Disorder-promoted $C_4$-symmetric magnetic order in iron-based superconductors

In most iron-based superconductors, the transition to the magnetically ordered state is closely linked to a lowering of structural symmetry from tetragonal $\left(C_4\right)$ to orthorhombic $\left(C_2\right)$. However, recently, a regime of $C_4$-symmetric magnetic order has been reported in certain hole-doped iron-based superconductors. This novel magnetic ground state can be understood as a double-$\mathbf{Q}$ spin density wave characterized by two order parameters ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ related to each of the two $\mathbf{Q}$ vectors. Depending on the relative orientations of the order parameters, either a noncollinear spin-vortex crystal or a nonuniform charge-spin density wave could form. Experimentally, Mössbauer spectroscopy, neutron scattering, and muon spin rotation established the latter as the magnetic configuration of some of these optimally hole-doped iron-based superconductors. Theoretically, low-energy itinerant models do support a transition from single-$\mathbf{Q}$ to double-$\mathbf{Q}$ magnetic order, but with nearly degenerate spin-vortex crystal and charge-spin density wave states. In fact, extensions of these low-energy models including additional electronic interactions tip the balance in favor of the spin-vortex crystal, in apparent contradiction with the recent experimental findings. In this paper we revisit the phase diagram of magnetic ground states of low-energy multiband models in the presence of weak disorder. We show that impurity scattering not only promotes the transition from $C_2$ to $C_4$-magnetic order, but it also favors the charge-spin density wave over the spin-vortex crystal phase. Additionally, in the single-$\mathbf{Q}$ phase, our analysis of the nematic coupling constant in the presence of disorder supports the experimental finding that the splitting between the structural and stripe-magnetic transition is enhanced by disorder.

Figure 1

Illustration of the two double-$\mathbf{Q}$ magnetically ordered states (upper panels) as a superposition of two single-$\mathbf{Q}$ stripe-magnetic states (lower panels). (a) Aligning the order parameters ${\mathbf{M}}_1=\pm {\mathbf{M}}_2$ (anti)parallel yields a charge-spin density wave. This order is favorable if $g<\left|w\right|$ and $w<0$. (b) Aligning the order parameters ${\mathbf{M}}_1\perp {\mathbf{M}}_2$ perpendicular to each other leads to the formation of a spin-vortex crystal. This state is favorable if $g<0$ and $w>0$. Otherwise, single-$\mathbf{Q}$ stripe order is favored.

Figure 2

Evolution of the phase diagrams of the possible magnetic ground states of our three-band minimal model of iron-based superconductors upon increase of the scattering rate. Here we used ${\mathrm{\Gamma }}_{\mathrm{e}-\mathrm{e}}^{\mathrm{inter}}=0.1{\mathrm{\Gamma }}_{\mathrm{h}}^{\mathrm{intra}}$, and the phase diagrams are obtained in the limit ${\delta }_{\mu }\ll 2\pi T$ and ${\delta }_m\ll 2\pi T$. The regime of single-$\mathbf{Q}$ stripe order (SM) is shown in green, the double-$\mathbf{Q}$ spin-vortex crystal (SVC) order is indicated by blue, and the yellow region represents the double-$\mathbf{Q}$ charge-spin density wave (CSDW). In the clean regime, where all scattering rates are zero, SVC and CSDW order are degenerate and we indicated this region with $w=0$ in red. The crosses mark the points in the phase diagram at which we plotted $g$ and $w$ as a function of scattering rate in Figs. 5 and 7, respectively.

Figure 3

Quartic diagrams in the absence of disorder. (a) Sketch of a generic quartic diagram before performing the spin trace: Each vertex couples the hole band to one of the electron bands, and the dashed lines indicate that scattering or additional interactions could alter the diagram. (b) Contribution to the planar coupling constant $w$ if the incipient hole pocket at $M$ is taken into account.

Figure 4

Leading-order diagrams contributing to the quartic coefficients that determine the magnetic ground state. Double lines indicate that the respective propagators acquire a finite lifetime due to impurity scattering, whereas single lines are used for propagators in bands that, within our model, are not affected by impurity scattering. Additional scattering processes in (b) and (c) are indicated by a dashed line corresponding to the scattering rates ${\mathrm{\Gamma }}_{\mathrm{h}}^{\mathrm{intra}}$ and ${\mathrm{\Gamma }}_{\mathrm{e}-\mathrm{e}}^{\mathrm{inter}}$, respectively. (a) and (b) ${\mathcal{G}}_i^{\left(1\right)}$ and ${\mathcal{G}}_i^{\left(2\right)}\left(i\in \{1,2\}\right)$ are the contributions to $g$ (as well as to $u$) in the presence of intraband scattering in the hole band which is the dominant scattering mechanism in FeSC. (c) $\mathcal{W}$ is the contribution to $w$ which is finite owing to interband scattering between the two electron bands, and in the presence of the dominant intraband scattering in the hole band.

Figure 5

Nematic coupling constant $g$ in the presence of intraband scattering in the hole band, characterized by the scattering rate ${\mathrm{\Gamma }}_{\mathrm{h}}^{\mathrm{intra}}$. We chose ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0.2$ (blue dotted line) as an example of small ellipticity and detuning which guarantees $w<0$ and $g>0$, and ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0.35$ (green dashed line) as an example where disorder can tune $g$ and $w$ to be either positive or negative. The red line represents the result at particle-hole symmetry ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0$.

Figure 6

Dependence of the dimensionless nematic coupling constant $g/u$ on disorder. (a) Close to particle-hole symmetry, $g/u$ decreases monotonically with increasing scattering rate. (b) and (c) With increasing distance to particle-hole symmetry, an initial increase of the dimensionless nematic coupling constant is found for small scattering rates, but for stronger disorder, the ratio $g/u$ decreases again.

Figure 7

Planar coupling $w$ as a function of intraband scattering rate in the hole band ${\mathrm{\Gamma }}_{\mathrm{h}}^{\mathrm{intra}}$, where we set ${\mathrm{\Gamma }}_{\mathrm{e}-\mathrm{e}}^{\mathrm{inter}}=0.1{\mathrm{\Gamma }}_{\mathrm{h}}^{\mathrm{intra}}$. We chose ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0.2$ (blue dotted line) as an example of small ellipticity and detuning which guarantees $w<0$ and $g>0$, and ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0.35$ (green dashed line) as an example where disorder can tune $w$ and $g$ to be either positive or negative. The red line represents the result at particle-hole symmetry ${\delta }_{\mu }/\left(2\pi T\right)={\delta }_m/\left(2\pi T\right)=0$.