耦合线模型与 Chern-Simons 理论的对应

耦合线模型是一种通过相互耦合的一维量子线序列来实现各种拓扑物态的微 观模型。而 Chern-Simons 理论作为一种低能有效场论，只描述系统在宏观上的拓 扑自由度，不涉及系统的微观细节。如何从系统的微观细节演生出宏观自由度一 直是有待揭示的课题。在过去的研究中，研究者给出了 𝑣 = 1/𝑚 的费米 Laughlin 态与 Abelian Chern-Simons 的联系，但目前还没有研究涉及 Non-Abelian 的情况。

本文研究了 𝑣 = 1/2 玻色态与 𝑆𝑈(2)Non-Abelian Chern-Simons 理论的联系。 通过分析 𝑣 = 1/2 态中的 𝑆𝑈(2) 对称性，我们用玻色化形式写出了体系中流 的表达式，并借助这些流算符将规范场强与耦合线模型中的玻色场算符对应起 来。在考虑连续性极限后，我们证明了我们构造的场强算符满足 Yang-Mills-ChernSimons 对易关系。同时我们还给出了保证对应成立的合适耦合参数条件，对 YangMills-Chern-Simons 的哈密顿量也进行了计算。这一研究弥补了人们在 Non-Abelian Chern-Simons 理论与耦合线模型之间联系的空白，将会为更复杂的 Non-Abelian 分 数量子霍尔效应的研究打下基础。

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