Spontaneous symmetry breaking in free theories with boundary potentials

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.

17 pages, 3 figures + Mathematica notebook included

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