标题： $\mathbb{Z}_2$ 对$XX$模型及其相关模型的规约和自对偶性 显示英文标题

标题： $\mathbb{Z}_2$ gauging and self-dualities of the $XX$ model and its cousins

摘要： 在本工作中，我们通过动量和绕数$U(1)$对称性的视角来研究一维$XX$晶格模型及其相关模型。 我们根据它们与$U(1)$对称性的关系，区分两种密切相关的$\mathbb{Z}_2$对称性，并建立这些模型之间的$\mathbb{Z}_2$规范化关系网络，其基础是两个基本种子：$ XY \pm YX$模型。 这两个种子在各自$\mathbb{Z}_2$对称性的规范下各自自对偶，具有明显对称的守恒电荷，从而清晰地揭示了非可逆对称性与 Kramer-Wannier 对偶性之间的联系。 通过利用这两个种子模型的自对偶性，我们通过适当的规范化过程推导出它们的亲戚模型的自对偶性，包括$XX$模型和 Levin-Gu 模型。 此外，在这些规范方案下，晶格 T 对偶矩阵的形式为单位矩阵。 这些晶格模型流向$c =1$紧致标量共形场理论，其扭曲依赖于晶格尺寸模四的结果。 最后，我们统一了这些模型中局部守恒荷的映射结构，提供了一个全面的框架来理解它们的对称性和对偶性。 显示英文摘要

摘要： In this work, we investigate the one-dimensional $XX$ lattice model and its cousins through the lens of momentum and winding $U(1)$ symmetries. We distinguish two closely related $\mathbb{Z}_2$ symmetries based on their relation to the $U(1)$ symmetries, and establish a web of $\mathbb{Z}_2$-gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective $\mathbb{Z}_2$-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the $XX$ model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the $c =1$ compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.