Phase transition of four-dimensional lattice $\phi^4$ theory with tensor renormalization group

TL;DR

This paper uses tensor renormalization group methods to analyze the phase transition in four-dimensional lattice $^4$ theory, revealing a weak first-order transition and its relation to the 3D Ising universality class.

Contribution

It is the first to apply tensor renormalization group to four-dimensional $^4$ lattice theory and compare its phase transition properties with the 3D Ising model.

Findings

Weak first-order phase transition at critical hopping parameter

Consistency with 3D Ising universality class

Large volume simulations up to $1024^4$

Abstract

We investigate the phase transition of the four-dimensional single-component $\phi^4$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $\langle\phi\rangle$ at a finite quartic coupling $\lambda$ on large volumes up to $V=1024^4$ in order to detect the spontaneous breaking of the $\mathbb{Z}_2$ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional $\phi^4$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.