Intertwined spin-orbital coupled orders in the iron-based superconductors

The underdoped phase diagram of the iron-based superconductors exemplifies the complexity common to many correlated materials. Indeed, multiple ordered states that break different symmetries but display comparable transition temperatures are present. Here we argue that such a complexity can be understood within a simple unifying framework. This framework, built to respect the symmetries of the nonsymmorphic space group of the FeAs/Se layer, consists of primary magnetically ordered states and their vestigial phases that intertwine spin and orbital degrees of freedom. All vestigial phases have Ising-like and zero wave-vector order parameters, described in terms of composite spin order and exotic orbital-order patterns such as spin-orbital loop currents, staggered atomic spin-orbit coupling, and emergent Rashba- and Dresselhaus-type spin-orbit interactions. Moreover, they host unusual phenomena, such as the electronematic effect, by which electric fields act as transverse fields to the nematic order parameter, and the ferro-Néel effect, by which a uniform magnetic field induces Néel order. We discuss the experimental implications of our findings to iron-based superconductors and possible extensions to other correlated compounds with similar space groups.

Figure 1

Schematic representation of the cascade of symmetry breakings leading to the emergence of intertwined orders in the iron-based superconductors. The starting point is the magnetic SO(6) supervector, presented in Eq. (3). Space-group symmetries only allow certain combinations of the supervector components to order, giving rise to the primary magnetically ordered states ${\mathit{\eta }}_i$. These, in turn, support different types of vestigial spin-orbital coupled phases, characterized by Ising-like, zero wave-vector order parameters that transform as the $B_{2g}, A_{2u}$, and $B_{2u}$ irreducible representations of the tetragonal point group.

Figure 2

(a) Illustration of the 1-Fe (dotted square) and 2-Fe (dashed square) unit cells. To correctly capture the puckering of the As/Se atoms, two Fe atoms and two As/Se atoms are required in the unit cell. Here sharp purple spheres denote As/Se atoms above the Fe layer while blurred purple spheres denote As/Se atoms below the Fe layer. (b) Crystallographic 2-Fe Brillouin zone (dashed square) and idealized 1-Fe Brillouin zone (dotted square).

Figure 3

Illustration of the possible magnetic orders for each order parameter ${\mathit{\eta }}_i$, with transforms as a two-dimensional irreducible representation of the space group. The double-$\mathbf{Q}$ structures for ${\mathit{\eta }}_1$ and ${\mathit{\eta }}_2$ are the in-plane hedgehog- and loop-SVC phases, respectively. For ${\mathit{\eta }}_3$, the double-$\mathbf{Q}$ phase is denoted the charge-spin density-wave phase. All single-Q phases are stripe magnetically ordered states.

Figure 4

Effects of $B_{2g}$ vestigial nematic order. In (a) we illustrate the ferro-orbital order involving the $d_{xz}$ and $d_{yz}$ Fe $3d$ orbitals (${\mathrm{\Delta }}_{B_{2g}}^{\left(1\right)}\ne 0$) while in (b), the bond-nematic order (hopping anisotropy) involving the $d_{xy}$ orbitals (${\mathrm{\Delta }}_{B_{2g}}^{\left(3\right)}\ne 0$). The band structure and Fermi surface in the vestigial phase are shown in (c) and (d), respectively. The dotted lines show the band structure in the nonordered phase. Note the Pomeranchuk-type distortion of the Fermi surface. The parameters used here are given in Appendix pp1.

Figure 5

Effects of $A_{2u}$ vestigial spin-current order. In (a) the bonds between next-nearest-neighbor Fe atoms become inequivalent in a staggered pattern as a result of ${\mathrm{\Delta }}_{A_{2u}}^{\left(1\right)}\ne 0$. This renders the As/Se atoms above and below the Fe plane inequivalent as well. In (b) we illustrate the formation of staggered spin currents involving the $d_{xy}$ orbitals (${\mathrm{\Delta }}_{A_{2u}}^{\left(2\right)}\ne 0$). The band structure and Fermi surface in the vestigial phase are shown in (c) and (d), respectively. Note the doubling of the number of bands due to the lifting of spin degeneracy. The parameters used here are given in Appendix pp1.

Figure 6

Effects of the $B_{2u}$ vestigial charge order. In (a) we show the induced charge order in the $d_{xy}$ orbital (${\mathrm{\Delta }}_{B_{2u}}^{\left(1\right)}\ne 0$) and in (b) the emergence of a staggered $L_z{S}_z$ spin-orbital coupling between the $d_{xz}$ and $d_{yz}$ orbitals (${\mathrm{\Delta }}_{B_{2u}}^{\left(2\right)}\ne 0$). The band structure and Fermi surfaces in the vestigial phase are shown in (c) and (d), respectively. Note the doubling of the number of bands due to the lifting of spin degeneracy. The parameters used here are given in Appendix pp1.

Figure 7

Illustration of the induced magnetic configurations in the $A_{2u}$ and $B_{2u}$ paramagnetic vestigial phases induced by the presence of an external magnetic field $\mathbf{H}$. Besides the usually induced ferromagnetic order, Néel order is also induced in this case, resulting in ferrimagnetic or canted antiferromagnetic configurations. For out-of-plane fields, currents are induced around the As/Se atoms. These are omitted in the figure.