Four-Spin Interactions as a Route to Multiple-Q Topological Magnetic Order

TL;DR

This work develops a momentum-space inverse-design framework that links microscopic real-space four-spin couplings to effective momentum-space interactions $K_\beta$ to stabilize complex multiple-$Q$ topological spin textures. It demonstrates two concrete routes to SkX2: (i) constructing a frustrated Heisenberg model with three nearest-neighbor four-spin terms tailored to yield $K_1>0$ and $K_2<0$, verified by simulated annealing, and (ii) leveraging ring-exchange with $K_1>0$, $K_2<0$ to promote SkX2, confirmed by Monte Carlo simulations. The methodology provides a general pathway to engineer and understand higher-order magnetic orders beyond bilinear models, with potential extensions to SkX2-like states, skyrmioniums, and hopfions. Together, these results establish a systematic, two-stage design paradigm for targeting complex topological magnetism via multi-spin interactions and offer insight into how microscopic couplings shape emergent spin textures.

Abstract

We investigate the role of four-spin interactions in stabilizing exotic multiple-$Q$ topological spin textures and demonstrate their ability to realize a skyrmion crystal. While such higher-order interactions are known to be important, their intricate nature makes systematic model construction significantly challenging. To address this issue, we develop a theoretical framework that connects microscopic real-space four-spin couplings to their effective interactions in momentum space, providing a clear route to engineer target magnetic phases. Applying this framework to a frustrated Heisenberg model with designed four-spin interactions, we identify the stabilization of the zero-field skyrmion crystal with a topological number of two via simulated annealing. Furthermore, our momentum-space analysis reveals the intrinsic mechanism by which the well-known ring-exchange interaction also favors the skyrmion crystal. Our findings not only present a concrete model for a higher-order skyrmion crystal but also offer a general methodology for understanding and designing a wide range of complex multiple-$Q$ magnetic orders driven by multi-spin interactions.

Figures (9)

• Figure 1: Schematic illustrations of the seven types of four-body interactions considered in this study, characterized by the coefficients $L_1$ to $L_7$ from (a) to (g).
• Figure 2: Energy contributions of each four-spin interaction term for (a) the SkX2 state and (b) the spiral state as a function of the ordering wave vector magnitude $|\bm{Q}_{\nu}|$.
• Figure 3: Energy contribution difference between the SkX2 and spiral states, $\Delta I_{\alpha} = I^{\text{SkX2}}_{\alpha} - I^{\text{Spiral}}_{\alpha}$, for each four-spin interaction term as a function of the ordering wave vector magnitude $|\bm{Q}_{\nu}|$.
• Figure 4: Momentum-space couplings $K_{\beta}$ generated by each real-space four-spin interaction term $L_\alpha$ ($\alpha=1,\ldots,7$) as a function of the ordering wave vector magnitude $|\bm{Q}_{\nu}|$, assuming $L_\alpha=1$ for each respective term. Each panel (a) to (g) corresponds to the contributions from $L_1$ to $L_7$, respectively.
• Figure 5: Phase diagram showing the skyrmion number $N_{\rm Sk}$ and the magnetization $M$ as a function of the external magnetic field $H$ on the Hamiltonian with four-spin interactions between the nearest-neighbor bonds.