Phase Transitions in the Simplicial Ising Model on Hypergraphs

We study the phase transitions in the simplicial Ising model on hypergraphs, in which the energy within each hyperedge (group) is lowered only when all the member spins are unanimously aligned. The Hamiltonian of the model is equivalent to a weighted sum of lower-order interactions, evoking an Ising model defined on a simplicial complex. Using the Landau free energy approach within the mean-field theory, we identify diverse phase transitions depending on the sizes of hyperedges. Specifically, when all hyperedges have the same size $q$, the nature of the transitions shifts from continuous to discontinuous at the tricritical point $q=4$, with the transition temperatures varying nonmonotonically, revealing the ambivalent effects of group size $q$. Furthermore, if both pairwise edges and hyperedges of size $q>2$ coexist in a hypergraph, novel scenarios emerge, including mixed-order and double transitions, particularly for $q>8$. Adopting the Bethe--Peierls method, we investigate the interplay between pairwise and higher-order interactions in achieving global magnetization, illuminating the multiscale nature of the higher-order dynamics.