论文标题
2+1d $ \ mathbb {z} _n $拓扑顺序通过拓扑芯旋转的边界相变
The boundary phase transitions of the 2+1D $\mathbb{Z}_N$ topological order via topological Wick rotation
论文作者
论文摘要
在这项工作中，我们表明，在$ \ mathbb {z} _n $拓扑顺序的两个间隙边界之间的1D自偶联边界相变的临界点可以用称为丰富融合类别的数学结构来描述。边界相变的临界点可以看作是$ \ mathbb {z} _n $拓扑顺序的可靠的非手续无间隙边界。 Kong and Zheng（Arxiv：1905.04924 and arxiv：1912.01760）开发的2D拓扑顺序的无间隙界限的数学理论，告诉我们所有宏观的可观察物都告诉我们，所有宏观的边界上都可以通过这种富有融合的融合类别来获得这种旋转的旋转，以旋转的方式旋转，旋转的旋转是奇异的，又可以旋转智慧。获取丰富的融合类别，描述了$ \ mathbf {e} $ - 凝结边界和$ \ mathbf {m} $之间的关键点 - $ \ mathbb {z}表明边界的分类对称性是由大容量中的拓扑缺陷确定的，这表明全息原理是间接的，这是一个具体的例子，即2+1D拓扑阶的无间隙边界的数学理论是研究一般相位过渡的强大工具。
In this work, we show that a critical point of a 1d self-dual boundary phase transition between two gapped boundaries of the $\mathbb{Z}_N$ topological order can be described by a mathematical structure called an enriched fusion category. The critical point of a boundary phase transition can be viewed as a gappable non-chiral gapless boundary of the $\mathbb{Z}_N$ topological order. A mathematical theory of the gapless boundaries of 2d topological orders developed by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760) tells us that all macroscopic observables on the gapless boundary form an enriched unitary fusion category, which can be obtained by a holographic principle called the ``topological Wick rotation." Using this method, we obtain the enriched fusion category that describes a critical point of the phase transition between the $\mathbf{e}$-condensed boundary and the $\mathbf{m}$-condensed boundary of the $\mathbb{Z}_N$ topological order. To verify this idea, we also construct a lattice model to realize the critical point and recover the mathematical data of this enriched fusion category. The construction further shows that the categorical symmetry of the boundary is determined by the topological defects in the bulk, which indicates the holographic principle indirectly. This work shows, as a concrete example, that the mathematical theory of the gapless boundaries of 2+1D topological orders is a powerful tool to study general phase transitions.
