Competing magnetic double-$Q$ phases and superconductivity-induced reentrance of $C_2$ magnetic stripe order in iron pnictides

We perform a microscopic theoretical study of the generic properties of competing magnetic phases in iron pnictides. As a function of electron filling and temperature, the magnetic stripe (single-$Q$) order forms a dome, but competing noncollinear and nonuniform double-$Q$ phases exist at the foot of the dome, in agreement with recent experiments. We compute and compare the electronic properties of the different magnetic phases, investigate the role of competing superconductivity, and show how disorder may stabilize double-$Q$ order. Superconductivity is shown to compete more strongly with double-$Q$ magnetic phases, which can lead to reentrance of the $C_2$ (single-$Q$) order, in agreement with recent thermal expansion measurements on K-doped Ba-122 crystals.

Figure 1

Spin and charge order of the (a) $1Q$ MS (${\mathbf{M}}_{\mathbf{2}}=0$), (b) $2Q$ OM (${\mathbf{M}}_{\mathbf{1}}\perp {\mathbf{M}}_{\mathbf{2}}$), and (c) SCO (${\mathbf{M}}_{\mathbf{1}}={\mathbf{M}}_{\mathbf{2}}$). The black circles in (c) represent the ${\mathbf{Q}}_{\mathbf{1}}+{\mathbf{Q}}_{\mathbf{2}}=(\pi ,\pi )$ charge order. (d), (e) Magnetic phase diagrams as obtained from Eq. (1) as a function of $T$ and filling $n_{\text{opt}}-x$ for (d) $U=0.85$ eV and (e) $U=0.95$ eV. The green (purple) area indicates regions of 2Q phases with $0<M_1<M_2$ with collinear ${\mathbf{M}}_{\mathbf{1}}\left|\right|{\mathbf{M}}_{\mathbf{2}}$ (coplanar ${\mathbf{M}}_{\mathbf{1}}\perp {\mathbf{M}}_{\mathbf{2}}$) spin structure.

Figure 2

(a) $T$ evolution at $x=-0.09$ of the magnetic components $M_1$ ($▾$), $M_2$ ($▴$), and ${\overline{M}}_r=\frac{1}{2}\sqrt{M_1^2+M_2^2}$ ($\circ$), and (c) the entropy $\mathcal{S}\left(T\right)$. A zoom of $\mathcal{S}$ at $T_1$ is shown in the inset. The color changes in ${\overline{M}}_r$ and $\mathcal{S}$ represent the magnetic phase transitions shown also in Fig. 1 with $U=0.95$ eV. (b), (d) The same as (a), (c) but for $x=0.02$ with SCO order. (e) and (f) summarize the $T$ dependence of the magnetic structure and the expected associated lattice symmetry.

Figure 3

Fermi surface in the folded BZ, $-\pi /2<k_x,k_y<\pi /2$, in the PM state for (a) $x=-0.09$, (b) $x=-0.04$, and (c) $x=0.02$, and in the magnetic states (d) OM (${\mathbf{M}}_{\mathbf{1}}=0.2\widehat{z}$; ${\mathbf{M}}_{\mathbf{2}}=0.2\widehat{x}$), (e) MS ($M_1=0.52$; $M_2=0$), and (f) SCO ($M_1=M_2=0.43$) for $U=0.95$ eV. (g)–(i) Band structure along the momentum path $\tilde{X}-\mathrm{\Gamma }-\tilde{Y}$ shown in (d) for the (g) OM, (h) MS, and (i) SCO phase. The main orbital contributions are shown by purple: $d_{xz}$; green: $d_{yz}$; orange: $d_{xy}$; cyan: $d_{z^2}$.

Figure 4

(a) Modified phase diagram from the region indicated by the dashed box in Fig. 1 in the presence of SC order. (b) Magnetic anisotropy ${\delta }_M=(M_1-M_2)/(M_1+M_2)$ vs $T$ for four different $x$ with (without) SC order shown by the solid (open) symbols. The arrows mark $T_c$. (c), (d) Fermi surface at the cross in (a) for MS (c) and SCO (d) order (without SC) using the same orbital color code as in Fig. 3.

Figure 5

(a), (c) Magnetization at low $T$ and $x=-0.09$ in the absence of an impurity, and (b), (d) in the presence of a single impurity at the center (red dot) with (b) $V=5$ eV and (d) $V=2$ eV. The associated Fourier components ${\mathbf{M}}_{\mathbf{1}}$ and ${\mathbf{M}}_{\mathbf{2}}$ are also shown [the vectors are normalized to (a) $M=0.16$, (b) $M=0.43$, (c) $M=0.2$, and (d) $M=0.68]$.