One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of \({\mathsf {G}}\)- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

表现出拓扑顺序的带隙量子相最显着的特征之一是存在长程纠缠，而任何地方的顺序参数都无法检测到这种纠缠。投影纠缠对状态的形式主义是带隙基态波函数参数化的自然框架，它允许人们根据编码波函数的局部张量的虚拟对称性来表征拓扑顺序。以其最一般的形式，这些对称性由作用在虚拟水平上的矩阵乘积运算符表示，这导致了一组代数规则，这些规则表征了具有拓扑量子顺序的状态。此构造概括了\（{\ mathsf {G}} \）的概念-内射性扭曲；相应的矩阵乘积运算符对理论的所有拓扑特征进行编码，并提供圆环上基态流形的完整图片。我们将说明Levin和Wen的字符串网络模型如何适合这种形式主义，并以此为矩阵符号运算符通过字符串网络张量网络的拉动提供F符号五边形方程的特别直观的解释。我们的方法为在弦网之外寻找新的拓扑阶段铺平了道路，并根据缠结哈密顿量和边缘理论阐明了拓扑阶段的描述。