发表机构
Université Libre de Bruxelles; Ruhr University Bochum; Racah Institute of Physics, The Hebrew University; CERN（布鲁塞尔自由大学; 波鸿鲁尔大学; 希伯来大学拉卡物理研究所; 欧洲核子研究组织）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究探究耦合$O(N)$矢量场的多标量共形场论中的热有序，通过泛函重整化群等方法分析不同维度下的不动点、重整化群流与共形流形，揭示了一类大N共形场论的非零温持续对称性破缺新模式。
AI 中文摘要
我们研究了$d=4-ε$维和$d=3$维下、多个标量场耦合$O(N)$矢量场的共形场论（CFT）中的热有序。在$(4-ε)$维中，我们考虑将具有$\mathbb{Z}_2^M\rtimes S_M$对称性（对应标量的符号翻转与置换）的$M$标量场立方模型，耦合到$O(N)$矢量模型的理论。对于任意$M$，我们发现存在一个$N$的取值区间，其中存在两个类威尔逊-费希尔不动点。我们证明，在该区间内，当满足$M=2$且$N$足够大时，$\mathbb{Z}_2^M\rtimes S_M$对称性在任意非零温度下都会发生自发破缺；而当满足$M>2$时，该对称性保持不破缺。利用泛函重整化群将这些不动点向三维延拓，我们发现它们在到达$d=3$之前就会发生碰撞并进入复平面，这表明将其延拓到$d=3$时无法得到幺正共形场论。随后我们直接在三维下研究大而有限的$N$情形，其中$M\ll N$，初始阶段不假设$M$个标量之间存在任何置换对称性。通过分析重整化群流，我们证明这些模型存在一个共形流形，其成立范围到β函数的$1/N$展开的领头非平凡阶（即$\mathcal{O}(1/N)$阶）为止。在该流形的每一点上，标量场会根据其与$O(N)$矢量场的耦合方式分为两类。接着我们将研究限制在共形流形的一个子空间，该子空间中每一类内部的标量置换都对应额外的对称性。我们证明，在该子空间的一个区域内，所有$M$个标量都会获得热期望值，使得标量符号翻转对称性在所有非零温度下自发破缺。这提供了一类新型的大$N$共形场论，它们在非零温度下呈现出丰富的持续对称性破缺模式。
英文摘要
We study thermal order in conformal field theories (CFTs) in $d=4-ε$ and $d=3$ dimensions where several scalars are coupled to an $O(N)$ vector field. In $(4-ε)$ dimensions, we consider models coupling a cubic model of $M$ scalars with $\mathbb{Z}_2^M\rtimes S_M$ symmetry (corresponding to sign flips and permutations of the scalars) to an $O(N)$ vector model. For any $M$, we find a window of $N$ in which two Wilson-Fisher-like fixed points exist. We show that the $\mathbb{Z}_2^M\rtimes S_M$ symmetry is spontaneously broken at arbitrary nonzero temperatures for $M=2$ and sufficiently large $N$ within this window, but it remains unbroken for $M>2$. Using the functional renormalisation group to continue these fixed points towards three dimensions, we find that they collide and move into the complex plane well before reaching $d=3$, suggesting that their continuation to $d=3$ does not yield unitary CFTs. We then work directly in three dimensions at large but finite $N$, with $M\ll N$, initially without assuming any permutation symmetry among the $M$ scalars. By studying the RG flow, we show that these models possess a conformal manifold up to the leading nontrivial order in the $1/N$ expansion of the beta functions (which is $\mathcal{O}(1/N)$). At each point on this manifold, the scalars split into two classes that are distinguished by how they couple to the $O(N)$ vector field. We then restrict to a subspace of the conformal manifold where there is an additional symmetry under permutations of the scalars within each class. We prove that in a domain of this subspace, all $M$ scalars acquire thermal expectation values such that the symmetry under sign flips of the scalars is spontaneously broken at all nonzero temperatures. This provides a novel class of large $N$ CFTs that exhibit a rich pattern of persistent symmetry breaking at nonzero temperatures.
Comments61 pages
