Wave localization is a fundamental phenomenon that appears universally in both natural materials and artificial structures and plays a crucial role in understanding the various physical properties of a system. Usually, a localized state has an exponential profile with a localization length independent of the system size. Here, we experimentally demonstrate a new class of localized states called scale-free localized states, which has an unfixed localization length scaling linearly with the system size. Using circuit lattices, we observe that a non-Hermitian defect added to a Hermitian lattice induces an extensive number of states with scale-free localization. Furthermore, we demonstrate that, in a lattice with a parity-time-symmetric non-Hermitian defect, the scale-free localization emerges because of spontaneous parity-time symmetry breaking. Our results uncover a new type of localized states and extend the study of defect physics to the non-Hermitian regime.
We develop a systematically general theory of one-dimensional (1D) non-Hermitian systems, elaborating on the energy bands, the band degeneracy, and the defectiveness of eigenstates under open boundary conditions. We analyze the band degeneracy and defectiveness of two typical 1D non-Hermitian models. We obtain the unusual presence and absence of the exceptional points in the generalized non-Hermitian Su-Schrieffer-Heeger model under open boundary conditions. Beyond the general theory, we discover that infernal points exist in 1D non-Hermitian systems, where the energy spectra under open bounda…
It was known that for non-Hermitian topological systems due to the non-Hermitian skin effect, the bulk-edge correspondence is broken down. In this paper, by using one-dimensional Su-SchriefferHeeger model and two-dimensional (deformed) Qi-Wu-Zhang model as examples, we focus on a special type of non-Hermitian topological system without non-Hermitian skin effect-topological systems under non-Hermitian similarity transformation. In these non-Hermitian systems, the defective edge states and the breakdown of bulk-edge correspondence are discovered. To characterize the topological properties, we in…
Non-Hermitian topological systems show quite different properties as their Hermitian counterparts. An important, puzzled issue on non-Hermitian topological systems is the existence of defective edge states beyond usual bulk-boundary correspondence (BBC) that localize either on the left edge or the right edge of the one-dimensional system. In this paper, to understand the existence of the defective edge states, the theory of anomalous bulk-boundary correspondence (A-BBC) is developed that distinguishes the non-Bloch bulk-boundary correspondence (NB-BBC) from non-Hermitian skin effect. By using …
We classify topological defects in non-Hermitian systems with point gap, real gap and imaginary gap for all the Bernard-LeClair classes or generalized Bernard-LeClair classes in all dimensions. The defect Hamiltonian $H(\bf{k}, {\bf r})$ is described by a non-Hermitian Hamiltonian with spatially modulated adiabatical parameter ${\bf r}$ surrounding the defect. While the non-Hermitian system with point gap belongs to any of 38 symmetry classes (Bernard-LeClair classes), for non-Hermitian systems with line-like gap we get 54 non-equivalent generalized Bernard-LeClair classes as a natural general…
We investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions. In one-dimensional lattices, we show that the proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions. Depending on the non-local nature of the control interactions, the resulting non-reciprocity occurs in multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is also investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself thro…
Radiative damping is a strong dissipation source for the quantum emitters hybridized with propagating photons, electrons, or phonons, which is not easily avoidable for on-chip magnonic emitters as well that can radiate via the surface acoustic waves of the substrate. Here we demonstrate in an array of on-chip nano-magnets coupled in a long range via exchanging the surface acoustic waves that a point defect in the array, which can be introduced by the local magnon frequency shift by a local biased magnetic field or the absence of a magnetic wire, strongly localizes the magnons, in contrast to t…
We study the emergence and disappearance of defect states in the complex Su-Schrieffer-Heeger (cSSH) model, a non-Hermitian one-dimensional lattice model containing gain and loss on alternating sites. Previous studies of this model have focused on the existence of a non-Hermitian defect state that is localized to the interface between two cSSH domains, and is continuable to the topologically protected defect state of the Hermitian Su-Schrieffer-Heeger (SSH) model. For large gain/loss magnitudes, we find that these defect states can disappear into the continuum, or undergo pairwise spontaneous …
Defects usually play an important role in tuning and modifying various properties of semiconducting or insulating materials. Therefore we study the impact of point and line defects on the electronic structure and optical properties of MoS2 monolayers using density-functional methods. The different types of defects form electronic states that are spatially localized on the defect. The strongly localized nature is reflected in weak electronic interactions between individual point or line defect and a weak dependence of the defect formation energy on the defect concentration or line defect separa…
The Aubry-André-Harper (AAH) model with a self-dual symmetry plays an important role in studying the Anderson localization. Here we find a self-dual symmetry determining the quantum phase transition between extended and localized states in a non-Hermitian AAH model and show that the eigenenergies of these states are characterized by two types of winding numbers. By constructing and studying a non-Hermitian generalized AAH model, we further generalize the notion of the mobility edge, which separates the localized and extended states in the energy spectrum of disordered systems, to the non-Hermi…
We study spatial photoluminescence characteristics of neutral and charged excitons across extended monolayer MoS$_2$ synthesized by chemical vapor deposition. Using two-dimensional hyperspectral photoluminescence mapping at cryogenic temperatures we identify regions with increased emission from charged excitons associated with both spin-orbit split valence subbands. Such regions are attributed to unintentional doping at defect sites with excess charge that bind neutral excitons to form defect-pinned trions. Our findings imply comparable timescales for the formation, relaxation, and radiative d…
We demonstrate that dislocations in two-dimensional non-Hermitian systems can give rise to density accumulation or depletion through the localization of an extensive number of states. These effects are shown by numerical simulations in a prototype lattice model and expose a different face of non-Hermitian skin effect, by disentangling it from the need for boundaries. We identify a topological invariant responsible for the dislocation skin effect, which takes the form of a ${\mathbb Z}_2$ Hopf index that depends on the Burgers vector characterizing the dislocations. Remarkably, we find that thi…
A $d$-dimensional second-order topological insulator (SOTI) can host topologically protected $(d - 2)$-dimensional gapless boundary modes. Here we show that a 2D non-Hermitian SOTI can host zero-energy modes at its corners. In contrast to the Hermitian case, these zero-energy modes can be localized only at one corner. A 3D non-Hermitian SOTI is shown to support second-order boundary modes, which are localized not along hinges but anomalously at a corner. The usual bulk-corner (hinge) correspondence in the second-order 2D (3D) non-Hermitian system breaks down. The winding number (Chern number) …
Manipulating and reserving the valley pseudospin of excitons is one core aim in the two-dimensional transition metal dichalcogenides (TMDs). However, due to the strong electron-hole exchange and spin-orbit coupling interactions, the exciton recombination lifetime is subject to picosecond timescale intrinsically, and the valley polarization is hardly modulated by a moderate magnetic field. It is fortunate that interlayer and defect-localized excitons promise to overcome these difficulties by suppressing the above interactions. Here we clearly reveal that the valley polarization can be reversed …
Non-Hermitian degeneracies are classified as defective and nondefective exceptional points~(EP). While in defective EPs, both eigenvalues and eigenenergies coalesce, nondefective EPs are characterized merely by the occurrence of degeneracies in their eigenvalues. It is also known that all degeneracies are either symmetry-protected or accidental. In this paper, I prove that anti-unitary symmetries protect all nondefective twofold EPs. By developing a 2D non-Hermitian tight-binding model, I have demonstrated that these symmetries are composite of various symmetry operations such as discrete or s…
Single-photon emitters play a key role in present and emerging quantum technologies. Several recent measurements have established monolayer WSe$_2$ as a promising candidate for a reliable single photon source. The origin and underlying microscopic processes have remained, however, largely elusive. We present a multi-scale tight-binding simulation for the optical spectra of WSe$_2$ under non-uniform strain and in the presence of point defects employing the Bethe-Salpeter equation. Strain locally shifts excitonic energy levels into the band gap where they overlap with localized intra-gap defect …
Topological phases have recently witnessed a rapid progress in non-Hermitian systems. Here we study a one-dimensional non-Hermitian Aubry-André-Harper model with imaginary periodic or quasiperiodic modulations. We demonstrate that the non-Hermitian off-diagonal AAH models can host zero-energy modes at the edges. In contrast to the Hermitian case, the zero-energy mode can be localized only at one edge. Such a topological phase corresponds to the existence of a quarter winding number defined by eigenenergy in momentum space. We further find the coexistence of a zero-energy mode located only at o…
Quasiparticle states in Dirac systems with complex impurity potentials are investigated. It is shown that an impurity site with loss leads to a nontrivial distribution of the local density of states (LDOS). While the real part of defect potential induces a well-pronounced peak in the density of states (DOS), the DOS is either weakly enhanced at small frequencies or even forms a peak at the zero frequency for a lattice in the case of non-Hermitian impurity. As for the spatial distribution of the LDOS, it is enhanced in the vicinity of impurity but shows a dip at a defect itself when the potenti…
Coupling of an atom-like emitter to surface plasmons provides a path toward significant optical nonlinearity, which is essential in quantum information processing and quantum networks. A large coupling strength requires nanometer-scale positioning accuracy of the emitter near the surface of the plasmonic structure, which is challenging. We demonstrate the coupling of single localized defects in a tungsten diselenide (WSe2) monolayer self-aligned to the surface plasmon mode of a silver nanowire. The silver nanowire induces a strain gradient on the monolayer at the overlapping area, leading to t…
Non-Hermitian skin effect exhibits the collapse of the extended bulk modes into the extensive number of localized boundary states in open boundary conditions. Here we demonstrate the disorder-driven phase transition of the trivial non-Hermitian system to the higher-order non-Hermitian skin effect phase. In contrast to the clean systems, the disorder-induced boundary modes form an arc in the complex energy plane, which is the manifestation of the disorder-driven dynamical phase transition. At the phase transition, the localized corner modes and bulk modes characterized by trivial Hamiltonian co…
Symmetries associated with complex conjugation and Hermitian conjugation, such as time-reversal symmetry and pseudo-Hermiticity, have great impact on eigenvalue spectra of non-Hermitian random matrices. Here, we show that time-reversal symmetry and pseudo-Hermiticity lead to universal level statistics of non-Hermitian random matrices on and around the real axis. From the extensive numerical calculations of large random matrices, we obtain the five universal level-spacing and level-spacing-ratio distributions of real eigenvalues, each of which is unique to the symmetry class. Furthermore, we an…
We theoretically study two-dimensional exciton-polariton lattices and predict that non-Hermitian topological corner modes can be formed under non-resonant pumping. As a generalization of the non-Hermitian skin effect, all eigenstates are localized at the two corners in our model. This is also a higher dimensional topology compared to other proposals in exciton-polariton systems and we find that it allows propagating signals in the bulk of the system to travel around defects, which is not possible in one-dimensional topological lattices or two-dimensional lattices with Hermitian edge states. Fu…
Non-Hermitian generalizations of the Su-Schrieffer-Heeger (SSH) models with higher periods of the hopping coefficients, called the SSH3 and SSH4 models, are analyzed. The conventional construction of the winding number fails for the Hermitian SSH3 model, but the non-Hermitian generalization leads to a topological system due to a point gap on the complex plane. The non-Hermitian SSH3 model thus has a winding number and exhibits the non-Hermitian skin effect. Moreover, the SSH3 model has two types of localized states and a zero-energy state associated with special symmetries. The total Zak phase…
We investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we …
Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a non-Hermitian hamiltonian are localized at the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodic-boundary spectrum of the hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic non-Hermitian hamiltonian, and, unlike in one dimension, is compa…
Non-Hermitian skin-edge states emerge only at one edge in one-dimensional nonreciprocal chains, where all states are localized at the edge irrespective of eigenvalues. The bulk topological number is the winding number associated with the complex energy spectrum, which is well defined for metals. We study non-Hermitian nonreciprocal systems in higher dimensions, and propose to realize them with the use of electric diode circuits. We first investigate one-dimensional interface states between two domains carrying different topological numbers, where all states are localized at the interface. They…
In this paper, we study the localized states of a generic quadratic fermionic chain with finite-range couplings and an inhomogeneity in the hopping (defect) that breaks translational invariance. When the hopping of the defect vanishes, which represents an open chain, we obtain a simple bulk-edge correspondence: the zero-energy modes localized at the ends of the chain are related to the roots of a polynomial determined by the couplings of the Hamiltonian of the bulk. From this result, we define an index that characterizes the different topological phases of the system and can be easily computed…
We study the localization transition in periodically driven one-dimensional non-Hermitian lattices where the piece-wise two-step drive is constituted by uniform coherent tunneling and incommensurate onsite gain and loss. We find that the system can be in localized, delocalized, or mixed-phase depending on the driving frequency and the phase shift of complex potential. Two critical driving frequencies of the system are identified, the first one corresponds to the largest phase shift of the complex potential so that the quasi-energy spectrum is still real and all the states are extended, the sec…
A non-Hermitian extension of a Chern insulator and its bulk-boundary correspondence are investigated. It is shown that in addition to the robust chiral edge states that reflect the nontrivial topology of the bulk (nonzero Chern number), anomalous helical edge states localized only at one edge can appear, which are unique to the non-Hermitian Chern insulator.
Alice string is a topological defect with a very peculiar feature. When a defect with a monopole charge encircles an Alice string, the monopole charge changes sign. In this work, we generalize this notion to momentum space of periodic media with loss and gain. In particular, we find that the generic band-structure node for a three-dimensional non-Hermitian crystalline system acts as an Alice string, which can flip the Chern number charge carried by Weyl points and by exceptional-line rings. We discuss signatures of this topological structure for a lattice model with one tuning parameter, inclu…
The traditional bulk-boundary correspondence assuring robust gapless modes at the edges and surfaces of insulating and nodal topological materials gets masked in non-Hermitian (NH) systems by the skin effect, manifesting an accumulation of a macroscopic number of states near such interfaces. Here we show that dislocation lattice defects are immune to such skin effect or at most display a \emph{weak} skin effect (depending on its relative orientation with the Burgers vector), and as such they support robust topological modes in the bulk of a NH system, specifically when the parent Hermitian pha…
A non-Hermitian topological insulator is fundamentally different from conventional topological insulators. The non-Hermitian skin effect arises in a nonreciprocal tight binding lattice with open edges. In this case, not only topological states but also bulk states are localized around the edges of the nonreciprocal system. We discuss that controllable switching from topological edge states into topological extended states in a chiral symmetric non-Hermitian system is possible. We show that the skin depth decreases with non-reciprocity for bulk states but increases with it for topological zero …
