[论文解读] Topological order: from long-range entangled quantum matter to an unification of light and electrons

Wen, Xiao-Gang|arXiv (Cornell University)|Oct 3, 2012

Quantum many-body systems被引用 119

一句话总结

本文提出，由长程量子纠缠产生的拓扑序可统一光与电子：规范场（光子）作为纠缠自旋网络的涨落而涌现，而费米子（电子）则作为此类网络中的缺陷出现。其核心贡献是一个统一的场论框架，其中通过弦网凝聚描述的拓扑序，为规范相互作用和费米统计提供了微观起源。

ABSTRACT

In primary school, we were told that there are four states of matter: solid, liquid, gas, and plasma. In college, we learned that there are much more than four states of matter. For example, there are ferromagnetic states as revealed by the phenomenon of magnetization and superfluid states as defined by the phenomenon of zero-viscosity. The various phases in our colorful world are so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. In this paper, we will review the progress in last 20 -- 30 years, during which we discovered that there are even more interesting phases that are beyond Landau symmetry breaking theory. We discuss new "topological" phenomena, such as topological degeneracy, that reveal the existence of those new phases - topologically ordered phases. Just like zero-viscosity defines the superfluid order, the new "topological" phenomena define the topological order at macroscopic level. As a new type of order, topological order requires a new mathematical frame work, such as fusion category and group cohomology, to describe it. More recently, we find that, at microscopical level, topological order is due to long-range quantum entanglements, just like fermion superfluid is due to fermion-pair condensation. Long-range quantum entanglements lead to many amazing emergent phenomena, such as fractional quantum numbers, fractional/non-Abelian statistics, and perfect conducting boundary channels. Long-range quantum entanglements can even provide a unified origin of light and electrons (or more generally, gauge interactions and Fermi statistics): light waves (gauge fields) are fluctuations of long-range entanglements, and electrons (fermions) are defects of long-range entanglements.

研究动机与目标

• 将拓扑序确立为超越朗道对称性自发破缺理论的新型量子物相分类范式。
• 证明长程量子纠缠是拓扑序的微观起源。
• 通过单一基本机制——弦网凝聚，统一规范玻色子（光）与费米子（电子）的涌现。
• 提供一个场论与数学框架——使用融合范畴与群上同调——以描述拓扑序。
• 表明分数统计、任意子与拓扑简并性是拓扑序态中长程纠缠的直接结果。

提出的方法

• 通过宏观特征定义拓扑序：拓扑简并的基态简并度与简并基态的非交换几何相位。
• 采用弦网凝聚作为微观机制：一种凝聚为拓扑序相的定向或非定向弦的量子液体。
• 应用融合范畴与群上同调等数学工具，对拓扑序及其涌现规范理论进行分类。
• 推导出规范场作为长程纠缠弦网的集体涨落，规范对称性源于局域弦动力学。
• 将费米子识别为弦网中的涡旋或缺陷，从而导致分数统计与非阿贝尔统计。
• 提出可证伪的预测：在某些自旋液体中存在非阿贝尔统计，可通过拓扑量子计算进行检验。

实验结果

研究问题

• RQ1 能否在朗道对称性自发破缺范式之外系统地分类拓扑序？
• RQ2 长程量子纠缠如何导致规范场与费米统计的涌现？
• RQ3 在量子自旋液体中，分数统计与拓扑简并性的微观起源是什么？
• RQ4 能否从单一纠缠量子态中涌现出统一的场论，从而同时产生规范玻色子与费米子？
• RQ5 需要哪些数学结构（如融合范畴、群上同调）来对拓扑序进行分类与描述？

主要发现

• 拓扑序由拓扑简并的基态简并度与非阿贝尔几何相位定义，对局域微扰具有鲁棒性。
• 长程量子纠缠是拓扑序的根本起源，类似于超导性中的库珀对。
• 弦网凝聚导致涌现的U(1)规范理论与费米统计：光子作为集体模式出现，电子作为涡旋出现。
• 在某些弦网态中出现非阿贝尔统计，具备实现拓扑量子计算的可能性。
• 该理论预测某些自旋液体——如手征自旋液体——会存在具有非阿贝尔编织统计的任意子。
• 该框架提出了可证伪的预测：若在量子自旋液体中观测到非阿贝尔统计，将证实弦网机制。