Band strcucture, symmetry and homotopy description of non-Hermitian topology*

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. To describe the band structures of non-Hermitian systems, we consider the notions of non-Hermitian band gaps and separation gaps. With these concepts, we provide a unified and systematic classification of both gapped and nodal systems in the presence of physically relevant parity-time (PT ) and pseudo-Hermitian symmetries using homotopy theory. We reveal different Abelian and non-Abelian phases in PT-symmetric systems, described by frame and braid topology. They exibit a new type of stable and fragile topology depending on the symmetry properties of the bands. The corresponding invariants are robust to symmetry-preserving perturbations that do not close band gaps, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by a Chern-Euler description, a fingerprint of unprecedented non-Hermitian topology overlooked in existing study. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platform