论文标题

2+1D费米子拓扑订单的边界

The boundaries of 2+1D fermionic topological orders

论文作者

Chen, Chang-Han, Wen, Xiao-Gang

论文摘要

$ 2+1 $ d的玻感拓扑订单可以以编码拓扑激发统计的$ s,t $矩阵来表征。特别是,$ s,t $矩阵可用于系统地获得骨拓扑订单的差距。但是,这种方法并不天真地适用于费米子拓扑秩序(FTOS)。在这项工作中,我们提出了一种系统的方法,以获取$ 2+1 $ d Abelian FTO的间隙边界。主要诀窍是构造玻感扩展,其中将费米子激发“凝结”以形成相关的FTO。在这里,我们选择父授予拓扑命令作为$ \ mathbb {z} _2 $拓扑顺序,它确实具有fermionic激发。这样的结构使我们能够在Abelian FTO(用Odd $ K $ -Matrix $ K_F $)和“ Fermion-” Condensed $ \ Mathbb {Z} _2 $拓扑订单(甚至由$ K $ -MATRIX $ K_B $描述)之间找到明确的对应关系。这提供了一种系统的算法,以获得模块化协变边界分区函数以及Abelian FTOS的边界拓扑激发。例如,当$ν= 1- \ frac {1} {m} $ lughlin的状态完全具有一种类型的边界,当$ m $是一个正方形时,其边界激发形成$ \ mathbb {z} _ {2} _ {2} \ times \ times \ times \ times \ mathbb {z} _} _ {\ s {\ sqrt {\ sqrt} $我们的方法很容易被推广,以获得非亚伯式福音拓扑秩序的间隙和无间隙的边界。

$2+1$D bosonic topological orders can be characterized by the $S,T$ matrices that encode the statistics of topological excitations. In particular, the $S,T$ matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of $2+1$D abelian FTOs. The main trick is to construct a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the parent bosonic topological order to be the $\mathbb{Z}_2$ topological order, which indeed has a fermionic excitation. Such a construction allows us to find an explicit correspondence between abelian FTOs (described by odd $K$-matrix $K_F$) and the "fermion-" condensed $\mathbb{Z}_2$ topological orders (described by even $K$-matrix $K_B$). This provides a systematic algorithm to obtain the modular covariant boundary partition functions as well as the boundary topological excitations of abelian FTOs. For example, the $ν=1-\frac{1}{m}$ Laughlin's states have exactly one type of gapped boundary when $m$ is a square, whose boundary excitations form a $\mathbb{Z}_{2}\times\mathbb{Z}_{\sqrt{m}}$ fusion ring. Our approach can be easily generalized to obtain gapped and gapless boundaries of non-abelian fermionic topological orders.

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