$O(N)$模型的大$N$分析的不完全性：有限$N$上的非微扰尖点不动点及其非平凡同伦
Incompleteness of the large-$N$ analysis of the $O(N)$ models: Nonperturbative cuspy fixed points and their nontrivial homotopy at finite $N$
摘要：
我们总结了$O(N)$模型的大$N$极限的通常实现，并详细说明了为什么以及如何当它们在极限$N\to\infty$中变得奇异时，它们会错过一些物理上重要的不动点。使用威尔逊的重整化群在其功能的非微扰版本，我们展示了如何建立的奇异性作为$N$增加。在威尔逊-波尔钦斯基版本的非微扰重整化群，我们表明，奇点的尖点，成为边界层有限，但大值的$N$。相应的不动点从来没有接近高斯，是超出了通常的微扰方法。我们发现了四个新的不动点，并研究它们在所有维度和所有$N>0$，并表明它们发挥了重要作用的三临界物理的$O(N)$模型。最后，我们证明了这些不动点中的一些是双值的，当它们被认为是$d$和$N$的函数时，从而揭示了重要的和非平凡的同伦结构。发生在$N=\infty$和$d=3$的Bardeen-Moshe-Bander现象被证明对我们所有结果的内部一致性起着至关重要的作用。
英文摘要：
We summarize the usual implementations of the large $N$ limit of $O(N)$ models and show in detail why and how they can miss some physically important fixed points when they become singular in the limit $N\to\infty$. Using Wilson's renormalization group in its functional nonperturbative versions, we show how the singularities build up as $N$ increases. In the Wilson-Polchinski version of the nonperturbative renormalization group, we show that the singularities are cusps, which become boundary layers for finite but large values of $N$. The corresponding fixed points being never close to the Gaussian, are out of reach of the usual perturbative approaches. We find four new fixed points and study them in all dimensions and for all $N>0$ and show that they play an important role for the tricritical physics of $O(N)$ models. Finally, we show that some of these fixed points are bi-valued when they are considered as functions of $d$ and $N$ thus revealing important and nontrivial homotopy structures. The Bardeen-Moshe-Bander phenomenon that occurs at $N=\infty$ and $d=3$ is shown to play a crucial role for the internal consistency of all our results.
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