多极拓扑相的高级形式规范对称性
Higher-form Gauge Symmetries in Multipole Topological Phases
摘要：
本文研究多极拓扑绝缘体的场论问题。先前的研究已经表明，这样的系统自然地耦合到更高秩的张量规范场，这是由于测量偶极或子系统$U(1)$对称性而产生的。在这里，我们提出了一个互补的框架，使用电更高形式的对称性。我们利用的事实，规范1-形式的电对称性的结果在2-形式的规范场自然耦合到扩展的线状物体：威尔逊线。在我们的上下文中，威尔逊线是与系统的电极化的强度相关联的电通量线。这使得我们可以定义一个广义的2-形式Peierls'替换偶极子，表明秩-2张量规范场$A_{ij}$的非对角分量可以作为背景反对称2-形式规范场产生的格点Peierls因子而出现。这个框架有直接的应用：（i）它允许我们构造一个由Dixmier-Douady不变量给出的明显的拓扑四极响应作用量--一个2-形式规范场的Chern数的推广--这使得在某些晶体对称性存在下四极矩的量子化变得简单;（ii）它允许一个更清晰的解释秩-2贝里相位计算的四极矩;（iii）它允许一个通用的李伯-舒尔茨-马蒂斯定理的偶极守恒系统的证明。
英文摘要：
In this article we study field-theoretical aspects of multipolar topological insulators. Previous research has shown that such systems naturally couple to higher-rank tensor gauge fields that arise as a result of gauging dipole or subsystem $U(1)$ symmetries. Here we propose a complementary framework using electric higher-form symmetries. We utilize the fact that gauging 1-form electric symmetries results in a 2-form gauge field which couples naturally to extended line-like objects: Wilson lines. In our context the Wilson lines are electric flux lines associated to the electric polarization of the system. This allows us to define a generalized 2-form Peierls' substitution for dipoles that shows that the off-diagonal components of a rank-2 tensor gauge field $A_{ij}$ can arise as a lattice Peierls factor generated by the background antisymmetric 2-form gauge field. This framework has immediate applications: (i) it allows us to construct a manifestly topological quadrupolar response action given by a Dixmier-Douady invariant -- a generalization of a Chern number for 2-form gauge fields -- which makes plain the quantization of the quadrupole moment in the presence of certain crystal symmetries; (ii) it allows for a clearer interpretation of the rank-2 Berry phase calculation of the quadrupole moment; (iii) it allows for a proof of a generic Lieb-Schultz-Mattis theorem for dipole-conserving systems.
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