The Lee-Yang model and its generalizations through the lens of long-range deformations

In two dimensions, the non-unitary class of conformal minimal models, $\mathcal{M}(2,2m+1)$, has been recently conjectured to arise as renormalization-group fixed points of scalar field theories with complex $i{\phi }^{2m-1}$ interaction, $m\in \mathbb{N}$, $m\ge 2$. We test a variation of this conjecture through the perturbative study of two separate long-range constructions based on respectively the minimal model and its potential Landau-Ginzburg formalism. For $m>2$, inconsistencies are found when subsequently relating both constructions. In contrast, the long-range Lee-Yang model, the $m=2$ case, is shown to be analogue to the long-range Ising model.