Lattice construction of exotic invertible topological phases
Lattice construction of exotic invertible topological phases
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DOI：
10.1103/physrevb.105.035153
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发表时间：
2021-06
影响因子：
3.7
通讯作者：
R. Kobayashi
中科院分区：
文献类型：
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作者：
R. Kobayashi
在这篇文章中，我们给出了奇异可逆拓扑相的状态和路径积分定义。奇异相具有时间反转($T$)对称性，并且取决于对被称为Wu结构的时空结构的选择。任何玻色子或费米子拓扑相的分类都不能捕捉到奇异相，从而给出了一类新的可逆拓扑相。当$T$对称性缺陷允许自旋结构时，我们的结构退化为一种装饰磁区壁结构，根据玻色子理论，$T$对称性缺陷用费米子相装饰，该费米子相取决于$T$对称性缺陷的自旋结构。利用我们的路径积分，我们提出了一种奇异相的晶格结构，它在Wu结构的基础上产生了(3+1)d可逆相的分类。这推广了Fidkowski和Kitaev提出的$T对称(1+1)维拓扑超导体的$\mathbb{Z}_8$分类。在定向时空中，这种具有特定选择的Wu结构的(3+1)d可逆相简化为具有拓扑有序状态的玻色子Crane-Yetter TQFT，其边界上有一个Semion。此外，我们还提出了一个$G$-SPT相的子类，它基于由一般时空维中的一对上同调数据标记的Wu结构。这推广了费米子SPT相的Gu-wen子类。
In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian. The exotic phase has time reversal ($T$) symmetry, and depends on a choice of the spacetime structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases, and thus gives a novel class of invertible topological phases. When the $T$ symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with $T$ symmetry defects decorated with a fermionic phase that depends on a spin structure of the $T$ symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the $\mathbb{Z}_8$ classification of the (3+1)d invertible phase based on the Wu structure. This generalizes the $\mathbb{Z}_8$ classification of the $T$-symmetric (1+1)d topological superconductor proposed by Fidkowski and Kitaev. On oriented spacetime, this (3+1)d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of $G$-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic spacetime dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases.