二维经典$ϕ^4$理论完备性的系统研究
Systematic study of the completeness of two-dimensional classical $ϕ^4$ theory
摘要：
一些经典统计力学（SM）模型的完备性是最近由配分函数的量子形式主义发展起来的结果。在本文中，我们考虑一个二维经典的$ϕ^4$场理论，其完备性已在[V.Karimipour and et al，Phys.Rev.A85，032316]中得到证明。我们给出了这样一个模型的完整性一般系统的证明，通过几个简单的步骤，我们展示了如何从一个2D经典$ϕ^4$模型的任意经典场论的配分函数。为此，我们开始从各种经典场论包含模型的任意晶格和$U(1)$格点规范理论。然后，我们将它们转换为一个新的经典场模型的非平面二部图与虚动力学项。之后，我们证明了通过向二部图添加足够数量的顶点，可以将相应Hamilton量中的域的任何多项式函数近似转换为$ϕ^4$项。在接下来的步骤中，我们给出了一些图形转换，将最终的非平面图转换为2D矩形格。我们还表明，应添加的顶点数的增长多项式与原模型中的顶点数。
英文摘要：
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 $ϕ^4$ 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 $ϕ^4$ 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 $ϕ^4$ 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.
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