Phase transition of four-dimensional lattice ϕ4 theory with tensor renormalization group

Abstract

We investigate the phase transition of the four-dimensional single-component ${\ensuremath{\phi}}^{4}$ 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 $⟨\ensuremath{\phi}⟩$ at a finite quartic coupling $\ensuremath{\lambda}$ on large volumes up to $V=102{4}^{4}$ 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 ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.