The bulk-corner correspondence of time-reversal symmetric insulators

Abstract

The topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.

Topological Zero-Dimensional Defect and Flux States in Three-Dimensional Insulators

“…Using first-principles calculations, we have shown that PbTe monolayers [124][125][126] have a nontrivial first-order defect-state response (Appendix F 1), and that the 3D TCI and HOTI SnTe (but not PbTe) [26,31,32,[127][128][129] carries a nontrivial HEND-state dislocation response (Appendix F 2). HEND dislocation states may also be observable in "fragile" topological insulators (TIs) [33,68,96,[103][104][105][106][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147] and obstructed atomic limits [148][149][150][151][152]. Numerous candidate fragile TIs have recently been predicted through high-throughput material searches [128,147], and 3D OAL phases have recently been identified in electrides [153][154][155] and other stoichiometric insulators [156]…”

“…In this section, we will employ k • p theory to demonstrate the presence of HEND bound states on the corners and ends of crystal defects and flux tubes in two classes of HOTIs: inversion-(I-) symmetric axion insulators (AXIs) [30-33, 92, 93, 96, 97, 102-106, 151, 168-171], which are 3D magnetic insulators with anomalous, gapped 2D surfaces and gapless hinges with 1D chiral modes [7,8,10,13,25,33,92,102,[215][216][217][218], and time-reversal-(T -) and I-symmetric HOTIs [30,96,[99][100][101] with anomalous, gapped (T -symmetric) surfaces [25] and gapless hinges with 1D helical modes (see Appendix B 1 c for more details). We will also show that crystal defects, but crucially not π-flux tubes, bind anomalous HEND states in weak stacks of I-symmetric fragile topological insulators (FTIs) [33,68,96,[103][104][105][106][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147] and obstructed atomic limits (OALs) …”

“…We emphasize that M F ν does not distinguish between 3D insulators for which both the k i = 0, π planes have topologically equivalent 2D Hamiltonians and insulators for which k i indexes a pumping cycle between a 2D trivial insulator at k i = 0 and a 2D FTI at k i = π [33,[103][104][105][106]. Furthermore, FTIs can be trivialized by the introduction of appropriately chosen trivial bands [33,68,96,[103][104][105][106][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147]. Therefore, we conclude that the dislocation response of 3D insulators does not distinguish between strong and weak (fragile) topology.…”

“…33 and 96), then we will in this work associate M F ν with fragile topology. However and crucially, because FTIs can be trivialized into OALs through the addition of appropriately chosen trivial bands [33,68,96,[103][104][105][106][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145], and because there also exist simple models of 2D I-symmetric OALs with anomalous corner charges [172], then we note for completeness that a nontrivial M F ν does not necessarily imply that the valence manifold taken as a whole exhibits fragile topology. Like the established index M ν for weak TIs [7][8][9]207], M F ν has three components:…”

“…, 68, 91, 94-96, 137-139, 141-145, 152, 172, 225-230] and fragile[33, 68,96,[103][104][105][106][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147] topology, we will show below that filling anomalies (Refs. 33, 68, 96, and 172 and Appendix D) can be exploited to further extend Eq.…”