Connected Green function approach to ground state symmetry breaking in   $\Phi^4_{1+1}$-theory
  Using the cluster expansions for n-point Green functions we derive a closed set of dynamical equations of motion for connected equal-time Green functions by neglecting all connected functions higher than $4^{th}$ order for the $\lambda \Phi^4$-theory in $1+1$ dimensions. We apply the equations to the investigation of spontaneous ground state symmetry breaking, i.e. to the evaluation of the effective potential at temperature $T=0$. Within our momentum space discretization we obtain a second order phase transition (in agreement with the Simon-Griffith theorem) and a critical coupling of $\lambda_{crit}/4m^2=2.446$ as compared to a first order phase transition and $\lambda_{crit}/4m^2=2.568$ from the Gaussian effective potential approach. 
J.M. Haeuser, W. Cassing, A. Peter and M.H. Thoma