Symmetry-based classification of non-Bloch bands under open boundary conditions

Determine the role of symmetries in the topological classification of non-Bloch bands of non-Hermitian multiband systems under open boundary conditions, specifically whether and how symmetry constraints refine or extend the homotopy and braiding-topology characterization of open-boundary energy bands associated with generalized Brillouin zones (sub-GBZs).

Background

The paper develops a continuity criterion and a homotopy-based braiding-topology framework for non-Bloch bands of non-Hermitian systems under open boundary conditions, including multiband cases with sub-GBZs. This characterization focuses on band vorticity and braid-group structures, and is constructed without invoking symmetry constraints.

In the conclusions, the authors explicitly point out that their analysis does not rely on symmetry assumptions. They then highlight as an open question how symmetries (e.g., crystalline or antiunitary symmetries that are known to affect non-Hermitian topology under periodic boundaries) might participate in and further refine the topological classification of non-Bloch bands under open boundaries. Addressing this would connect the proposed braiding-topology characterization with symmetry-protected or symmetry-enriched classifications in the broader non-Hermitian band-theory literature.

References

Our analysis also does not depend on symmetries, whose participation in further topological classifications of the non-Bloch bands is an interesting open question for future studies.

— Braiding Topology of Non-Hermitian Open-Boundary Bands  (2405.11832 - Fu et al., 2024) in Discussions and conclusions (end of main text)

Whether this structure supports additional pump invariants, and which spectral features such invariants enforce in the SIBC/OBC geometries, remains an interesting question for further study.

— Universal aspects of bulk density of states in non-Hermitian lattices  (2608.14508 - Pavliuk et al., 14 Aug 2026) in Section 3.4, “Symmetry-protected spectra”

We leave a dedicated analysis of such prospective equivariant pump topology in non-Hermitian Hamiltonians open to future studies.

— Universal aspects of bulk density of states in non-Hermitian lattices  (2608.14508 - Pavliuk et al., 14 Aug 2026) in Section 3.4, “Symmetry-protected spectra”