Topological characterization of non-Hermitian multiband systems using Majorana's stellar representation

For topological characterization of non-Hermitian multiband systems, Majorana's stellar representation (MSR) is applied to one-dimensional multiband models consisting of asymmetric nearest-neighbor hopping and imaginary on-site potentials. The number of edge states isolated from the continuous bulk bands in the complex energy plane is successfully linked with a topological invariant constructed from MSR. Specifically, the number of isolated edge states can be obtained from a winding number defined for the Majorana stars, which also allows for a geometric visualization of the topology related to the isolated edge modes. A remarkable success of our approach is that our winding number characterization remains valid even in the presence of exceptional points of the continuous bulk bands, where the Hamiltonian becomes nondiagonalizable and hence conventional topological invariants such as the Zak phase and the Chern number cannot be properly defined. Furthermore, cases with the so-called non-Hermitian skin effect are also studied, showing that the bulk-boundary correspondence between our defined winding numbers and isolated edge states can be restored. Of particular interest is a four-band example with an odd number of isolated edge states, where the Zak phase approach necessarily fails upon removing the skin effect, but our MSR-based characterization works equally well. For these reasons, our study is expected to be widely useful in topological studies of non-Hermitian multiband systems, regardless of the skin effect or the presence of the exceptional points in non-Hermitian systems.