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<p>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 <i>q</i>-uniform hypergraphs, the nature of transitions shifts from continuous to discontinuous at the tricritical point <i>q</i>&#xa0;=&#xa0;4. When both pairwise edges and hyperedges of size <i>q</i>&#xa0;&gt;&#xa0;2 coexist, diverse scenarios emerge, including mixed-order and double transitions, for <i>q</i>&#xa0;&gt;&#xa0;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.</p>

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Phase transitions in the simplicial Ising model on hypergraphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月27日

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  • Gangmin Son,
  • Deok-Sun Lee,
  • K.-I. Goh

摘要

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.