The study of topological band insulators has revealed fascinating phases characterized by band topology indices and anomalous boundary modes protected by global symmetries. In strongly correlated systems, where the traditional notion of electronic bands becomes obsolete, it has been established that topological insulator phases persist as stable phases, separate from the trivial insulators. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wave functions for numerical simulations is highly desirable. In this paper, we tackle this challenge for two-dimensional topological insulators by developing a comprehensive framework for fermionic tensor network states. Starting from simple assumptions, we obtain possible sets of tensor equations for any given symmetry group, capturing consistent relations governing symmetry transformation rules on tensor legs. We then examine the connection between these tensor equations and non-chiral topological insulators by constructing edge theories and extracting quantum anomaly data from each set of tensor equations. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of non-chiral topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for the corresponding topological insulator phases. Our methodology provides an important step toward simulating topological insulators in strongly correlated systems. We discuss the limitations and potential generalizations of our results, paving the way for further advancements in this field.

中文翻译：

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通过费米子张量网络状态揭示相关的二维拓扑绝缘体——分类、边缘理论和变分波函数


拓扑带绝缘体的研究揭示了以带拓扑指数和受全局对称性保护的反常边界模式为特征的迷人相。在强相关系统中，电子能带的传统概念已经过时，已经确定拓扑绝缘体相作为稳定相持续存在，与普通绝缘体分开。然而，由于无法将此类系统的基态表达为 Slater 行列式，因此非常需要制定用于数值模拟的通用变分波函数。在本文中，我们通过开发费米子张量网络状态的综合框架来应对二维拓扑绝缘体的这一挑战。从简单的假设开始，我们获得任何给定对称群的可能的张量方程组，捕获控制张量腿上的对称变换规则的一致关系。然后，我们通过构建边缘理论并从每组张量方程中提取量子异常数据来检查这些张量方程与非手性拓扑绝缘体之间的联系。通过详尽地探索所有可能的方程组，我们实现了非手性拓扑绝缘体相的系统分类。将给定方程组的解施加到局部张量上，我们获得相应拓扑绝缘体相的通用变分波函数。我们的方法为模拟强相关系统中的拓扑绝缘体提供了重要的一步。我们讨论了我们结果的局限性和潜在的概括，为该领域的进一步进步铺平了道路。