We investigate the phase transition of the four-dimensional single-component ? 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 ? ? ? at a finite quartic coupling λ on large volumes up to V = 102 4 4 in order to detect the spontaneous breaking of the 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 ? 4 theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.
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