Effect of interaction range on the phase diagram of magnetic hedgehog lattices

Abstract

Topological spin textures have attracted considerable attention through their emergent electromagnetic properties, including the topological Hall effect and the topological Nernst effect. Among them, magnetic hedgehog lattices (HLs) are three-dimensional topological spin textures that host emergent magnetic monopoles and antimonopoles. HLs have been identified in metallic compounds, such as $\mathrm{MnSi}_{1-x}\mathrm{Ge}_x$ and $\mathrm{SrFeO_3}$. Various theoretical models have been proposed, with either short-range or long-range interactions, which are typically regarded as effective descriptions of insulating and metallic systems, respectively. However, the role of the interaction range in stabilizing the HLs remains only partially understood. To address this gap, we investigate the stability of HLs by systematically varying the range and spatial decay of the exchange interactions, thereby interpolating between interaction profiles commonly associated with localized-spin insulating models and itinerant-electron metallic models. Based on extensive Monte Carlo simulations, we find that HLs are stabilized over a broad interaction range. Intriguingly, among the interaction profiles examined, intermediate-range interactions are particularly favorable for stabilizing HLs. They give rise to two distinct types of $3Q$-HLs that differ in the number of monopole-antimonopole pairs per magnetic unit cell, and their relative stability depends sensitively on the interaction range. Our results provide a unified framework for tuning phase diagrams from microscopic interaction models, serving as a guideline for interpreting experiments in candidate materials.