标题： 二极对称性保护拓扑相的分类：矩阵乘积态、稳定子哈密顿量和有限张量规范理论 显示英文标题

标题： Classification of Dipolar Symmetry-Protected Topological Phases: Matrix Product States, Stabilizer Hamiltonians and Finite Tensor Gauge Theories

摘要： 我们对由偶极对称性保护的一维对称性保护拓扑（SPT）相进行分类。 偶极对称性包含两组对称性生成元：电荷算符和偶极算符，它们与平移一起形成一个非平凡的代数。 使用矩阵乘积态（MPS），我们表明对于一个$G$偶极对称性，其中$G$是一个有限阿贝尔群，一维偶极SPT相由群$H^2[G\times G,U(1)]/H^2[G,U(1)]^2$分类。 由于对称性代数，MPS张量表现出一种不寻常的性质，禁止在边界处对电荷算符进行分数化。 对于分类中的每个相，我们显式构造一个稳定器哈密顿量来实现SPT相，并通过将偶极对称性耦合到背景张量规范场来推导响应场理论。 这些场理论将Dijkgraaf-Witten理论推广到扭曲的有限张量规范理论。 显示英文摘要

摘要： We classify one-dimensional symmetry-protected topological (SPT) phases protected by dipole symmetries. A dipole symmetry comprises two sets of symmetry generators: charge and dipole operators, which together form a non-trivial algebra with translations. Using matrix product states (MPS), we show that for a $G$ dipole symmetry with $G$ a finite abelian group, the one-dimensional dipolar SPTs are classified by the group $H^2[G\times G,U(1)]/H^2[G,U(1)]^2$. Because of the symmetry algebra, the MPS tensors exhibit an unusual property, prohibiting the fractionalization of charge operators at the edges. For each phase in the classification, we explicitly construct a stabilizer Hamiltonian to realize the SPT phase and derive the response field theories by coupling the dipole symmetry to background tensor gauge fields. These field theories generalize the Dijkgraaf-Witten theories to twisted finite tensor gauge theories.