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Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
Max Planck Inst Phys Komplexer Syst, D-01187 Dresden, Germany;Harvard Univ, Dept Phys, Cambridge, MA 02138 USA.
2019 (English)In: Physical Review B, ISSN 2469-9950, E-ISSN 2469-9969, Vol. 100, no 19, article id 195135Article in journal (Refereed) Published
Abstract [en]

We present a general methodology toward the systematic characterization of crystalline topological insulating phases with time-reversal symmetry. In particular, taking the two-dimensional spinful hexagonal lattice as a proof of principle, we study windings of Wilson loop spectra over cuts in the Brillouin zone that are dictated by the underlying lattice symmetries. Our approach finds a prominent use in elucidating and quantifying the recently proposed "topological quantum chemistry" concept. Namely, we prove that the split of an elementary band representation (EBR) by a band gap must lead to a topological phase. For this we first show that in addition to the Fu-Kane-Mele Z(2) classification, there is C2T-symmetry-protected Z classification of two-band subspaces that is obstructed by the other crystalline symmetries, i.e., forbidding the trivial phase. This accounts for all nontrivial Wilson loop windings of split EBRs that are independent of the parametrization of the flow of Wilson loops. Then, by systematically embedding all combinatorial four-band phases into six-band phases, we find a refined topological feature of split EBRs. Namely, we show that while Wilson loop winding of split EBRs can unwind when embedded in higher-dimensional band space, two-band subspaces that remain separated by a band gap from the other bands conserve their Wilson loop winding, hence revealing that split EBRs are at least "stably trivial," i.e., necessarily nontrivial in the nonstable (few-band) limit but possibly trivial in the stable (many-band) limit. This clarifies the nature of fragile topology that has appeared very recently. We then argue that in the many-band limit, the stable Wilson loop winding is only determined by the Fu-Kane-Mele Z(2) invariant implying that further stable topological phases must belong to the class of higher-order topological insulators.

Place, publisher, year, edition, pages
AMER PHYSICAL SOC , 2019. Vol. 100, no 19, article id 195135
National Category
Condensed Matter Physics
Identifiers
URN: urn:nbn:se:uu:diva-398430DOI: 10.1103/PhysRevB.100.195135ISI: 000498057300001OAI: oai:DiVA.org:uu-398430DiVA, id: diva2:1375929
Funder
Swedish Research Council, 621-2014-3721Knut and Alice Wallenberg FoundationAvailable from: 2019-12-06 Created: 2019-12-06 Last updated: 2019-12-06Bibliographically approved

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Bouhon, AdrienBlack-Schaffer, Annica M.

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