Intertwined charge, spin, and pairing orders in doped iron ladders

Motivated by recent experimental progress on iron-based ladder compounds, we study the doped two-orbital Hubbard model for the two-leg ladder ${\mathrm{BaFe}}_2{\mathrm{S}}_3$. The model is constructed by using ab initio hopping parameters and the ground state properties are investigated using the density matrix renormalization group method. We show that the $(\pi ,0)$ magnetic ordering at half filling, with ferromagnetic rungs and antiferromagnetic legs, becomes incommensurate upon hole doping. Moreover, depending on the strength of the Hubbard $U$ coupling, other magnetic patterns, such as $(0,\pi )$, are also stabilized. We found that the binding energy for two holes becomes negative for intermediate Hubbard interaction strength, indicating hole pairing. Due to the crystal-field split among orbitals, the holes primarily reside in one orbital, with the other one remaining half filled. This resembles orbital selective Mott states. The formation of tight hole pairs continues with increasing hole density, as long as the magnetic order remains antiferromagnetic in one direction. The study of pair-pair correlations indicates the dominance of the intraorbital spin-singlet channel, as opposed to other pairing channels. Although in a range of hole doping pairing correlations decay slowly, our results can also be interpreted as corresponding to a charge density wave made of pairs, a precursor of eventual superconductivity after interladder couplings are included. Such a scenario of intertwined orders has been extensively discussed before in the cuprates, and our results suggest a similar physics could exist in ladder iron-based superconductors. Finally, we also show that a robust Hund's coupling is needed for pairing to occur.

Figure 1

Schematic representation of a two-leg ladder and the two-orbital Hubbard model used in this work. At each site, the red circle represents orbital $a$ and the blue square orbital $b$. The $2\times 2$ hopping matrix along the legs is indicated by $t^x$ and along the rung by $t^y$. We have also considered plaquette diagonal hopping matrices $t^{x+y}$ and $t^{x-y}$. For details, see text.

Figure 2

Spin structure factors (a) $S(k_x,0)$ and (b) $S(k_x,\pi )$ vs wave vector $k_x$ for various values of hole doping. (c), (d) Spin-spin correlation at a fixed and site-projected arrangement of holes along the rungs of the ladder. (c) is for four holes projected at the four red circles indicated, using only orbital $a$ because this is the orbital populated by hole doping due to the crystal-field splitting. (d) is for eight holes projected at the eight red circles (again using only orbital $a$). Blue color lines represent AFM bonds, whereas FM bonds are represented with red color. These results were obtained using DMRG at $U/W=2.0$ and $J_H/U=0.25$, for a cluster size $L=2\times 12$.

Figure 3

Spin structure factor (a) $S(k_x,0)$ and (b) $S(k_x,\pi )$ vs wave vector $k_x$ for various values of interaction strength $U/W$ at a fixed number of holes $N_h=8$. The inset shows site averaged $\langle S^2\rangle$ vs the number of holes $N_h$ at $U/W=3.0$. (c) Sketch of the magnetic phase diagram at fixed $J_H/U=0.25$ and $N_h=8$.

Figure 4

(a), (b) Spin-spin correlation at a fixed and site-projected arrangement of $N_h=8$ holes along the rungs of the ladder, using only orbital $a$ because this is the orbital that is the most relevant for the location of the holes. (a) At $U/W=3$, (b) at $U/W=5$. Blue color lines represent AFM bonds, whereas FM bonds are represented with red color. (c) Real-space density $\langle n_a\left(i\right)\rangle$ vs site index $i$ with $N_h=8$ holes and for various values of $U/W$.

Figure 5

(a) Binding energy $\mathrm{\Delta }E$ vs interaction strength $U/W$ at $J_H/U=0.25$ for cluster sizes $L=2\times 8$ and $2\times 12$ for comparison. (b) Real-space charge density $\langle n_{\alpha }\rangle$ vs site index $i$ for the case of $N_h=2$ holes and at $U/W=3.0$. Bottom: Schematic representation of a two-leg ladder with the snakelike counting of sites appropriate for DMRG. Orbital $a$ is represented by red circles (this is the orbital populated by holes) while orbital $b$ is represented by blue squares (this orbital is mainly undoped at low doping).

Figure 6

Real-space density profile at $U/W=3.0$ for (a) $N_h=4$ holes and for (b) $N_h=8$ holes. (a) contains two hole pairs, while (b) contains four hole pairs. (c) Real-space density profile at weak coupling $U/W=0.2$ with $N_h=8$ holes. The results suggest no hole pairs at this small Hubbard $U$. All these numerical calculations were performed using DMRG for a cluster size $L=2\times 12$ and at fixed $J_H/U=0.25$.

Figure 7

Singlet and triplet pair correlations, $P_a^{\text{rung}}\left(S\right)$ and $P_a^{\text{rung}}\left(T\right)$, calculated along the vertical rung, whereas $P_a^{\text{diag}}\left(S\right)$ and $P_a^{\text{diag}}\left(T\right)$ are calculated along the diagonal of the ladder, as indicated in the sketch at the bottom of the figure, and using orbital $a$ which is the orbital where holes are located. All results obtained at $U/W=3.0$ and $J_H/U=0.25$ for a cluster size $L=2\times 12$. Bottom: Schematic diagrams of vertical $O_{\text{rung}}$ and diagonal $O_{\text{diag}}$ pairing operators involving orbital $a$ (red ovals) and orbital $b$ (blue squares).

Figure 8

Comparison of spin-spin $S{p}_a^{\text{rung}}\left(d\right)$ correlations, singlet pair-pair $P_a^{\text{rung}}\left(d\right)$ and $P_a^{\text{diag}}\left(d\right)$ correlations, and charge-charge $N_a^{\text{rung}}\left(d\right)$ correlations. The various panels correspond to (a) $N_h=2$, (b) $N_h=4$, (c) $N_h=8$, and (d) $N_h=12$ holes, all at $U/W=3.0$ and $J_H/U=0.25$ and using a system size $L=2\times 12$. We use orbital $a$ in all cases because this is the orbital where holes are located due to the crystal-field splitting.

Figure 9

(a) Rung-singlet pair-pair $P_a^{\text{rung}}\left(d\right)$ at $U/W=3.0$ with $N_h=8$ holes, for different values of $J_H/U$. Orbital $a$ is used. (b) Binding energy $\mathrm{\Delta }E$ vs interaction strength $U/W$ for different values of Hund's coupling $J_H/U$, using a cluster size $L=2\times 8$.

Figure 10

Results using an alternative set of hoppings. (a) Binding energy $\mathrm{\Delta }E$ vs interaction strength $U/W$ at $J_H/U=0.25$ for cluster sizes $L=2\times 8$ and $2\times 12$. (b) Real-space charge density $\langle n_{\alpha }\rangle$ vs site index $i$ with $N_h=4$ holes and at $U/W=8.0$.