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Non-Hermitian skin effect (NHSE) is a unique phenomenon studied intensively in non-Hermitian systems during the past few years. In this work, we discuss the energy dependence of NHSE by introducing nonreciprocity beyond the nearest-neighboring hopping in the one-dimensional lattices. The direction of NHSE reverses as the eigenenergy of the system under open boundary conditions (OBCs) sweeps across some critical energies. To characterize such a phenomenon, we introduce the concept of non-Hermitian skin effect edges, which separate the eigenstates localized at opposite ends of the lattice in the OBC spectrum. We find the skin effect edges are determined by the self-intersections in the spectrum under periodic boundary conditions (PBCs), which are topological as the winding number of the PBC spectrum changes sign when crossing them. Moreover, the NHSE edges will disappear when the self-intersections merge into a single point by tuning the system parameters. Our work reveals the intricate interplay between NHSE and nonreciprocal hopping in non-Hermitian systems., 12 pages, including Appendix

We introduce the one-dimensional quasireciprocal lattices where the forward hopping amplitudes between nearest neighboring sites $\{ t+t_{jR} \}$ are chosen to be a random permutation of the backward hopping $\{ t+t_{jL} \}$ or vice versa. The values of $\{ t_{jL} \}$ (or $\{t_{jR} \}$) can be periodic, quasiperiodic, or randomly distributed. We show that the Hamiltonian matrices are pseudo-Hermitian and the energy spectra are real as long as $\{ t_{jL} \}$ (or $\{t_{jR} \}$) are smaller than the threshold value. While the non-Hermitian skin effect is always absent in the eigenstates due to the global cancellation of local nonreciprocity, the competition between the nonreciprocity and the accompanying disorders in hopping amplitudes gives rise to energy-dependent localization transitions. Moreover, in the quasireciprocal Su-Schrieffer-Heeger models with staggered hopping $t_{jL}$ (or $t_{jR}$), topologically nontrivial phases are found in the real-spectra regimes characterized by nonzero winding numbers. Finally, we propose an experimental scheme to realize the quasireciprocal models in electrical circuits. Our findings shed new light on the subtle interplay among nonreciprocity, disorder, and topology., Comment: 14 pages

We introduce a one-dimensional lattice model whose hopping amplitudes are modulated for equally spaced sites. Such mosaic lattice exhibits many interesting topological and localization phenomena that do not exist in the regular off-diagonal lattices. When the mosaic modulation is commensurate with the underlying lattice, topologically nontrivial phases with zero- and nonzero-energy edge modes are observed as we tune the modulation, where the nontrivial regimes are characterized by quantized Berry phases. If the mosaic lattice becomes incommensurate, Anderson localization will be induced purely by the quasiperiodic off-diagonal modulations. The localized eigenstate is found to be centered on two neighboring sites connected by the quasiperiodic hopping terms. Furthermore, both the commensurate and incommensurate off-diagonal mosaic lattices can host Chern insulators in their two-dimensional generalizations. Our work provides a platform for exploring topological phases and Anderson localization in low-dimensional systems., Comment: 9 pages including supplementary

We study topological superconductor in one-dimensional (1D) mosaic lattice whose on-site potentials are modulated for equally spaced sites. When the system is topologically nontrivial, Majorana zero modes appear at the two ends of the 1D lattice. By calculating energy spectra and topological invariant of the system, we find the interval of the mosaic modulation of the on-site potential, whether it is periodic, quasiperiodic, or randomly distributed, can influence the topological properties significantly. For even interval of the mosaic potential, the system will always exist in the topological superconducting phase for any finite on-site potentials. When the interval is odd, the system undergoes a topological phase transition and enters into the trivial phase as the on-site potentials become stronger than a critical value, except for some special cases in the commensurate lattices. These conclusions are proven and the phase boundaries determined analytically by exploiting the method of transfer matrix. They reveal that robust Majorana zero modes can arise in 1D mosaic lattice independent of the strength of the spatially modulated potentials., 5+9 pages, including supplementary

Non-Hermitian topological systems have been shown to exhibit many interesting properties. However, due to the presence of complex energy spectra and non-Hermitian skin effect (NHSE), the topological edge states will be dynamically unstable and unresolvable from the bulk states which are also localized at the boundaries. Here we propose a one-dimensional nonreciprocal superlattice model in which topological phases are solely induced by the modulated asymmetric hopping amplitude. Most importantly, we can obtain robust real energy spectra, and in the meanwhile, eliminate the NHSE both in the commensurate and incommensurate lattices by tuning the modulation. Thus the topological zero and nonzero energy edge states are stable and distinguishable from the bulk states, which would definitely facilitate the experimental identifications of non-Hermitian topological phases. We further propose a practical scheme to realize this nonreciprocal model by exploiting electrical circuits.

Higher-order topological phases of matter have been extensively studied in various areas of physics. While the Aubry-Andr\'e-Harper model provides a paradigmatic example to study topological phases, it has not been explored whether a generalized Aubry-Andr\'e-Harper model can exhibit a higher-order topological phenomenon. Here, we construct a two-dimensional higher-order topological insulator with chiral symmetry based on the Aubry-Andr\'e-Harper model. We find the coexistence of zero-energy and nonzero energy corner-localized modes. The former is protected by the quantized quadrupole moment, while the latter by the first Chern number of the Wannier band. The nonzero-energy mode can also be viewed as the consequence of a Chern insulator localized on a surface. More interestingly, the non-zero energy corner mode can lie in the continuum of extended bulk states and form a bound state in the continuum of higher-order topological systems. We finally propose an experimental scheme to realize our model in electric circuits. Our study opens a door to further study higher-order topological phases based on the Aubry-Andr\'e-Harper model., Comment: 10 pages, including supplemental material

We study the conductance and interference effects through an AB interferometer with an embedded quantum dot (QD) side-coupled to a single Majorana bound state (MBS) by using non-equilibrium Green's function method. The energy levels appearing in the QD are calculated by diagonalizing the Hamiltonian of the embedded QD–MBS system. When the single QD energy level ϵ 0 is set to 0, there are three discrete energy levels in the QD appearing at around ω = 0 , ± ϵ M 2 + 2 λ 2 due to the coupling with MBS where ϵ M is the coupling strength between the two MBSs at the two ends of the nanowire and λ is the coupling strength between the MBS and the QD. Asymmetric Fano lineshapes are found around these levels in the conductance due to the interference between electrons traversing through different paths. The phase shift of electrons through the QD changes from π / 2 to − π / 2 at each of these three energy values. However, the phase does not vary smoothly between these three energy levels but shows severe changes from − π / 2 to π / 2 at ω = ± ϵ M 2 + λ 2 . As a comparison, we also study the similar AB interferometer in which the QD–MBS system is replaced by a normal QD–QD system or a simple single QD system, which shows only two or one Fano peak and the phase shifts from π / 2 to − π / 2 only at the Fano peaks. These differences reflect the distinct influences of Majorana bound state on the transport properties of AB interferometer.

We investigate the Anderson localization in non-Hermitian Aubry-Andr\'e-Harper (AAH) models with imaginary potentials added to lattice sites to represent the physical gain and loss during the interacting processes between the system and environment. By checking the mean inverse participation ratio (MIPR) of the system, we find that different configurations of physical gain and loss have very different impacts on the localization phase transition in the system. In the case with balanced physical gain and loss added in an alternate way to the lattice sites, the critical region (in the case with p-wave superconducting pairing) and the critical value (both in the situations with and without p-wave pairing) for the Anderson localization phase transition will be significantly reduced, which implies an enhancement of the localization process. However, if the system is divided into two parts with one of them coupled to physical gain and the other coupled to the corresponding physical loss, the transition process will be impacted only in a very mild way. Besides, we also discuss the situations with imbalanced physical gain and loss and find that the existence of random imaginary potentials in the system will also affect the localization process while constant imaginary potentials will not., Comment: 6 pages, 4 figures

We investigate electron transport inside a ring system composed of a quantum dot (QD) coupled to two Majorana bound states confined at the ends of a one-dimensional topological superconductor nanowire. By tuning the magnetic flux threading through the ring, the model system we consider can be switched into states with or without zero-energy modes when the nanowire is in its topological phase. We find that the Fano profile in the conductance spectrum due to the interference between bound and continuum states exhibits markedly different features for these two different situations, which consequently can be used to detect the Majorana zero-energy mode. Most interestingly, as a periodic function of magnetic flux, the conductance shows $2\pi$ periodicity when the two Majorana bound states are nonoverlapping (as in an infinitely long nanowire) but displays $4\pi$ periodicity when the overlapping becomes nonzero (as in a finite length nanowire). We map the model system into a QD--Kitaev ring in the Majorana fermion representation and affirm these different characteristics by checking the energy spectrum., Comment: 8 pages, 8 figures

We investigate a generalized Aubry-Andr\'e-Harper (AAH) model with p-wave superconducting pairing. Both the hopping amplitudes between the nearest neighboring lattice sites and the on-site potentials in this system are modulated by a cosine function with a periodicity of $1/\alpha$. In the incommensurate case [$\alpha=(\sqrt{5}-1)/2$], due to the modulations on the hopping amplitudes, the critical region of this quasiperiodic system is significantly reduced and the system becomes more easily to be turned from extended states to localized states. In the commensurate case ($\alpha = 1/2$), we find that this model shows three different phases when we tune the system parameters: Su-Schrieffer-Heeger (SSH)-like trivial, SSH-like topological, and Kitaev-like topological phases. The phase diagrams and the topological quantum numbers for these phases are presented in this work. This generalized AAH model combined with superconducting pairing provides us with a useful testfield for studying the phase transitions from extended states to Anderson localized states and the transitions between different topological phases., Comment: 9 pages, 5 figures

We investigate non-Hermitian Kitaev chains in which imaginary potentials are added to all or only the two end lattice sites to represent the physical gain and loss during their interacting processes with the environment. By analyzing the energy eigenvalues of the non-Hermitian Hamiltonians both analytically and numerically, we find that if the imaginary potentials are added to every lattice site, the topologically nontrivial phase region will be gradually narrowed down as the imaginary potentials increase. The zero-energy modes in the nontrivial phase region remain real and thus the Majorana bound states (MBSs) at the ends of the chain are presenting steadily until the imaginary potentials become strong enough to close the energy gap and change the bulk properties. However, if the imaginary potentials are only added to the two ends of the chain, the zero-energy modes will always be real and thus the MBSs exist stably all the time even when the imaginary potentials become very strong since the corresponding bulk properties cannot be changed by these two local perturbations. Our results explicitly demonstrate the evolution and robustness of the nontrivial topological state of Kitaev chain against external perturbations.