Three orbital model for the iron-based superconductors

The theoretical need to study the properties of the Fe-based high-$T_c$ superconductors using reliable many-body techniques has highlighted the importance of determining what is the minimum number of orbital degrees of freedom that will capture the physics of these materials. While the shape of the Fermi surface (FS) obtained with the local-density approximation (LDA) can be reproduced by a two-orbital model, it has been argued that the bands that cross the chemical potential result from the strong hybridization of three of the $\text{Fe}3d$ orbitals. For this reason, a three orbital Hamiltonian for LaOFeAs obtained with the Slater-Koster formalism by considering the hybridization of the $\text{As}p$ orbitals with the $\text{Fe}{d}_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals is discussed here. This model reproduces qualitatively the FS shape and orbital composition obtained by LDA calculations for undoped LaOFeAs when four electrons per Fe are considered. Within a mean-field approximation, its magnetic and orbital properties in the undoped case are here described for intermediate values of $J/U$. Increasing the Coulomb repulsion $U$ at zero temperature, four different regimes are obtained: (1) paramagnetic, (2) magnetic $\left(\pi ,0\right)$ spin order, (3) the same $\left(\pi ,0\right)$ spin order but now including orbital order, and finally (4) a magnetic and orbital ordered insulator. The spin-singlet pairing operators allowed by the lattice and orbital symmetries are also constructed. It is found that for pairs of electrons involving up to diagonal nearest-neighbors sites, the only fully gapped and purely intraband spin-singlet pairing operator is given by $\Delta \left(\mathbf{k}\right)=f\left(\mathbf{k}\right){\sum }_{\alpha }{d}_{\mathbf{k},\alpha ,\uparrow }{d}_{-\mathbf{k},\alpha ,\downarrow }$ with $f\left(\mathbf{k}\right)=1$ or $\text{cos}{k}_x\text{cos}{k}_y$ which would arise only if the electrons in all different orbitals couple with equal strength to the source of pairing.

Figure 1

(Color online) (a) Band structure and (b) Fermi surface of the tight-binding (i.e., noninteracting) three orbital model, with parameters from Table and in the unfolded BZ. The diagonal thin solid line in (b) indicates the boundary of the folded BZ. In panels (c)–(e), the orbital contributions to the two hole and one of the electron pockets are given. The winding angle $\theta$ is measured with respect to the $k_y$ axis. The second electron pocket is analogous to the one shown simply replacing $xz$ by $yz$. In all panels, the dashed lines refer to the $xz$ orbital, the solid to $yz$, and the dotted to $xy$.

Figure 2

(Color online) Cartoons representing the spin- and orbital-order configurations considered in this effort. Since four electrons occupy three orbitals, perfectly ordered states have one doubly occupied orbital and the unpaired electrons are located in the other two orbitals, each singly occupied, forming a net spin $S=1$. In this cartoon, the orbital drawn indicates the doubly occupied one and the arrows indicate the orientation of the total spin at that site. (a) $\left(\pi ,0\right)$-spin and FO order favoring the $yz$ orbital, (b) $\left(\pi ,0\right)$-spin and FO order favoring $\left(xz+yz\right)/\sqrt{2}$, (c) $\left(\pi ,\pi \right)$-spin and FO order. (d)–(f): states with the same ordering vector for the spins and the orbitals: $\left(\pi ,0\right)$ for (d) and (e), $\left(\pi ,\pi \right)$ in (f). (g) $\left(\pi ,0\right)$ for spins and $\left(0,\pi \right)$ for orbitals; (h) $\left(\pi ,0\right)$ for spins and $\left(\pi ,\pi \right)$ for orbitals; (i) $\left(\pi ,\pi \right)$ for spins and $\left(\pi ,0\right)$ for orbitals; (b) and (e) illustrate phases where the orbital order does not feature alternating $xz$ and $yz$ orbitals, but the combinations $\left(xz+yz\right)/\sqrt{2}$ and $\left(xz-yz\right)/\sqrt{2}$ do.

Figure 3

(Color online) Cartoons for the spin ferromagnetic phases taken into consideration in this effort. Results are depicted with the convention of Fig. 2, but for ferromagnetic spins combined with (a) FO, (b) $\left(\pi ,0\right)$, and (c) $\left(\pi ,\pi \right)$ orbital orders.

Figure 4

(Color online) Qualitative phase diagram in the $J/U$ vs $U$ plane. The shaded area denotes the stability region of the realistic $\left(\pi ,0\right)$ magnetic ordering. The lines are guides to the eyes, the dashed line approximately indicates the metal-insulator transition. The data points were obtained by comparing the energies of the various phases within the mean-field approximation described in Appendix . The meaning of the symbols is the following: $+$: $\left(\pi ,0\right)$-AF state without pronounced orbital order (orbital disorder, OD); $\times$: $\left(\pi ,0\right)$-AF state with $\left(\pi ,\pi \right)$ AO order with $\varphi =0$; empty squares: $\left(\pi ,0\right)$-AF state with $\left(0,\pi \right)$ SO $\left(\varphi =0\right)$; ${}^{\ast }$: $\left(\pi ,0\right)$-AF state with FO order $\left(\varphi =0\right)$; filled squares: FM with FO ordering tendencies, $\varphi =\pi /4$. This FO order is weaker for small $U$ and larger $J\approx 0.2$. The filled circles denote parameters that do not support magnetically ordered states. For small $J$, some FO order with $\varphi =\pi /4$ is found, similar to the FM phase. The empty circles at large $U$ and small $J$ denote similar states without magnetic ordering, but with extreme orbital order, where the $xy$ orbital is (almost) empty, while $xz$ and $yz$ are (almost) filled.

Figure 5

(Color online) (a) Orbital magnetization and (b) occupation number as a function of the Coulomb repulsion strength $U$, obtained with a mean-field approximation. The colors indicate the different phases (for increasing $U$): uncorrelated metal, itinerant $\left(\pi ,0\right)$ antiferromagnet without orbital order, itinerant $\left(\pi ,0\right)$ antiferromagnet with alternating orbital order [small white window, spin-orbital order as in Fig. 2h], and a ferro-orbitally ordered $\left(\pi ,0\right)$ antiferromagnetic insulator [spin-orbital order as in Fig. 2a]. Hopping parameters are from Table , and $J=U/4$. For the phase with alternating orbital order, the thin lines show (a) $m\pm q$ and (b) $2n\pm p$.

Figure 6

(Color online) Spectral density $A\left(\mathbf{k},\omega \right)$ for the antiferromagnet orbital-disordered metallic phase at (a) $U=0.7$, (b) $U=0.9$, and (c) $U=1.1$. The BZ is for the one-Fe unit cell, and $J=U/4$ was used. The uncorrelated $A\left(\mathbf{k},\omega \right)$ for $U=0$ is included as solid lines for comparison.

Figure 7

(Color online) Fermi surface in the orbital-disordered spin-antiferromagnetic metallic phase with [(a) and (b)] $U=0.7$, [(c) and (d)] $U=0.9$, and [(e) and (f)] $U=1.1$. (a), (c), and (e) show the unfolded BZ containing one Fe, for the antiferromagnetic ordering vector $q=\left(\pi ,0\right)$. (b), (d), and (f) depict the superposition of the FSs for $\mathbf{q}=\left(\pi ,0\right)$ and $\left(0,\pi \right)$ in the (rotated) folded BZ corresponding to two Fe atoms. The ratio $J=U/4$ was used. The color scale is the same as in Fig. 6.

Figure 8

(Color online) Spectral density $A\left(\mathbf{k},\omega \right)$ for (a) the orbitally ordered spin-$\left(\pi ,0\right)$ antiferromagnetic metallic phase at $U=1.36$ and (b) the orbitally polarized spin-$\left(\pi ,0\right)$ antiferromagnetic insulator at $U=1.6$. The ratio $J=U/4$ was used, and the unfolded BZ is for the one-Fe unit cell.

Figure 9

(Color online) Magnetic order and orbital occupation at large $U$ shown for four sites. The magnetic ordering vector is $\left(\pi ,0\right)$, i.e., the spin stripes run along the $y$ direction. For each site, the $xy$, $xz$, and $yz$ orbitals are shown: the $xy$ is the one below the other two and the $yz$ is doubly occupied. Dashed (continuous) lines indicate interorbital hopping $t_7$ connecting the $xy$ orbital to $xz\left(yz\right)$ along the $x\left(y\right)$ direction.

Figure 10

(Color online) The intensity of the points represents the values of the spectral function $A\left(\mathbf{k},\omega \right)$ for the three orbital model with pairing interaction (a) $V=0$; (b) $V=0.2$ for the $s\pm$ pairing operator given in the text.

Figure 11

(Color online) (a) The intensity of the points represents the values of the spectral function $A\left(\mathbf{k},\omega \right)$ for the three orbital model with pairing interaction $V=0.2$ for the pairing state $s_{IB}$ along the indicated high-symmetry directions in the folded BZ. (b) Ratio $R$ between the gaps for pairing $s_{IB}$ and pairing $s\pm$ for $V=0.05$ in the unfolded BZ.

Figure 12

(Color online) The intensity of the points represent the values of the spectral function $A\left(\mathbf{k},\omega \right)$ for the three orbital model with the pairing interaction $V=0.2$, for the pairing operators (a) $B_{2g}$ and (b) $B_{2g}^{\text{ext}}$ discussed in the text.