Misunderstanding that the Effective Action is Convex under Broken Symmetry

TL;DR

This paper refutes the common belief that the effective action is convex under broken symmetry, demonstrating that non-convexity is inherent in spontaneous symmetry breaking and exploring its implications in quantum field theory and statistical mechanics.

Contribution

It proves that the effective action is necessarily non-convex under broken symmetry and clarifies the associated physical consequences, including translational invariance breaking and phase coexistence.

Findings

Effective action is non-convex under broken symmetry.

Spontaneous symmetry breaking involves non-convexity in the effective action.

Translational invariance breaks spontaneously in the non-convex region.

Abstract

The widespread belief that the effective action is convex and has a flat bottom under broken global symmetry is shown to be wrong. We show spontaneous symmetry breaking necessarily accompanies non-convexity in the effective action for quantum field theory, or in the free energy for statistical mechanics, and clarify the magnitude of non-convexity. For quantum field theory, it is also proved that translational invariance breaks spontaneously when the system is in the non-convex region, and that different vacua of spontaneously broken symmetry cannot be superposed. As applications of non-convexity, we study the first-order phase transition which happens at the zero field limit of spontaneously broken symmetry, and we propose a simple model of phase coexistence which obeys the Born rule.