标题： 三维有限温度量子拓扑序 显示英文标题

标题： Finite-temperature quantum topological order in three dimensions

摘要： 我们识别出一个三维系统，在足够小但非零的温度下表现出长程纠缠，因此它在有限温度下构成了量子拓扑序。感兴趣的模型称为费米子环码，是通常的三维环码的一种变体，它能产生涌现的费米子点状激发。重要的是，费米子环码具有与费米子激发的空间类似威尔逊环相关的一个异常的2形式对称性。我们认为正是这种对称性赋予低温热态一种新颖的拓扑序和长程纠缠。根据目前对三维拓扑序的分类，我们预计费米子环码的低温热态属于只在非零温度下存在的平衡物质相。我们猜想，具有异常2形式对称性的离散规范理论进一步给出了非零温度下拓扑序的例子。因此，我们的工作为在物理上现实的维度中研究非零温度下的量子拓扑序打开了大门。 显示英文摘要

摘要： We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.