Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice

Bradlyn, Barry; Wang, Zhijun; Cano, Jennifer; Bernevig, B. Andrei

doi:10.1103/physrevb.99.045140

Title: Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice

Journal Article · 25 January 2019 · Physical Review. B

DOI: https://doi.org/10.1103/physrevb.99.045140 · OSTI ID:1612431

Bradlyn, Barry ^[1]; Wang, Zhijun ^[2]; Cano, Jennifer ^[3]; Bernevig, B. Andrei ^[4]

Univ. of Illinois at Urbana-Champaign, IL (United States); Donostia International Physics Center (Spain); DOE/OSTI
Chinese Academy of Sciences (CAS), Beijing (China); Princeton Univ., NJ (United States)
Stony Brook Univ., NY (United States); Flatiron Institute, New York, NY (United States)
Princeton Univ., NJ (United States); Freie Univ. Berlin (Germany); Max Planck Inst. of Microstructure Physics, Halle (Germany)

In this study, 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.

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Research Organization:: Princeton Univ., NJ (United States)

Sponsoring Organization:: USDOE Office of Science (SC); National Science Foundation

Grant/Contract Number:: SC0016239

OSTI ID:: 1612431

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 4 Vol. 99; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

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Title: Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice

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