Real Spectra and Phase Transition of Skin Effect in Nonreciprocal Systems

Abstract

We study the one-dimensional nonreciprocal lattices with real nearest neighboring hopping and find that the energy spectra under open boundary conditions can be entirely real or imaginary without imposing any symmetries. We further investigate the spectral properties and the non-Hermitian skin effect in the one-dimensional mosaic lattices with real nonreciprocal hopping introduced at equally spaced sites. The eigenenergies of such lattices undergo a real-complex-imaginary or realcomplex transition as the nonreciprocity varies. Moreover, the skin effect exhibits phase transitions depending on the period of the mosaic nonreciprocity. The bulk states are abruptly shifted from one end of the lattice to the opposite one by crossing the critical points, accompanied by the closing and reopening of point gaps in the spectra under periodic boundary conditions. The phase diagrams of the transition are presented and the critical boundaries are analytically determined. Our work unveils the intriguing properties of the energy spectrum and skin effect in non-Hermitian systems.The study on non-Hermitian systems has attracted much attention during the past few years [1-5]. Non-Hermiticity may appear due to the interactions between the systems and the external environment, which are commonly represented by physical gain and loss in the model Hamiltonian and can exist in both classical [6][7][8][9][10][11][12][13][14][15][16] and quantum systems [17][18][19][20][21][22][23][24][25][26]. The energy spectra of non-Hermitian systems are normally complex but can be entirely real when or pseudo-Hermiticity [30][31][32][33][34][35] is respected. The existence of real spectra ensures the stabilities of non-Hermitian systems, which facilitates their applications in various fields. However, whether real spectra exist in general non-Hermitian systems with no symmetry constraints remains elusive.Recently, nonreciprocal systems have inspired considerable research interest [36][37][38]. The energy spectra of such systems are very sensitive to the change of boundary conditions [39], which can be utilized to design new types of quantum sensors [40,41]. For systems with nonreciprocal hopping and under open boundary conditions (OBCs), the non-Hermitian skin effect (NHSE) will emerge with all the bulk states localized at the boundaries. The NHSE impacts a multitude of physical phenomena significantly [42][43][44][45][46][47][48][49][50][51][52][53][54], such as the bulk-boundary correspondence of topological phases [43,44,[55][56][57][58][59][60][61] and Anderson localization phenomenon [62][63][64][65][66][67][68]. Though the NHSE is found to be connected to the point gap in the spectra under periodic boundary conditions (PBCs) [69][70][71], how the behavior of NHSE is influenced by the variation of nonreciprocity is less studied.