Phase transitions in the simplicial Ising model on hypergraphs

Complex higher-order interactions can reshape higher-order dynamics in complex systems. Systematic understanding of the repertoire of phase transitions that higher-order interactions can trigger remains incomplete. Here we study the phase transitions in the simplicial Ising model on hypergraphs, in which the energy within each hyperedge is lowered only when all the member spins are unanimously aligned. We show that its Hamiltonian is equivalent to a weighted sum of lower-order interactions and use the Landau free energy to identify diverse phase transitions depending on the hyperedge sizes. For q-uniform hypergraphs, the nature of transitions shifts from continuous to discontinuous at the tricritical point q = 4. When both pairwise edges and hyperedges of size q > 2 coexist, diverse scenarios emerge, including mixed-order and double transitions, for q > 8. We further adopt the Bethe–Peierls method to investigate the interplay between pairwise and higher-order interactions in achieving global magnetization, illuminating the multiscale nature of double transitions. The simplicial Ising model on hypergraphs addresses the complex interactions within hyperedges, where energy is minimized only with unanimous spin alignment. The authors reveal diverse phase transitions influenced by hyperedge size, uncovering novel mixed-order and double transitions, which enhance understanding of higher-order dynamics in complex systems.