Index Theorems, Generalized Hall Currents, and Topology for Gapless Defect Fermions

Area of Science:

• Condensed matter physics
• Topological materials science

Background:

• Topological insulators and superconductors host unique electronic states.
• Understanding boundary phenomena is crucial for topological material applications.

Purpose of the Study:

• To demonstrate a novel method for detecting gapless modes on defects.
• To connect topological properties in momentum and real space.

Main Methods:

• Utilizing the index of the fermion operator derived from the Euclidean action.
• Analyzing 1-loop Feynman diagrams to compute the topological index.
• Investigating systems with and without conserved currents or chiral anomalies.

Main Results:

• The fermion operator index successfully identifies gapless modes on defects like edges and vortices.
• A quantum Hall current analog is observed flowing on and off these defects.
• The method explicitly reveals the interplay between momentum-space and real-space topology.

Conclusions:

• The fermion operator index provides a powerful tool for uncovering defect-localized topological states.
• This approach offers new insights into the behavior of topological materials at boundaries.
• The findings are applicable to various topological systems, including insulators and superconductors.