标题： 一维中对称保护拓扑相的检测 显示英文标题

标题： Detection of Symmetry Protected Topological Phases in 1D

摘要： 拓扑相是不能通过局部序参量来表征的物质相。 已经证明，在1D系统中，有能隙的相可以完全通过与对称群的投影表示相关的工具来表征。 我们展示了一种简单的方法来确定这些表示的矩阵，以便直接区分不同的相。 从这些矩阵中，我们还指出如何推导出时间反演、宇称对称性和$Z_2 \times Z_2$对称性的多种不同类型的非局部序参量，以及一些更一般的情况（其中一些之前已通过其他方法获得）。 利用这些概念，Haldane相的普通弦序可以与一个在临界点发生变化的选择规则相关联。 此外，我们指出一个更复杂的内部对称性的例子，其中普通的弦序无法应用。 显示英文摘要

摘要： A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped phases in 1D systems can be completely characterized using tools related to projective representations of the symmetry groups. We show how to determine the matrices of these representations in a simple way in order to distinguish between different phases directly. From these matrices we also point out how to derive several different types of non-local order parameters for time reversal, inversion symmetry and $Z_2 \times Z_2$ symmetry, as well as some more general cases (some of which have been obtained before by other methods). Using these concepts, the ordinary string order for the Haldane phase can be related to a selection rule that changes at the critical point. We furthermore point out an example of a more complicated internal symmetry for which the ordinary string order cannot be applied.