Completeness of classical $\phi^4$ theory on 2D lattices

TL;DR

This paper demonstrates that the 2D lattice $\,phi^4$ theory is a universal model capable of representing the partition functions of any other discretized scalar field theories through an extended quantum formalism.

Contribution

It extends a quantum formalism from the Ising model to continuous variables, establishing the completeness of 2D $\,phi^4$ theory for scalar field models.

Findings

Proves the completeness of 2D $\,phi^4$ theory in representing other models

Extends quantum formalism to continuous variables

Shows universality of the $\,phi^4$ lattice model

Abstract

We formulate a quantum formalism for the statistical mechanical models of discretized field theories on lattices and then show that the discrete version of $\phi^4$ theory on 2D square lattice is complete in the sense that the partition function of any other discretized scalar field theory on an arbitrary lattice with arbitrary interactions can be realized as a special case of the partition function of this model. To achieve this, we extend the recently proposed quantum formalism for the Ising model \cite{quantum formalism} and its completeness property \cite{completeness} to the continuous variable case.