Spontaneous symmetry breaking via inhomogeneities and the differential surface tension

We discuss spontaneously broken quantum field theories with a continuous symmetry group via the constraint effective potential. Employing lattice simulations with constrained values of the order parameter, we demonstrate explicitly that the path integral is dominated by inhomogeneous field configurations and that these are unambiguously related to the flatness of the effective potential in the broken phase. We determine characteristic features of these inhomogeneities, including their topology and the scaling of the associated excess energy with their size. Concerning the latter we introduce the differential surface tension -- the generalization of the concept of a surface tension pertaining to discrete symmetries. Within our approach, spontaneous symmetry breaking is captured merely via the existence of inhomogeneities, i.e. without the inclusion of an explicit breaking parameter and a careful double limiting procedure to define the order parameter. While here we consider the three-dimensional $O(2)$ model, we also elaborate on possible implications of our findings for the chiral limit of QCD.