AI 中文总结
本文提出一种构造整数自旋对称保护相矩阵乘积态代表的拟设，覆盖所有拓扑指标组合，并验证其与现有分类一致。
AI 中文摘要
我们提供了一个简单的拟设，用于寻找整数自旋 $SO(3)\times\Z_2^T\times\Z_2^R$ 不变的矩阵乘积态，这些态代表文献中 Tasaki \cite{tasakiheistop}\cite{tasakitop} 和 Ogata \cite{ogatatime}\cite{ogatareflection} 研究的每个 $\Z_2$ 拓扑指标三元组（对应于时间反演和反射对称性）所对应的相。平凡和非平凡指标的每种组合都有一个自旋-$1$ $SU(2)$、时间反演和位点反射对称的 MPS 代表，其注入长度 $l\leq 8$。我们提供了一些自旋-$1$ 的精确例子，以及一个 Fortran 程序，该程序可以为任意整数自旋找到任何三元组的代表，还提供了一些小自旋随机采样的数据。这类态的存在性与 Chen、Gu 和 Wen \cite{chen} 的分类一致，也与 Tasaki \cite{tasakinew} 的最新结果一致，该结果将隐藏序和 Ogata 的一个指标等同起来，其中 Ogata 指标与二面体 $D_2$ 对称性相关。
英文摘要
We provide a simple ansatz for finding integer spin $SO(3)\times\Z_2^T\times\Z_2^R$-invariant matrix product states representing a phase for each triple of $\Z_2$ topological indices studied in the literature by Tasaki \cite{tasakiheistop}\cite{tasakitop} and Ogata (corresponding to time-reversal and reflection symmetries) \cite{ogatatime}\cite{ogatareflection}. Every combination of trivial and non-trivial indices has a spin-$1$ $SU(2)$, time-reversal, and site-reflection symmetric MPS representative with injectivity length $l\leq 8$. We provide some exact examples for spin-$1$ and a Fortran program which will find representatives of any triple for any integer spin, as well as some data from random sampling for small spin. The existence of such states agrees with the classification of Chen, Gu, and Wen \cite{chen}, and with the recent results of Tasaki \cite{tasakinew} equating the hidden-order and one of Ogata's indices, wherein the Ogata index is that associated to the dihedral $D_2$ symmetry.
