This is an introductory review of the physics of topological quantum matter with cold atoms. Topological quantum phases, originally discovered and investigated in condensed matter physics, have recently been explored in a range of different systems, which produced both fascinating physics findings and exciting opportunities for applications. Among the physical systems that have been considered to realize and probe these intriguing phases, ultracold atoms become promising platforms due to their high flexibility and controllability. Quantum simulation of topological phases with cold atomic gases is a rapidly evolving field, and recent theoretical and experimental developments reveal that some toy models originally proposed in condensed matter physics have been realized with this artificial quantum system. The purpose of this article is to introduce these developments. The article begins with a tutorial review of topological invariants and the methods to control parameters in the Hamiltonians of neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail, especially several celebrated models, such as the Su–Schrieffer–Heeger model, the Hofstadter–Harper model, the Haldane model and the Kane–Mele model. The theoretical proposals and experimental implementations of these models are discussed. Notably, many of these models cannot be directly realized in conventional solid-state experiments. The newly developed methods for probing the intrinsic properties of the topological phases in cold-atom systems are also reviewed. Finally, some topological phases with cold atoms in the continuum and in the presence of interactions are discussed, and an outlook on future work is given.

中文翻译：

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具有冷原子的拓扑量子物质

这是对冷原子拓扑量子物质物理学的介绍性评论。拓扑量子相最初是在凝聚态物理中发现和研究的，最近在一系列不同的系统中进行了探索，产生了引人入胜的物理发现和令人兴奋的应用机会。在被认为可以实现和探测这些有趣相的物理系统中，超冷原子因其高度的灵活性和可控性而成为有前途的平台。具有冷原子气体的拓扑相的量子模拟是一个快速发展的领域，最近的理论和实验发展表明，最初在凝聚态物理中提出的一些玩具模型已经通过这种人工量子系统实现。本文的目的是介绍这些发展。本文首先对拓扑不变量和控制中性原子哈密顿量参数的方法进行了教程回顾。接下来，详细介绍了光学晶格中的拓扑量子相，特别是一些著名的模型，如 Su-Schrieffer-Heeger 模型、Hofstadter-Harper 模型、Haldane 模型和 Kane-Mele 模型。讨论了这些模型的理论建议和实验实施。值得注意的是，许多这些模型不能在传统的固态实验中直接实现。还回顾了新开发的用于探测冷原子系统中拓扑相的内在特性的方法。最后，