Topological properties and localization transition in a one-dimensional non-Hermitian lattice with a slowly varying potential

We explore the eigenenergy distribution, localization transition, and topological property of a one-dimensional non-Hermitian lattice with a slowly varying potential, which does not possess parity-time (PT) symmetry. We found that, in contrast to PT-symmetric systems whose eigenenergies can stay real until the PT-symmetry breaking point, the eigenenergies in the present system are fully complex and their distribution in the complex plane depends on the parameters. As non-Hermiticity is introduced into the system, also depending on the parameters, the eigenstates may show a crossover behavior from fully extended to fully localized as non-Hermiticity is increased, or there will be a sharp localization transition at finite non-Hermiticity. Nevertheless, this non-Hermitian system is proven to be topologically trivial, thus we conclude that in general non-Hermitian systems, the localization transition is not always of a topological nature.