We study the boundary extraordinary transition of a three-dimensional (3D) tricritical $O(N)$ model. We first compute the mean-field Green's function with a general coupling of $|\vec \phi|^{2n}$ (with $n=3$ corresponding to the tricritical model) at the extraordinary phase transition. Then, using layer susceptibility, we obtain the boundary operator expansion for the transverse and longitudinal modes within the $\epsilon=3 - d$ expansion. Based on these results, we demonstrate that the tricritical point exhibits an extraordinary transition characterized by an ordered boundary for any $N$. This provides the first nontrivial example of continuous symmetry breaking in 2D in the context of boundary criticality.
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We sincerely thank the editors and all referees for their thorough and thoughtful reviews. We have carefully considered each comment and suggestion. For each referee, a detailed point-by-point response is provided in the corresponding attachment. We hope our revisions and explanations address all concerns and help clarify the contributions of our manuscript.
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We have carefully revised the manuscript in response to the comments from the editors and referees. The specific changes and detailed responses are listed in the attachments to the corresponding replies for each referee.
