标题： 冷原子超流体中的量子双结构 显示英文标题

标题： Quantum double structure in cold atom superfluids

摘要： 拓扑量子计算的理论建立在两个重要的模型类别之上。 一个是基于非阿贝尔陈-西蒙斯理论，它产生所谓的$\rm{SU}(2)_k$任意子模型，这些模型经常出现在带电量子流体的背景下。 另一个的物理特性由无陈-西蒙斯项的对称性破缺杨-米尔斯理论所描述，并导致所谓的量子双代数模型。 已经投入了大量资源来寻找$\rm{SU}(2)_k$任意子准粒子；特别是所谓的伊辛任意子（$k=2$），其中马约拉纳零模被认为是一种表现形式。 与$\rm{SU}(2)_k$模型相比，尽管量子双代数在纠错理论中起着关键作用，但它们在实验中却很少受到关注。 除了拓扑纠错码之外，量子双代数的出现仅限于数学物理领域，因此似乎对于实验可实现系统的研究几乎没有相关性。 然而，最近的研究表明，量子双代数任意子可能存在于旋量玻色-爱因斯坦凝聚体中。 鉴于此，本文的核心目的是通过构造各种基态对称群的量子双代数并讨论其实验可行性，来提供一个关于量子双代数结构的自洽阐述。 我们还从量子双代数 Clebsch-Gordan 系数导出了一个解析方程，从而可以计算出相关的辫矩阵。 最后，在这一背景下揭示并阐明了粒子-涡旋对偶性的存在。 显示英文摘要

摘要： The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern-Simons theory, which yields the so-called $\rm{SU}(2)_k$ anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang-Mills theory in the absence of a Chern-Simons term, and results in the so-called quantum double models. Extensive resources have been invested into the search for $\rm{SU}(2)_k$ anyon quasi-particles; in particular the so-called Ising anyons ($k=2$) of which Majorana zero modes are believed to be an incarnation. In contrast to the $\rm{SU}(2)_k$ models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose-Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch-Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.