We provide a generalization of the matrix product operator formalism for string-net projected entangled pair states (PEPS) to include nonunitary solutions of the pentagon equation. These states provide the explicit lattice realization of the Galois conjugated counterparts of ($2+1$)-dimensional topological quantum field theories, based on tensor fusion categories. Although the parent Hamiltonians of these renormalization group fixed point states are gapless, these states can still be the topological ground states of a gapped non-Hermitian Hamiltonian. We show by example that the topological sectors of the Yang-Lee theory (the nonunitary counterpart of the Fibonacci fusion category) can be constructed, even in the absence of closure under Hermitian conjugation of the basis elements of the Ocneanu tube algebra. The topological sector construction is demonstrated by applying the concept of strange correlators to the Yang-Lee model, giving rise to a nonunitary version of the classical hard hexagon model in the Yang-Lee universality class and obtaining all generalized twisted boundary conditions on a finite cylinder of the Yang-Lee edge singularity. Finally, we construct the PEPS transfer matrix and show that taking the Hermitian conjugate changes the topological phase for these nonunitary string-net models.

中文翻译：

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Galois共轭张量融合类别和非unit形共形场理论

我们为弦网投影纠缠对状态（PEPS）提供了矩阵乘积算子形式化的概括，以包括五边形方程的非unit解。这些状态提供了（）的Galois共轭对应物的明确晶格实现$2+\mathrm{1个}$）维拓扑量子场论，基于张量融合类别。尽管这些重归一化组不动点状态的父哈密顿量是无间隙的，但这些状态仍然可以是带间隙的非厄米特哈密顿量的拓扑基态。通过示例显示，即使在Ocneanu管代数的基本元素的Hermitian共轭不封闭的情况下，也可以构造Yang-Lee理论的拓扑部分（斐波纳契融合类别的非unit对应）。通过将奇异相关器的概念应用于Yang-Lee模型来演示拓扑部门的构建，在Yang-Lee普遍性类中产生了经典硬六边形模型的非unit形式，并在Yang-Lee边奇异性的有限圆柱上获得了所有广义扭曲边界条件。最后，我们构造了PEPS传递矩阵，并表明采用Hermitian共轭会改变这些非unit一字符串网络模型的拓扑相位。