Non-Bloch Band Theory of Non-Hermitian Systems

Dynamic protected states in the non-Hermitian system

The non-Hermitian skin effect and nonreciprocal behavior are sensitive to the boundary conditions, which are unique features of non-Hermitian systems. In such systems, eigenenergies can become complex, and all eigenstates tend to localize at the boundary, a phenomenon that contrasts with Hermitian topologies. In this work, we theoretically study the dynamic behavior of the propagation of Gaussian wavepackets inside a non-Hermitian lattice and analyze the self-acceleration process of bulk state or Gaussian wavepackets toward the system's boundary. The initial wavepackets will not only propagate toward the side where the eigenstates are localized, but also their momentum will approach to a specific value. This value corresponds to the maximum imaginary components of the energy dispersion. In addition, if the wavepackets in the momentum space cover this specific momentum, they will eventually exhibit exponentially increasing amplitudes with time evolution, maintaining the dynamic protected condition for an extended period of time until they approach the boundary. We also take two widely used toy models as examples in one and two dimensions to verify the correspondence of the non-Hermitian skin effect and the dynamic protected state.

Non-Hermitian Mott Skin Effect

We propose a novel type of skin effects in non-Hermitian quantum many-body systems that we dub a "non-Hermitian Mott skin effect." This phenomenon is induced by the interplay between strong correlations and the non-Hermitian point-gap topology. The Mott skin effect induces extreme sensitivity to the boundary conditions only in the spin degree of freedom (i.e., the charge distribution is not sensitive to boundary conditions), which is in sharp contrast to the ordinary non-Hermitian skin effect in noninteracting systems. Concretely, we elucidate that a bosonic non-Hermitian chain exhibits the Mott skin effect in the strongly correlated regime by closely examining an effective Hamiltonian. The emergence of the Mott skin effect is also supported by numerical diagonalization of the bosonic chain. The difference between the ordinary non-Hermitian skin effect and the Mott skin effect is also reflected in the time evolution of physical quantities; under the time evolution spin accumulation is observed while the charge distribution remains spatially uniform.

Observation of Topological Transition in Floquet Non-Hermitian Skin Effects in Silicon Photonics

Non-Hermitian physics has greatly enriched our understanding of nonequilibrium phenomena and uncovered novel effects such as the non-Hermitian skin effect (NHSE) that has profoundly revolutionized the field. NHSE has been predicted in systems with nonreciprocal couplings which, however, are challenging to realize in experiments. Without nonreciprocal couplings, the NHSE can also emerge in systems with coexisting gauge fields and loss or gain (e.g., in Floquet non-Hermitian systems). However, such Floquet NHSE remains largely unexplored in experiments. Here, we realize the Floquet NHSEs in periodically modulated optical waveguides integrated on a silicon photonic platform. By engineering the artificial gauge fields induced by the periodical modulation, we observe various Floquet NHSE phases and unveil their rich topological transitions. Remarkably, we discover the transitions between the unipolar NHSE phases and an unconventional bipolar NHSE phase, which is accompanied by the directional reversal of the NHSEs. The underlying physics is revealed by the band winding in complex quasienergy space which undergoes a topology change from isolated loops with the same winding to linked loops with opposite windings. Our work unfolds a new route toward Floquet NHSEs originating from the interplay between gauge fields and dissipation effects, and thus offers fundamentally new ways for steering light and other waves.

Observation of Non-Hermitian Edge Burst in Quantum Dynamics

The non-Hermitian skin effect, by which the eigenstates of the Hamiltonian are predominantly localized at the boundary, has revealed a strong sensitivity of non-Hermitian systems to the boundary condition. Here we experimentally observe a striking boundary-induced dynamical phenomenon known as the non-Hermitian edge burst, which is characterized by a sharp boundary accumulation of loss in non-Hermitian time evolutions. In contrast to the eigenstate localization, the edge burst represents a generic non-Hermitian dynamical phenomenon that occurs in real time. Our experiment, based on photonic quantum walks, not only confirms the prediction of the phenomenon, but also unveils its complete space-time dynamics. Our observation of edge burst paves the way for studying the rich real-time dynamics in non-Hermitian topological systems.

Kadanoff-Baym Equations for Interacting Systems with Dissipative Lindbladian Dynamics

The extraordinary quantum properties of nonequilibrium systems governed by dissipative dynamics have become a focal point in contemporary scientific inquiry. The nonequilibrium Green's functions (NEGF) theory provides a versatile method for addressing driven nondissipative systems, utilizing the powerful diagrammatic technique to incorporate correlation effects. We here present a second-quantization approach to the dissipative NEGF theory, reformulating Keldysh ideas to accommodate Lindbladian dynamics and extending the Kadanoff-Baym equations accordingly. Generalizing diagrammatic perturbation theory for many-body Lindblad operators, the formalism enables correlated and dissipative real-time simulations for the exploration of transient and steady-state changes in the electronic, transport, and optical properties of materials.

Experimental observation of exceptional bound states in a classical circuit network

Exceptional bound (EB) states represent a unique new class of robust bound states protected by the defectiveness of non-Hermitian exceptional points. Conceptually distinct from the more well-known topological states and non-Hermitian skin states, they were recently discovered as a novel source of negative entanglement entropy in the quantum entanglement context. Yet, EB states have been physically elusive, being originally interpreted as negative probability eigenstates of the propagator of non-Hermitian Fermi gases. In this work, we show that EB states are in fact far more ubiquitous, also arising robustly in broad classes of systems whether classical or quantum. This hinges crucially on a newly-discovered spectral flow that rigorously justifies the EB nature of small candidate lattice systems. As a highlight, we present their first experimental realization through an electrical circuit, where they manifest as prominent stable resonant voltage profiles. Our work brings a hitherto elusive but fundamentally distinctive quantum phenomenon into the realm of classical metamaterials, and provides a novel pathway for the engineering of robust modes in otherwise sensitive systems..

Topological origin of non-Hermitian skin effect in higher dimensions and uniform spectra

The non-Hermitian skin effect is an iconic phenomenon characterized by the aggregation of eigenstates near the system boundaries in non-Hermitian systems. While extensively studied in one dimension, understanding the skin effect and extending the non-Bloch band theory to higher dimensions encounter a formidable challenge, primarily due to infinite lattice geometries or open boundary conditions. This work adopts a point-gap perspective and unveils that non-Hermitian skin effect in all spatial dimensions originates from point gaps. We introduce the concept of uniform spectra and reveal that regardless of lattice geometry, their energy spectra are universally given by the uniform spectra, even though their manifestations of skin modes may differ. Building on the uniform spectra, we demonstrate how to account for the skin effect with generic lattice cuts and establish the connections of skin modes across different geometric shapes via momentum-basis transformations. Our findings highlight the pivotal roles point gaps play, offering a unified understanding of the topological origin of non-Hermitian skin effect in all dimensions.

Construction and Observation of Flexibly Controllable High-Dimensional Non-Hermitian Skin Effects

Non-Hermitian skin effect (NHSE) is one of the most fundamental phenomena in non-Hermitian physics. It is established that 1D NHSE originates from the nontrivial spectral winding topology. However, the topological origin behind the higher-dimensional NHSE remains unclear, which poses a substantial challenge in constructing and manipulating high-dimensional NHSEs. Here, an intuitive bottom-to-top scheme to construct high-dimensional NHSEs is proposed, through assembling multiple independent 1D NHSEs. Not only the elusive high-dimensional NHSEs can be effectively predicted from the well-defined 1D spectral winding topologies, but also the high-dimensional generalized Brillouin zones can be directly synthesized from the 1D counterparts. As examples, two 2D nonreciprocal acoustic metamaterials are experimentally implemented to demonstrate highly controllable multi-polar NHSEs and hybrid skin-topological effects, where the sound fields can be frequency-selectively localized at any desired corners and boundaries. These results offer a practicable strategy for engineering high-dimensional NHSEs, which can boost advanced applications such as selective filters and directional amplifiers.

Pseudomagnetic suppression of non-Hermitian skin effect

It has recently been shown that the non-Hermitian skin effect can be suppressed by magnetic fields. In this work, using a two-dimensional tight-binding lattice, we demonstrate that a pseudomagnetic field can also lead to the suppression of the non-Hermitian skin effect. With an increasing pseudomagnetic field, the skin modes are found to be pushed into the bulk, accompanied by the reduction of skin topological area and the restoration of Landau level energies. Our results provide a time-reversal invariant route to localization control and could be useful in various classical wave devices that are able to host the non-Hermitian skin effect but inert to magnetic fields.

Self acceleration from spectral geometry in dissipative quantum-walk dynamics

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry.

Observation of Non-Hermitian Edge Burst Effect in One-Dimensional Photonic Quantum Walk

Non-Hermitian systems can exhibit unique quantum phases without any Hermitian counterparts. For example, the latest theoretical studies predict a new surprising phenomenon that bulk bands can localize and dissipate prominently at the system boundary, which is dubbed the non-Hermitian edge burst effect. Here we realize a one-dimensional non-Hermitian Su-Schrieffer-Heeger lattice with bulk translation symmetry implemented with a photonic quantum walk. Employing time-resolved single-photon detection to characterize the chiral motion and boundary localization of bulk bands, we determine experimentally that the dynamics underlying the non-Hermitian edge burst effect is due to the interplay of non-Hermitian skin effect and imaginary band gap closing. This new non-Hermitian physical effect deepens our understanding of quantum dynamics in open quantum systems.

Incoherent non-Hermitian skin effect in photonic quantum walks

The non-Hermitian skin effect describes the concentration of an extensive number of eigenstates near the boundaries of certain dissipative systems. This phenomenon has raised a huge interest in different areas of physics, including photonics, deeply expanding our understanding of non-Hermitian systems and opening up new avenues in both fundamental and applied aspects of topological phenomena. The skin effect has been associated to a nontrivial point-gap spectral topology and has been experimentally demonstrated in a variety of synthetic matter systems, including photonic lattices. In most of physical models exhibiting the non-Hermitian skin effect full or partial wave coherence is generally assumed. Here we push the concept of skin effect into the fully incoherent regime and show that rather generally (but not universally) the non-Hermitian skin effect persists under dephasing dynamics. The results are illustrated by considering incoherent light dynamics in non-Hermitian photonic quantum walks.

Probing PT-Symmetry Breaking of Non-Hermitian Topological Photonic States via Strong Photon-Magnon Coupling

Light-matter interaction is crucial to both understanding fundamental phenomena and developing versatile applications. Strong coupling, robustness, and controllability are the three most important aspects in realizing light-matter interactions. Topological and non-Hermitian photonics have provided frameworks for robustness and control flexibility, respectively. How to engineer the properties of the edge state such as photonic density of state by using non-Hermiticity while ensuring topological protection has not been fully studied. Here we construct a parity-time-symmetric dimerized photonic lattice and probe the spontaneous PT-symmetry breaking of the edge states by utilizing the strong coupling between the photonic mode and a spin ensemble. Our Letter presents an accurate and almost noninvasive approach for investigating non-Hermitian topological states, while also offering methodologies for the implementation and manipulation of topological light-matter interactions.

Non-Hermitian Moiré Valley Filter

A valley filter capable of generating a valley-polarized current is a crucial element in valleytronics, yet its implementation remains challenging. Here, we propose a valley filter made of a graphene bilayer which exhibits a 1D moiré pattern in the overlapping region of the two layers controlled by heterostrain. In the presence of a lattice modulation between layers, electrons propagating in one layer can have valley-dependent dissipation due to valley asymmetric interlayer coupling, thus giving rise to a valley-polarized current. Such a process can be described by an effective non-Hermitian theory, in which the valley filter is driven by a valley-resolved non-Hermitian skin effect. Nearly 100% valley polarization can be achieved within a wide parameter range and the functionality of the valley filter is electrically tunable. The non-Hermitian topological scenario of the valley filter ensures high tolerance against imperfections such as disorder and edge defects. Our work opens a new route for efficient and robust valley filters while significantly relaxing the stringent implementation requirements.

Topological phases and non-Hermitian topology in tunable nonreciprocal cyclic three-mode optical systems

We propose a method for simulating a 1D non-Hermitian Su-Schrieffer-Heeger model with modulated nonreciprocal hopping using a cyclic three-mode optical system. The current system exhibits different localization of topologically nontrivial phases, which can be characterized by the winding number. We find that the eigenenergies of such a system undergo a real-complex transition as the nonreciprocal hopping changes, accompanied by a non-Bloch parity-time symmetry breaking. We explain this phase transition by considering the evolution of saddle points on the complex energy plan and the ratio of complex eigenenergies. Additionally, we demonstrate that the skin states resulting from the non-Hermitian skin effect possess higher-order exceptional points under the critical point of the non-Bloch parity-time phase transition. Furthermore, we investigate the non-Hermitian skin phase transition by the directional mean inverse participation ratio and the generalized Brillouin zone. This work provides an alternative way to investigate the novel topological and non-Hermitian effects in nonreciprocal optical systems.

Bulk-Boundary Correspondence in Point-Gap Topological Phases

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the K theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.

Bulk-Edge Correspondence for Nonlinear Eigenvalue Problems

Although topological phenomena attract growing interest not only in linear systems but also in nonlinear systems, the bulk-edge correspondence under the nonlinearity of eigenvalues has not been established so far. We address this issue by introducing auxiliary eigenvalues. We reveal that the topological edge states of auxiliary eigenstates are topologically inherited as physical edge states when the nonlinearity is weak but finite (i.e., auxiliary eigenvalues are monotonic as for the physical one). This result leads to the bulk-edge correspondence with the nonlinearity of eigenvalues.

Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems

We present a pedagogical review of the periodically driven non-Hermitian systems, particularly on the rich interplay between the non-Hermitian skin effect and the topology. We start by reviewing the non-Bloch band theory of the static non-Hermitian systems and discuss the establishment of its generalized bulk-boundary correspondence (BBC). Ultimately, we focus on the non-Bloch band theory of two typical periodically driven non-Hermitian systems: harmonically driven non-Hermitian system and periodically quenched non-Hermitian system. The non-Bloch topological invariants were defined on the generalized Brillouin zone and the real space wave functions to characterize the Floquet non-Hermtian topological phases. Then, the generalized BBC was established for the two typical periodically driven non-Hermitian systems. Additionally, we review novel phenomena in the higher-dimensional periodically driven non-Hermitian systems, including Floquet non-Hermitian higher-order topological phases and Floquet hybrid skin-topological modes. The experimental realizations and recent advances have also been surveyed. Finally, we end with a summarization and hope this pedagogical review can motivate further research on Floquet non-Hermtian topological physics.

Exact Solution of the Bose-Hubbard Model with Unidirectional Hopping

A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.

Topological spin textures in electronic non-Hermitian systems

Non-Hermitian systems have been discussed mostly in the context of open systems and nonequilibrium. Recent experimental progress is much from optical, cold-atomic, and classical platforms due to the vast tunability and clear identification of observables. However, their counterpart in solid-state electronic systems in equilibrium remains unmasked although highly desired, where a variety of materials are available, calculations are solidly founded, and accurate spectroscopic techniques can be applied. We demonstrate that, in the surface state of a topological insulator with spin-dependent relaxation due to magnetic impurities, highly nontrivial topological soliton spin textures appear in momentum space. Such spin-channel phenomena are delicately related to the type of non-Hermiticity and correctly reveal the most robust non-Hermitian features detectable spectroscopically. Moreover, the distinct topological soliton objects can be deformed to each other, mediated by topological transitions driven by tuning across a critical direction of doped magnetism. These results not only open a solid-state avenue to exotic spin patterns via spin- and angle-resolved photoemission spectroscopy, but also inspire non-Hermitian dissipation engineering of spins in solids.

Photonic Floquet Skin-Topological Effect

Non-Hermitian skin effect and photonic topological edge states are of great interest in non-Hermitian physics and optics. However, the interplay between them is largely unexplored. Here, we propose and demonstrate experimentally the non-Hermitian skin effect constructed from the nonreciprocal flow of Floquet topological edge states, which can be dubbed "Floquet skin-topological effect." We first show the non-Hermitian skin effect can be induced by structured loss when the one-dimensional (1D) system is periodically driven. Next, based on a two-dimensional (2D) Floquet topological photonic lattice with structured loss, we investigate the interaction between the non-Hermiticity and the topological edge states. We observe that all the one-way edge states are imposed onto specific corners, featuring both the non-Hermitian skin effect and topological edge states. Furthermore, a topological switch for the skin-topological effect is presented by utilizing the phase-transition mechanism. Our experiment paves the way for realizing non-Hermitian topological effects in nonlinear and quantum regimes.

Geometric Origin of Non-Bloch PT Symmetry Breaking

The parity-time (PT) symmetry of a non-Hermitian Hamiltonian leads to real (complex) energy spectrum when the non-Hermiticity is below (above) a threshold. Recently, it has been demonstrated that the non-Hermitian skin effect generates a new type of PT symmetry, dubbed the non-Bloch PT symmetry, featuring unique properties such as high sensitivity to the boundary condition. Despite its relevance to a wide range of non-Hermitian lattice systems, a general theory is still lacking for this generic phenomenon even in one spatial dimension. Here, we uncover the geometric mechanism of non-Bloch PT symmetry and its breaking. We find that non-Bloch PT symmetry breaking occurs by the formation of cusps in the generalized Brillouin zone (GBZ). Based on this geometric understanding, we propose an exact formula that efficiently determines the breaking threshold. Moreover, we predict a new type of spectral singularities associated with the symmetry breaking, dubbed non-Bloch van Hove singularity, whose physical mechanism fundamentally differs from their Hermitian counterparts. This singularity is experimentally observable in linear responses.

Effective enhancement of the non-Hermitian corner skin effect in reciprocal photonic crystals

With the rich physical phenomena arising from non-Hermitian systems, the non-Hermitian skin effect (NHSE) has become a current research hotspot. Nowadays, the corner skin effect based on non-reciprocal photonic crystals has been proposed. Considering the complexity of realizing non-reciprocity, the corner skin effect based on reciprocal photonic crystals is well worth investigating. In this Letter, a non-Hermitian reciprocal geometry-dependent corner skin effect based on two-dimensional photonic crystals is presented, which is manifested as the distribution of eigenstates on the corners of a particular geometry by applying open boundary conditions in both directions of photonic crystals. For the better application of the NHSE in the future, such as highly sensitive sensors and lasers, a new, to the best of our knowledge, method that can effectively enhance the performance of the NHSE in photonic crystals is proposed. The method introduces both gain and loss in an ideal photonic crystal to enhance the non-Hermitian specificity of the system, which improves the performance of the non-Hermitian corner skin effect of photonic crystals by 64.5%. Furthermore, this geometry-dependent corner skin effect is corroborated with the spectral topology.

Synthetic Non-Abelian Gauge Fields for Non-Hermitian Systems

Non-Abelian gauge fields are versatile tools for synthesizing topological phenomena, but have so far been mostly studied in Hermitian systems, where gauge flux has to be defined from a closed loop in order for vector potentials, whether Abelian or non-Abelian, to become physically meaningful. We show that this condition can be relaxed in non-Hermitian systems by proposing and studying a generalized Hatano-Nelson model with imbalanced non-Abelian hopping. Despite lacking gauge flux in one dimension, non-Abelian gauge fields create rich non-Hermitian topological consequences. With SU(2) gauge fields, the braiding degrees that can be achieved are twice the highest hopping order of a lattice model, indicating the utility of spinful freedom to attain high-order nontrivial braiding. At both ends of an open chain, non-Abelian gauge fields lead to the simultaneous presence of non-Hermitian skin modes, whose population can be effectively tuned near the exceptional points. Generalizing to two dimensions, the gauge invariance of Wilson loops can also break down in non-Hermitian lattices dressed with non-Abelian gauge fields. Toward realization, we present a concrete experimental proposal for non-Abelian gauge fields in non-Hermitian systems via the synthetic frequency dimension of a polarization-multiplexed fiber ring resonator.

Role of Topology in Relaxation of One-Dimensional Stochastic Processes

Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions.

Universal Platform of Point-Gap Topological Phases from Topological Materials

Whereas point-gap topological phases are responsible for exceptional phenomena intrinsic to non-Hermitian systems, their realization in quantum materials is still elusive. Here, we propose a simple and universal platform of point-gap topological phases constructed from Hermitian topological insulators and superconductors. We show that (d-1)-dimensional point-gap topological phases are realized by making a boundary in d-dimensional topological insulators and superconductors dissipative. A crucial observation of the proposal is that adding a decay constant to boundary modes in d-dimensional topological insulators and superconductors is topologically equivalent to attaching a (d-1)-dimensional point-gap topological phase to the boundary. We furthermore establish the proposal from the extended version of the Nielsen-Ninomiya theorem, relating dissipative gapless modes to point-gap topological numbers. From the bulk-boundary correspondence of the point-gap topological phases, the resultant point-gap topological phases exhibit exceptional boundary states or in-gap higher-order non-Hermitian skin effects.

Terminal-coupling induced critical eigenspectrum transition in closed non-Hermitian loops

A hallmark feature of non-Hermitian (NH) systems is the non-Hermitian skin effect (NHSE), in which the eigenenergy spectra of the system under open boundary conditions (OBC) and periodic boundary conditions (PBC) differ markedly from each other. In particular, the critical NHSE occurs in systems consisting of multiple non-Hermitian chains coupled in parallel where even an infinitesimally small inter-chain coupling can cause the thermodynamic-limit eigenenergy spectrum of the system to deviate significantly from the OBC spectra of the individual component chains. We overturn the conventional wisdom that multiple chains are required for such critical transitions by showing that such a critical effect can also be induced in a single finite-length non-Hermitian chain where its two ends are connected together by a weak terminal coupling to form a closed loop. An infinitesimally small terminal coupling can induce the thermodynamic-limit energy spectrum of the closed loop to switch from the OBC to the PBC spectrum of the chain. Similar to the critical NHSE, this switch occurs abruptly when the chain length exceeds a critical size limit. We explain analytically the underlying origin of the effect in a Hatano-Nelson chain system, and demonstrate its generality in more complex one-dimensional non-Hermitian chains. Our findings illustrate the generality of critical size-dependent effects in finite NH systems that arise from the interplay between the interfacial boundary conditions and the influence of edge localization.

Photonic skin-topological effects in microring lattices

We investigate the non-Hermitian Hofstadter-Harper model composed of microring resonators, in which the non-Hermitian skin effect (NHSE) is particularly analyzed. The effect is achieved through the interaction between well-designed gain-loss layouts and artificial gauge fields. Remarkably, we reveal the emergence of a hybrid skin-topological effect (HSTE), where only the original topological edge modes convert to skin modes while bulk modes remain extended. By changing the distributions of gauge fields, we show the NHSE can manifest itself in bulk modes and be localized at specific edges. Using the equivalence of sites in the bulk or at boundaries to 1D SSH chains, we analyze the potential cancellation of NHSE in these configurations. Additionally, we demonstrate a new, to the best of our knowledge, type of HSTE in topological insulators which emerge at any gain-loss interfaces. The study may improve the understanding of the NHSE behavior in 2D topological systems and provide a promising avenue for tuning light propagation and localization.

Observation of the geometry-dependent skin effect and dynamical degeneracy splitting

The non-Hermitian skin effect is a distinctive phenomenon in non-Hermitian systems, which manifests as the anomalous localization of bulk states at the boundary. To understand the physical origin of the non-Hermitian skin effect, a bulk band characterization based on the dynamical degeneracy on an equal frequency contour is proposed, which reflects the strong anisotropy of the spectral function. In this paper, we report the experimental observation of a newly-discovered geometry-dependent non-Hermitian skin effect and dynamical degeneracy splitting in a two-dimensional acoustic crystal and reveal their remarkable correspondence by performing single-frequency excitation measurements. Our work not only provides a controllable experimental platform for studying the non-Hermitian physics, but also confirms the unique correspondence between the non-Hermitian skin effect and the dynamical degeneracy splitting, paving a new way to characterize the non-Hermitian skin effect.

Making Topologically Trivial Non-Hermitian Systems Nontrivial via Gauge Fields

Non-Hermiticity significantly enriches the concepts of symmetry and topology in physics. Particularly, non-Hermiticity gives rise to the ramified symmetries, where the non-Hermitian Hamiltonian H is transformed to H^{†}. For time-reversal (T) and sublattice symmetries, there are six ramified symmetry classes leading to novel topological classifications with various non-Hermitian skin effects. As artificial crystals are the main experimental platforms for non-Hermitian physics, there exists the symmetry barrier for realizing topological physics in the six ramified symmetry classes: while artificial crystals are in spinless classes with T^{2}=1, nontrivial classifications dominantly appear in spinful classes with T^{2}=-1. Here, we present a general mechanism to cross the symmetry barrier. With an internal parity symmetry P, the square of the combination T[over ˜]=PT can be modified by appropriate gauge fluxes. Using the general mechanism, we systematically construct spinless models for all non-Hermitian spinful topological phases in one and two dimensions, which are experimentally realizable. Our Letter suggests that gauge structures may significantly enrich non-Hermitian physics at the fundamental level.

Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

Manipulating directional flow in a two-dimensional photonic quantum walk under a synthetic magnetic field

Matter transport is a fundamental process in nature. Understanding and manipulating flow in a synthetic media often have rich implications for modern device design. Here we experimentally demonstrate directional transport of photons in a two-dimensional quantum walk, where the light propagation is highly tunable through dissipation and synthetic magnetic flux. The directional flow hereof underlies the emergence of the non-Hermitian skin effect, with its orientation continuously adjustable through the photon-loss parameters. By contrast, the synthetic magnetic flux originates from an engineered geometric phase, which, by inducing localized cyclotron orbits, suppresses the bulk flow through magnetic confinement. We further demonstrate how the directional flow and synthetic flux impact the dynamics of the Floquet topological edge modes along an engineered boundary. Our results exemplify an intriguing strategy for engineering directed light transport, highlighting the interplay of non-Hermiticity and gauge fields in synthetic systems of higher dimensions.

Non-Hermitian Floquet Topological Matter-A Review

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird's-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

Enhancement of Second-Order Non-Hermitian Skin Effect by Magnetic Fields

The non-Hermitian skin effect is a unique phenomenon in which an extensive number of eigenstates are localized at the boundaries of a non-Hermitian system. Recent studies show that the non-Hermitian skin effect is significantly suppressed by magnetic fields. In contrast, we demonstrate that the second-order skin effect (SOSE) is robust and can even be enhanced by magnetic fields. Remarkably, SOSE can also be induced by magnetic fields from a trivial non-Hermitian system that does not experience any skin effect at zero field. These properties are intimately related to to the persistence and emergence of topological line gaps in the complex energy spectrum in the presence of magnetic fields. Moreover, we show that a magnetic field can drive a non-Hermitian system from a hybrid skin effect, where the first-order skin effect and SOSE coexist, to pure SOSE. Our results describe a qualitatively new magnetic field behavior of the non-Hermitian skin effect.

Observation of a Topological Edge State Stabilized by Dissipation

Robust states emerging at the boundary of a system constitute a hallmark for topological band structures. Other than in closed systems, topologically protected states can occur even in systems with a trivial band structure, if exposed to suitably modulated losses. Here, we study the dissipation-induced emergence of a topological band structure in a non-Hermitian one-dimensional lattice system, realized by arrays of plasmonic waveguides with tailored loss. We obtain direct evidence for a topological edge state that resides in the center of the band gap. By tuning dissipation and hopping, the formation and breakdown of an interface state between topologically distinct regions is demonstrated.

Emulating Non-Hermitian Dynamics in a Finite Non-Dissipative Quantum System

We discuss the emulation of non-Hermitian dynamics during a given time window using a low-dimensional quantum system coupled to a finite set of equidistant discrete states acting as an effective continuum. We first emulate the decay of an unstable state and map the quasi-continuum parameters, enabling the precise approximation of non-Hermitian dynamics. The limitations of this model, including in particular short- and long-time deviations, are extensively discussed. We then consider a driven two-level system and establish criteria for non-Hermitian dynamics emulation with a finite quasi-continuum. We quantitatively analyze the signatures of the finiteness of the effective continuum, addressing the possible emergence of non-Markovian behavior during the time interval considered. Finally, we investigate the emulation of dissipative dynamics using a finite quasi-continuum with a tailored density of states. We show through the example of a two-level system that such a continuum can reproduce non-Hermitian dynamics more efficiently than the usual equidistant quasi-continuum model.

Dimensional Transmutation from Non-Hermiticity

Dimensionality plays a fundamental role in the classification of novel phases and their responses. In generic lattices of 2D and beyond, however, we found that non-Hermitian couplings do not merely distort the Brillouin zone (BZ), but can in fact alter its effective dimensionality. This is due to the fundamental noncommutativity of multidimensional non-Hermitian pumping, which obstructs the usual formation of a generalized complex BZ. As such, basis states are forced to assume "entangled" profiles that are orthogonal in a lower dimensional effective BZ, completely divorced from any vestige of lattice Bloch states unlike conventional skin states. Characterizing this reduced dimensionality is an emergent winding number intimately related to the homotopy of noncontractible spectral paths. We illustrate this dimensional transmutation through a 2D model whose topological zero modes are protected by a 1D, not 2D, topological invariant. Our findings can be readily demonstrated via the bulk properties of nonreciprocally coupled platforms such as circuit arrays, and provokes us to rethink the fundamental role of geometric obstruction in the dimensional classification of topological states.

Observation of geometry-dependent skin effect in non-Hermitian phononic crystals with exceptional points

Exceptional points and skin effect, as the two distinct hallmark features unique to the non-Hermitian physics, have each attracted enormous interests. Recent theoretical works reveal that the topologically nontrivial exceptional points can guarantee the non-Hermitian skin effect, which is geometry-dependent, relating these two unique phenomena. However, such novel relation remains to be confirmed by experiments. Here, we realize a non-Hermitian phononic crystal with exceptional points, which exhibits the geometry-dependent skin effect. The exceptional points connected by the bulk Fermi arcs, and the skin effects with the geometry dependence, are evidenced in simulations and experiments. Our work, building an experimental bridge between the exceptional points and skin effect and uncovering the unconventional geometry-dependent skin effect, expands a horizon in non-Hermitian physics.

Unconventional quantum criticality in a non-Hermitian extended Kitaev chain

We investigate the nature of quantum criticality and topological phase transitions near the critical lines obtained for the extended Kitaev chain with next nearest neighbor hopping parameters and non-Hermitian chemical potential. We surprisingly find multiple gap-less points, the locations of which in the momentum space can change along the critical line unlike the Hermitian counterpart. The interesting simultaneous occurrences of vanishing and sign flipping behavior by real and imaginary components, respectively of the lowest excitation is observed near the topological phase transition. Introduction of non-Hermitian factor leads to an isolated critical point instead of a critical line and hence, reduced number of multi-critical points as compared to the Hermitian case. The critical exponents obtained for the multi-critical and critical points show a very distinct behavior from the Hermitian case.

Dynamical Degeneracy Splitting and Directional Invisibility in Non-Hermitian Systems

In this Letter, we introduce the concept of dynamical degeneracy splitting to describe the anisotropic decay behaviors in non-Hermitian systems. We demonstrate that systems with dynamical degeneracy splitting exhibit two distinctive features: (i) the system shows frequency-resolved non-Hermitian skin effect; (ii) Green's function exhibits anomalous behavior at given frequency, leading to uneven broadening in spectral function and anomalous scattering. As an application, we propose directional invisibility based on wave packet dynamics to investigate the geometry-dependent skin effect in higher dimensions. Our work elucidates a faithful correspondence between non-Hermitian skin effect and Green's function, offering a guiding principle for exploration of novel physical phenomena emerging from this effect.

Non-Hermitian topology in static mechanical metamaterials

The combination of broken Hermiticity and band topology in physical systems unveils a novel bound state dubbed as the non-Hermitian skin effect (NHSE). Active control that breaks reciprocity is usually used to achieve NHSE, and gain and loss in energy are inevitably involved. Here, we demonstrate non-Hermitian topology in a mechanical metamaterial system by exploring its static deformation. Nonreciprocity is introduced via passive modulation of the lattice configuration without resorting to active control and energy gain/loss. Intriguing physics such as the reciprocal and higher-order skin effects can be tailored in the passive system. Our study provides an easy-to-implement platform for the exploration of non-Hermitian and nonreciprocal phenomena beyond conventional wave dynamics.

Topological Metamaterials

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Active metamaterials for realizing odd mass density

Solids built out of active components have exhibited odd elastic stiffness tensors whose active moduli appear in the antisymmetric part and which give rise to non-Hermitian static and dynamic phenomena. Here, we present a class of active metamaterial featured with an odd mass density tensor whose asymmetric part arises from active and nonconservative forces. The odd mass density is realized using metamaterials with inner resonators connected by asymmetric and programmable feed-forward control on acceleration and active forces along the two perpendicular directions. The active forces produce unbalanced off-diagonal mass density coupling terms, leading to non-Hermiticity. The odd mass is then experimentally validated through a one-dimensional nonsymmetric wave coupling where propagating transverse waves are coupled with longitudinal ones whereas the reverse is forbidden. We reveal that the two-dimensional active metamaterials with the odd mass can perform in either energy-unbroken or energy-broken phases separated by exceptional points along principal directions of the mass density. The odd mass density contributes to the wave anisotropy in the energy-unbroken phase and directional wave energy gain in the energy-broken phase. We also numerically illustrate and experimentally demonstrate the two-dimensional wave propagation phenomena that arise from the odd mass in active solids. Finally, the existence of non-Hermitian skin effect is discussed in which boundaries host an extensive number of localized modes. It is our hope that the emergent concept of the odd mass can open up a new research platform for mechanical non-Hermitian system and pave the ways for developing next-generation wave steering devices.

Quantum-Squeezing-Induced Point-Gap Topology and Skin Effect

We theoretically predict the squeezing-induced point-gap topology together with a symmetry-protected Z_{2} "skin effect" in a one-dimensional (1D) quadratic-bosonic system. Protected by a time-reversal symmetry, such a topology is associated with a novel Z_{2} invariant (similar to quantum spin-Hall insulators), which is fully capable of characterizing the occurrence of the Z_{2} skin effect. Focusing on zero energy, the parameter regime of this skin effect in the phase diagram just corresponds to a "real- and point-gap coexisting topological phase." Moreover, this phase associated with the symmetry-protected Z_{2} skin effect is experimentally observable by detecting the steady-state power spectral density. Our Letter is of fundamental interest in enriching non-Bloch topological physics by introducing quantum squeezing and has potential applications for the engineering of symmetry-protected sensors based on the Z_{2} skin effect.

Mixed state behavior of Hermitian and non-Hermitian topological models with extended couplings

Geometric phase is an important tool to define the topology of the Hermitian and non-Hermitian systems. Besides, the range of coupling plays an important role in realizing higher topological indices and transition among them. With a motivation to understand the geometric phases for mixed states, we discuss finite temperature analysis of Hermitian and non-Hermitian topological models with extended range of couplings. To understand the geometric phases for the mixed states, we use Uhlmann phase and discuss the merit-limitation with respect extended range couplings. We extend the finite temperature analysis to non-Hermitian models and define topological invariant for different ranges of coupling. We include the non-Hermitian skin effect, and provide the derivation of topological invariant in the generalized Brillouin zone and their mixed state behavior also. We also adopt mixed geometric phases through interferometric approach, and discuss the geometric phases of extended-range (Hermitian and non-Hermitian) models at finite temperature.

Tensile strain and finite size modulation of low lattice thermal conductivity in monolayer TMDCs (HfSe2 and ZrS2) from first-principles: a comparative study

With excellent physical and chemical properties, 2D TMDC materials have been widely used in engineering applications, but they inevitably suffer from the dual effects of strain and device size. As typical 2D TMDCs, HfSe₂ and ZrS₂ are reported to have excellent thermoelectric properties. Thermal transport properties have great significance for exerting the performance of materials, ensuring device lifetime and stable operation, but current research is not detailed enough. Here, first-principles combined with the phonon Boltzmann transport equation are used to study the phonon transport inside monolayer HfSe₂ and ZrS₂ under tensile strain and finite size, and explore the band structure properties. Our research shows that they have similar phonon dispersion curve structures, and the band gap of HfSe₂ increases monotonically with the increase of tensile strain, while the bandgap of ZrS₂ increases and then decreases with the increase of tensile strain. Thermal conductivity has obvious strain dependence: with the increase of tensile strain, the thermal conductivity of HfSe₂ gradually decreases, while that of ZrS₂ increases slightly, and then gradually decreases. Reducing the system size can limit the contribution of phonons with a long mean free path, significantly decreasing thermal conductivity through the controlling effect of tensile strain. The mode contribution of thermal conductivity is systematically investigated, and anharmonic properties including mode and frequency-level scattering rates, group velocity and Grüneisen parameters are used to explain the associated mechanism. Phonon scattering processes and channels in various cases are discussed in detail. Our research provides a detailed understanding of the phonon transport and electronic structural properties of low thermal conductivity monolayers of HfSe₂ and ZrS₂, and further completes the study of thermal transport of the two materials under strain and size tuning, which will provide a foundation for further popularization and application.

Continuum of Bound States in a Non-Hermitian Model

In a Hermitian system, bound states must have quantized energies, whereas free states can form a continuum. We demonstrate how this principle fails for non-Hermitian systems, by analyzing non-Hermitian continuous Hamiltonians with an imaginary momentum and Landau-type vector potential. The eigenstates, which we call "continuum Landau modes" (CLMs), have Gaussian spatial envelopes and form a continuum filling the complex energy plane. We present experimentally realizable 1D and 2D lattice models that host CLMs; the lattice eigenstates are localized and have other features matching the continuous model. One of these lattices can serve as a rainbow trap, whereby the response to an excitation is concentrated at a position proportional to the frequency. Another lattice can act a wave funnel, concentrating an input excitation onto a boundary over a wide frequency bandwidth. Unlike recent funneling schemes based on the non-Hermitian skin effect, this requires a simple lattice design with reciprocal couplings.

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.

Delocalization of interacting directed polymers on a periodic substrate: Localization length and critical exponents from non-Hermitian spectra

We study a classical model of thermally fluctuating polymers confined to two dimensions, experiencing a grooved periodic potential, and subject to pulling forces both along and transverse to the grooves. The equilibrium polymer conformations are described by a mapping to a quantum system with a non-Hermitian Hamiltonian and with fermionic statistics generated by noncrossing interactions among polymers. Using molecular dynamics simulations and analytical calculations, we identify a localized and a delocalized phase of the polymer conformations, separated by a delocalization transition which corresponds (in the quantum description) to the breakdown of a band insulator when driven by an imaginary vector potential. We calculate the average tilt of the many-body system, at arbitrary shear values and filling density of polymer chains, in terms of the complex-valued non-Hermitian band structure. We find the critical shear value, the localization length, and the critical exponent by which the shear modulus diverges in terms of the branch points (exceptional points) in the band structure at which the bandgap closes. We also investigate the combined effects of non-Hermitian delocalization and localization due to both periodicity and disorder, uncovering preliminary evidence that while disorder favors localization at high values, it encourages delocalization at lower values.

Properties of the non-Hermitian SSH model: role ofPTsymmetry

The present work addresses the distinction between the topological properties ofPTsymmetric and non-PTsymmetric scenarios for the non-Hermitian Su-Schrieffer-Heeger model. The non-PTsymmetric case is represented by non-reciprocity in both the inter- and the intra-cell hopping amplitudes, while the one withPTsymmetry 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-PTsymmetric 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 thePTsymmetric 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.