Spontaneous symmetry breaking in free theories with boundary potentials

TL;DR

This paper investigates how boundary potentials in free $O(N)$ models induce spontaneous symmetry breaking, leading to boundary RG flows and phase transitions, with detailed analysis using the $psilon$-expansion and effective potential calculations.

Contribution

It provides a detailed analysis of boundary-induced symmetry breaking in free $O(N)$ models, including a general formula for one-loop effective potentials and the RG flow diagram in models with boundary interactions.

Findings

Boundary RG flow ends with $N-1$ Neumann modes in IR.

Existence of an IR-stable critical point with conformal boundary conditions.

Identification of a phase boundary between symmetric and symmetry-broken phases.

Abstract

Patterns of symmetry breaking induced by potentials at the boundary of free $O(N)$ models in $d=3- \epsilon$ dimensions are studied. We show that the spontaneous symmetry breaking in these theories leads to a boundary RG flow ending with $N - 1$ Neumann modes in the IR. The possibility of fluctuation-induced symmetry breaking is examined and we derive a general formula for computing one-loop effective potentials at the boundary. Using the $\epsilon-$expansion we test these ideas in an $O(N)\oplus O(N)$ model with boundary interactions. We determine the RG flow diagram of this theory and find that it has an IR-stable critical point satisfying conformal boundary conditions. The leading correction to the effective potential is computed and we argue the existence of a phase boundary separating the region flowing to the symmetric fixed point from the region flowing to a symmetry-broken phase with a combination of Neumann and Dirchlet boundary conditions.