Magnetic order without tetragonal-symmetry-breaking in iron arsenides: Microscopic mechanism and spin-wave spectrum

Most iron-based superconductors undergo a transition to a magnetically ordered state characterized by staggered stripes of parallel spins. With ordering vectors $(\pi ,0)$ or $(0,\pi )$, this magnetic state breaks the high-temperature tetragonal symmetry of the system, which is manifested by a splitting of the lattice Bragg peaks. Remarkably, recent experiments in hole-doped iron arsenides reported an ordered state that displays magnetic Bragg peaks at $(\pi ,0)$ and $(0,\pi )$ but remains tetragonal. Despite being inconsistent with a magnetic stripe configuration, this unusual magnetic phase can be described in terms of a double-$\mathbf{Q}$ magnetic structure consisting of an equal-weight superposition of the ordering vectors $(\pi ,0)$ and $(0,\pi )$. Here we show that a noncollinear double-$\mathbf{Q}$ magnetic configuration, dubbed orthomagnetic, arises naturally within an itinerant three-band microscopic model for the iron pnictides. In particular, we find that strong deviations from perfect nesting and residual interactions between the electron pockets favor the orthomagnetic over the stripe magnetic state. Using an effective low-energy model, we also calculate the spin-wave spectrum of the orthomagnetic state. In contrast to the stripe state, there are three Goldstone modes, manifested in all diagonal and one off-diagonal components of the spin-spin correlation function. The total magnetic structure factor displays two anisotropic gapless spin-wave branches emerging from both $(\pi ,0)$ and $(0,\pi )$ momenta, in contrast to the case of domains of stripe order, where only one gapless spin-wave branch emerges from each momentum. We propose that these unique features of the orthomagnetic state can be used to unambiguously distinguish it from the stripe state via neutron scattering experiments, and discuss the implications of its existence for the nature of the magnetism of the iron arsenides.

Figure 1

Magnetic ground state configurations for (a) ${\mathbf{Q}}_1=\left(\pi ,0\right)$ stripe order, (b) ${\mathbf{Q}}_2=(0,\pi )$ stripe order, (c) double-Q noncollinear (orthomagnetic) order, and (d) double-Q nonuniform magnetic order. The dashed rectangle denotes the magnetic unit cell in each case. While (a) and (b) are orthorhombic, (c) and (d) are tetragonal with a unit cell four times larger than in the paramagnetic phase.

Figure 2

Quartic Ginzburg-Landau coefficients $u$ (red) and $g$ (blue) of the free energy (9) as a function of $|{\delta }_{\mu }|/\left(4\pi T\right)$ for ${\delta }_m/\left(4\pi T\right)=0.2$ (top) and ${\delta }_m/\left(4\pi T\right)=0.5$ (bottom). The insets show the shape of the Fermi pockets for $\frac{{\delta }_{\mu }}{4\pi T}=-0.05\text{and}-0.2$ in the top panel, $\frac{{\delta }_{\mu }}{4\pi T}=-0.1\text{and}-0.4$ in the bottom panel. Note that $u$ and $g$ are normalized by their values at ${\delta }_{\mu }=0$. For $g>0$, the magnetic ground state is the stripe state, which lowers the tetragonal symmetry to orthorhombic. For $g<0$, tetragonal symmetry is preserved, and either a noncollinear or a nonuniform double-$\mathbf{Q}$ magnetic state arises. Notice that the free energy remains bounded as long as $u>0$.

Figure 3

Top panel: The vertex that couples the magnetic order parameter to the low-energy fermionic states, which has an ${\mathbf{M}}_i·\sigma$ structure. Solid lines refer to the noninteracting electronic Green's functions. Bottom panel: The Feynman diagrams containing the leading-order corrections to the free energy arising from the residual $U_7$ interaction.

Figure 4

Spin-wave dispersions for the stripe phase with ordering vector ${\mathbf{Q}}_1=\left(\pi ,0\right)$ (a) and ${\mathbf{Q}}_2=(0,\pi )$ (b). Both are doubly degenerate modes. The energies are in units of $4J_{2}S$ and the parameters used are $J_1=0.8J_2$ and $K=0.1J_2$.

Figure 5

The spin-wave dispersions of the noncollinear orthomagnetic order in the magnetic Brillouin zone. The four dispersions are linked by a shift of the momentum coordinate system by the orthomagnetic ordering vectors. Here, the energies are in units of $4J_{2}S$ and the parameters are $J_1=0.8J_2$ and $K=-0.1J_2$.

Figure 6

The total spin-spin structure factor $\mathcal{S}={\sum }_i{\mathcal{S}}_{ii}=2S_{xx}$ for the ${\mathbf{Q}}_1$ (top panel) and ${\mathbf{Q}}_2$ (middle panel) stripe phases. Bottom panel is the structure factor assuming equal domains of ${\mathbf{Q}}_1$ and ${\mathbf{Q}}_2$ stripes. The vertical axis is the energy measured in units of $4J_{2}S$, whereas the horizontal axis displays momentum cuts in the Fe-square-lattice Brillouin zone. The intensity is highest at the ordering vectors ${\mathbf{Q}}_1$ and ${\mathbf{Q}}_2$. The parameters used here are $J_1=0.8J_2$ and $K=0.1J_2$.

Figure 7

The structure factors ${\mathcal{S}}_{xx}={\mathcal{S}}_{zz}$ (top panel), ${\mathcal{S}}_{yy}$ (middle panel), and $\mathcal{S}={\sum }_i{\mathcal{S}}_{ii}$ (bottom panel) for the orthomagnetic phase. The vertical axis is the energy measured in units of $4J_{2}S$, whereas the horizontal axis displays momentum cuts in the Fe-square-lattice Brillouin zone. The parameters used here are $J_1=0.8J_2$ and $K=-0.1J_2$. Two gapless spin-wave branches emerge from the ordering vectors ${\mathbf{Q}}_1=\left(\pi ,0\right)$ and ${\mathbf{Q}}_2=(0,\pi )$, in sharp contrast to the case of domains of stripes shown in Fig. 6.