Phase transitions in non-Hermitian superlattices

We investigate the energy spectral phase transitions arising in one-dimensional superlattices under an imaginary gauge field and possessing M sites in each unit cell in the large M limit. It is shown that in models displaying nearly flat bands a smooth phase transition, from quasi entirely real to complex energies, can be observed as the imaginary gauge field is increased, and that the phase transition becomes sharper and sharper (exact) as M is increased. In this limiting case, for superlattices with random or incommensurate disorder the spectral phase transition corresponds to a localization-delocalization transition of the eigenfunctions within each unit cell, dubbed nonHermitian delocalization transition and originally predicted by Hatano and Nelson. However, it is shown here that in superlattices without disorder a spectral phase transition can be observed as well, which does not correspond to a non-Hermitian delocalization phase transition. The predicted phenomena could be observed in non-Hermitian photonic quantum walks, where synthetic superlattices with controllable M and imaginary gauge fields can be realized with existing experimental apparatus.