Fragile Topology and Wannier Obstructions

Area of Science:

• Condensed Matter Physics
• Materials Science
• Quantum Chemistry

Background:

• Topological phases, like Chern insulators, are characterized by stable topological indices.
• These topological properties obstruct the formation of Wannier functions, which are localized electronic states.
• Wannier functions are crucial for modeling real materials and understanding electronic properties.

Purpose of the Study:

• To investigate whether obstructions to Wannier representation necessarily imply stable band topology.
• To explore the existence and implications of alternative topological classifications beyond stable topology.
• To provide a physical model and counterexample for existing theories on band representations.

Main Methods:

• Theoretical analysis of band structures and topological invariants.
• Construction of a physical model exhibiting fragile topology on a honeycomb lattice.
• Investigation of Wannier function realizability for different band structures.

Main Results:

• Identified a new class of bands termed "fragile topology."
• Demonstrated that fragile topology obstructs Wannier representation unless trivial bands are added.
• Constructed a honeycomb lattice model where half the split bands are trivial and half exhibit fragile topology.
• Showcased that splitting an elementary band representation does not guarantee individual topological bands.

Conclusions:

• Obstructions to Wannier representation do not exclusively indicate stable band topology.
• Fragile topology represents a distinct and significant topological classification of electronic bands.
• The findings challenge assumptions in topological quantum chemistry and highlight the importance of Wannier functions in band theory.