We study quantized non-local order parameters, constructed by using partial time-reversal and partial reflection, for fermionic topological phases of matter in one spatial dimension protected by an orientation reversing symmetry, using topological quantum field theories (TQFTs). By formulating the order parameters in the Hilbert space of state sum TQFT, we establish the connection between the quantized non-local order parameters and the underlying field theory, clarifying the nature of the order parameters as topological invariants. We also formulate several entanglement measures including the entanglement negativity on state sum spin TQFT, and describe the exact correspondence of the entanglement measures to path integrals on a closed surface equipped with a specific spin structure.

中文翻译：

---

费米子拓扑场论的非局域序参数和量子纠缠

我们使用拓扑量子场论 (TQFT) 研究了通过使用部分时间反转和部分反射构造的量化非局部有序参数，用于受方向反转对称保护的一个空间维度中物质的费米子拓扑相。通过在状态和 TQFT 的希尔伯特空间中制定阶次参数，我们建立了量化的非局部阶次参数与基础场论之间的联系，阐明了阶次参数作为拓扑不变量的性质。我们还制定了几种纠缠测度，包括状态和自旋 TQFT 上的纠缠负性，并描述了纠缠测度与配备特定自旋结构的封闭表面上的路径积分的精确对应关系。