[论文解读] Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems

Meng Cheng, Nathan Seiberg|arXiv (Cornell University)|Nov 22, 2022

Cold Atom Physics and Bose-Einstein Condensates被引用 22

一句话总结

本文通过耦合背景规范场并扭曲边界条件，开发了一个用于1+1维晶格系统的统一基于异常的框架，揭示 ’t Hooft 异常如何约束晶格对称性、LSM 型结果和Luttinger 约束。

ABSTRACT

We analyze lattice Hamiltonian systems whose global symmetries have 't Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of "on-site action.") In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the $c=1$ compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an 't Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.

研究动机与目标

• 澄清 ’t Hooft 异常如何在具有离散空间结构的晶格系统中显现。
• 展示扭曲/缺陷如何编码背景规范场并揭示对称性混合与投影表示。
• 演示晶格模型中与已知连续理论（例如 c=1 压缩玻色子）对应的异常匹配。
• 在异常匹配范式内重新表述 LSM 型定理和 Luttinger 约束。

提出的方法

• 通过扭曲边界条件将晶格系统耦合到内部对称的平坦背景规范场。
• 插入空间扭曲并将其转化为拓扑缺陷，以研究扭曲希尔伯特空间上的对称性作用。
• 计算带有对称性插入的配分函数，以检测在规范变换下的相位模糊（异常）。
• 通过外流异常来解释异常，将其扩展到更高维度的体积作为数学工具。
• 使用连续极限（如 c=1 压缩玻色子）来引导保持完整连续性对称性和异常的晶格构造。
• 将晶格平移分析为 emanant/内部对称性，并将 LSM 与 Luttinger 型约束与异常考量联系起来。

实验结果

研究问题

• RQ1 当存在晶格平移且微观作用不是 on-site 时，内部对称性的 ’t Hooft 异常如何显现？
• RQ2 扭曲边界条件和拓扑缺陷如何捕获具有异常的晶格系统的背景规范场？
• RQ3 LSM 型约束和 Luttinger 定理能否被理解为异常匹配或晶格模型中相关的 emanant 对称性？
• RQ4 哪些晶格构造在保持异常与对偶性的同时收敛到已知连续理论（例如 c=1 压缩玻色子）？
• RQ5 由晶格平移产生的 emergent/emanant 对称性如何参与到异常结构与谱约束中？

主要发现

• 给出一个晶格系统，对任意半径均收敛到 c=1 压缩玻色子，保持完整的连续内部对称性及其异常，包括 T-duality。
• 给出一些示例，其中异常的内部对称性在晶格上并非就地作用，但可以通过平坦背景规范场和扭曲边界条件进行分析。
• 将 LSM 定理重新表述为自旋链中的 ’t Hooft 异常匹配条件，并利用连续理论洞见提取了精确的有限尺寸谱信息。
• 在某些 XXZ 型链中鉴定出 emanant（平移诱导的）Z2 对称性，并解释了尽管源自晶格，在低能下的精确性。
• 阐明了何时 Luttinger 型填充约束反映异常，何时不反映，凸显了填充、emanant 对称性与异常之间的微妙关系。