Convex effective potential of $O(N)$-symmetric $phi^4$ theory for large $N$

TL;DR

This paper derives the convex effective potential for large N $O(N)$-symmetric $^4$ theory starting from finite lattices, analyzing the thermodynamic and continuum limits, and studying the flattening of the potential barrier.

Contribution

It provides a detailed analysis of the effective potential's convexity, flatness, and renormalization effects in large N $O(N)$-symmetric $^4$ theory, including numerical and continuum limit insights.

Findings

Effective potential is convex and real-valued in the thermodynamic limit.

The barrier height of the effective potential decreases with system size, following a power law with exponent about -2.

In finite systems, the effective potential is nonconvex, but becomes convex as size increases.

Abstract

We obtain effective potential of $O(N)$-symmetric $\phi^4$ 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 power law and the exponent is about $-2$. The result is clearly understood from dominance of configurations with slowly-rotating field in one direction.