Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice
Abstract
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.
- Authors:
-
- Univ. of Illinois at Urbana-Champaign, IL (United States); Donostia International Physics Center (Spain)
- 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)
- Publication Date:
- Research Org.:
- Princeton Univ., NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); National Science Foundation (NSF)
- OSTI Identifier:
- 1612431
- Alternate Identifier(s):
- OSTI ID: 1492107
- Grant/Contract Number:
- SC0016239
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review. B
- Additional Journal Information:
- Journal Volume: 99; Journal Issue: 4; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics; Topological phases of matter; Topological materials Band structure methods; Symmetries in condensed matter
Citation Formats
Bradlyn, Barry, Wang, Zhijun, Cano, Jennifer, and Bernevig, B. Andrei. Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice. United States: N. p., 2019.
Web. doi:10.1103/physrevb.99.045140.
Bradlyn, Barry, Wang, Zhijun, Cano, Jennifer, & Bernevig, B. Andrei. Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice. United States. https://doi.org/10.1103/physrevb.99.045140
Bradlyn, Barry, Wang, Zhijun, Cano, Jennifer, and Bernevig, B. Andrei. Fri .
"Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice". United States. https://doi.org/10.1103/physrevb.99.045140. https://www.osti.gov/servlets/purl/1612431.
@article{osti_1612431,
title = {Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice},
author = {Bradlyn, Barry and Wang, Zhijun and Cano, Jennifer and Bernevig, B. Andrei},
abstractNote = {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.},
doi = {10.1103/physrevb.99.045140},
journal = {Physical Review. B},
number = 4,
volume = 99,
place = {United States},
year = {Fri Jan 25 00:00:00 EST 2019},
month = {Fri Jan 25 00:00:00 EST 2019}
}
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