Second-order topological insulators (SOTIs) are the topological phases of matter in d dimensions that manifest (d-2)-dimensional localized modes at the intersection of the edges. We show that SOTIs can be designed via stacked Chern insulators with opposite chiralities connected by interlayer coupling. To characterize the bulk-corner correspondence, we establish a Jacobian-transformed nested Wilson loop method and an edge theory that are applicable to a wider class of higher-order topological systems. The corresponding topological invariant admits a filling anomaly of the corner modes with fractional charges. The system manifests a fragile topological phase characterized by the absence of a Wannier gap in the Wilson loop spectrum. Furthermore, we argue that the proposed approach can be generalized to multilayers. Our work offers perspectives for exploring and understanding higher-order topological phenomena.
We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadra…
By means of the first-principles calculations and magnetic topological quantum chemistry, we demonstrate that the low energy physics in the checkerboard antiferromagnetic (AFM) monolayer FeSe, very close to an AFM topological insulator that hosts robust edge states, can be well captured by a double-degenerate fragile topologically flat band just below the Fermi level. The Wilson loop calculations identify that such fragile topology is protected by the $S_{4z}$ symmetry, which gives rise to an AFM higher-order topological insulator that support the bound state with fractional charge $e/2$ at th…
Topological insulator (TI) nanoribbons with proximity-induced superconductivity are a promising platform for Majorana bound states (MBSs). In this work, we consider a detailed modeling approach for a TI nanoribbon in contact with a superconductor via its top surface, which induces a superconducting gap in its surface-state spectrum. The system displays a rich phase diagram with different numbers of end-localized MBSs as a function of chemical potential and magnetic flux piercing the cross section of the ribbon. These MBSs can be robust or fragile upon consideration of electrostatic disorder. W…
Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop
本文研究了受反转对称性保护的易碎拓扑绝缘体（TI）的体积和边界性质，主要集中在Altland-Zirnbauer分类的A类。首先，我们提出了一种有效的方法来诊断脆弱的带拓扑结构，利用对称性数据在动量空间。使用这种方法，我们表明，在所有可能的宇称配置的反转对称绝缘体，至少有17%的人有脆弱的拓扑结构在2D中，而脆弱的TI小于3%，在3D中。其次，我们研究了由反转对称性保护的脆弱TI的体边界对应。特别地，我们将通常为$0<k\le d$定义的$d$维（$d$ D）$k$阶TI的概念推广到具有$k>d$的情形，并证明了它们都具有脆弱拓扑.根据狄拉克哈密顿量，$d$ D $k$阶TI具有$(k-1)$边界质量项。我们证明了具有填充异常的最小脆弱TI可以被认为是$d$ D $(d+1)$阶TI，并且所有其他具有$k>(d+1)$的$d$ D $k$阶TI可以通过堆叠$d$ D $(d+1)$阶TI来构造。虽然$d$ D $(d+1)$阶TI没有隙内态，但边界质量项沿边界携带奇数缠绕数，这在边界质量项突变的位置处在边界上感应出局域电荷。在有$k>(d+1)$的情况下，我们证明了有边界系统的净宇称可以区分拓扑绝缘体和平凡绝缘体。同时，通过研究（嵌套）威尔逊圈谱，确定了解决Wannier阻塞的最小谱带数，这与我们的脆弱拓扑诊断方法的预测是一致的.最后，我们证明了通过绝热泵浦$d$ D $k$阶TI…
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
本文提出了一种系统地表征具有时间反转对称性的晶体拓扑绝缘相的一般方法。特别地，以二维自旋六角晶格作为原理的证明，我们研究了威尔逊圈谱在布里渊区中的切口上的缠绕，这些切口由潜在的晶格对称性决定。我们的方法在阐明和量化最近提出的“拓扑量子化学”（TQC）概念方面有着突出的用途。也就是说，我们证明了带隙对基本能带表示（EBR）的分裂必然导致拓扑相。为此，我们首先表明，除了Fu-Kane-Mele $\mathbb{Z}_2$分类外，还有$C_2\mathcal{T}$ -对称性保护的$\mathbb{Z}$分类，它被其他晶体对称性所阻碍，即~禁止琐碎的阶段。这说明了分裂式EBRs \textit{that are independent of the parameterization of the flow of Wilson loops}的所有非平凡威尔逊圈绕组。~然后，我们证明了分裂EBRs的威尔逊圈缠绕在嵌入高维能带空间时可以展开，而与其他能带保持带隙分离的两个能带子空间保持其威尔逊圈缠绕，从而揭示了分裂EBRs至少是“稳定平凡的”，即在非稳定（少带）极限中必然是非平凡的，但在稳定（多带）极限中可能是平凡的。这阐明了最近出现的\textit{fragile}拓扑的本质。~然后，我们认为，在多带极限的稳定威尔逊圈绕组是唯一的Fu-Kane-Mele $\mathbb{Z}_2$不变量…
Symmetries play a crucial role in the classification of topological phases of matter. Although recent studies have established a powerful framework to search for and classify topological phases based on symmetry indicators, there exists a large class of fragile topology beyond the description. The Euler class characterizing the topology of two-dimensional real wave functions is an archetypal fragile topology underlying some important properties, such as non-Abelian braiding of crossing nodes and higher-order topology. However, as a minimum model of fragile topology, the two-dimensional topolog…
We consider the stability of fragile topological bands protected by space-time inversion symmetry in the presence of strong electron-electron interactions. At the single-particle level, the topological nature of the bands prevents the opening of a gap between them. In contrast, we show that when the fragile bands are half filled, interactions can open a gap in the many-body spectrum without breaking any symmetry or mixing degrees of freedom from remote bands. Furthermore, the resulting ground state is not topologically ordered. Thus, a fragile topological band structure does not present an obs…
Excitonic insulators (EI) arise from the formation of bound electron-hole pairs (excitons) in semiconductors and provide a solid-state platform for quantum many-boson physics. Strong exciton-exciton repulsion is expected to stabilize condensed superfluid and crystalline phases by suppressing both density and phase fluctuations. Although spectroscopic signatures of EIs have been reported, conclusive evidence for strongly correlated EI states has remained elusive. Here, we demonstrate a strongly correlated spatially indirect two-dimensional (2D) EI ground state formed in transition metal dichalc…
Fragile topology for six-fold rotation symmetry indicated by the concentric Wilson loop spectrum
Institute of Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）；Institute for Theoretical Physics Amsterdam, University of Amsterdam（阿姆斯特丹大学理论物理研究所）
Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially-localized Wannier representation, each Wannier function cannot be completely localized to a single primitive unit cell in the bulk. Such multicellular topology is shown to be neither stable, nor fragile, but delicate, i.e., the topology can be nullified by adding trivial bands to eith…
Ocean University of China（中国海洋大学）；Qingdao Key Laboratory of Advanced Optoelectronics（青岛先进光电学重点实验室）；Engineering Research Center of Advanced Marine Physical Instruments and Equipment of MOE（教育部海洋高端物理仪器和设备工程研究中心）
Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle
The interband effect is an important concept both in traditional and modern solid-state physics. In this paper, we present a theory of the nontrivial interband effect, which cannot be removed without breaking given rules. We define the general nontrivial interband effect by regarding a property of the set of the total bands of a tight-binding Hamiltonian as the triviality. As examples of the source of the nontrivial interband effect, we consider several topological concepts: stable topological insulator, symmetry-based indicator, fragile topological insulator, and multipole/higher-order topolo…
In condensed matter physics, symmetry profoundly governs the fundamentals of topological matter. The emergence of new topological phase is typically linked to the enrichment of symmetries. Different parity-time symmetry relations distinguish between spinless and spinful physical systems. In spinless systems, creating pseudo-spins can realize fragile topological phase but not break the time-reversal symmetry. Therefore, growing attentions were recently focused on the strong topological phase in spinless systems. Here we break the framework of crystallographic symmetry groups by utilizing the pr…
In this work we discuss particular effects that take place in systems of quantum emitters coupled to two-dimensional bosonic topological insulators. For a single emitter coupled to the Haldane model, we find a "fragile" quasibound state that makes the emitter dynamics very sensitive to the model's parameters, and gives rise to effective long-range interactions that break time-reversal symmetry. We then discuss one-dimensional arrangements of emitters, emitter line defects, and how the topology of the bath affects the effective polariton models that appear in the weak-coupling regime when the e…
