It is widely expected that at sufficiently high temperatures order is always lost, e.g. magnets loose their ferromagnetic properties. We pose the question of whether this is always the case in the context of quantum field theory in $d$ space dimensions. More concretely, one can ask whether there exist critical points (CFTs) which break some global symmetry at arbitrary finite temperature. The most familiar CFTs do not exhibit symmetry breaking at finite temperature, and moreover, in the context of the AdS/CFT correspondence, critical points at finite temperature are described by an uncharged black brane which obeys a no-hair theorem. Yet, we show that there exist CFTs which have some of their internal symmetries broken at arbitrary finite temperature. Our main example is a vector model which we study both in the epsilon expansion and arbitrary rank as well as the large rank limit (and arbitrary dimension). The large rank limit of the vector model displays a conformal manifold, a moduli space of vacua, and a deformed moduli space of vacua at finite temperature. The appropriate Nambu-Goldstone bosons including the dilaton-like particle are identified. Using these tools we establish symmetry breaking at finite temperature for finite small $\epsilon$. We also prove that a large class of other fixed points, which describe some of the most common quantum magnets, indeed behave as expected and do not break any global symmetry at finite temperature. We discuss some of the consequences of finite temperature symmetry breaking for the spectrum of local operators. Finally, we propose a class of fixed points which appear to be possible candidates for finite temperature symmetry breaking in $d=2$.

中文翻译：

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共形理论中的热序

人们普遍预计，在足够高的温度下，秩序总是会丢失，例如磁铁失去其铁磁特性。我们提出的问题是，在 $d$ 空间维度的量子场论背景下，这是否总是如此。更具体地说，人们可以问是否存在在任意有限温度下破坏某些全局对称性的临界点 (CFT)。最熟悉的 CFT 在有限温度下不会表现出对称性破坏，而且，在 AdS/CFT 对应的上下文中，有限温度下的临界点由不带电的黑膜描述，该黑膜遵循无毛定理。然而，我们表明存在 CFT，它们的一些内部对称性在任意有限温度下被破坏。我们的主要示例是一个向量模型，我们在 epsilon 扩展和任意秩以及大秩限制（和任意维度）中都研究了该模型。矢量模型的大秩极限显示了一个共形流形、一个真空模空间和一个在有限温度下变形的真空模空间。确定了合适的 Nambu-Goldstone 玻色子，包括类似膨胀的粒子。使用这些工具，我们在有限温度下为有限的小 $\epsilon$ 建立对称破坏。我们还证明了一大类其他不动点，它们描述了一些最常见的量子磁体，确实表现得如预期，并且在有限温度下不会破坏任何全局对称性。我们讨论了局部算子谱的有限温度对称性破坏的一些后果。最后，