Multiple-$Q$ spin textures induced by spiral--staggered interference in one-dimensional itinerant magnets

TL;DR

This work addresses how symmetry-unrelated multiple-$Q$ spin textures can emerge in a centrosymmetric, one-dimensional itinerant system. It develops two phenomenological models—momentum- and real-space formulations—that include bilinear and biquadratic exchanges, and analyzes their ground states via simulated annealing to reveal a robust double-$Q$ texture arising from spiral–staggered interference. The double-$Q$ state produces antisymmetric spin-split electronic bands even without spin–orbit coupling and exhibits field-induced asymmetric band modulations, highlighting novel spin-electronic couplings in 1D magnets. These insights provide a general framework for unconventional multiple-$Q$ orders, potentially shedding light on broken-helix states observed in materials like $ $EuIn_{2}As_{2}$ and suggesting avenues for complex textures in higher dimensions without relying on Dzyaloshinskii–Moriya interactions.

Abstract

We theoretically investigate multiple-$Q$ magnetic states emerging from the interference between finite-$Q$ spiral and staggered spin modulations in a one-dimensional itinerant electron system. The multiple-$Q$ spin textures are characterized by a superposition of symmetry-unrelated ordering wave vectors in the same direction with distinct periodicities rather than rotationally symmetry-related ones. Motivated by recent experimental observations of broken helix magnetic structures in EuIn$_2$As$_2$, we focus on the microscopic interaction conditions in stabilizing such multiple-$Q$ states.We employ two effective spin models: One is the momentum-space-based model, and the other is the real-space-based model, both of which include bilinear and biquadratic easy-plane anisotropic interactions. By analyzing their ground state via simulated annealing, we find that a superposition of a spiral and a staggered modulation yields a robust double-$Q$ magnetic structure. Moreover, we demonstrate that the obtained double-$Q$ spin configuration exhibits an antisymmetric spin-split band structure even without the relativistic Dzyaloshinskii-Moriya interaction, and further reveals asymmetric band modulations when the magnetic field is applied along the out-of-plane direction. Our results provide a theoretical framework for understanding unconventional multiple-$Q$ magnetic textures in one-dimensional systems.

Figures (6)

• Figure 1: (Color online) Ground-state phase diagram in the plane of $J_{\rm sta}$ and $Q_{\rm sp}$ at (a) $K=0$, (b) $K=0.2$, (c) $K=0.4$, and (d) $K=0.6$. The thick gray lines denote the phase boundaries. 1$Q$ and 2$Q$ stand for the single-$Q$ and double-$Q$ states, respectively.
• Figure 2: Real-space spin configurations of (a) the single-$Q$ spiral state at $K=0$, $J_{\rm sta}=0.4$, and $Q_{\rm sp}=0.3\pi$, (b) the single-$Q$ staggered state at $K=0$, $J_{\rm sta}=0.6$, and $Q_{\rm sp}=0.3\pi$, (c) the double-$Q$ state at $K=0.2$, $J_{\rm sta}=0.44$, and $Q_{\rm sp}=0.5\pi$, (d) the double-$Q$ state at $K=0.2$, $J_{\rm sta}=0.6$, and $Q_{\rm sp}=0.3\pi$, and (e) the double-$Q$ state at $K=0.6$, $J_{\rm sta}=0.4$, and $Q_{\rm sp}=0.2\pi$, which are obtained by simulated annealing. The spin configuraion in (f) stands for the simply superposed spin configuration as $\bm{S}_i= (\cos Q_{\rm sp} x_i + 0.5 \cos Q_{\rm sta} x_i, \sin Q_{\rm sp} x_i)$ without normalization ($|\bm{S}_i| \neq 1$) at $Q_{\rm sp}=0.5\pi$ for reference.
• Figure 3: (Color online) $Q_{\rm sp}$ dependence of the intensity plot of $(m_q)^2$ at (a) $J_{\rm sta}=0.3$ and $K=0$, (b) $J_{\rm sta}=0.46$ and $K=0.2$, (c) $J_{\rm sta}=0.5$ and $K=0.2$, and (d) $J_{\rm sta}=0.4$ and $K=0.6$.
• Figure 4: (Color online) Contour plots of (a) $(m_{Q_{\rm sta}}/m_{Q_{\rm sp}})^2$ and (b) $K_{\rm sp}$ against the real-space four-spin interactions $K_{1212}$ and $K_{1223}$. The dashed lines denote the phase boundary between the single-$Q$ and double-$Q$ states.
• Figure 5: (Color online) Electronic band structure in the absence of the magnetic field ($H=0$) under (a) the single-$Q$ spiral state at $Q_{\rm sp}=0.1\pi$ and (b) the double-$Q$ state at $Q_{\rm sp}=0.1\pi$, $K=0.6$, and $J_{\rm sta}=0.4$. The color represents the $z$-spin polarization.