We propose a new type of quantum liquids, dubbed Stiefel liquids, based on ($2+1$)-dimensional nonlinear sigma models on target space $SO(N)/SO(4)$, supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies. We show that the well-known deconfined quantum critical point and $U(1)$ Dirac spin liquid are unified as two special examples of Stiefel liquids, $N=5$ and $N=6$, respectively. Furthermore, we conjecture that Stiefel liquids with $N>6$ are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian. Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics. The intrinsic absence of (conventional or partonlike) mean-field construction also means that, within the traditional approaches, will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system). For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems. Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or kagome lattice, through the intertwinement between noncoplanar magnetic orders and valence-bond-solid orders.

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Stiefel Liquids：来自交织订单的可能的非拉格朗日量子临界性

我们提出了一种新型的量子液体，称为 Stiefel 液体，基于（$2+1$) 维非线性西格玛模型在目标空间 $秒哦(N)/秒哦(4)$，辅以 Wess-Zumino-Witten 项。我们认为 Stiefel 液体形成了一类具有非凡特性的临界量子液体，例如大的涌现对称性、级联结构和非平凡的量子异常。我们证明了众所周知的去限制量子临界点和$你(1)$ Dirac 自旋液体被统一为 Stiefel 液体的两个特殊例子， $N=5$ 和 $N=6$， 分别。此外，我们推测 Stiefel 液体具有$N>6$是非拉格朗日的，从某种意义上说，在重整化群下，它们流向红外（共形不变）固定点，这些固定点无法用任何可重整化的连续拉格朗日函数来描述。这种非拉格朗日状态超出了凝聚态物理中奇异量子液体研究中所熟悉的部分规范平均场理论范式。（传统的或类似parton 的）平均场构造的内在缺失也意味着，在传统方法中，将难以确定非拉格朗日状态是否真的可以从特定的 UV 系统（例如晶格自旋系统）中出现。为此，我们假设，如果量子态的量子异常与（广义的）Lieb-Schultz-Mattis 定理的约束相匹配，则该量子态可从晶格系统中出现。基于这个假设，