctuations, whereas using a full decoupling scheme in the NLSWT produces much more accurate results, and agrees favorably with the GSWT and the DM methods. In (b), we plot the evolution of the normalized maximum of the ground-state dispersion [at point $\left(\pi ,\pi \right)$] for the $J-K$ model, comparing the results from all our methods to those of Ref. [49] (Stanek et al). Again, our results from the simply decoupled NLSWT overlap with those obtained via SB also in Ref. [49] (known to also underestimate fluctuations), while the rest of the methods account for all the possible correlations to a similar extent.

Figure 7

Comparison of the calculated (left) and experimentally measured (right) dynamical structure factors at low temperature ($T=7$ K in Ref. [52]). Calculations were performed using GSWT at $K=0.8$ for energy cuts of (a) $\omega =2J_2$, (b) $\omega =3J_2$, (c) $\omega =5J_2$, and (d) $\omega =6J_2$. A value of $J_2=25$ meV was used with broadening $\gamma =0.5J_2$ in Eq. (11).

Figure 8

$T$ vs $K$ phase diagrams for the case of $S=1$ with values of the interlayer coupling of (a) $J_c=0.1J_2$ and (b) $0.01J_2$, respectively. The colored area corresponds to the Ising-nematic region. Both temperatures decrease rapidly with $K$ and coincide fast, with wider nematic regions still appreciable for smaller interlayer couplings.