Ground states in two-dimensional boson quantum field-theory

We consider λ:P(?):₂ models of Euclidean quantum field theory, P(?) = Σ_k=1²ⁿa_k?^k and define for them translation-invariant ground states. It appears that the number of them is not more than two provided $\text{λ}\text{m}{}_0{}^2$ is large.