# Topological classification of defects in non-Hermitian systems

## TL;DR

This paper provides a comprehensive topological classification of defects in non-Hermitian systems across all symmetry classes and dimensions, extending known Hermitian classifications to include point, real, and imaginary gaps.

## Contribution

It introduces a complete classification scheme for topological defects in non-Hermitian systems, generalizing the Hermitian ten-fold way to include new symmetry classes and gap types.

## Key findings

- Classifies defects in 38 and 54 non-Hermitian symmetry classes.
- Provides explicit topological invariants for non-trivial classes.
- Establishes correspondence between topological numbers and zero modes.

## Abstract

We classify topological defects in non-Hermitian systems with point gap, real gap and imaginary gap for all the Bernard-LeClair classes or generalized Bernard-LeClair classes in all dimensions. The defect Hamiltonian $H(\bf{k}, {\bf r})$ is described by a non-Hermitian Hamiltonian with spatially modulated adiabatical parameter ${\bf r}$ surrounding the defect. While the non-Hermitian system with point gap belongs to any of 38 symmetry classes (Bernard-LeClair classes), for non-Hermitian systems with line-like gap we get 54 non-equivalent generalized Bernard-LeClair classes as a natural generalization of point gap classes. Although the classification of defects in Hermitian systems has been explored in the context of standard ten-fold Altland-Zirnbauer symmetry classes, a complete understanding of the role of the general non-Hermitian symmetries on the topological defects and their associated classification are still lacking. By continuous transformation and homeomorphic mapping, these non-Hermitian defect systems can be mapped to topologically equivalent Hermitian systems with associated symmetries, and we get the topological classification by classifying the corresponding Hermitian Hamiltonians. We discuss some non-trivial classes with point gap according to our classification table, and give explicitly the topological invariants for these classes. We also study some lattice or continuous models and discuss the correspondence between the topological number and zero modes at the topological defect.