无异常拓扑顺序与边界相的独特关系

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"这篇学术论文探讨了无异常的n+1维拓扑顺序与n维间隙边界相之间的关系，强调了这种关系的唯一性，并提出了'体-边界'对应的概念。作者通过研究揭示了n+1维拓扑序的'体相'是由n维边界相的'中心'决定的理论，进一步深化了对拓扑序的理解。" 在物理学领域，拓扑序是一种重要的概念，它描述了物质的非平凡量子态，这些状态在拓扑上是不变的，即不受平滑连续变化的影响。无异常的n+1维拓扑顺序是指在该维度下没有违反物理定律的拓扑结构。而"间隙边界相"则指的是系统中存在一个能量间隙的边界状态，这些状态通常与体内的拓扑性质密切相关。文章的核心在于阐述了给定的n维间隙边界相与唯一的n+1维体无异常拓扑顺序之间的对应关系。这种唯一性意味着，对于特定的n维边界相，存在着一种特定的n+1维拓扑序与其匹配。这种关系可以被理解为"体-边界对应"，类似于几何中的体积与表面的关系，其中"体"代表n+1维的拓扑序，"边界"则指n维的间隙边界相。 "中心"这一概念在这里具有关键意义，它可能指的是边界相中的某种对称性保护的、具有代表性的性质或子系统。当n+1维的拓扑序被其n维边界相的"中心"所确定时，意味着体内的拓扑特性可以从边界的状态和性质中推导出来，这是一种非常深刻的理论联系。文章的贡献不仅在于提出了体-边界的对应关系，还在于为理解和分类拓扑序提供了新的视角和工具。这对于研究高维量子系统的性质、设计新型量子材料以及发展量子计算等领域具有深远影响。此外，由于文章是开放获取的，科研人员可以自由访问和引用，有助于促进相关领域的国际交流和合作。总结来说，这篇由Liang Kong、Xiao-Gang Wen和Hao Zheng共同撰写的论文深入探讨了拓扑序的边界与体的相互作用，揭示了两者间的独特对应关系，为理解拓扑量子态提供了新的理论框架。这项工作对于推动量子物理、凝聚态物理以及相关交叉学科的发展具有重要意义。

64 L. Kong et al. / Nuclear Physics B 922 (2017) 62–76

we show that there is a canonical morphism ρ : P

))



→ C

, where



denotes

the stacking of two topological orders of the same dimension.

3. We show that the pair (P

)), ρ), satisﬁes the universal property of the center of an

algebra in mathematics (see Theorem 5.1). This implies that bulk

= center.

This result is independent of how we describe the boundary/bulk phase mathematically. It is a

non-trivial result that leads to concrete physical predictions (see Remark 5.4).

We denote n+1D topological orders by A

n+1

, B

n+1

, C

n+1

, D

n+1

, etc. If the spacetime di-

mension n+1is clear from the context, we abbreviate C

n+1

as C. We denote the trivial n+1D

topological order by 1

n+1

. In physics, the trivial topological order 1

n+1

corresponds to the equiv-

alence class of the product states [10].

This paper is written for wo

rking condensed matter physicists, especially for those working

in the ﬁeld of topological phases of matters. In order to convey the simply idea, we try to keep

the categorical language to the minimum. In particular, we collect all mathematically technical

parts in Remarks and Examples. The main te

xt should be readable to those who do not have any

extra background in category theory beyond those basics that have already been widely used in

condensed matter physics (see [24]).

Acknowledgments: X-G.W is supported by NSF Grant No. DMR-1506475 and NSFC

11274192. HZ is supported by NSFC under Grant No. 11131008.

2. Basics of topological orders

In this section, we recall some basic facts about topological orders, state our key assumptions

and set our notations.

A potentially anomalous nD topological order C

can always be realized as a gapped boundary

of an anomaly-free n+1D topological order E

n+1

[28]. In physics, it is generally believed that the

bulk anomaly-free topological order E

n+1

is uniquely determined by its gapped boundary phase

. One way to see this is to note that there is no preferred length scale. In order to deﬁne the

boundary phase C

, we need deﬁne the equivalence class of quantum states by allowing proper

deformation of the state in an arbitrary large neighborhood of the boundary (without closing the

gap). As a consequence, E

n+1

should be unique (see [28] for details). This uniqueness is the key

assumption of this paper. We highlight it here.

Unique-bulk h

ypothesis: for any given nD potentially anomalous topological order C

, there

is a unique anomaly-free n+1D topological order, denoted by Z

), such that C

can be

realized as a gapped boundary of Z

We denote E

n+1

by Z

) and refer to it as the bulk of C

. It is clear that Z

) = 1

n+1

. In this

work, we often use the following picture:

to illustrate the geometric relation between the boundary phase C

and the bulk phase Z

More precisely, the n+1D bulk

phase Z

) (deﬁned on an open n-disk as the space manifold)

is depicted by an open interval and the nD boundary phase C

(on an open n−1-disk) is depicted

by one of the end point of the open interval.

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