Systematic study of the completeness of two-dimensional classical   $\phi^4$ theory
  The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical $\phi^4$ filed theory whose completeness has been proved in [V. Karimipour and et al, Phys. Rev. A 85, 032316]. We give a general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical $\phi^4$ model. To this end, we start from various classical field theories containing models on arbitrary lattices and also $U(1)$ lattice gauge theories. Then we convert them to a new classical field model on a non-planar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a $\phi^4$ term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final non-planar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model. 
Mohammad Hossein Zarei, Yahya Khalili
hep-lat cond-mat.stat-mech quant-ph