A Variational Expansion for the Free Energy of a Bosonic System
  In this paper, a variational perturbation scheme for nonrelativistic many-Fermion systems is generalized to a Bosonic system. By calculating the free energy of an anharmonic oscillator model, we investigated this variational expansion scheme for its efficiency. Using the modified Feynman rules for the diagrams, we obtained the analytical expression of the free energy up to the fourth order. Our numerical results at various orders are compared with the exact and other relevant results. 
Wen-Fa Lu, Sang Koo You, Jino Bak, Chul Koo Kim and Kyun Nahm