Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice
Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice
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DOI：
10.1103/physrevb.99.045140
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发表时间：
2019-01-25
影响因子：
3.7
通讯作者：
Bernevig, B. Andrei
中科院分区：
文献类型：
--
作者：
Bradlyn, Barry;Wang, Zhijun;Bernevig, B. Andrei
在这项工作中，我们研究了具有不连接的初等带表示的三角形晶格中可能出现的拓扑相。我们表明，虽然这些相可能是“脆弱的”相对于额外的带的增加，他们的拓扑性质是在某些非平凡的完整（威尔逊环）在非平凡带的空间中表现出来的。我们引入了脆弱拓扑的特征值指标，并证明了该指标的非平凡值如何表现为六边形威尔逊环的缠绕；即使在没有时间反转或六重旋转对称的情况下，这也是正确的。此外，当时间反转和双重旋转对称存在时，我们直接证明了在更传统的威尔逊环中存在一个受保护的非平凡绕组。至关重要的是，我们强调，如果不关闭非平凡带的间隙，这些威尔逊环就不能改变。通过研究脆性带的纠缠谱，我们评论了Bradlyn等人的脆性拓扑与“受阻原子极限”之间的关系[Nature (London) 547, 298(2017)]。我们总结了一些超越k理论分类的拓扑物质的观点。
In this work, we examine the topological phases that can arise in triangular lattices with disconnected elementary band representations. We show that, although these phases may be "fragile" with respect to the addition of extra bands, their topological properties are manifest in certain nontrivial holonomies (Wilson loops) in the space of nontrivial bands. We introduce an eigenvalue index for fragile topology, and we show how a nontrivial value of this index manifests as the winding of a hexagonal Wilson loop; this remains true even in the absence of time-reversal or sixfold rotational symmetry. Additionally, when time-reversal and twofold rotational symmetry are present, we show directly that there is a protected nontrivial winding in more conventional Wilson loops. Crucially, we emphasize that these Wilson loops cannot change without closing a gap to the nontrivial bands. By studying the entanglement spectrum for the fragile bands, we comment on the relationship between fragile topology and the "obstructed atomic limit" of Bradlyn et al. [Nature (London) 547, 298 (2017)]. We conclude with some perspectives on topological matter beyond the K-theory classification.