We study the entanglement structure of topological orders subject to decoherence on the bipartition boundary. Focusing on the toric codes in $d$ space dimensions for $d=2,3,4$, we explore whether the boundary decoherence may be able to induce a disentangling transition, characterized by the destruction of mixed-state long-range entanglement across the bipartition, measured by topological entanglement negativity. A key insight of our approach is the connection between the negativity spectrum of the decohered mixed states and emergent symmetry-protected topological orders under certain symmetry-preserving perturbation localized on the bipartition boundary. This insight allows us to analytically derive the exact results of entanglement negativity and characterize the universality class of the disentangling transition without using a replica trick.

中文翻译：

---

解开边界退相干引起的拓扑顺序转变


我们学习 $d$ y 在二分边界上受退相干影响的拓扑序的纠缠结构。关注 d 空间维度中的环面代码 $d=2,3,4$ ，我们探讨了边界退相干是否能够引起解缠结转变，其特征是跨二分的混合态长程纠缠的破坏，通过拓扑纠缠负性来测量。我们的方法的一个关键见解是，在二分边界上的某些保对称扰动下，退相干混合态的负谱与涌现的对称性保护的拓扑顺序之间的联系。这种见解使我们能够分析得出纠缠负性的确切结果，并在不使用复制技巧的情况下表征解缠结转变的普遍性类别。