Spontaneous symmetry breaking via inhomogeneities and the differential surface tension

Abstract

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.

Inhomogeneities in the $2$-Flavor Chiral Gross-Neveu Model

“…Models with a continuous symmetry react differently to the constraints. For example, in the 3-dimensional O(2) model with a Mexican hat potential for a complex scalar field ∆ the constraint | ∆| < |∆| is met by inhomogeneous spin-wave-like configurations with |∆(x )| ≈ |∆| [45]. These configurations resemble the chiral spiral in the cGN model, for which the modulus of ∆ can be much smaller than |∆| .…”

“…In a typical configuration the modulus of ∆(x ) is near the minimum ρ of the effective potential -in order to minimize the bulk energy -but the real part σ and imaginary part π have vanishing expectation values caused by large phase fluctuations about the relevant chiral spiral. The main difference between the 3-dimensional O(2) model and the 2-dimensional cGN model is that in the former model the wavelength of the inhomogeneity is given by the box size [45] and in the latter by the inverse chemical potential. here are taken on a 72 × 255 lattice, with g −2 = 1.0540 and µ = 0.…”