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1. Twisted bulk-boundary correspondence of fragile topology

   https://doi.org/10.1126/science.aaz7650  

   Song, Zhi-Da; Elcoro, Luis; Bernevig, B. Andrei (February 2020, Science)

   null (Ed.)

   A topological insulator reveals its nontrivial bulk through the presence of gapless edge states: This is called the bulk-boundary correspondence. However, the recent discovery of “fragile” topological states with no gapless edges casts doubt on this concept. We propose a generalization of the bulk-boundary correspondence: a transformation under which the gap between the fragile phase and other bands must close. We derive specific twisted boundary conditions (TBCs) that can detect all the two-dimensional eigenvalue fragile phases. We develop the concept of real-space invariants, local good quantum numbers in real space, which fully characterize these phases and determine the number of gap closings under the TBCs. Realizations of the TBCs in metamaterials are proposed, thereby providing a route to their experimental verification. 

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2. Observation of the topological Anderson insulator in disordered atomic wires

   https://doi.org/10.1126/science.aat3406  

   Meier, Eric J.; An, Fangzhao Alex; Dauphin, Alexandre; Maffei, Maria; Massignan, Pietro; Hughes, Taylor L.; Gadway, Bryce (October 2018, Science)

   Topology and disorder have a rich combined influence on quantum transport. To probe their interplay, we synthesized one-dimensional chiral symmetric wires with controllable disorder via spectroscopic Hamiltonian engineering, based on the laser-driven coupling of discrete momentum states of ultracold atoms. Measuring the bulk evolution of a topological indicator after a sudden quench, we observed the topological Anderson insulator phase, in which added disorder drives the band structure of a wire from topologically trivial to nontrivial. In addition, we observed the robustness of topologically nontrivial wires to weak disorder and measured the transition to a trivial phase in the presence of strong disorder. Atomic interactions in this quantum simulation platform may enable realizations of strongly interacting topological fluids. 

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3. Fragility of surface states in topological superfluid 3He

   https://doi.org/10.1038/s41467-021-21831-y  

   Heikkinen, P. J.; Casey, A.; Levitin, L. V.; Rojas, X.; Vorontsov, A.; Sharma, P.; Zhelev, N.; Parpia, J. M.; Saunders, J. (December 2021, Nature Communications)

   null (Ed.)

   Abstract Superfluid 3 He, with unconventional spin-triplet p-wave pairing, provides a model system for topological superconductors, which have attracted significant interest through potential applications in topologically protected quantum computing. In topological insulators and quantum Hall systems, the surface/edge states, arising from bulk-surface correspondence and the momentum space topology of the band structure, are robust. Here we demonstrate that in topological superfluids and superconductors the surface Andreev bound states, which depend on the momentum space topology of the emergent order parameter, are fragile with respect to the details of surface scattering. We confine superfluid 3 He within a cavity of height D comparable to the Cooper pair diameter ξ 0 . We precisely determine the superfluid transition temperature T c and the suppression of the superfluid energy gap, for different scattering conditions tuned in situ, and compare to the predictions of quasiclassical theory. We discover that surface magnetic scattering leads to unexpectedly large suppression of T c , corresponding to an increased density of low energy bound states. 

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4. Band Representations and Topological Quantum Chemistry

   https://doi.org/10.1146/annurev-conmatphys-041720-124134  

   Cano, Jennifer; Bradlyn, Barry (March 2021, Annual Review of Condensed Matter Physics)

   null (Ed.)

   In this article, we provide a pedagogical review of the theory of topological quantum chemistry and topological crystalline insulators. We begin with an overview of the properties of crystal symmetry groups in position and momentum space. Next, we introduce the concept of a band representation, which quantifies the symmetry of topologically trivial band structures. By combining band representations with symmetry constraints on the connectivity of bands in momentum space, we show how topologically nontrivial bands can be cataloged and classified. We present several examples of new topological phases discovered using this paradigm and conclude with an outlook toward future developments. 

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5. Hidden non-collinear spin-order induced topological surface states

   https://doi.org/10.1038/s41467-024-47340-2  

   Huang, Zengle; Yi, Hemian; Kaplan, Daniel; Min, Lujin; Tan, Hengxin; Chan, Ying-Ting; Mao, Zhiqiang; Yan, Binghai; Chang, Cui-Zu; Wu, Weida (April 2024, Nature Communications)

   Abstract Rare-earth monopnictides are a family of materials simultaneously displaying complex magnetism, strong electronic correlation, and topological band structure. The recently discovered emergent arc-like surface states in these materials have been attributed to the multi-wave-vector antiferromagnetic order, yet the direct experimental evidence has been elusive. Here we report observation of non-collinear antiferromagnetic order with multiple modulations using spin-polarized scanning tunneling microscopy. Moreover, we discover a hidden spin-rotation transition of single-to-multiple modulations 2 K below the Néel temperature. The hidden transition coincides with the onset of the surface states splitting observed by our angle-resolved photoemission spectroscopy measurements. Single modulation gives rise to a band inversion with induced topological surface states in a local momentum region while the full Brillouin zone carries trivial topological indices, and multiple modulation further splits the surface bands via non-collinear spin tilting, as revealed by our calculations. The direct evidence of the non-collinear spin order in NdSb not only clarifies the mechanism of the emergent topological surface states, but also opens up a new paradigm of control and manipulation of band topology with magnetism. 

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