Sign-Reversing Orbital Polarization in the Nematic Phase of FeSe due to the $C_2$ Symmetry Breaking in the Self-Energy

To understand the nematicity in Fe-based superconductors, nontrivial $\mathit{k}$ dependence of the orbital polarization [$\mathrm{\Delta }{E}_{xz}(\mathit{k})$, $\mathrm{\Delta }{E}_{yz}(\mathit{k})$] in the nematic phase, such as the sign reversal of the orbital splitting between $\mathrm{\Gamma }$ and $X$, $Y$ points in FeSe, provides significant information. To solve this problem, we study the spontaneous symmetry breaking with respect to the orbital polarization and spin susceptibility self-consistently. In FeSe, due to the sign-reversing orbital order, the hole and electron pockets are elongated along the $k_y$ and $k_x$ axes, respectively, consistently with experiments. In addition, an electron pocket splits into two Dirac cone Fermi pockets while increasing the orbital polarization. The orbital order in Fe-based superconductors originates from the strong positive feedback between the nematic orbital order and spin susceptibility.

Figure 1

Band structures of the eight-orbital models for (a) FeSe and (b) LaFeAsO in the unfolded Brillouin zone at $T=50\mathrm{meV}$. FSs for the FeSe and LaFeAsO models are shown in (c) and (d), respectively. The colors correspond to 2 (green), 3 (red), and 4 (blue), respectively.

Figure 2

(a) Orbital sign-reversing polarization [$\mathrm{\Delta }{E}_{xz}(\mathit{k})$, $\mathrm{\Delta }{E}_{yz}(\mathit{k})$] and (b) ${\chi }^s(\mathit{q})$ in the FeSe model for ${\alpha }_S=0.868$ ($r=0.257$). The realized FSs are shown by the green lines. The splitting $E_{xz}-E_{yz}$ at the $\mathrm{\Gamma }$ point is positive, as observed in Ref. [38]. (c) The polarization at $\mathrm{\Gamma }$, $X$, and $Y$ points. (d) $\mathrm{\Delta }n\equiv n_{xz}-n_{yz}$, and (e) ${\alpha }_S$ for $0.261>r>0.251$.

Figure 3

Orbital polarizations and band structures in the FeSe model along $Y\to \mathrm{\Gamma }\to X$ for (a) ${\alpha }_S=0.854$ and (b) ${\alpha }_S=0.868$. The corresponding FSs at $T=50\mathrm{meV}$ are shown in (c) and (d), respectively.

Figure 4

(a) Expansion of the self-energy with respect to $\mathrm{\Delta }E$. The MT term (AL term) is included in ${\widehat{\mathrm{\Sigma }}}^{\mathrm{MT}(\mathrm{AL})}$. (b) $r$ dependences of $\mathrm{\Delta }{E}_{yz}^{\mathrm{AL}}$ and $\mathrm{\Delta }{E}_{yz}^{\mathrm{MT}}$ at the $X$ point. $\mathrm{\Delta }{E}_{yz}^{\mathrm{all}}$ is the orbital polarization derived from Eq. (3) shown in Fig. 2 2. (c) ${\alpha }_S$ due to ${\widehat{\mathrm{\Sigma }}}^{\mathrm{AL}}$, and (d) ${\alpha }_S$ due to ${\widehat{\mathrm{\Sigma }}}^{\mathrm{MT}}$.

Figure 5

(a) Obtained sign-preserving orbital polarization [$\mathrm{\Delta }{E}_{xz}(\mathit{k}),\mathrm{\Delta }{E}_{yz}(\mathit{k})$] in the LaFeAsO model for ${\alpha }_S=0.98$ ($r=0.376$). The FSs are shown by the green lines. The corresponding spin susceptibility and the band structure are shown in (b) and (c), respectively.