Convex effective potential of O(N)-symmetricø4 theory for large N Original Research Article

We obtain an effective potential of the O(N)-symmetric ø4 theory for large N starting with a finite lattice system and taking the thermodynamic limit with great care. In the thermodynamic limit, it is globally real-valued and convex in both the symmetric and the broken phases. In particular, it has a flat bottom in the broken phase. Taking the continuum limit, we discuss renormalization effects to the flat bottom and exhibit the effective potential of the continuum theory in three and four dimensions. On the other hand, the effective potential is nonconvex in a finite lattice system. Our numerical study shows that the barrier height of the effective potential flattens as a linear size of the system becomes large. It decreases obeying a power law and the exponent is about −2. The result is clearly understood from the dominance of configurations with a slowly-rotating field in one direction.