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

Abstract

We investigate the phase transition of the four-dimensional single-component ${\varphi }^4$ 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 $⟨\varphi ⟩$ 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 ${\varphi }^4$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.

Comments: 7 pages, 11 figures