The Simplicial Bridge: Mapping quantum multi-spin exchange to higher-order topological networks in continuous magnetic fields

TL;DR

This paper introduces the 'Simplicial Bridge,' an analytical framework that maps quantum multi-spin exchange interactions in magnetic materials onto higher-order topological networks, revealing new stabilization mechanisms for topological solitons.

Contribution

The work provides a novel analytical approach linking high-dimensional Landau-Lifshitz equations to simplicial complex networks, demonstrating intrinsic stabilization of topological solitons without DMI.

Findings

Higher-order topological forces arise from spatial overlap in continuous limits.

Spatial compression of skyrmion tails generates 3-simplices (tetradic forces).

Multi-spin interactions create energetic barriers that stabilize 2D solitons without DMI.

Abstract

The macroscopic dynamics of topological defects in magnetic materials are traditionally modeled using pairwise interactions. However, higher-order quantum exchange mechanisms - such as biquadratic and 4-spin ring exchange-play a critical role in strongly correlated systems. In this work, we introduce the "Simplicial Bridge," an exact analytical framework that maps these high-dimensional, non-linear Landau-Lifshitz partial differential equations onto generalized Kuramoto phase-oscillator networks operating on abstract simplicial complexes. We rigorously demonstrate that spatial overlap in the continuous limit natively generates higher-order topological forces without requiring a supportive discrete atomic lattice. Specifically, the overlap of 1D helimagnetic kinks generates 2-simplices (triadic forces), while the spatial compression of 2D skyrmion tails - governed by modified Bessel function asymptotics - generates true 3-simplices (tetradic forces). Furthermore, we establish that the higher-order spatial derivatives inherent to these multi-spin interactions provide an intrinsic energetic barrier that bypasses Derrick's Theorem, stabilizing 2D topological solitons without the strict need for Dzyaloshinskii-Moriya Interaction (DMI).