Euler topology in classical spin liquids

Classical spin liquids have recently been analyzed in view of the single-gap homotopy classification of their dispersive eigenvectors. We show that the recent progress in defining multi-gap topologies, notably exemplified by the Euler class, can be naturally included in these homotopy-based classification schemes and present phases that change topology by band node braiding. This process alters the topology of the pinch points in the spin structure factor and consequently their stability. Furthermore, we discuss how these notions also pertain to models discussed previously in the literature and have a broader range of application beyond our specific results. Our work thus opens up an uncharted avenue in the understanding of spin liquids.