Spin-orbital interplay and topology in the nematic phase of iron pnictides

The origin of the nematic state is an important puzzle to be solved in iron pnictides. Iron superconductors are multiorbital systems and these orbitals play an important role at low energy. The singular $C_4$ symmetry of $d_{zx}$ and $d_{yz}$ orbitals has a profound influence at the Fermi surface since the $\mathrm{\Gamma }$ pocket has vortex structure in the orbital space and the $X/Y$ electron pockets have $yz/zx$ components, respectively. We propose a low-energy theory for the spin-nematic model derived from a multiorbital Hamiltonian. In the standard spin-nematic scenario, the ellipticity of the electron pockets is a necessary condition for nematicity. In the present model, nematicity is essentially due to the singular $C_4$ symmetry of $yz$ and $zx$ orbitals. By analyzing the ($\pi ,0$) spin susceptibility in the nematic phase, we find a spontaneous generation of orbital splitting, extending previous calculations in the magnetic phase. We also find that the ($\pi ,0$) spin susceptibility has an intrinsic anisotropic momentum dependence due to the nontrivial topology of the $\mathrm{\Gamma }$ pocket.

Figure 1

(a)–(c) $w_{\mathrm{\Gamma }X}^{{\eta }_1}{w}_{\mathrm{\Gamma }X}^{{\eta }_2}$ factors weighting the k integrals of the magnetic bubble Eq. (7) computed using the same model parameters of [56]. The dominant orbital component is the $w_{\mathrm{\Gamma }X}^1{w}_{\mathrm{\Gamma }X}^1$. The contribution from the orbital 2 is considerably smaller as well as the mixed one $w_{\mathrm{\Gamma }X}^1{w}_{\mathrm{\Gamma }X}^2$. By drawing the $\mathbf{k}$ dependence of the orbital weights for the magnetic fluctuations $\mathrm{\Gamma }Y$, we would find the same results rotated by $\pi /2$ as expected by symmetry. (d) ${\tau }_3$ weight component $w_{\mathrm{\Gamma }X}^3=\frac{1}{2}(w_{\mathrm{\Gamma }X}^1{w}_{\mathrm{\Gamma }X}^1-w_{\mathrm{\Gamma }X}^2{w}_{\mathrm{\Gamma }X}^2)$. The ${\tau }_3$ weight is similar to $\sim w_{\mathrm{\Gamma }X}^1{w}_{\mathrm{\Gamma }X}^1/2$ as found in the simplified model where we assume the contribution of the orbital 2 being zero for the $X$ pocket.

Figure 2

Band structure of the two-orbital $d_{yz}{d}_{zx}$ model for the same Hamiltonian parameters used in Ref. [56]: $t_1=-1,t_2=1.3,t_3=t_4=-0.85$, and $\mu =1.85$ (dashed line) in units of $|t_1|$. Green (red) stands for the $yz$ ($zx$) orbital weight. Both orbitals contribute to the $\mathrm{\Gamma }$ pockets while the $X/Y$ pocket is mostly $yz/zx$.

Figure 3

Vortex in the orbital space. The arrows represent the vector field $\vec{n}=\vec{h}\left(\mathbf{k}\right)/\left|\vec{h}\left(\mathbf{k}\right)\right|$ in the BZ. The FSs of the two-orbital model, whose orbital contribution is shown using the same color code than before ($yz$ green, $zx$ red), are superimposed. Notice the vortex around the $\mathrm{\Gamma }$ point. The vector field around the Fermi surface can be identified in the continuum limit $\vec{n}\sim {\vec{n}}_{\mathrm{\Gamma }}=(sin\left(2{\varphi }_{\mathbf{k}}\right),cos\left(2{\varphi }_{\mathbf{k}}\right))$, while around the $X/Y$ pockets $\vec{n}\sim {\vec{n}}_{X/Y}=\pm (0,1)$. We used the same Hamiltonian parameters of Fig. 2.