Coalescing Majorana edge modes in non-Hermitian $${\mathscr{P}}{\mathscr{T}}$$PT -symmetric Kitaev chain

A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We present an exact solvable non-Hermitian finite-size Kitaev chain with $${\mathscr{P}}{\mathscr{T}}$$-symmetric chemical potentials at the symmetric point. The straightforward calculation shows that there are two kinds of Majorana edge modes in this model divided by $${\mathscr{P}}{\mathscr{T}}$$ symmetry-broken and unbroken. The one appeared in the $${\mathscr{P}}{\mathscr{T}}$$ symmetry-unbroken region can be seen as the finite-size projection of the conventional degenerate zero modes in a Hermitian infinite system with the open boundary condition. It indicates a possible variant of the bulk-edge correspondence: The number of Majorana edge modes in a finite non-Hermitian system can be the topological invariant to identify the topological phase of the corresponding bulk Hermitian system.