Topological Band Theory for Non-Hermitian Hamiltonians

Resilience of quantum spin fluctuations against Dzyaloshinskii-Moriya interaction

In low-dimensional systems, the lack of structural inversion symmetry combined with the spin-orbit coupling gives rise to an anisotropic antisymmetric superexchange known as the Dzyaloshinskii-Moriya interaction (DMI). Various features have been reported due to the presence of DMIs in quantum systems. We here study the one-dimensional spin-1/2 transverse field XY chains with a DMI at zero temperature. Our focus is on the quantum fluctuations of the spins measured by the spin squeezing and the entanglement entropy. We find that these fluctuations are resistant to the effect of the DMI in the system. This resistance will fail as soon as the system is placed in the chiral phase where its state behaves as a squeezed state, suggesting the merit of the chiral phase to be used for quantum metrology. Remarkably, we prove that the central charge vanishes on the critical lines between gapless chiral and ferromagnetic/paramagnetic phases where there is no critical scaling versus the system size for the spin squeezing parameter. Our phenomenal results provide a further understanding of the effects of the DMIs in the many-body quantum systems which may be testable in experiments.

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.

Non-Hermitian Strongly Interacting Dirac Fermions

Exotic quantum phases and phase transition in the strongly interacting Dirac systems have attracted tremendous interests. On the other hand, non-Hermitian physics, usually associated with dissipation arising from the coupling to environment, emerges as a frontier of modern physics in recent years. In this Letter, we investigate the interplay between non-Hermitian physics and strong correlation in Dirac-fermion systems. We generalize the projector quantum Monte-Carlo (PQMC) algorithm to the non-Hermitian interacting fermionic systems. Employing PQMC simulation, we decipher the ground-state phase diagram of the honeycomb Hubbard model with spin resolved non-Hermitian asymmetric hopping processes. The antiferromagnetic (AFM) ordering induced by Hubbard interaction is enhanced by the non-Hermitian asymmetric hopping. Combining PQMC simulation and renormalization group analysis, we reveal that the quantum phase transition between Dirac semi-metal and AFM phases belongs to Hermitian chiral XY universality class, implying that a Hermitian Gross-Neveu transition is emergent at the quantum critical point although the model is non-Hermitian.

Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Quantum Hall systems host chiral edge states extending along the one-dimensional boundary of any two-dimensional sample. In solid state materials, the edge states serve as perfectly robust transport channels that produce a quantized Hall conductance; due to their chirality, and the topological protection by the Chern number of the bulk band structure, they cannot be spatially localized by defects or disorder. Here, we show experimentally that the chiral edge states of a lossy quantum Hall system can be localized. In a gyromagnetic photonic crystal exhibiting the quantum Hall topological phase, an appropriately structured loss configuration imparts the edge states' complex energy spectrum with a feature known as point-gap winding. This intrinsically non-Hermitian topological invariant is distinct from the Chern number invariant of the bulk (which remains intact) and induces mode localization via the "non-Hermitian skin effect." The interplay of the two topological phenomena-the Chern number and point-gap winding-gives rise to a non-Hermitian generalization of the paradigmatic Chern-type bulk-boundary correspondence principle. Compared to previous realizations of the non-Hermitian skin effect, the skin modes in this system have superior robustness against local defects and disorders.

Exponentially Enhanced Non-Hermitian Cooling

Certain non-Hermitian systems exhibit the skin effect, whereby the wave functions become exponentially localized at one edge of the system. Such exponential amplification of wavefunction has received significant attention due to its potential applications in, e.g., classical and quantum sensing. However, the opposite edge of the system, featured by exponentially suppressed wave functions, remains largely unexplored. Leveraging this phenomenon, we introduce a non-Hermitian cooling mechanism, which is fundamentally distinct from traditional refrigeration or laser cooling techniques. Notably, non-Hermiticity will not amplify thermal excitations, but rather redistribute them. Hence, thermal excitations can be cooled down at one edge of the system, and the cooling effect can be exponentially enhanced by the number of auxiliary modes, albeit with a lower bound that depends on the dissipative interaction with the environment. Non-Hermitian cooling does not rely on intricate properties such as exceptional points or nontrivial topology, and it can apply to a wide range of excitations.

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.

Lieb-Schultz-Mattis Theorem in Open Quantum Systems

The Lieb-Schultz-Mattis (LSM) theorem provides a general constraint on quantum many-body systems and plays a significant role in the Haldane gap phenomena and topological phases of matter. Here, we extend the LSM theorem to open quantum systems and establish a general theorem that restricts the steady state and spectral gap of Liouvillians based solely on symmetry. Specifically, we demonstrate that the unique gapped steady state is prohibited when translation invariance and U(1) symmetry are simultaneously present for noninteger filling numbers. As an illustrative example, we find that no dissipative gap is open in the spin-1/2 dissipative Heisenberg model, while a dissipative gap can be open in the spin-1 counterpart-an analog of the Haldane gap phenomena in open quantum systems. Furthermore, we show that the LSM constraint manifests itself in a quantum anomaly of the dissipative form factor of Liouvillians. We also find the LSM constraints due to symmetry intrinsic to open quantum systems, such as Kubo-Martin-Schwinger symmetry. Our work leads to a unified understanding of phases and phenomena in open quantum 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.

Resolving the topology of encircling multiple exceptional points

Non-Hermiticity has emerged as a new paradigm for controlling coupled-mode systems in ways that cannot be achieved with conventional techniques. One aspect of this control that has received considerable attention recently is the encircling of exceptional points (EPs). To date, most work has focused on systems consisting of two modes that are tuned by two control parameters and have isolated EPs. While these systems exhibit exotic features related to EP encircling, it has been shown that richer behavior occurs in systems with more than two modes. Such systems can be tuned by more than two control parameters, and contain EPs that form a knot-like structure. Control loops that encircle this structure cause the system's eigenvalues to trace out non-commutative braids. Here we consider a hybrid scenario: a three-mode system with just two control parameters. We describe the relationship between control loops and their topology in the full and two-dimensional parameter space. We demonstrate this relationship experimentally using a three-mode mechanical system in which the control parameters are provided by optomechanical interaction with a high-finesse optical cavity.

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.

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.

Creating pairs of exceptional points for arbitrary polarization control: asymmetric vectorial wavefront modulation

Exceptional points (EPs) can achieve intriguing asymmetric control in non-Hermitian systems due to the degeneracy of eigenstates. Here, we present a general method that extends this specific asymmetric response of EP photonic systems to address any arbitrary fully-polarized light. By rotating the meta-structures at EP, Pancharatnam-Berry (PB) phase can be exclusively encoded on one of the circular polarization-conversion channels. To address any arbitrary wavefront, we superpose the optical signals originating from two orthogonally polarized -yet degenerate- EP eigenmodes. The construction of such orthogonal EP eigenstates pairs is achieved by applying mirror-symmetry to the nanostructure geometry flipping thereby the EP eigenmode handedness from left to right circular polarization. Non-Hermitian reflective PB metasurfaces designed using such EP superposition enable arbitrary, yet unidirectional, vectorial wavefront shaping devices. Our results open new avenues for topological wave control and illustrate the capabilities of topological photonics to distinctively operate on arbitrary polarization-state with enhanced performances.

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.

Experimental Realization of Stable Exceptional Chains Protected by Non-Hermitian Latent Symmetries Unique to Mechanical Systems

Lines of exceptional points are robust in the three-dimensional non-Hermitian parameter space without requiring any symmetry. However, when more elaborate exceptional structures are considered, the role of symmetry becomes critical. One such case is the exceptional chain (EC), which is formed by the intersection or osculation of multiple exceptional lines (ELs). In this Letter, we investigate a non-Hermitian classical mechanical system and reveal that a symmetry intrinsic to second-order dynamical equations, in combination with the source-free principle of ELs, guarantees the emergence of ECs. This symmetry can be understood as a non-Hermitian generalized latent symmetry, which is absent in prevailing formalisms rooted in first-order Schrödinger-like equations and has largely been overlooked so far. We experimentally confirm and characterize the ECs using an active mechanical oscillator system. Moreover, by measuring eigenvalue braiding around the ELs meeting at a chain point, we demonstrate the source-free principle of directed ELs that underlies the mechanism for EC formation. Our Letter not only enriches the diversity of non-Hermitian exceptional point configurations, but also highlights the new potential for non-Hermitian physics in second-order dynamical systems.

Experimental Realization of Geometry-Dependent Skin Effect in a Reciprocal Two-Dimensional Lattice

Recent studies of non-Hermitian periodic lattices unveiled the non-Hermitian skin effect (NHSE), in which the bulk modes under the periodic boundary conditions (PBC) become skin modes under open boundary conditions. The NHSE is a topological effect owing to the nontrivial spectral winding, and such spectral behaviors appear naturally in nonreciprocal systems. Hence prevailing approaches rely on nonreciprocity to achieve the NHSE. Here, we report the experimental realization of the geometry-dependent skin effect in a two-dimensional reciprocal system, in which the skin effect occurs only at boundaries whose macroscopic symmetry mismatches with the lattice symmetry. The role of spectral reciprocity and symmetry is revealed by connecting reflective channels at given boundaries with the spectral topology of the PBC spectrum. Our work highlights the vital role of reciprocity, symmetry, and macroscopic geometry on the NHSE in dimensionality larger than one and opens new routes for wave structuring using non-Hermitian effects.

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.

Broad and colossal edge supercurrent in Dirac semimetal Cd3As2 Josephson junctions

Edge supercurrent has attracted great interest recently due to its crucial role in achieving and manipulating topological superconducting states. Proximity-induced superconductivity has been realized in quantum Hall and quantum spin Hall edge states, as well as in higher-order topological hinge states. Non-Hermitian skin effect, the aggregation of non-Bloch eigenstates at open boundaries, promises an abnormal edge channel. Here we report the observation of broad edge supercurrent in Dirac semimetal Cd3As2-based Josephson junctions. The as-grown Cd3As2 nanoplates are electron-doped by intrinsic defects, which enhance the non-Hermitian perturbations. The superconducting quantum interference indicates edge supercurrent with a width of ~1.6 μm and a magnitude of ~1 μA at 10 mK. The wide and large edge supercurrent is inaccessible for a conventional edge system and suggests the presence of non-Hermitian skin effect. A supercurrent nonlocality is also observed. The interplay between band topology and non-Hermiticity is beneficial for exploiting exotic topological matter.

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.

Realizing tight-binding Hamiltonians using site-controlled coupled cavity arrays

Analog quantum simulators rely on programmable and scalable quantum devices to emulate Hamiltonians describing various physical phenomenon. Photonic coupled cavity arrays are a promising alternative platform for realizing such simulators, due to their potential for scalability, small size, and high-temperature operability. However, programmability and nonlinearity in photonic cavities remain outstanding challenges. Here, using a silicon photonic coupled cavity array made up of [Formula: see text] high quality factor ([Formula: see text] up to[Formula: see text]) resonators and equipped with specially designed thermo-optic island heaters for independent control of cavities, we demonstrate a programmable photonic cavity array in the telecom regime, implementing tight-binding Hamiltonians with access to the full eigenenergy spectrum. We report a [Formula: see text] reduction in the thermal crosstalk between neighboring sites of the cavity array compared to traditional heaters, and then present a control scheme to program the cavity array to a given tight-binding Hamiltonian. The ability to independently program high-Q photonic cavities, along with the compatibility of silicon photonics to high volume manufacturing opens new opportunities for scalable quantum simulation using telecom regime infrared photons.

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.

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.

Topological Microlaser with a Non-Hermitian Topological Bulk

Bulk-edge correspondence, with quantized bulk topology leading to protected edge states, is a hallmark of topological states of matter and has been experimentally observed in electronic, atomic, photonic, and many other systems. While bulk-edge correspondence has been extensively studied in Hermitian systems, a non-Hermitian bulk could drastically modify the Hermitian topological band theory due to the interplay between non-Hermiticity and topology, and its effect on bulk-edge correspondence is still an ongoing pursuit. Importantly, including non-Hermicity can significantly expand the horizon of topological states of matter and lead to a plethora of unique properties and device applications, an example of which is a topological laser. However, the bulk topology, and thereby the bulk-edge correspondence, in existing topological edge-mode lasers is not well defined. Here, we propose and experimentally probe topological edge-mode lasing with a well-defined non-Hermitian bulk topology in a one-dimensional (1D) array of coupled ring resonators. By modeling the Hamiltonian with an additional degree of freedom (referred to as synthetic dimension), our 1D structure is equivalent to a 2D non-Hermitian Chern insulator with precise mapping. Our Letter may open a new pathway for probing non-Hermitian topological effects and exploring non-Hermitian topological device applications.

Multi-dimensional band structure spectroscopy in the synthetic frequency dimension

The concept of synthetic dimensions in photonics provides a versatile platform in exploring multi-dimensional physics. Many of these physics are characterized by band structures in more than one dimensions. Existing efforts on band structure measurements in the photonic synthetic frequency dimension however are limited to either one-dimensional Brillouin zones or one-dimensional subsets of multi-dimensional Brillouin zones. Here we theoretically propose and experimentally demonstrate a method to fully measure multi-dimensional band structures in the synthetic frequency dimension. We use a single photonic resonator under dynamical modulation to create a multi-dimensional synthetic frequency lattice. We show that the band structure of such a lattice over the entire multi-dimensional Brillouin zone can be measured by introducing a gauge potential into the lattice Hamiltonian. Using this method, we perform experimental measurements of two-dimensional band structures of a Hermitian and a non-Hermitian Hamiltonian. The measurements reveal some of the general properties of point-gap topology of the non-Hermitian Hamiltonian in more than one dimensions. Our results demonstrate experimental capabilities to fully characterize high-dimensional physical phenomena in the photonic synthetic frequency dimension.

Coexistence of stable and unstable population dynamics in a nonlinear non-Hermitian mechanical dimer

Non-Hermitian two-site dimers serve as minimal models in which to explore the interplay of gain and loss in dynamical systems. In this paper, we experimentally and theoretically investigate the dynamics of non-Hermitian dimer models with nonreciprocal hoppings between the two sites. We investigate two types of non-Hermitian couplings; one is when asymmetric hoppings are externally introduced, and the other is when the nonreciprocal hoppings depend on the population imbalance between the two sites, thus introducing the non-Hermiticity in a dynamical manner. We engineer the models in our synthetic mechanical setup comprised of two classical harmonic oscillators coupled by measurement-based feedback. For fixed nonreciprocal hoppings, we observe that, when the strength of these hoppings is increased, there is an expected transition from a PT-symmetric regime, where oscillations in the population are stable and bounded, to a PT-broken regime, where the oscillations are unstable and the population grows/decays exponentially. However, when the non-Hermiticity is dynamically introduced, we also find a third intermediate regime in which these two behaviors coexist, meaning that we can tune from stable to unstable population dynamics by simply changing the initial phase difference between the two sites. As we explain, this behavior can be understood by theoretically exploring the emergent fixed points of a related dimer model in which the nonreciprocal hoppings depend on the normalized population imbalance. Our study opens the way for the future exploration of non-Hermitian dynamics and exotic lattice models in synthetic mechanical networks.

A second wave of topological phenomena in photonics and acoustics

Light and sound are the most ubiquitous forms of waves, associated with a variety of phenomena and physical effects such as rainbows and echoes. Light and sound, both categorized as classical waves, have lately been brought into unexpected connections with exotic topological phases of matter. We are currently witnessing the onset of a second wave of active research into this topic. The past decade has been marked by fundamental advances comprising two-dimensional quantum Hall insulators and quantum spin and valley Hall insulators, whose topological properties are characterized using linear band topology. Here, going beyond these conventional topological systems, we focus on the latest frontiers, including non-Hermitian, nonlinear and non-Abelian topology as well as topological defects, for which the characterization of the topological features goes beyond the standard band-topology language. In addition to an overview of the current state of the art, we also survey future research directions for valuable applications.

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.

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.

Spontaneous Formation of Exceptional Points at the Onset of Magnetism

We reveal how symmetry-protected nodal points in topological semimetals may be promoted to pairs of generically stable exceptional points (EPs) by symmetry-breaking fluctuations at the onset of long-range order. This intriguing interplay between non-Hermitian (NH) topology and spontaneous symmetry breaking is exemplified by a magnetic NH Weyl phase spontaneously emerging at the surface of a strongly correlated three-dimensional topological insulator, when entering the ferromagnetic regime from a high-temperature paramagnetic phase. Here, electronic excitations with opposite spin acquire significantly different lifetimes, thus giving rise to an anti-Hermitian structure in spin that is incompatible with the chiral spin texture of the nodal surface states, and hence facilitate the spontaneous formation of EPs. We present numerical evidence of this phenomenon by solving a microscopic multiband Hubbard model nonperturbatively in the framework of dynamical mean-field theory.

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.

Nonreciprocal photon blockade in a spinning optomechanical system with nonreciprocal coupling

A scheme is presented to achieve quantum nonreciprocity by manipulating the statistical properties of the photons in a composite device consisting of a double-cavity optomechanical system with a spinning resonator and nonreciprocal coupling. It can be found that the photon blockade can emerge when the spinning device is driven from one side but not from the other side with the same driving amplitude. Under the weak driving limit, to achieve the perfect nonreciprocal photon blockade, two sets of optimal nonreciprocal coupling strengths are analytically obtained under different optical detunings based on the destructive quantum interference between different paths, which are in good agreement with the results obtained from numerical simulations. Moreover, the photon blockade exhibits thoroughly different behaviors as the nonreciprocal coupling is altered, and the perfect nonreciprocal photon blockade can be achieved even with weak nonlinear and linear couplings, which breaks the orthodox perception.

Enhancement of Quantum Heat Engine by Encircling a Liouvillian Exceptional Point

Quantum heat engines are expected to outperform the classical counterparts due to quantum coherences involved. Here we experimentally execute a single-ion quantum heat engine and demonstrate, for the first time, the dynamics and the enhanced performance of the heat engine originating from the Liouvillian exceptional points (LEPs). In addition to the topological effects related to LEPs, we focus on thermodynamic effects, which can be understood by the Landau-Zener-Stückelberg process under decoherence. We witness a positive net work from the quantum heat engine if the heat engine cycle dynamically encircles a LEP. Further investigation reveals that a larger net work is done when the system is operated closer to the LEP. We attribute the enhanced performance of the quantum heat engine to the Landau-Zener-Stückelberg process, enabled by the eigenenergy landscape in the vicinity of the LEP, and the exceptional point-induced topological transition. Therefore, our results open new possibilities toward LEP-enabled control of quantum heat engines and of thermodynamic processes in open quantum systems.

Topological skyrmion phases of matter

We introduce topological phases of matter defined by skyrmions in the ground state spin-or pseudospin-expectation value textures in the Brillouin zone, the chiral and helical topological skyrmion phases of matter. These phases are protected by a symmetry present in centrosymmetric superconductors. We consider a tight-binding model for spin-triplet superconductivity in transition metal oxides and find it realizes each of these topological skyrmion phases. The chiral phase is furthermore realized for a parameter set characterizing Sr2RuO4with spin-triplet superconductivity. We also find two types of topological phase transitions by which the skyrmion number can change. The second type occurs without the closing of energy gaps in a system described by a quadratic Hamiltonian without breaking the protecting symmetries when atomic spin-orbit coupling is non-negligible and there is a suitable additional degree of freedom. This contradicts the 'flat band' limit assumption important in use of entanglement spectrum and Wilson loops, and in construction of the ten-fold way classification scheme of topological phases of matter. We furthermore predict two kinds of bulk-boundary correspondence signatures-one for measurements which execute a partial trace over degrees of freedom other than spin, which yields quantized transport signatures-and a second resulting from skyrmions trapping defects with their own non-trivial topology that is discussed in a second work, which yields generalizations of unpaired Majorana zero-modes.

Hyperbolic band topology with non-trivial second Chern numbers

Topological band theory establishes a standardized framework for classifying different types of topological matters. Recent investigations have shown that hyperbolic lattices in non-Euclidean space can also be characterized by hyperbolic Bloch theorem. This theory promotes the investigation of hyperbolic band topology, where hyperbolic topological band insulators protected by first Chern numbers have been proposed. Here, we report a new finding on the construction of hyperbolic topological band insulators with a vanished first Chern number but a non-trivial second Chern number. Our model possesses the non-abelian translational symmetry of {8,8} hyperbolic tiling. By engineering intercell couplings and onsite potentials of sublattices in each unit cell, the non-trivial bandgaps with quantized second Chern numbers can appear. In experiments, we fabricate two types of finite hyperbolic circuit networks with periodic boundary conditions and partially open boundary conditions to detect hyperbolic topological band insulators. Our work suggests a new way to engineer hyperbolic topological states with higher-order topological invariants.

Non-Hermitian kagome photonic crystal with a totally topological spatial mode selection

Recently, the study of non-Hermitian topological edge and corner states in sonic crystals (SCs) and photonic crystals (PCs) has drawn much attention. In this paper, we propose a Wannier-type higher-order topological insulator (HOTI) model based on the kagome PC containing dimer units and study its non-Hermitian topological corner states. When balanced gain and loss are introduced into the dimer units with a proper parity-time symmetric setting, the system will show asymmetric Wannier bands and can support two Hermitian corner states and two pairs of complex-conjugate or pseudo complex-conjugate non-Hermitian corner states. These topological corner states are solely confined at three corners of the triangular supercell constructed by the trivial and non-trivial kagome PCs, corresponding to a topological spatial mode selection effect. As compared to the non-Hermitian quadrupole-type HOTIs, the non-Hermitian Wannier-type HOTIs can realize totally topological spatial mode selection by using much lower coefficients of gain and loss. Our results pave the way for the development of novel non-Hermitian photonic topological devices based on Wannier-type HOTIs.

Investigating exceptional points in dark-bright mode-coupled plasmonic systems

Exceptional points (EPs) of non-Hermitian systems are gaining more and more attention due to their important applications in unidirectional transmitters, sensors, etc. However, theoretical studies on EPs of reflection, transmission, and absorption spectra are less available. In this paper, in the dark-bright mode-coupled plasmonic systems, the variations of EPs of reflection, transmission, and absorption spectra are numerically investigated using temporal coupled-mode theory, and an assumption is given using the representation transformation theory. The intermediate representation (IR) is firstly proposed and related to the reflection spectrum, while the normal representation (NR) is associated with the absorption spectrum. In the region far from EPs, the IR (or NR) describes the reflection (or absorption) spectrum well. Near EPs, modified formulas similar to the representation transformation theory are given. In order to verify the correctness of the assumption, two metasurfaces are designed. And the simulation results are in good agreement with the assumption and it is found in the near-infrared and visible-light band that the absorption loss of the dark mode is linearly related to the EPs of reflection, transmission, and absorption spectra, while the radiation loss of the bright mode is only linearly related to the EPs of the absorption spectrum. These laws can help to manipulate the splitting of spectral lines for reflection, transmission, and absorption by adjusting the radiation loss and absorption loss of bright mode, the absorption loss of dark mode, and the coupling coefficients between two resonant modes. This research provides a guiding scheme for the design of micro and nano photonics devices.

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.

Topological Unidirectional Guided Resonances Emerged from Interband Coupling

Unidirectional guided resonances (UGRs) are optical modes in photonic crystal slabs that radiate toward one side without the need for mirrors on the other. In this Letter, we report a mechanism to realize UGRs by tuning the interband coupling effect originating from up-down symmetry breaking. We theoretically find that UGRs that reside along high-symmetric lines correspond to phase singularities of far-field radiation, depicted by phase winding numbers as a type of topological indices. We investigate the phase dislocation lines in three-dimensional parameter space and elaborate on the interplay between UGRs and non-Hermitian degeneracies accordingly. Our findings reveal the topological nature of UGRs about their generation, evolution, and annihilation in general parameter spaces, thus paving the way to new possibilities of light manipulation.

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.

Realization of exceptional points along a synthetic orbital angular momentum dimension

Exceptional points (EPs), at which more than one eigenvalue and eigenvector coalesce, are unique spectral features of non-Hermiticity (NH) systems. They exist widely in open systems with complex energy spectra. We experimentally demonstrate the appearance of paired EPs in a periodical-driven degenerate optical cavity along the synthetic orbital angular momentum dimension with a tunable parameter. The complex-energy band structures and the key features of EPs, i.e., their bulk Fermi arcs, parity-time symmetry breaking transition, energy swapping, and half-integer band windings, are directly observed by detecting the wavefront angle-resolved transmission spectrum. Our results demonstrate the flexibility of using the photonic synthetic dimensions to implement NH systems beyond their geometric dimension and EP-based sensing.

Observation of Acoustic Non-Hermitian Bloch Braids and Associated Topological Phase Transitions

Topological features embedded in ancient braiding and knotting arts endow significant impacts on our daily life and even cutting-edge science. Recently, fast growing efforts are invested to the braiding topology of complex Bloch bands in non-Hermitian systems. This new classification of band topology goes far beyond those established in Hermitian counterparts. Here, we present the first acoustic realization of the topological non-Hermitian Bloch braids, based on a two-band model easily accessible for realizing any desired knot structure. The non-Hermitian bands are synthesized by a simple binary cavity-tube system, where the long-range, complex-valued, and momentum-resolved couplings are accomplished by a well-controlled unidirectional coupler. In addition to directly visualizing various two-band braiding patterns, we unambiguously observe the highly elusive topological phase transitions between them. Not only do our results provide a direct demonstration for the non-Hermitian band topology, but also the experimental techniques open new avenues for designing unconventional acoustic metamaterials.

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.

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.