Orbitally and magnetically induced anisotropy in iron-based superconductors

Recent experimental developments in the iron pnictides have unambiguously demonstrated the existence of in-plane electronic anisotropy in the absence of the long-range magnetic order. Such anisotropy can arise from orbital ordering, which is described by an energy splitting between the two otherwise degenerate $d_{xz}$ and $d_{yz}$ orbitals. By including this phenomenological orbital order into a five-orbital Hubbard model, we obtain the mean-field solutions where the magnetic order is determined self-consistently. Despite sensitivity of the resulting states to the input parameters, we find that a weak orbital order that places the $d_{yz}$ orbital slightly higher in energy than the $d_{xz}$ orbital, combined with intermediate on-site interactions, produces band dispersions that are compatible with the photoemission results. In this regime, the stripe antiferromagnetic order is further stabilized and the resistivity displays the observed anisotropy. We also calculate the optical conductivity and show that it agrees with the temperature evolution of the anisotropy seen experimentally.

Figure 1

Fermi surfaces in terms of (a), (b) the pseudocrystal momentum $\tilde{\mathit{k}}$ and (c), (d) the crystal momentum $\mathit{k}$. We plot the Fermi surfaces in the $xy$ plane and set the $z$ component in each figure to (a) ${\tilde{k}}_z=0$, (b) ${\tilde{k}}_z=\pi$, (c) $k_z=0$, and (d) $k_z=\pi$. (e) Unpolarized dispersions and (f) polarized dispersions along the crystal momentum line of $X$-$\Gamma$-$X$. In (f), only the components of the $d_{yz}$ and $d_{xy}$ orbitals are shown. We represent the values of the spectral function $A(k,\omega )$ by the color scale, which is used consistently for all of the figures in this paper.

Figure 2

Fermi surfaces in the plane for (a) $k_z=0$ and (b) $k_z=\pi$. Polarized dispersions along the line of (c) $X$-$\Gamma$-$X$ and (d) $Y$-$\Gamma$-$Y$. We have set the orbital nematic order parameter $\Delta =-0.08$, explicitly breaking $C_4$ symmetry.

Figure 3

Optical conductivity ${\sigma }_{xx}$ and ${\sigma }_{yy}$ as a function of the frequency $\omega$ for the orbital nematic order parameter (a) $\Delta =0$ and (b) $\Delta =-0.08$. The inset of (a) displays the ratio of the Drude weight, ${\sigma }_{xx}\left(0\right)/{\sigma }_{yy}\left(0\right)$, as a function of $\Delta$. $\sigma \left(\omega \right)$ is plotted in an arbitrary unit, which is kept the same in this paper. The two arrows in (b) denote the two characteristic frequencies ${\omega }_1\approx 0.2$ and ${\omega }_2\approx 0.7$, where $\sigma \left(\omega \right)$ exhibits a peak structure.

Figure 4

The total staggered magnetic moment $m$ as a function of the Coulomb repulsion $U$ for different Hund's exchanges $J$. We set the energy splitting $\Delta =0$ in order to find the regime of parameters that are of interest in the context of iron-based superconductors.

Figure 5

Polarized dispersions along (a) $X$-$\Gamma$-$X$ and (b) $Y$-$\Gamma$-$Y$, for $U=1.08$, $J=0.20U$, and $\Delta =0$. The corresponding magnetic moment $m=0.46$.

Figure 6

The total staggered magnetic moment as a function of the Coulomb repulsion $U$ and orbital nematic order parameter $\Delta$. We set Hund's exchange $J=0.20U$.

Figure 7

Fermi surfaces in the plane of (a) $k_z=0$ and (b) $k_z=\pi$. Polarized dispersions along the line of (c) $X$-$\Gamma$-$X$ and (d) $Y$-$\Gamma$-$Y$. The parameters used here are $U=1.08$, $J=0.20U$, and $\Delta =-0.08$, with the total mean-field staggered moment $m=0.62$.

Figure 8

Polarized dispersions along (a) $X$-$\Gamma$-$X$ and (b) $Y$-$\Gamma$-$Y$, for $U=1.13$, $J=0.20U$, and $\Delta =0$. The corresponding magnetic moment $m=0.61$, which is close to $m=0.62$ of Fig. 7.

Figure 9

(a) Optical conductivity ${\sigma }_{xx}\left(\omega \right)$ and ${\sigma }_{yy}\left(\omega \right)$ calculated using $U=1.08$, $J=0.20U$, and $\Delta =-0.08$. They are plotted in the same unit as Fig. 3. (b) Anisotropy of Drude weight, ${\sigma }_{xx}\left(0\right)/{\sigma }_{yy}\left(0\right)$, as a function of $U$ and $\Delta$. The black line separates the regions where ${\sigma }_{xx}\left(0\right)>{\sigma }_{yy}\left(0\right)$ and where ${\sigma }_{xx}\left(0\right)<{\sigma }_{yy}\left(0\right)$. We choose $J=0.20U$ here.