Existence of a second-order phase transition in a two-dimensional ${\phi }^4$ field theory

We study the stability of the vacuum in a two-dimensional ${\phi }^4$ field theory with a fixed ${\mathrm{}\left(\mathrm{mass}\right)\mathrm{}}^2$ term and a variable ${\phi }^4$ coupling term. We make the theory finite by normal-ordering the Hamiltonian with respect to a fixed mass. As the ${\phi }^4$ coupling strength increases, we can show that the system will undergo a second-order phase transition from a normal vacuum to an abnormal vacuum. There is no contradiction between the existence of a second-order transition in the absence of an external field and the Simon-Griffiths theorems which forbid any possible phase transition in the presence of an external field.