Spin excitations in the nematic phase and the metallic stripe spin-density wave phase of iron pnictides

We present a general study of the magnetic excitations within a weak-coupling five-orbital model relevant to itinerant iron pnictides. As a function of enhanced electronic correlations, the spin excitations in the symmetry-broken spin-density wave phase evolve from broad low-energy modes in the limit of weak interactions to sharply dispersing spin waves prevailing to higher energies at larger interaction strengths. We show how the resulting spin response at high energies depends qualitatively on the magnitude of the interactions. We also calculate the magnetic excitations in the nematic phase by including an orbital splitting, and find a pronounced $C_2$-symmetric excitation spectrum right above the transition to long-range magnetic order. Finally, we discuss the $C_2$ versus $C_4$ symmetry of the spin excitations as a function of energy for both the nematic and the spin-density wave phase.

Figure 1

Self-consistent orbitally resolved magnetization (a) and electron density (b) versus interaction parameter $U$ for $J=U/6$.

Figure 2

(a) The diagrammatic RPA series of the transverse spin susceptibility in the SDW state where the propagators correspond to the full Green's functions in orbital and $\mathbf{q},{\mathbf{q}}^{\prime }$ space and ${\widehat{g}}^{\sigma \overline{\sigma }}=g^{\sigma \overline{\sigma }}{\delta }_{{\mathbf{k}}_1+{\mathbf{k}}_2,{\mathbf{k}}_1^{\prime }+{\mathbf{k}}_2^{\prime }}$ is a diagonal matrix in $\mathbf{q},{\mathbf{q}}^{\prime }$ space. The paramagnetic normal state expression is obtained by considering normal propagators only. (b) The diagrammatic RPA series of the longitudinal spin susceptibility with both bubble- and ladder-type diagrams. The same expression applies to both the normal and the SDW phases in the longitudinal channel because the umklapp processes do not contribute in this case.

Figure 3

Imaginary part of the physical transverse dynamical RPA susceptibility, $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$, in the SDW phase along the momentum-space path $Y\to \Gamma \to X\to Y$ for different values of $U=1.1,1.2,1.3,1.4,1.8,1.9,2.0$ eV (top to bottom) shown in each panel.

Figure 4

Imaginary part of the bare physical transverse spin susceptibility, $\text{Im}{\chi }_0^{+-}(\mathbf{q},\omega )$, for the same values of $U$ and momenta as in Fig. 3.

Figure 5

Evolution of the Fermi surface as a function of the SDW order parameter for the values of Coulomb $U$ stated in the insets. Each panel shows the spectral weight in a window of $\pm 5$ meV around the Fermi level.

Figure 6

Constant-energy $\omega$ cuts of $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$ in the SDW phase with $U=1.1$ eV and $J=U/6$ shown at $\omega =20$ meV (a), $\omega =40$ meV (b), $\omega =80$ meV (c), $\omega =100$ meV (d), $\omega =120$ meV (e), and $\omega =140$ meV (f).

Figure 7

Constant-energy $\omega$ cuts of $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$ in the SDW phase with $U=2.0$ eV and $J=U/6$ shown at $\omega =60$ meV (a), $\omega =120$ meV (b), $\omega =160$ meV (c), $\omega =200$ meV (d), $\omega =220$ meV (e), and $\omega =280$ meV (f).

Figure 8

Imaginary part of the physical transverse dynamical RPA susceptibility, $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$, in the SDW phase along the perpendicular momentum cut $\Gamma \to X\to M$ for (a) $U=1.1$ eV and (b) $U=2.0$ eV on a logarithmic scale. (c) and (d) reveal the spectral weight of the physical matrix components $\text{Im}{\left[{\chi }_{\text{RPA}}^{+-}\right]}_{sp}^{sp}$ (the orbital $s$ is represented by color and $p$ can be read off the $x$ axis) for $U=1.1$ eV (c) and $U=2.0$ eV (d) contributing to the entire spectrum shown in (a) and (b), respectively. (e) Momentum-integrated physical transverse dynamical RPA susceptibility, $\text{Im}{\chi }_{\text{RPA}}^{+-}\left(\omega \right)$ in the low-$U$ (circles, solid curve) and large-$U$ (crosses, dashed curve) limits.

Figure 9

(a) Spin susceptibility $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$ in the paramagnetic nematic phase at ${\mathbf{Q}}_1=(\pi ,0)$ and ${\mathbf{Q}}_2=(0,\pi )$ versus temperature $T$ for representative energies shown in the legend and an orbital splitting of $\delta =65$ meV. (b) Two-dimensional (2D) map of the normalized spin anisotropy $[\text{Im}{\chi }_{\text{RPA}}({\mathbf{Q}}_1,\omega )-\text{Im}{\chi }_{\text{RPA}}({\mathbf{Q}}_2,\omega )]/\mathcal{N}$ where $\mathcal{N}=\text{Max}[\text{Im}{\chi }_{\text{RPA}}({\mathbf{Q}}_1,\omega )-\text{Im}{\chi }_{\text{RPA}}({\mathbf{Q}}_2,\omega )]$ at $T=1.05T_N^{\delta }$.

Figure 10

Constant-energy cuts of the spin susceptibility $\text{Im}{\chi }_{\text{RPA}}^{+-}(\mathbf{q},\omega )$ in the paramagnetic nematic phase at temperature $T=1.04T_N^{\delta }$ for $U=1.0$ eV, $J=U/4$, and $\delta =65$ meV shown at $\omega =2$ meV (a), $\omega =15$ meV (b), $\omega =25$ meV (c), $\omega =40$ meV (d), $\omega =65$ meV (e), and $\omega =100$ meV (f).