Non-Hermitian SSH3 model in 1D and Hopf model in 3D

We present non-Hermitian generalizations of the SSH3 model with period-3 patterns of the hopping coefficients in 1D and the two-band Hopf model in 3D. The Hermitian SSH3 model does not have a well-defined winding number, despite its resemblance to the SSH model with alternating hopping coefficient. Adding non-Hermitian terms to the SSH3 model leads to a well-defined winding number due to its complex-valued spectrum. Moreover, we found two types of localized states associated with a special symmetry. The Hermitian 3D two-band Hopf model, on the other hand, is a topological insulator but does not fall into the standard classification. The modulus of the Hopf index remains quantized in the non-Hermitian generalizations. Depending on the types of non-Hermitian terms, the non-Hermitian Hopf insulator exhibits complex-valued spectra with a bulk-boundary correspondence or non-Hermitian skin effect with skewed profiles of the bulk states.