Motivation of $p$-spin glass model

Question

In general, the $p$-spin glass model focuses on $p$-body interactions like $${\displaystyle H({\boldsymbol {\sigma }})=\sum_{i_1,...,i_p} J_{i_{1},\ldots i_{p}} \sigma _{i_{1}}\cdots \sigma _{i_{p}}.}$$ But what is the physical significance of this Hamiltonian? Does there exist any system that has this p-body interaction among p spins? It makes sense to me only when it looks like the Ising model, i.e. p=2 and sum over all adjacent sites...