Vestigial chiral and charge orders from bidirectional spin-density waves: Application to the iron-based superconductors

Recent experiments in optimally hole-doped iron arsenides have revealed a novel magnetically ordered ground state that preserves tetragonal symmetry, consistent with either a charge-spin density wave (CSDW), which displays a nonuniform magnetization, or a spin-vortex crystal (SVC), which displays a noncollinear magnetization. Here we show that, similarly to the partial melting of the usual stripe antiferromagnet into a nematic phase, either of these phases can also melt in two stages. As a result, intermediate paramagnetic phases with vestigial order appears: a checkerboard charge density wave for the CSDW ground state, characterized by an Ising-like order parameter, and a remarkable spin-vorticity density wave for the SVC ground state—a triplet $d$-density wave characterized by a vector chiral order parameter. We propose experimentally detectable signatures of these phases, show that their fluctuations can enhance the superconducting transition temperature, and discuss their relevance to other correlated materials.

Figure 1

Magnetic ground states of the iron pnictides: (a) stripe antiferromagnet, (b) charge-spin density wave (CSDW), and (c) spin-vortex crystal (SVC). The first is orthorhombic with a doubled unit cell; the latter two remain tetragonal but with a quadrupled unit cell. ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ are the magnetic order parameters corresponding to the ordering vectors ${\mathbf{Q}}_1=\left(\pi ,0\right)$ and ${\mathbf{Q}}_2=\left(0,\pi \right)$. The left panels are the actual spin density-wave patterns in real space, whereas the right panels are schematic representations focusing on the magnetization at the lattice sites.

Figure 2

The vestigial composite states associated with (a) the stripe antiferromaget, (b) the CSDW state, and (c) the SVC state. The real-space spins (in gray) and the magnetic order parameters in spin space (red and blue arrows, representing ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$, respectively) should be understood as fluctuating, i.e., $〈{\mathbf{M}}_i〉=0$ in all cases. In (a), the vestigial state is nematic (unequal blue and red bonds), associated with selecting between ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ fluctuations in spin space. The original unit cell is shown as a dashed square. In (b), the vestigial state breaks translational symmetry via a checkerboard charge density wave (unequal blue and red sites). ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ are locked to be collinear in spin space. In (c), the vestigial spin-vorticity density-wave state breaks inversion and translational symmetries via a staggered pattern of spin vortices in the center of the plaquettes (unequal blue and red plaquettes). The corresponding spin-current pattern is shown by the green arrows. Because ${\mathbf{M}}_1$ and ${\mathbf{M}}_2$ are locked to be orthogonal in spin space, the residual spin-rotational symmetry is $O\left(2\right)$ instead of $O\left(3\right)$. Both (b) and (c) preserve tetragonal symmetry, as shown by the dashed-line unit cell.

Figure 3

Phase diagram, within the saddle-point approximation, of the coupled SVDW paramagnetic and SVC magnetic transitions. In the shaded area, where the out-of-plane anisotropy is strong, the two transitions are split. The tuning parameters are the Ginzburg-Landau coefficients $u$ and $w$ [see Eq. (1)] and the magnetic coupling between layers $J_z$. The parameter $\overline{w}$ is given by $w=T_{N,0}w/2\pi$, as discussed in Appendix pp2.

Figure 4

Schematic Fermi surface of the iron pnictides, with a central hole pocket and elliptical electron pockets. The wavy lines represent the interpocket pairing interactions generated by the magnetic fluctuations (repulsive $V>0$) and by fluctuations of the vestigial CDW and SVDW states (attractive $U<0$).