三维格子二聚体模型拓扑性质的神经网络研究
Probing topological properties of 3D lattice dimer model with neural networks
摘要：
机器学习方法正被积极考虑作为描述许多身体物理的新工具。然而，到目前为止，它们的能力仅在以前研究的模型中得到证明，例如伊辛模型。在这里，我们考虑一个简单的问题，证明神经网络可以成功地用于提供统计物理学的新见解。具体来说，我们考虑三维晶格二聚体模型，它由形成晶格的位点和连接相邻位点的键组成，以这样的方式，每个键可以是空的或填充有二聚体，并且在一个位点结束的二聚体的总数固定为1。如果二聚体构型通过一系列局部翻转连接，即一对平行相邻二聚体的同时“旋转”，则它们可以被视为等同的。事实证明，在一个给定的三维晶格上的二聚体配置的整个集合可以被分成不同的拓扑类，这样属于不同类的二聚体配置是不等价的。在本文中，我们通过使用神经网络来识别这些类。更具体地说，我们训练神经网络，以区分二聚体配置从两个已知的拓扑类，之后，我们测试他们对二聚体配置从未知的拓扑类。我们证明了在二分格上的三维格二聚体模型可以用一个整数拓扑不变量（“霍普夫数”）来描述，而在非二分格上的格二聚体模型可以用$Z_2$不变量来描述.因此，我们证明了神经网络可以成功地用于识别凝聚态系统中新的拓扑相，其存在可以稍后通过其他（例如分析）技术进行验证。
英文摘要：
Machine learning methods are being actively considered as a new tool of describing many body physics. However, so far, their capabilities has been only demonstrated in previously studied models, such as e.g. Ising model. Here, we consider a simple problem, demonstrating that neural networks can be successfully used to give new insights in statistical physics. Specifically, we consider 3D lattice dimer model, which consists of sites forming a lattice and bonds connecting the neighboring sites, in such a way that every bond can be either empty or filled with a dimer, and the total number of dimers ending at one site is fixed to be one. Dimer configurations can be viewed as equivalent if they are connected through a series of local flips, i.e. simultaneous 'rotation' of a pair of parallel neighboring dimers. It turns out that the whole set of dimer configurations on a given 3D lattice can be split into distinct topological classes, such that dimer configurations belonging to different classes are not equivalent. In this paper we identify these classes by using neural networks. More specifically, we train the neural networks to distinguish dimer configurations from two known topological classes, and after it, we test them on dimer configurations from unknown topological classes. We demonstrate that 3D lattice dimer model on a bipartite lattice can be described by an integer topological invariant ('Hopf number'), whereas lattice dimer model on a non-bipartite lattice is described by $Z_2$ invariant. Thus, we demonstrate that neural networks can be successfully used to identify new topological phases in condensed matter systems, whose existence can be later verified by other (e.g. analytical) techniques.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
