Topological Defects and Gapless Modes in Insulators and Superconductors

TL;DR

This paper presents a unified topological framework to classify defects in insulators and superconductors, linking topological invariants to protected gapless modes and broadening understanding of defect-related phenomena.

Contribution

It introduces a comprehensive classification scheme for topological defects in various symmetry classes and generalizes the bulk-boundary correspondence to defect Hamiltonians.

Findings

Classifies topological defects across ten symmetry classes.

Provides explicit formulas for topological invariants.

Links defects to protected gapless modes and zero modes.

Abstract

We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified, and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions and helical Majorana fermions, as well as zero dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.