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.

中文翻译：

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具有边界势的自由理论中的自发对称性破缺

研究了由 $d=3-\epsilon$ 维度中自由 $O(N)$-模型边界处的电位引起的对称破坏模式。我们表明，这些理论中的自发对称性破缺导致边界 RG 流以 IR 中的 $N - 1$ Neumann 模式结束。检查了波动引起的对称性破坏的可能性，我们推导出了计算边界处单环有效势的一般公式。使用 $\epsilon$-expansion，我们在具有边界交互作用的 $O(N)\oplus O(N)$-模型中测试这些想法。我们确定了该理论的 RG 流程图，并发现它具有满足共形边界条件的 IR 稳定临界点。