non-Hermitian Skin Effect Research Articles

Non-Hermiticity induced exceptional points and skin effect in the Haldane model on a dice lattice

The interplay of topology and non-Hermiticity has led to diverse, exciting manifestations in a plethora of systems. In this work, we systematically investigate the role of non-Hermiticity in the Chern insulating Haldane model on a dice lattice. Due to the presence of a non-dispersive flat band, the dice-Haldane model hosts a topologically rich phase diagram with the non-trivial phases accommodating Chern numbers $\pm 2$. We introduce non-Hermiticity into this model in two ways -- through balanced non-Hermitian gain and loss, and by non-reciprocal hopping in one direction. Both these types of non-Hermiticity induce higher-order exceptional points of order three. We substantiate the presence and the order of these higher-order exceptional points using the phase rigidity and its scaling. Further, we construct a phase diagram to identify and locate the occurrence of these exceptional points in the parameter space. Non-Hermiticity has yet more interesting consequences on a finite-sized lattice. Unlike for balanced gain and loss, in the case of non-reciprocal hopping, the nearest-neighbour dice lattice system under periodic boundary conditions accommodates a finite, non-zero spectral area in the complex plane. This manifests as the non-Hermitian skin effect when open boundary conditions are invoked. In the more general case of the dice-Haldane lattice model, the non-Hermitian skin effect can be caused by both gain and loss or non-reciprocity. Fascinatingly, the direction of localization of the eigenstates depends on the nature and strength of the non-Hermiticity. We establish the occurrence of the skin effect using the local density of states, inverse participation ratio and the edge probability, and demonstrate its robustness to disorder. Our results place the dice-Haldane model as an exciting platform to explore non-Hermitian physics.

Emergent non-Hermitian localization phenomena in the synthetic space of zero-dimensional bosonic systems

Phase transitions in non-Hermitian systems are at the focus of cutting edge theoretical and experimental research. On the one hand, parity-time- $(\mathcal{PT}\ensuremath{-})$ and $\mathrm{anti}\ensuremath{-}\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ physics have gained ever-growing interest, due to the existence of non-Hermitian spectral singularities called exceptional points (EPs). On the other hand, topological and localization transitions in non-Hermitian systems reveal new phenomena, e.g., the non-Hermitian skin effect and the absence of conventional bulk-boundary correspondence. The great majority of previous studies exclusively focus on non-Hermitian Hamiltonians, whose realization requires an a priori fine-tuned extended lattice to exhibit topological and localization transition phenomena. In this work, we show how the non-Hermitian localization phenomena can naturally emerge in the synthetic field moment space of zero-dimensional bosonic systems, e.g., in $\mathrm{anti}\ensuremath{-}\mathcal{PT}$- and $\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ quantum dimers. This offers an opportunity to simulate localization transitions in low-dimensional systems, without the need to construct complex arrays of, e.g., coupled cavities or waveguides. Indeed, the field moment equations of motion can describe an equivalent (quasi)particle moving in a one-dimensional (1D) synthetic lattice. This synthetic field moment space can exhibit nontrivial localization phenomena, such as non-Hermitian skin effect, induced by the presence of highly degenerate EPs. We demonstrate our findings on the example of an $\mathrm{anti}\ensuremath{-}\mathcal{PT}\ensuremath{-}\mathrm{symmetric}$ two-mode system, whose higher-order field moment eigenspace is emulated by a synthetic 1D non-Hermitian Hamiltonian having a Sylvester matrix shape. Our results can be directly verified in state-of-the-art optical setups, such as superconducting circuits and toroidal resonators, by measuring photon moments or correlation functions.

Edge states and skin effect dependent electron transport properties of non-Hermitian Su-Schrieffer-Heeger chain

In the non-reciprocal Su-Schrieffer-Heeger (SSH) chain, the hopping amplitude of an electron in the intra-cell depends on its hopping direction. Consequently, the non-Hermitian SSH chain has both non-trivial topological edge state and non-Hermitian skin effect. However, how to detect the non-trivial topological edge states and non-Hermitian skin effect has become an important topic in non-Hermitian physics. In this paper, we study the relationships of the non-trivial topological edge states and the non-Hermitian skin effect of non-Hermitian SSH chain with their electron transport properties in the vicinity of the zero energy. It is demonstrated that when the peak value of the electron transmission probability in the vicinity of the zero energy is much smaller than 1, the non-Hermitian SSH chain has a left-non-Hermitian skin effect; while that in the vicinity of the zero energy is much larger than 1, the non-Hermitian SSH chain has a right-non-Hermitian skin effect. In particular, the skin effect of non-Hermitian SSH chain can be further enhanced in the region of non-trivial topological edge states. Moreover, with the increase of the electron tunneling coupling amplitudes between the non-Hermitian SSH chain and the left and right leads from the weak coupling regime to the strong coupling one, the number of the dips of reflection probability in the vicinity of the zero energy will change from two to zero. Therefore, these results theoretically provide an alternative scheme for detecting non-trivial topological edge states and non-Hermitian skin effect types of the non-Hermitian SSH chain.

Bulk-boundary correspondence in disordered non-Hermitian systems

The bulk-boundary correspondence (BBC) refers to the consistency between eigenvalues calculated under open and periodic boundary conditions. This consistency can be destroyed in systems with non-Hermitian skin effect (NHSE). In spite of the great success of the generalized Brillouin zone (GBZ) theory in clean non-Hermitian systems, the applicability of GBZ theory is questionable when the translational symmetry is broken. Thus, it is of great value to rebuild the BBC for disordered samples, which extends the application of GBZ theory in non-Hermitian systems. Here, we propose a scheme to reconstruct BBC, which can be regarded as the solution of an optimization problem. By solving the optimization problem analytically, we reconstruct the BBC and obtain the modified GBZ theory in several prototypical disordered non-Hermitian models. The modified GBZ theory provides a precise description of the fantastic NHSE, which predicts the asynchronous-disorder-reversed NHSE’s directions.

Dissipation-induced recurrence of non-Hermitian edge burst

In quantum mechanics, the Hermitian Hamiltonian is generally used to describe the ideal closed quantum system, but in reality, the physical system is closely related to the environment, and the open quantum system coupled to the environment can be described by the equivalent non-Hermitian Hamiltonian to a certain extent. Among them, the dissipation intensity is closely related to the dynamic properties of non-Hermitian quantum systems. Therefore, it is of great practical significance to study how dissipation affects particle loss. In this paper, the dynamic law related to dissipation intensity in a one-dimensional non-Hermitian system under open boundary condition is studied, and it is found that dissipation can induce the recurrence of edge burst. After the time-dependent evolution of the particles in the one-dimensional non-Hermitian dissipative lattice system with open boundary condition, there is an edge burst in the system, that is, there is a large probability of particle loss at the edge, and the edge burst disappears after increasing the intracell hopping. It is found that if the dissipation intensity increases or decreases, the edge burst will reappear. This kind of reappearance is different from the original edge burst, which is mainly manifested in the loss probability distribution of particles from the edge distribution to the bulk distribution, which is due to the difference in probability of particle motion direction between the two cases. Under the re-induced edge burst, the particles move leftward and rightward from their initial positions, and rebound from the left after having reached the boundary, forming a more obvious loss probability at the edge and gradually decreasing to the body area. In the original edge burst, the particles only move to the left with a greater probability, and are ‘trapped’ at the edge to completely dissipated, forming a distribution with an independent loss peak at the edge. The movement to the left is due to the non-Hermitian skin effect. The deeper reason for different movement directions is related to parity-time symmetry. Under the parameter near the parity-time symmetry breaking point, the loss probability of the particle is of unilateral distribution, and the loss probability of the particle moving to both sides is of bilateral distribution when it is far away. This is the description of the dissipation-induced edge burst recurrence phenomenon and its characteristics. In addition, this paper also studies the influence of impurity barrier on the probability distribution of particle loss in non-Hermitian dynamics. The results show that placing a small barrier on the non-dissipative <i>A</i>-site can obviously hinder the particle motion, and when the barrier increases to a certain height, its influence on the particle motion tends to be unchanged. And the barrier at the dissipative <i>B</i> lattice has little effect on the dynamics.

Electrical circuit simulation of non-Hermitian lattice models

<sec>Quantum simulation serves as a significant tool for studying and understanding novel phenomena in the quantum world. In recent years, it has be realized that apart from quantum platforms, classical systems like photonic crystals, phononic crystals, and mechanical oscillators can also be used to simulate quantum models by analogizing the Schrödinger equation. Among these systems, electrical circuits have emerged as a promising simulation platform owing to their low cost, technological maturity, and ease of scalability, successfully simulating numerous important quantum phenomena. Meanwhile, non-Hermitian physics breaks the Hermiticity of systems’ Hamiltonians in traditional quantum mechanics, providing a fresh perspective for understanding the physics of quantum systems, particularly open quantum systems. Non-Hermitian systems, owing to their manifestation of unique phenomena absent in Hermitian systems, have become emerging research subjects in various fields of physics. However, many non-Hermitian phenomena require specialized configurations that pose relatively high technical thresholds on quantum platforms. For instance, the non-Hermitian skin effect typically requires systems to possess non-reciprocal hopping between lattice sites. Therefore, utilizing flexible electrical circuits to simulate non-Hermitian physics becomes a natural choice.</sec><sec>This paper provides a short review of the current experimental progress in simulating non-Hermitian lattice models by using electrical circuits. It offers a brief introduction to the relevant knowledge of non-Hermitian physics, including mathematical concepts and novel phenomena, as well as the simulation theory of electrical circuits, including the mapping theory of the lattice models, the introduction of non-Hermiticity, and the measurement of physical quantities. The aim is to provide readers with a reference for better understanding or engagement in related researches, thus promoting further development in this field.</sec>

Properties of the non-Hermitian SSH model: role of symmetry

The present work addresses the distinction between the topological properties of symmetric and non- symmetric scenarios for the non-Hermitian Su–Schrieffer–Heeger model. The non- symmetric case is represented by non-reciprocity in both the inter- and the intra-cell hopping amplitudes, while the one with symmetry is modeled by a complex on-site staggered potential. In particular, we study the loci of the exceptional points, the winding numbers, band structures, and explore the breakdown of bulk-boundary correspondence (BBC). We further study the interplay of the dimerization strengths on the observables for these cases. The non- symmetric case denotes a more familiar situation, where the winding number abruptly changes by half-integer through tuning of the non-reciprocity parameters, and demonstrates a complete breakdown of BBC, thereby showing non-Hermitian skin effect. The topological nature of the symmetric case appears to follow closely to its Hermitian analogue, except that it shows unbroken (broken) regions with complex (purely real) energy spectra, while another variant of the winding number exhibits a continuous behavior as a function of the strength of the potential, while the conventional BBC is preserved.

Deterministic bulk-boundary correspondences for skin and edge modes in a general two-band non-Hermitian system

In isolated Hermitian systems, the bulk-boundary correspondence is generally thought of as a compelling principle guiding the occurrence of robust edge states through the topological invariants of the bulk. However, in open non-Hermitian systems that support non-Hermitian skin effects, the universality of the bulk-boundary correspondence has so far remained disputable in spite of being subjected to intensive studies. Here we provide analytical and numerical evidence of bulk-boundary correspondences for both skin and edge modes in 1D two-band non-Hermitian systems within the framework of a prototypical non-Hermitian Su-Schrieffer-Heeger model. Two different topological winding numbers, one on a Brillouin zone and the other on a generalized Brillouin zone, are defined to fully characterize the topological phases of matter and to understand such deterministic correspondences. In addition to the exact solutions for bulk states, we also obtain explicit solution forms for edge modes along with analytical conditions for their existence and localization, all exhibiting remarkable agreement with numerical results. Our results solve the elusive non-Hermitian topology and may facilitate experimental investigations of bulk-boundary correspondence in a wide range of non-Hermitian systems.

Dynamic skin effects in non-Hermitian systems

We study the time evolution processes of non-Hermitian systems under the open boundary condition and confirm that the dynamical skin effect exists in non-Hermitian systems analytically, and unveil the mechanism of its formation, which is caused by both the non-Hermitian skin effect and the Hermitian wave packet spreading. Furthermore, we find that in contrast to the uniform speed motion in Hermitian situations, the Gaussian wave packet can be accelerated and amplified during its time evolution in non-Hermitian systems. This additional motion is found to be responsible for the dynamic skin effect.

Non-Hermitian skin effect in a single trapped ion

Non-Hermitian skin effect (NHSE) describes the exponential localization of all eigenstates toward boundaries in non-Hermitian systems, and has attracted intense research interest of late. Here we theoretically propose a scheme in which the NHSE significantly impacts the external motion of a single trapped ion through complex spin-motion dynamics. On the one hand, we show the competition between the NHSE and the coherent Bloch dynamics. On the other hand, since the NHSE manifests as a non-reciprocal flow in occupied phonon modes, we demonstrate that such dynamics can have potential applications in cooling and sensing. Our proposal can be readily implemented using existing experimental techniques, and offers a scalable (in terms of the available ions and phonon modes) simulation platform for relevant non-Hermitian physics.

Extended State in a Localized Continuum.

Bound state in a continuum (BIC) is a spatially confined resonance with its energy embedded in a continuous spectrum of propagative modes, yet their coupling is prohibited. In this Letter, we report the discovery of a generic non-Hermitian phenomenon that we call an "extended state in a localized continuum" (ELC). As the name suggests, the ELC is the inversion of the BIC-a single extended state embedded in a continuous spectrum entirely consisting of localized modes, and its emergence rests in the interplay between the BIC and the non-Hermitian skin effect (NHSE). Herein, the BIC is a zero-energy corner mode that spectrally overlaps with a bulk band in a Hermitian kagome lattice. The ELC emerges with the introduction of the NHSE in a particular way, such that it turns all the bulk states into corner skin modes and simultaneously delocalizes the corner mode. We experimentally realize the ELC using an active mechanical lattice. Our findings not only demonstrate the rich potential of the NHSE but may also spark new wave-based applications.

Theoretical prediction of a non-Hermitian skin effect in ultracold-atom systems

The authors demonstrate a robust non-Hermitian skin effect in spin-orbit-coupled fermions with dissipation, which has been implemented in a recent experiment. They also reveal an anomalous dynamic phenomenon named the dynamic sticky effect, which shares the same physical origin as the non-Hermitian skin effect.

Fundamental constraints on the observability of non-Hermitian effects in passive systems

Utilizing scattering theory, we quantify the consequences of physical constraints that limit the visibility of non-Hermitian effects in passive devices. The constraints arise from the fundamental requirement that the system obeys causality, and can be captured concisely in terms of an internal time-delay operator, which furthermore provides a direct quantitative measure of the visibility of specific non-Hermitian phenomena in the density of states. We illustrate the implications by contrasting different symmetry classes and non-Hermitian effects, including exceptional points and the non-Hermitian skin effect, whose underlying extreme mode nonorthogonality turns out to be effectively disguised.

Relaxation dynamics in the driven-dissipative Su-Schrieffer-Heeger model

For a chain with an open boundary described by the non-Hermitian Hamiltonian, the combination of the algebraic damping and the non-Hermitian skin effect leads to an edge burst. It is interesting to ask whether these features remain when the system is described by a master equation with periodic drives. In this paper, taking the Su-Schrieffer-Heeger model as an example, we explore the relaxation dynamics of the system with time periodically modulated intracell tunneling and single-particle dissipations. We find that for systems with periodic boundary conditions, the relaxation in the infinite-frequency driving limit can be algebraic or exponential depending on the intracell and intercell tunneling amplitudes. Finite-frequency driving can generally open the Liouvillian gap regardless of its strength and can turn the dynamics from algebraic damping to an exponential one. The Liouvillian gap increases with the increase of the driving period until a critical driving period is reached, and then it tends to a small, nonzero value for a sufficiently slow drive. For open boundary systems, we find that the dynamics of the localized excitation is still closely related to the Liouvillian gap before the excitation reaches the system boundary. The propagation speed of the localized excitation that controls the transition relaxation timescale can be slowed down by properly increasing the driving period and amplitude.

Non-Hermitian skin effect edge

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.

Universal characteristics of one-dimensional non-Hermitian superconductors

We establish a non-Bloch band theory for one-dimensional(1D) non-Hermitian topological superconductors. The universal physical properties of non-Hermitian topological superconductors are revealed based on the theory. According to the particle-hole symmetry, there exist reciprocal particle and hole loops of generalized Brillouin zone. The critical point of quantum phase transition, where the energy gap closes, appears when the particle and hole loops intersect at Bloch points. If the non-Hermitian system has non-Hermitian skin effects, the non-Hermitian skin effect should be the Z 2 skin effect: the corresponding eigenstates of particle and hole localize at opposite ends of an open chain, respectively. The non-Bloch band theory is applied to two examples, non-Hermitian p- and s-wave topological superconductors. In terms of Majorana Pfaffian, a Z 2 non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.

Transient non-Hermitian skin effect

The discovery of non-Hermitian skin effect (NHSE) has opened an exciting direction for unveiling unusual physics and phenomena in non-Hermitian system. Despite notable theoretical breakthroughs, actual observation of NHSE’s whole evolvement, however, relies mainly on gain medium to provide amplified mode. It typically impedes the development of simple, robust system. Here, we show that a passive system is fully capable of supporting the observation of the complete evolution picture of NHSE, without the need of any gain medium. With a simple lattice model and acoustic ring resonators, we use complex-frequency excitation to create virtual gain effect, and experimentally demonstrate that exact NHSE can persist in a totally passive system during a quasi-stationary stage. This results in the transient NHSE: passive construction of NHSE in a short time window. Despite the general energy decay, the localization character of skin modes can still be clearly witnessed and successfully exploited. Our findings unveil the importance of excitation in realizing NHSE and paves the way towards studying the peculiar features of non-Hermitian physics with diverse passive platforms.

Flux-controlled skin effect and topological transition in a dissipative two-leg ladder model

The synthetic gauge field and dissipation are of crucial importance in both fundamental physics and applications. Here, we investigate the interplay of the uniform flux and the on-site gain and loss by considering a dissipative two-leg ladder model. By calculating the spectral winding number and the generalized Brillouin zone, we predict the non-Hermitian skin effect, whose skin modes display the bipolar localization. This skin effect emerges when the flux is not an integer multiple of $\ensuremath{\pi}$. We further demonstrate the breakdown of the chiral currents due to the presence of the skin effect by studying single-particle dynamics. Moreover, we show that the non-Hermiticity can drive a flux-dependent topological transition characterized by a hidden Chern number. Our results provide a scheme to manipulate the non-Hermitian skin effect and topological phase transitions, which may find potential applications in lasing, light manipulating, and signal processing.

Non-Hermitian physics in magnetic systems

Non-Hermitian Hamiltonians provide an alternative perspective on the dynamics of quantum and classical systems coupled non-conservatively to an environment. Once primarily an interest of mathematical physicists, the theory of non-Hermitian Hamiltonians has solidified and expanded to describe various physically observable phenomena in optical, photonic, and condensed matter systems. Self-consistent descriptions of quantum mechanics based on non-Hermitian Hamiltonians have been developed and continue to be refined. In particular, non-Hermitian frameworks to describe magnonic and hybrid magnonic systems have gained popularity and utility in recent years with new insights into the magnon topology, transport properties, and phase transitions coming into view. Magnonic systems are in many ways a natural platform in which to realize non-Hermitian physics because they are always coupled to a surrounding environment and exhibit lossy dynamics. In this Perspective, we review recent progress in non-Hermitian frameworks to describe magnonic and hybrid magnonic systems, such as cavity magnonic systems and magnon–qubit coupling schemes. We discuss progress in understanding the dynamics of inherently lossy magnetic systems as well as systems with gain induced by externally applied spin currents. We enumerate phenomena observed in both purely magnonic and hybrid magnonic systems which can be understood through the lens of non-Hermitian physics, such as PT and anti-PT-symmetry breaking, dynamical magnetic phase transitions, non-Hermitian skin effect, and the realization of exceptional points and surfaces. Finally, we comment on some open problems in the field and discuss areas for further exploration.

Observation of size-dependent boundary effects in non-Hermitian electric circuits

The non-Hermitian systems with the non-Hermitian skin effect (NHSE) are very sensitive to the imposed boundary conditions and lattice sizes, which lead to size-dependent non-Hermitian skin effects. Here, we report the experimental observation of NHSE with different boundary conditions and different lattice sizes in the unidirectional hopping model based on a circuit platform. The circuit admittance spectra and corresponding eigenstates are very sensitive to the presence of the boundary. Meanwhile, our experimental results show how the lattice sizes and boundary terms together affect the strength of NHSE. Therefore, our electric circuit provides a good platform to observe size-dependent boundary effects in non-Hermitian systems.