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 non-Hermitian delocalization transition and originally predicted by Hatano and Nelson [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996)]. 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.