Real spectra and phase transition of skin effect in nonreciprocal systems

We study the one-dimensional nonreciprocal lattices with real nearest-neighboring hopping and find that energy spectra under open boundary conditions can be entirely or imaginary. further investigate spectral properties non-Hermitian skin effect in mosaic introduced at equally spaced sites. The eigenenergies of such undergo a real-complex-imaginary real-complex transition as nonreciprocity varies. Moreover, exhibits phase transitions depending on period nonreciprocity. bulk states are abruptly shifted from one end lattice to opposite by crossing critical points, accompanied closing reopening point gaps periodic conditions. diagrams presented boundaries analytically determined. Our work unveils intriguing spectrum systems.