Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas

Anomalous pumping in the non-Hermitian Rice-Mele model

We study topological charge pumping in the Rice-Mele (RM) model with irreciprocal hopping. The non-Hermiticity gives rise to interesting pumping physics, owing to the presence of skin effect and exceptional points. In the static one-dimensional (1D) RM model, we find two independent tuning knobs that can drive the topological transition, namely, non-Hermitian parameterγand system sizeN. To elucidate the system-size dependency, we use a finite-size generalized Brillouin zone scheme to show that the edge modes can be distinguished from the non-hermiticity induced skin modes. Moreover, this scheme can capture the state pumping of topological edge modes as a function ofγin the static 1D RM model and it further provides insight into engineering novel gapless exceptional edge modes with the help of adiabatic drive. Furthermore, we show that the standard topological pumping due to the adiabatic and periodic variation of the model parameters survives even with finiteγ. However, we observe that it depends upon the driving protocols and strength of the non-Hermiticity (γ). With increasingγ, the adiabatic pumping for non-trivial protocols is destroyed first and then re-emerges as an anomalous pumping which does not have any Hermitian counterpart. Additionally, we observe that even a trivial adiabatic protocol can give rise to pumping as opposed to the Hermitian system. This is attributed to the inherent point gap physics of non-Hermitian system which we explain by reformulating a non-Bloch topological invariant for the 1+1D RM model. This invariant explains the number of pumped charges (in each period) for all the driving protocols.

Non-Hermitian Topology in Hermitian Topological Matter

Non-Hermiticity gives rise to distinctive topological phenomena absent in Hermitian systems. However, connection between such intrinsic non-Hermitian topology and Hermitian topology has remained largely elusive. Here, considering the bulk and boundary as an environment and system, respectively, we demonstrate that anomalous boundary states in Hermitian topological insulators exhibit non-Hermitian topology. We study the self-energy capturing the particle exchange between the bulk and boundary, and show that it detects Hermitian topology in the bulk and induces non-Hermitian topology at the boundary. As an illustrative example, we reveal non-Hermitian topology and concomitant skin effect inherently embedded within chiral edge states of Chern insulators. We also identify the emergence of hinge states within effective non-Hermitian Hamiltonians at surfaces of three-dimensional topological insulators. Furthermore, we comprehensively classify our correspondence across all the tenfold symmetry classes of topological insulators and superconductors. Our Letter uncovers hidden connection between Hermitian and non-Hermitian topology, and provides an approach to identifying non-Hermitian topology in quantum matter.

Non-Hermitian Theory of Valley Excitons in Two-Dimensional Semiconductors

Electron-hole exchange interaction in two-dimensional transition metal dichalcogenides is extremely strong due to the dimension reduction, which promises valley-superposed excitonic states with linearly polarized optical emissions. However, strong circular polarization reflecting valley-polarized excitonic states is commonly observed in helicity-resolved optical experiments. Here, we present a non-Hermitian theory of valley excitons by incorporating optical pumping and intrinsic decay, which unveils an anomalous valley-polarized excitonic state with elliptically polarized optical emission. This novel state arises from the non-Hermiticity induced parity-time (PT) symmetry breaking, which impedes the experimental observation of intervalley excitonic coherence effect. At large excitonic center-of-mass momenta, the PT symmetry is restored and the excitonic states recover their valley coherence. Interestingly, the linear polarization directions in optical emissions from these valley-superposed excitonic states are nonorthogonal and even become parallel at exceptional points. Our non-Hermitian theory also predicts a nonzero Berry curvature for valley excitons, which admits a topological excitonic Hall transport beyond the Hermitian predictions.

Non-Hermitian ideal Weyl photonic metamaterials and polarization-momentum resolved ultrahigh absorption

In the past decade, there has been a significant surge of interest in investigating non-Hermitian Hamiltonians, particularly in photonics. The eigenvalues of general non-Hermitian Hamiltonians are complex and possess unique topological features such as exceptional degeneracy. The introduction of non-Hermitian perturbations into Weyl semimetals can transform Weyl points into exceptional rings characterized by multiple topological invariants. However, the ideal realization of Weyl rings within practical three-dimensional structures has remained a significant challenge. In this work, we extend artificial photonic metamaterial structures that can transform ideal Weyl points into non-Hermitian exceptional rings. We show the associated intriguing polarization-momentum ultrahigh absorption, which enables what we believe to be a new device application in non-Hermitian photonics. Our study not only proposes the practical model for ideal non-Hermitian photonic Weyl exceptional rings but also opens the gate of non-Hermitian scattering characterization.

Anomalous Non-Hermitian Open-Boundary Spectrum

For a long time, it was presumed that continuum bands could be readily encompassed by open-boundary spectra, irrespective of the system's modest dimensions. However, our findings reveal a nuanced picture: under open-boundary conditions, the proliferation of complex eigenvalues progresses in a sluggish, oscillating manner as the system expands. Consequently, even in larger systems, the overlap between continuum bands and open-boundary eigenvalues becomes elusive, with the surprising twist that the count of these complex eigenvalues may actually diminish with increasing system size. This counterintuitive trend underscores that the pursuit of an ideal, infinite-sized system scenario does not necessarily align with enlarging the system size. Notably, despite the inherent non-Hermiticity of our system, the eigenstates distribute themselves in a manner reminiscent of Bloch waves. These discoveries hold potential significance for both theoretical explorations and experimental realizations of non-Hermitian systems.

Dynamical Transition Due to Feedback-Induced Skin Effect

The traditional dynamical phase transition refers to the appearance of singularities in an observable with respect to a control parameter for a late-time state or singularities in the rate function of the Loschmidt echo with respect to time. Here, we study the many-body dynamics in a continuously monitored free fermion system with conditional feedback under open boundary conditions. We surprisingly find a novel dynamical transition from a logarithmic scaling of the entanglement entropy to an area-law scaling as time evolves. The transition, which is noticeably different from the conventional dynamical phase transition, arises from the competition between the bulk dynamics and boundary skin effects. In addition, we find that while quasidisorder or disorder cannot drive a transition for the steady state, a transition occurs for the maximum entanglement entropy during the time evolution, which agrees well with the entanglement transition for the steady state of the dynamics under periodic boundary conditions.

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.

Homotopy, symmetry, and non-Hermitian band topology

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time (PT) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases inPT-symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneousPTsymmetry breaking is captured by Chern-Euler and Chern-Stiefel-Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

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.

Manipulation of Weyl Points in Reciprocal and Nonreciprocal Mechanical Lattices

We introduce feedback-measurement technologies to achieve flexible control of Weyl points and conduct the first experimental demonstration of Weyl type I-II transition in mechanical systems. We demonstrate that non-Hermiticity can expand the Fermi arc surface states from connecting Weyl points to Weyl rings, and lead to a localization transition of edge states influenced by the interplay between band topology and the non-Hermitian skin effect. Our findings offer valuable insights into the design and manipulation of Weyl points in mechanical systems, providing a promising avenue for manipulating topological modes in non-Hermitian systems.

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.

Occupation-Dependent Particle Separation in One-Dimensional Non-Hermitian Lattices

We unveil an exotic phenomenon arising from the intricate interplay between non-Hermiticity and many-body physics, namely, an occupation-dependent particle separation for hardcore bosons in a one-dimensional lattice driven by unidirectional non-Hermitian pumping. Taking hardcore bosons as an example, we find that a pair of particles occupying the same unit cell exhibit an opposite non-Hermitian pumping direction to that of unpaired ones occupying different unit cells. By turning on an intracell interaction, many-body eigenstates split in their real energies, forming separable clusters in the complex energy plane with either left-, right-, or bipolar-types of non-Hermitian skin effect (NHSE). The dependency of skin accumulating directions on particle occupation is further justified with local sublattice correlation and entanglement entropy of many-body eigenstates. Dynamically, this occupation-dependent NHSE manifests as uni- or bidirectional pumping for many-body initial states, allowing for spatially separating paired and unpaired particles. Our results unveil the possibility of designing and exploring novel non-Hermitian phases originated from particle nonconservation in subsystems (e.g., orbitals, sublattices, or spin species) and their spatial configurations.

Observation of continuum Landau modes in non-Hermitian electric circuits

Continuum Landau modes - predicted recently in a non-Hermitian Dirac Hamiltonian under a uniform magnetic field - are continuous bound states with no counterparts in Hermitian systems. However, they have still not been confirmed in experiments. Here, we report an experimental observation of continuum Landau modes in non-Hermitian electric circuits, in which the non-Hermitian Dirac Hamiltonian is simulated by non-reciprocal hoppings and the pseudomagnetic field is introduced by inhomogeneous complex on-site potentials. Through measuring the admittance spectrum and the eigenstates, we successfully verify key features of continuum Landau modes. Particularly, we observe the exotic voltage response acting as a rainbow trap or wave funnel through full-field excitation. This response originates from the linear relationship between the modes' center position and complex eigenvalues. Our work builds a bridge between non-Hermiticity and magnetic fields, and thus opens an avenue to explore exotic non-Hermitian physics.

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.

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.

Current-Voltage Characteristics of the Normal Metal-Insulator-PT-Symmetric Non-Hermitian Superconductor Junction as a Probe of Non-Hermitian Formalisms

We study theoretically a junction consisting of a normal metal, PT-symmetric non-Hermitian superconductor, and an insulating thin layer between them. We calculate current-voltage characteristics for this junction using left-right and right-right bases and compare the results. We find that in the left-right basis, the Andreev-scattered particles move in the opposite direction compared with the right-right basis and conventional Andreev scattering. This leads to profound differences in current-voltage characteristics. Based on this and other signatures, we argue that the left-right basis is not applicable in this case. Remarkably, we find that the growth and decay with time of the states with imaginary energies in the right-right basis are equilibrated.

Loss Difference Induced Localization in a Non-Hermitian Honeycomb Photonic Lattice

Non-Hermitian systems with complex-valued energy spectra provide an extraordinary platform for manipulating unconventional dynamics of light. Here, we demonstrate the localization of light in an instantaneously reconfigurable non-Hermitian honeycomb photonic lattice that is established in a coherently prepared atomic system. One set of the sublattices is optically modulated to introduce the absorptive difference between neighboring lattice sites, where the Dirac points in reciprocal space are extended into dispersionless local flat bands, with two shared eigenstates: low-loss (high-loss) one with fields confined at sublattice B (A). When these local flat bands are broad enough due to larger loss difference, the incident beam with its tangential wave vector being at the K point in reciprocal space is effectively localized at sublattice B with weaker absorption, namely, the commonly seen power exchange between adjacent channels in photonic lattices is effectively prohibited. The current work unlocks a new capability from non-Hermitian two-dimensional photonic lattices and provides an alternative route for engineering tunable local flat bands in photonic structures.

Non-Hermitian Chiral Heat Transport

Exceptional point (EP) has been captivated as a concept of interpreting eigenvalue degeneracy and eigenstate exchange in non-Hermitian physics. The chirality in the vicinity of EP is intrinsically preserved and usually immune to external bias or perturbation, resulting in the robustness of asymmetric backscattering and directional emission in classical wave fields. Despite recent progress in non-Hermitian thermal diffusion, all state-of-the-art approaches fail to exhibit chiral states or directional robustness in heat transport. Here we report the first discovery of chiral heat transport, which is manifested only in the vicinity of EP but suppressed at the EP of a thermal system. The chiral heat transport demonstrates significant robustness against drastically varying advections and thermal perturbations imposed. Our results reveal the chirality in heat transport process and provide a novel strategy for manipulating mass, charge, and diffusive light.

Nonlinear Exceptional Points with a Complete Basis in Dynamics

Exceptional points (EPs) are special spectral singularities at which two or more eigenvalues, and their corresponding eigenvectors, coalesce and become identical. In conventional wisdom, the coalescence of eigenvectors inevitably leads to the loss of completeness of the eigenbasis. Here, we show that this scenario breaks down in general at nonlinear EPs (NEPs). As an example, we realize a fifth-order NEP (NEP_{5}) within only three coupled resonators with both a theoretical model and simulations in circuits. One stable and another four auxiliary steady eigenstates of the nonlinear Hamiltonian coalesce at the NEP_{5}, and the response of eigenfrequency to perturbations demonstrates a fifth-order root law. Intriguingly, the biorthogonal eigenbasis of the Hamiltonian governing the system dynamics is still complete, and this fact is corroborated by a finite Petermann factor instead of a divergent one at conventional EPs. Consequently, the amplification of noise, which diverges at other EPs, converges at our NEP_{5}. Our finding transforms the understanding of EPs and shows potential for miniaturizing various key applications operating near EPs.

Observation of bulk quadrupole in topological heat transport

The quantized bulk quadrupole moment has so far revealed a non-trivial boundary state with lower-dimensional topological edge states and in-gap zero-dimensional corner modes. In contrast to photonic implementations, state-of-the-art strategies for topological thermal metamaterials struggle to achieve such higher-order hierarchical features. This is due to the absence of quantized bulk quadrupole moments in thermal diffusion fundamentally prohibiting possible band topology expansions. Here, we report a recipe for generating quantized bulk quadrupole moments in fluid heat transport and observe the quadrupole topological phases in non-Hermitian thermal systems. Our experiments show that both the real- and imaginary-valued bands exhibit the hierarchical features of bulk, gapped edge and in-gap corner states-in stark contrast to the higher-order states observed only on real-valued bands in classical wave fields. Our findings open up unique possibilities for diffusive metamaterial engineering and establish a playground for multipolar topological physics.

Topological phases and non-Hermitian topology in photonic artificial microstructures

In the past few decades, the discovery of topological matter states has ushered in a new era in topological physics, providing a robust framework for strategically controlling the transport of particles or waves. Topological photonics, in particular, has sparked considerable research due to its ability to construct and manipulate photonic topological states via photonic artificial microstructures. Although the concept of topology originates from condensed matter, topological photonics has given rise to new fundamental ideas and a range of potential applications that may lead to revolutionary technologies. Here, we review recent developments in topological photonics, with a focus on the realization and application of several emerging research areas in photonic artificial microstructures. We highlight the research trend, spanning from the photonic counterpart of topological insulator phases, through topological semimetal phases, to other emerging non-Hermitian topologies.

A symmetry-protected exceptional ring in a photonic crystal with negative index media

Non-Hermitian topological band structures such as symmetry-protected exceptional rings (SPERs) can emerge for systems described by the generalized eigenvalue problem (GEVP) with Hermitian matrices. In this paper, we numerically analyze a photonic crystal with negative index media, which is described by the GEVP with Hermitian matrices. Our analysis using COMSOL Multiphysics® demonstrates that a SPER emerges for photonic crystals composed of split-ring resonators and metal-wire structures. We expect that the above SPER can be observed in experiments as it emerges at a finite frequency.

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.

Probing Complex-Energy Topology via Non-Hermitian Absorption Spectroscopy in a Trapped Ion Simulator

Non-Hermitian systems generically have complex energies, which may host topological structures, such as links or knots. While there has been great progress in experimentally engineering non-Hermitian models in quantum simulators, it remains a significant challenge to experimentally probe complex energies in these systems, thereby making it difficult to directly diagnose complex-energy topology. Here, we experimentally realize a two-band non-Hermitian model with a single trapped ion whose complex eigenenergies exhibit the unlink, unknot, or Hopf link topological structures. Based on non-Hermitian absorption spectroscopy, we couple one system level to an auxiliary level through a laser beam and then experimentally measure the population of the ion on the auxiliary level after a long period of time. Complex eigenenergies are then extracted, illustrating the unlink, unknot, or Hopf link topological structure. Our work demonstrates that complex energies can be experimentally measured in quantum simulators via non-Hermitian absorption spectroscopy, thereby opening the door for exploring various complex-energy properties in non-Hermitian quantum systems, such as trapped ions, cold atoms, superconducting circuits, or solid-state spin systems.

Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems

Defective spectral degeneracy, known as exceptional point (EP), lies at the heart of various intriguing phenomena in optics, acoustics, and other nonconservative systems. Despite extensive studies in the past two decades, the collective behaviors (e.g., annihilation, coalescence, braiding, etc.) involving multiple exceptional points or lines and their interplay have been rarely understood. Here we put forward a universal non-Abelian conservation rule governing these collective behaviors in generic multiband non-Hermitian systems and uncover several counterintuitive phenomena. We demonstrate that two EPs with opposite charges (even the pairwise created) do not necessarily annihilate, depending on how they approach each other. Furthermore, we unveil that the conservation rule imposes strict constraints on the permissible exceptional-line configurations. It excludes structures like Hopf link yet permits novel staggered rings composed of noncommutative exceptional lines. These intriguing phenomena are illustrated by concrete models which could be readily implemented in platforms like coupled acoustic cavities, optical waveguides, and ring resonators. Our findings lay the cornerstone for a comprehensive understanding of the exceptional non-Abelian topology and shed light on the versatile manipulations and applications based on exceptional degeneracies in nonconservative systems.

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.

Non-Hermitian Spectral Flows and Berry-Chern Monopoles

We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit their topological nature. In the absence of a line gap, our system still displays a spectral flow whose topology can be captured by exploiting an emergent pseudo-Hermitian symmetry.

The Promise of Soft-Matter-Enabled Quantum Materials

The field of quantum materials has experienced rapid growth over the past decade, driven by exciting new discoveries with immense transformative potential. Traditional synthetic methods to quantum materials have, however, limited the exploration of architectural control beyond the atomic scale. By contrast, soft matter self-assembly can be used to tailor material structure over a large range of length scales, with a vast array of possible form factors, promising emerging quantum material properties at the mesoscale. This review explores opportunities for soft matter science to impact the synthesis of quantum materials with advanced properties. Existing work at the interface of these two fields is highlighted, and perspectives are provided on possible future directions by discussing the potential benefits and challenges which can arise from their bridging.

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.

Observation of Weyl Interface States in Non-Hermitian Synthetic Photonic Systems

Weyl medium has triggered remarkable interest owing to its nontrivial topological edge states in 3D photonic band structures that were mainly revealed as surface modes yet. It is undoubted that the connection of two different Weyl media will give rise to more fruitful physics at their interface, while they face extreme difficulty in high-dimensional lattice matching. Here, we successfully demonstrate the non-Hermitian Weyl interface physics in complex synthetic parameter space, which is implemented in a loss-controlled silicon waveguide array. By establishing non-Hermitian Hamiltonian in the parameter space, new Weyl interfaces with distinct topological origins are predicted and experimentally observed in silicon waveguides. Significantly, our Letter exploits the non-Hermitian parameter to create the synthetic dimension by manipulating the non-Hermitian order, which successfully circumvents the difficulty in lattice matching for high-dimensional interfaces. The revealed rich topological Weyl interface states and their phase transitions in silicon waveguide platform further imply potentials in chip-scale photonics integrations.

Transmission across non-HermitianPT-symmetric quantum dots and ladders

A non-Hermitian (NH) region connected to semi-infinite Hermitian lattices acts either as a source or as a sink and the probability current is not conserved in a scattering typically. Even aPT-symmetric region that contains both a source and a sink does not lead to current conservation plainly. We propose a model and study the scattering across a NHPT-symmetric two-level quantum dot (QD) connected to two semi-infinite one-dimensional lattices in a special way so that the probability current is conserved. Aharonov-Bohm type phases are included in the model, which arise from magnetic fluxes (ℏϕL/e, ℏϕR/e) through two loops in the system. We show that whenϕL=ϕR, the probability current is conserved. We find that the transmission across the QD can be perfect in thePT-unbroken phase (corresponding to real eigenenergies of the isolated QD) whereas the transmission is never perfect in thePT-broken phase (corresponding to purely imaginary eigenenergies of the QD). The two transmission peaks have the same width only for special values of the fluxes (being odd multiples ofπℏ/2e). In the broken phase, the transmission peak is surprisingly not at zero energy. We give an insight into this feature through a four-site toy model. We extend the model to aPT-symmetric ladder connected to two semi-infinite lattices. We show that the transmission is perfect in unbroken phase of the ladder due to Fabry-Pérot type interference, that can be controlled by tuning the chemical potential. In the broken phase of the ladder, the transmission is substantially suppressed.

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 theZ₂skin 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-Hermitianp- ands-wave topological superconductors. In terms of Majorana Pfaffian, aZ₂non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.

Anomalous Behaviors of Quantum Emitters in Non-Hermitian Baths

Both non-Hermitian systems and the behavior of emitters coupled to structured baths have been studied intensely in recent years. Here, we study the interplay of these paradigmatic settings. In a series of examples, we show that a single quantum emitter coupled to a non-Hermitian bath displays a number of unconventional behaviors, many without Hermitian counterpart. We first consider a unidirectional hopping lattice whose complex dispersion forms a loop. We identify peculiar bound states inside the loop as a manifestation of the non-Hermitian skin effect. In the same setting, emitted photons may display spatial amplification markedly distinct from free propagation, which can be understood with the help of the generalized Brillouin zone. We then consider a nearest-neighbor lattice with alternating loss. We find that the long-time emitter decay always follows a power law, which is usually invisible for Hermitian baths. Our Letter points toward a rich landscape of anomalous quantum emitter dynamics induced by non-Hermitian baths.

Non-Hermitian Absorption Spectroscopy

While non-Hermitian Hamiltonians have been experimentally realized in cold atom systems, it remains an outstanding open question of how to experimentally measure their complex energy spectra in momentum space for a realistic system with boundaries. The existence of non-Hermitian skin effects may make the question even more difficult to address given the fact that energy spectra for a system with open boundaries are dramatically different from those in momentum space; the fact may even lead to the notion that momentum-space band structures are not experimentally accessible for a system with open boundaries. Here, we generalize the widely used radio-frequency spectroscopy to measure both real and imaginary parts of complex energy spectra of a non-Hermitian quantum system for either bosonic or fermionic atoms. By weakly coupling the energy levels of a non-Hermitian system to auxiliary energy levels, we theoretically derive a formula showing that the decay of atoms on the auxiliary energy levels reflects the real and imaginary parts of energy spectra in momentum space. We further prove that measurement outcomes are independent of boundary conditions in the thermodynamic limit, providing strong evidence that the energy spectrum in momentum space is experimentally measurable. We finally apply our non-Hermitian absorption spectroscopy protocol to the Hatano-Nelson model and non-Hermitian Weyl semimetals to demonstrate its feasibility.

Experimental Realization of Weyl Exceptional Rings in a Synthetic Three-Dimensional Non-Hermitian Phononic Crystal

Weyl points-topological monopoles of quantized Berry flux-are predicted to spread to Weyl exceptional rings in the presence of non-Hermiticity. Here, we use a one-dimensional Aubry-Andre-Harper model to construct a Weyl semimetal in a three-dimensional parameter space comprising one reciprocal dimension and two synthetic dimensions. The inclusion of non-Hermiticity in the form of gain and loss produces a synthetic Weyl exceptional ring (SWER). The topology of the SWER is characterized by both its topological charge and non-Hermitian winding numbers. We experimentally observe the SWER and synthetic Fermi arc in a one-dimensional phononic crystal with the non-Hermiticity introduced by active acoustic components. Our findings pave the way for studying the high-dimensional non-Hermitian topological physics in acoustics.

Optoelectronic oscillator with improved sidemode suppression by joint use of spectral Vernier effect and parity-time symmetry

An optoelectronic oscillator (OEO) with improved sideband suppression by joint use of the spectral Vernier effect and parity-time (PT) symmetry is proposed and experimentally demonstrated. The spectral Vernier effect is implemented using two mutually coupled loops with different loop lengths, to increase the effective free spectral range (FSR). To further increase the mode selection capability to ensure stable single-frequency oscillation with an increased sidemode suppression ratio (SMSR), PT symmetry is implemented, in which the two mutually coupled loops are controlled with balanced gain and loss. Thanks to the combined effects, stable single-mode oscillation with a significantly increased SMSR is achieved. The proposed OEO is studied theoretically and evaluated experimentally. The results show that for a generated microwave signal at 10 GHz, the SMSR is 67.68 dB, which is increased by 11.20 dB or 26.05 dB, when using only the spectral Vernier effect or only the PT symmetry. Thanks to the long length of the longer loop, good phase noise performance is still maintained. The measurement shows that a phase noise as low as -124.5 dBc/Hz at an offset frequency of 10 kHz is achieved.

Non-Hermitian Spatial Symmetries and Their Stabilized Normal and Exceptional Topological Semimetals

We study non-Hermitian spatial symmetries-a class of symmetries that have no counterparts in Hermitian systems-and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. In deriving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermitian ones. First, we discover the nonlocal Hermitian conjugate pair of exceptional or normal band degeneracies that are enforced by non-Hermitian spatial symmetries. Remarkably, we find that these pairs lead to the symmetry-enforced violation of the Fermion doubling theorem in the long-time limit. Second, with the topological constraints, we unravel that a certain exceptional manifold is only compatible with and stabilized by non-Hermitian spatial symmetries but is intrinsically incompatible with Hermitian spatial symmetries. We illustrate these findings using two three-dimensional models of a non-Hermitian Weyl semimetal and an exceptional unconventional Weyl semimetal. Experimental cold-atom realizations of both models are also proposed.

Observation of Weyl exceptional rings in thermal diffusion

Thermal diffusion is dissipative and strongly related to non-Hermitian physics. At the same time, non-Hermitian Weyl systems have spurred tremendous interest across photonics and acoustics. This correlation has been long ignored and hence shed little light upon the question of whether the Weyl exceptional ring (WER) in thermal diffusion could exist. Intuitively, thermal diffusion provides no real parameter dimensions, thus prohibiting a topological nature and WER. This work breaks this perception by imitating synthetic dimensions via two spatiotemporal advection pairs. The WER is achieved in thermal diffusive systems. Both surface-like and bulk states are demonstrated by coupling two WERs with opposite topological charges. These findings extend topological notions to diffusions and motivate investigation of non-Hermitian diffusive and dissipative control.

A non-Hermitian Weyl equation indispensably requires a three-dimensional (3D) real/synthetic space, and it is thereby perceived that a Weyl exceptional ring (WER) will not be present in thermal diffusion given its purely dissipative nature. Here, we report a recipe for establishing a 3D parameter space to imitate thermal spinor field. Two orthogonal pairs of spatiotemporally modulated advections are employed to serve as two synthetic parameter dimensions, in addition to the inherent dimension corresponding to heat exchanges. We first predict the existence of WER in our hybrid conduction–advection system and experimentally observe the WER thermal signatures verifying our theoretical prediction. When coupling two WERs of opposite topological charges, the system further exhibits surface-like and bulk topological states, manifested as stationary and continuously changing thermal processes, respectively, with good robustness. Our findings reveal the long-ignored topological nature in thermal diffusion and may empower distinct paradigms for general diffusion and dissipation controls.

Topological delocalization transitions and mobility edges in the nonreciprocal Maryland model

Non-Hermitian effects could trigger spectrum, localization and topological phase transitions in quasiperiodic lattices. We propose a non-Hermitian extension of the Maryland model, which forms a paradigm in the study of localization and quantum chaos by introducing asymmetry to its hopping amplitudes. The resulting nonreciprocal Maryland model is found to possess a real-to-complex spectrum transition at a finite amount of hopping asymmetry, through which it changes from a localized phase to a mobility edge phase. Explicit expressions of the complex energy dispersions, phase boundaries and mobility edges are found. A topological winding number is further introduced to characterize the transition between different phases. Our work introduces a unique type of non-Hermitian quasicrystal, which admits exactly obtainable phase diagrams, mobility edges, and holding no extended phases at finite nonreciprocity in the thermodynamic limit.

Magnetic Suppression of Non-Hermitian Skin Effects

Skin effect, where macroscopically many bulk states are aggregated toward the system boundary, is one of the most important and distinguishing phenomena in non-Hermitian quantum systems. We discuss a new aspect of this effect whereby, despite its topological origin, applying a magnetic field can largely suppress it. Skin states are pushed back into the bulk, and the skin topological area, which we define, is sharply reduced. As seen from exact solutions of representative models, this is fundamentally rooted in the fact that the applied magnetic field restores the validity of the low-energy description that is rendered inapplicable in the presence of non-Bloch skin states. We further study this phenomenon using rational gauge fluxes, which reveals a unique irrelevance of the generalized Brillouin zone in the standard non-Bloch band theory of non-Hermitian systems.

Higher-Order Weyl-Exceptional-Ring Semimetals

For first-order topological semimetals, non-Hermitian perturbations can drive the Weyl nodes into Weyl exceptional rings having multiple topological structures and no Hermitian counterparts. Recently, it was discovered that higher-order Weyl semimetals, as a novel class of higher-order topological phases, can uniquely exhibit coexisting surface and hinge Fermi arcs. However, non-Hermitian higher-order topological semimetals have not yet been explored. Here, we identify a new type of topological semimetal, i.e., a higher-order topological semimetal with Weyl exceptional rings. In such a semimetal, these rings are characterized by both a spectral winding number and a Chern number. Moreover, the higher-order Weyl-exceptional-ring semimetal supports both surface and hinge Fermi-arc states, which are bounded by the projection of the Weyl exceptional rings onto the surface and hinge, respectively. Noticeably, the dissipative terms can cause the coupling of two exceptional rings with opposite topological charges, so as to induce topological phase transitions. Our studies open new avenues for exploring novel higher-order topological semimetals in non-Hermitian systems.

Symmetry and Higher-Order Exceptional Points

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic ∼k^{1/3} dispersion are protected by PT symmetry, while third-order EPs with a ∼k^{1/2} dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology.

Protected quantum coherence by gain and loss in a noisy quantum kicked rotor

We study the effects of non-Hermiticity on quantum coherence via a noisy quantum kicked rotor (NQKR). The random noise comes from the fluctuations in kick amplitude at each time. The non-Hermitian driving indicates the imaginary kicking potential, representing the environment-induced atom gain and loss. In the absence of gain and loss, the random noise destroys quantum coherence manifesting dynamical localization, which leads to classical diffusion. Interestingly, in the presence of non-Hermitian kicking potential, the occurrence of dynamical localization is highly sensitive to the gain and loss, manifesting the restoration of quantum coherence. Using the inverse participation ratio arguments, we numerically obtain a phase diagram of the classical diffusion and dynamical localization on the parameter plane of noise amplitude and non-Hermitian driving strength. With the help of analysis on the corresponding quasieigenstates, we achieve insight into dynamical localization, and uncover that the origin of the localization is interference between multiple quasi-eigenstates of the quantum kicked rotor. We further propose an experimental scheme to realize the NQKR in a dissipative cold atomic gas, which paves the way for future experimental investigation of an NQKR and its anomalous non-Hermitian properties.

Exceptional topological insulators

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter.

Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity

Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.

Entanglement entropy of non-Hermitian free fermions

We study the entanglement properties of non-Hermitian free fermionic models with translation symmetry using the correlation matrix technique. Our results show that the entanglement entropy has a logarithmic correction to the area law in both one-dimensional and two-dimensional systems. For any one-dimensional one-band system, we prove that each Fermi point of the system contributes exactly 1/2 to the coefficientcof the logarithmic correction. Moreover, this relation betweencand Fermi point is verified for more general one-dimensional and two-dimensional cases by numerical calculations and finite-size scaling analysis. In addition, we also study the single-particle and density-density correlation functions.

Exploration of the topological properties in a non-Hermitian long-range system

The asymmetrical long-range hopping amplitudes have a rich influence on the topological properties. Here, a non-Hermitian model including the long-range hopping amplitudes is constructed to explore those properties. It can be found that an extra topological invariantW= 2 emerges as a consequence of the long-range hopping amplitudes. Furthermore, we find that the phaseW= 2 can be directly characterized by the generalized Brillouin zone (GBZ) itself through the concept of the argument principle. Meanwhile, a gapless phase dubbed as topological semimetal phase can be induced by the asymmetrical long-range hopping. Moreover, the physical origin of the topological semimetal phase can be explained by the solutions of eigen-equation. It is also shown that the skin modes exist as long as the GBZ and the Brillouin zone differ from each other. These interesting phases may be realized in an electrical-circuit simulator.