Figure 1.

Example of gauge transformations between models with $n=4$ spins. Models are represented by diagrams (color online): spins are full dots • in presence of a local field, empty dots ∘ otherwise; blue lines are pairwise interactions (${\varphi }^{\mu }=s_i{s}_j$); orange triangles denote 3-spin interactions (${\varphi }^{\mu }=s_i{s}_j{s}_k$); and the 4-spin interaction ($s_1{s}_2{s}_3{s}_4$) is a filled blue triangle. Note that model a) has all its interactions grouped on 3 spins; the gauge transformations leading to this model are shown along the arrows. All the models belong to the same complexity class, with $\left|\mathcal{M}\right|=7$, $\lambda =4$ and a number of independent operators $n_{\mathcal{M}}=3$ (e.g., $s_1,s_2$ and $s_3$ in model a)—see tables in Section SM-6 of the Supplementary Material. The class contains in total 15 models, which are grouped, with respect to the permutation of the spins, behind the 4 representatives shown here with their multiplicity ($\times m$). Models a), b), c) and d) are respectively identified by the following sets of operators: (a) $\mathcal{M}=\{s_1,s_2,s_3,s_1{s}_2,s_1{s}_3,s_2{s}_3,s_1{s}_2{s}_3\}$; (b) $\mathcal{M}=\{s_1,s_2{s}_3,s_3{s}_4,s_4{s}_2,s_1{s}_2{s}_3,s_1{s}_3{s}_4,s_1{s}_2{s}_4\}$; (c) $\mathcal{M}=\{s_2,s_4,s_2{s}_4,s_1{s}_3,s_1{s}_2{s}_3,s_1{s}_3{s}_4,s_1{s}_2{s}_3{s}_4\}$; and (d) $\mathcal{M}=\{s_1{s}_2,s_1{s}_3,s_1{s}_4,s_2{s}_3,s_2{s}_4,s_3{s}_4,s_1{s}_2{s}_3{s}_4\}$.