Interplay between Magnetism, Superconductivity, and Orbital Order in 5-Pocket Model for Iron-Based Superconductors: Parquet Renormalization Group Study

We report the results of the parquet renormalization group (RG) analysis of the phase diagram of the most general 5-pocket model for Fe-based superconductors. We use as an input the orbital structure of excitations near the five pockets made out of $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals and argue that there are 40 different interactions between low-energy fermions in the orbital basis. All interactions flow under the RG, as one progressively integrates out fermions with higher energies. We find that the low-energy behavior is amazingly simple, despite the large number of interactions. Namely, at low energies the full 5-pocket model effectively reduces either to a 3-pocket model made of one $d_{xy}$ hole pocket and two electron pockets or a 4-pocket model made of two $d_{xz}/d_{yz}$ hole pockets and two electron pockets. The leading instability in the effective 4-pocket model is a spontaneous orbital (nematic) order, followed by $s^{+-}$ superconductivity. In the effective 3-pocket model, orbital fluctuations are weaker, and the system develops either $s^{+-}$ superconductivity or a stripe spin-density wave. In the latter case, nematicity is induced by composite spin fluctuations.

Figure 1

(Upper panel) Right: main orbital content of excitations near Fermi surfaces (presented by different colors); left: regions of different system behavior of the full 5-pocket model, indicated by the type of the effective model. In the ranges marked $4p_{1,2}$, the dominant interactions are between fermions near the $\mathrm{\Gamma }$-centered hole pockets and electron pockets. In the regions $3p_{1,2}$, the dominant interactions at low energies are within the subset of the two electron pockets and the $M=(\pi ,\pi )$-hole pocket. The index 1,2 distinguishes if interactions involving $d_{xz}/d_{yz}$ or $d_{xy}$ orbital components on the electron pockets are dominant. For illustration purposes, we used the bare model with local Hubbard and Hund interactions—intraorbital $U$, interorbital $U^{\prime }$, $J$, and $J^{\prime }$. We set $J=0.025/N_F$ and $J^{\prime }=0.03/N_F$, where $N_F$ is the density of states on the FSs (assumed to be equal on all FSs for simplicity), and varied $U$ and $U^{\prime }$ as two independent parameters. For this particular choice, $3p_2$ region does not exist. (Lower panel) Graphic representations of $3p_{1,2}$ and $4p_{1,2}$ models. Fermionic states, for which interactions become the largest in the process of pRG flow, are shown by solid lines.

Figure 2

Two different regions of system behavior indicated by fixed trajectories of the pRG flow for the toy model with electron pockets made entirely of $d_{xy}$, for different values of $U$, $U^{\prime }$ (treated as two independent parameters) and $J=J^{\prime }=0.03/N_F$. In the region labeled as $3p$, the interactions within the subset of the two electron pockets and the $M=(\pi ,\pi )$-hole pocket become dominant at low energies. In the region labeled as $4p$, interactions involving fermions from the two $\mathrm{\Gamma }$-centered hole pockets and the two electron pockets become dominant.

Figure 3

(a) Representative RG flow towards the $4p$ FT in the toy model for the interactions $u_1$ and $u_{1n}$. The inset shows the ten relevant interactions of the toy model, where double lines represent electron pockets, wavy lines the $M$-centered hole pocket, and solid single lines the $\mathrm{\Gamma }$-centered hole pockets. Bare values are $U=0.08/N_F$, $U^{\prime }=0.12/N_F$, and $J=J^{\prime }=0.03/N_F$. The RG parameter $L$ is $\log W/E$, where $W$ is the bandwidth and $E$ is running energy or temperature. The system undergoes an instability into an ordered state (SDW, SC, or orbital order) at $L=L_0$. (b) Corresponding flow of the SDW, SC $s^{+-}$ and orbital susceptibilities. Near $L=L_0$, the SC and the orbital susceptibilities keep increasing, while the SDW susceptibility remains finite. Inset: Orbital and SC susceptibilities near the end of the flow.