论文标题
$ \ mathbb {z} _2 $ gauge理论的持久同源性
Persistent Homology of $\mathbb{Z}_2$ Gauge Theories
论文作者
论文摘要
物质的拓扑阶段表现出许多独特的特征，包括可以解释为封闭字符串模式的基态。在本文中，我们考虑了从经典$ \ Mathbb {z} _2 $ gauge理论中检测和区分封闭字符串的问题。我们使用持久同源性的框架来解决这一问题，该框架通过形成几何复合物来计算自旋配置中通用环结构的大小和频率。在有限尺寸的晶格上实现，我们表明，越野河 - 里普斯复合体的第一个贝蒂数量在$ \ mathbb {z} _2 _2 $ gauge理论中在低温下达到了高密度。此外，它在三维理论的有限温度解料转换上显示了一个明显的信号。我们认为，持续的同源性应该能够解释各种系统中发生的突出循环结构，从而使其成为拓扑顺序的理论和实验搜索中有用的工具。
Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings. In this paper, we consider the problem of detecting and distinguishing closed strings in Ising spin configurations sampled from the classical $\mathbb{Z}_2$ gauge theory. We address this using the framework of persistent homology, which computes the size and frequency of general loop structures in spin configurations via the formation of geometric complexes. Implemented numerically on finite-size lattices, we show that the first Betti number of the Vietoris-Rips complexes achieves a high density at low temperatures in the $\mathbb{Z}_2$ gauge theory. In addition, it displays a clear signal at the finite-temperature deconfinement transition of the three-dimensional theory. We argue that persistent homology should be capable of interpreting prominent loop structures that occur in a variety of systems, making it an useful tool in theoretical and experimental searches for topological order.
