TY - JOUR
T1 - Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices
T2 - An example on the triangular lattice
AU - Bradlyn, Barry
AU - Wang, Zhijun
AU - Cano, Jennifer
AU - Bernevig, B. Andrei
N1 - Publisher Copyright:
© 2019 American Physical Society.
PY - 2019/1/25
Y1 - 2019/1/25
N2 - In this work, 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)NATUAS0028-083610.1038/nature23268]. We conclude with some perspectives on topological matter beyond the K-theory classification.
AB - In this work, 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)NATUAS0028-083610.1038/nature23268]. We conclude with some perspectives on topological matter beyond the K-theory classification.
UR - http://www.scopus.com/inward/record.url?scp=85060867668&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85060867668&partnerID=8YFLogxK
U2 - 10.1103/PhysRevB.99.045140
DO - 10.1103/PhysRevB.99.045140
M3 - Article
AN - SCOPUS:85060867668
SN - 2469-9950
VL - 99
JO - Physical Review B
JF - Physical Review B
IS - 4
M1 - 045140
ER -