Topological Origin of Non-Hermitian Skin Effects

Selective and tunable excitation of topological non-Hermitian quasi-edge modes

Non-Hermitian lattices under semi-infinite boundary conditions sustain an extensive number of exponentially localized states, dubbed non-Hermitian quasi-edge modes. Quasi-edge states arise rather generally in systems displaying the non-Hermitian skin effect and can be predicted from the non-trivial topology of the energy spectrum under periodic boundary conditions via a bulk-edge correspondence. However, the selective excitation of the system in one among the infinitely many topological quasi-edge states is challenging both from practical and conceptual viewpoints. In fact, in any realistic system with a finite lattice size most of the quasi-edge states collapse and become metastable states. Here we suggest a route toward the selective and tunable excitation of topological quasi-edge states that avoids the collapse problem by emulating semi-infinite lattice boundaries via tailored on-site potentials at the edges of a finite lattice. We illustrate such a strategy by considering a non-Hermitian topological interface obtained by connecting two Hatano–Nelson chains with opposite imaginary gauge fields, which is amenable for a full analytical treatment.

Curving the space by non-Hermiticity

Quantum systems are often classified into Hermitian and non-Hermitian ones. Extraordinary non-Hermitian phenomena, ranging from the non-Hermitian skin effect to the supersensitivity to boundary conditions, have been widely explored. Whereas these intriguing phenomena have been considered peculiar to non-Hermitian systems, we show that they can be naturally explained by a duality between non-Hermitian models in flat spaces and their counterparts, which could be Hermitian, in curved spaces. For instance, prototypical one-dimensional (1D) chains with uniform chiral tunnelings are equivalent to their duals in two-dimensional (2D) hyperbolic spaces with or without magnetic fields, and non-uniform tunnelings could further tailor local curvatures. Such a duality unfolds deep geometric roots of non-Hermitian phenomena, delivers an unprecedented routine connecting Hermitian and non-Hermitian physics, and gives rise to a theoretical perspective reformulating our understandings of curvatures and distance. In practice, it provides experimentalists with a powerful two-fold application, using non-Hermiticity to engineer curvatures or implementing synthetic curved spaces to explore non-Hermitian quantum physics.

Non-Hermitian laser arrays with tunable phase locking

Inspired by the idea of non-Hermitian spectral engineering and non-Hermitian skin effect, a novel, to the best of our knowledge, design for stable emission of coupled laser arrays with tunable phase locking and strong supermode competition suppression is suggested. We consider a linear array of coupled resonators with asymmetric mode coupling displaying the non-Hermitian skin effect and show that, under suitable tailoring of complex frequencies of the two edge resonators, the laser array can stably emit in a single extended supermode with tunable phase locking and with strong suppression of all other skin supermodes. The proposed laser array design offers strong robustness against both structural imperfections of the system and dynamical instabilities typical of semiconductor laser arrays.

Self-Healing of Non-Hermitian Topological Skin Modes

A unique feature of non-Hermitian (NH) systems is the NH skin effect, i.e., the edge localization of an extensive number of bulk-band eigenstates in a lattice with open or semi-infinite boundaries. Unlike extended Bloch waves in Hermitian systems, the skin modes are normalizable eigenstates of the Hamiltonian that originate from the intrinsic non-Hermitian point-gap topology of the Bloch band energy spectra. Here, we unravel a fascinating property of NH skin modes, namely self-healing, i.e., the ability to self-reconstruct their shape after being scattered off by a space-time potential.

Non-Hermitian Edge Burst

We unveil an unexpected non-Hermitian phenomenon, dubbed edge burst, in non-Hermitian quantum dynamics. Specifically, in a class of non-Hermitian quantum walk in periodic lattices with open boundary condition, an exceptionally large portion of loss occurs at the system boundary. The physical origin of this edge burst is found to be an interplay between two unique non-Hermitian phenomena: non-Hermitian skin effect and imaginary gap closing. Furthermore, we establish a universal bulk-edge scaling relation underlying the non-Hermitian edge burst. Our predictions are experimentally accessible in various non-Hermitian systems including quantum-optical and cold-atom platforms.

Topological triple phase transition in non-Hermitian Floquet quasicrystals

Phase transitions connect different states of matter and are often concomitant with the spontaneous breaking of symmetries. An important category of phase transitions is mobility transitions, among which is the well known Anderson localization¹, where increasing the randomness induces a metal–insulator transition. The introduction of topology in condensed-matter physics^2–4 lead to the discovery of topological phase transitions and materials as topological insulators⁵. Phase transitions in the symmetry of non-Hermitian systems describe the transition to on-average conserved energy⁶ and new topological phases^7–9. Bulk conductivity, topology and non-Hermitian symmetry breaking seemingly emerge from different physics and, thus, may appear as separable phenomena. However, in non-Hermitian quasicrystals, such transitions can be mutually interlinked by forming a triple phase transition¹⁰. Here we report the experimental observation of a triple phase transition, where changing a single parameter simultaneously gives rise to a localization (metal–insulator), a topological and parity–time symmetry-breaking (energy) phase transition. The physics is manifested in a temporally driven (Floquet) dissipative quasicrystal. We implement our ideas via photonic quantum walks in coupled optical fibre loops¹¹. Our study highlights the intertwinement of topology, symmetry breaking and mobility phase transitions in non-Hermitian quasicrystalline synthetic matter. Our results may be applied in phase-change devices, in which the bulk and edge transport and the energy or particle exchange with the environment can be predicted and controlled. 

A triple phase transition, where changing a single parameter simultaneously gives rise to metal–insulator, topological and a parity–time symmetry-breaking phase transitions, is observed in non-Hermitian Floquet quasicrystals.

A Review of Electromagnetic Shielding Fabric, Wave-Absorbing Fabric and Wave-Transparent Fabric

As the basic materials with specific properties, fabrics have been widely applied in electromagnetic (EM) wave protection and control due to their characteristics of low density, excellent mechanical properties as well as designability. According to the different mechanisms and application scenarios on EM waves, fabrics can be divided into three types: EM shielding fabric, wave-absorbing fabric and wave-transparent fabric, which have been summarized and prospected from the aspects of mechanisms and research status, and it is believed that the current research on EM wave fabrics are imperfect in theory. Therefore, in order to meet the needs of different EM properties and application conditions, the structure of fabrics will be diversified, and more and more attentions should be paid to the research on structure of fabrics that meets EM properties, which will be conductive to guiding the development and optimization of fabrics. Furthermore, the application of fabrics in EM waves will change from 2D to 3D, from single structure to multiple structures, from large to small, as well as from heavy to light.

Non-Hermitian topology in rock-paper-scissors games

Non-Hermitian topology is a recent hot topic in condensed matters. In this paper, we propose a novel platform drawing interdisciplinary attention: rock–paper–scissors (RPS) cycles described by the evolutionary game theory. Specifically, we demonstrate the emergence of an exceptional point and a skin effect by analyzing topological properties of their payoff matrix. Furthermore, we discover striking dynamical properties in an RPS chain: the directive propagation of the population density in the bulk and the enhancement of the population density only around the right edge. Our results open new avenues of the non-Hermitian topology and the evolutionary game theory.

Exceptional Bound States and Negative Entanglement Entropy

This Letter introduces a new class of robust states known as exceptional bound (EB) states, which are distinct from the well-known topological and non-Hermitian skin boundary states. EB states occur in the presence of exceptional points, which are non-Hermitian critical points where eigenstates coalesce and fail to span the Hilbert space. This eigenspace defectiveness not only limits the accessibility of state information but also interplays with long-range order to give rise to singular propagators only possible in non-Hermitian settings. Their resultant EB eigenstates are characterized by robust anomalously large or negative occupation probabilities, unlike ordinary Fermi sea states whose probabilities lie between 0 and 1. EB states remain robust after a variety of quantum quenches and give rise to enigmatic negative entanglement entropy contributions. Through suitable perturbations, the coefficient of the logarithmic entanglement entropy scaling can be continuously tuned. EB states represent a new avenue for robustness arising from geometric defectiveness, independent of topological protection or nonreciprocal pumping.

Detecting Non-Bloch Topological Invariants in Quantum Dynamics

Non-Bloch topological invariants preserve the bulk-boundary correspondence in non-Hermitian topological systems, and are a key concept in the contemporary study of non-Hermitian topology. Here we report the dynamic detection of non-Bloch topological invariants in single-photon quantum walks, revealed through the biorthogonal chiral displacement, and crosschecked with the dynamic spin textures in the generalized quasimomentum-time domain following a quantum quench. Both detection schemes are robust against symmetry-preserving disorders, and yield consistent results with theoretical predictions. Our experiments are performed far away from any boundaries, and therefore underline non-Bloch topological invariants as intrinsic properties of the system that persist in the thermodynamic limit. Our work sheds new light on the experimental investigation of non-Hermitian topology.

Universal Behavior of the Scattering Matrix Near Thresholds in Photonics

Scattering thresholds and their associated spectral square root branch points are ubiquitous in photonics. In this Letter, we show that the scattering matrix has a simple universal behavior near scattering thresholds. We use unitarity, reciprocity, and time-reversal symmetry to construct a two-parameter model for a two-port scattering matrix near a threshold. We demonstrate this universal behavior in three different optical systems, namely, a photonic crystal slab, a planar dielectric interface, and a junction between metallic waveguides of different widths.

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.

Bulk-edge correspondence and trapping at a non-Hermitian topological interface

In Hermitian systems, according to the bulk-edge correspondence, interfacing two topological optical media with different bulk topological numbers implies the existence of edge states, which can trap light at the interface. However, such a general scenario can be violated when dealing with non-Hermitian systems. Here we show that interfacing two semi-infinite Hatano-Nelson chains with different bulk topological numbers can result in the existence of infinitely many edge (interface) states; however, light waves cannot be rather generally trapped at the interface.

Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics

Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which persist over practical timescales.

Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits

Robust boundary states epitomize how deep physics can give rise to concrete experimental signatures with technological promise. Of late, much attention has focused on two distinct mechanisms for boundary robustness-topological protection, as well as the non-Hermitian skin effect. In this work, we report the experimental realizations of hybrid higher-order skin-topological effect, in which the skin effect selectively acts only on the topological boundary modes, not the bulk modes. Our experiments, which are performed on specially designed non-reciprocal 2D and 3D topolectrical circuit lattices, showcases how non-reciprocal pumping and topological localization dynamically interplays to form various states like 2D skin-topological, 3D skin-topological-topological hybrid states, as well as 2D and 3D higher-order non-Hermitian skin states. Realized through our highly versatile and scalable circuit platform, theses states have no Hermitian nor lower-dimensional analog, and pave the way for applications in topological switching and sensing through the simultaneous non-trivial interplay of skin and topological boundary localizations.

Nielsen-Ninomiya Theorem with Bulk Topology: Duality in Floquet and Non-Hermitian Systems

The Nielsen-Ninomiya theorem is a fundamental theorem on the realization of chiral fermions in static lattice systems in high-energy and condensed matter physics. Here we extend the theorem in dynamical systems, which include the original Nielsen-Ninomiya theorem in the static limit. In contrast to the original theorem, which is a no-go theorem for bulk chiral fermions, the new theorem permits them due to bulk topology intrinsic to dynamical systems. The theorem is based on duality enabling a unified treatment of periodically driven systems and non-Hermitian ones. We also present the extended theorem for nonchiral gapless fermions protected by symmetry. Finally, as an application of our theorem and duality, we predict a new type of chiral magnetic effect-the non-Hermitian chiral magnetic skin effect.

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.

Acoustic non-Hermitian skin effect from twisted winding topology

The recently discovered non-Hermitian skin effect (NHSE) manifests the breakdown of current classification of topological phases in energy-nonconservative systems, and necessitates the introduction of non-Hermitian band topology. So far, all NHSE observations are based on one type of non-Hermitian band topology, in which the complex energy spectrum winds along a closed loop. As recently characterized along a synthetic dimension on a photonic platform, non-Hermitian band topology can exhibit almost arbitrary windings in momentum space, but their actual phenomena in real physical systems remain unclear. Here, we report the experimental realization of NHSE in a one-dimensional (1D) non-reciprocal acoustic crystal. With direct acoustic measurement, we demonstrate that a twisted winding, whose topology consists of two oppositely oriented loops in contact rather than a single loop, will dramatically change the NHSE, following previous predictions of unique features such as the bipolar localization and the Bloch point for a Bloch-wave-like extended state. This work reveals previously unnoticed features of NHSE, and provides the observation of physical phenomena originating from complex non-Hermitian winding topology.

Non-Hermitian skin effect fundamentally challenges the conventional topological description of a system. Here the authors demonstrate a bipolar non-Hermitian skin effect, where bulk eigenstates localize towards two directions, in a one-dimensional non-reciprocal acoustic crystal with twisted topology.

Realization of active metamaterials with odd micropolar elasticity

Materials made from active, living, or robotic components can display emergent properties arising from local sensing and computation. Here, we realize a freestanding active metabeam with piezoelectric elements and electronic feed-forward control that gives rise to an odd micropolar elasticity absent in energy-conserving media. The non-reciprocal odd modulus enables bending and shearing cycles that convert electrical energy into mechanical work, and vice versa. The sign of this elastic modulus is linked to a non-Hermitian topological index that determines the localization of vibrational modes to sample boundaries. At finite frequency, we can also tune the phase angle of the active modulus to produce a direction-dependent bending modulus and control non-Hermitian vibrational properties. Our continuum approach, built on symmetries and conservation laws, could be exploited to design others systems such as synthetic biofilaments and membranes with feed-forward control loops.

Topological complex-energy braiding of non-Hermitian bands

Effects connected with the mathematical theory of knots¹ emerge in many areas of science, from physics^2,3 to biology⁴. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems⁵, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹₂. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension^6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems.

Non-Hermitian topological whispering gallery

In 1878, Lord Rayleigh observed the highly celebrated phenomenon of sound waves that creep around the curved gallery of St Paul's Cathedral in London^1,2. These whispering-gallery waves scatter efficiently with little diffraction around an enclosure and have since found applications in ultrasonic fatigue and crack testing, and in the optical sensing of nanoparticles or molecules using silica microscale toroids. Recently, intense research efforts have focused on exploring non-Hermitian systems with cleverly matched gain and loss, facilitating unidirectional invisibility and exotic characteristics of exceptional points^3,4. Likewise, the surge in physics using topological insulators comprising non-trivial symmetry-protected phases has laid the groundwork in reshaping highly unconventional avenues for robust and reflection-free guiding and steering of both sound and light^5,6. Here we construct a topological gallery insulator using sonic crystals made of thermoplastic rods that are decorated with carbon nanotube films, which act as a sonic gain medium by virtue of electro-thermoacoustic coupling. By engineering specific non-Hermiticity textures to the activated rods, we are able to break the chiral symmetry of the whispering-gallery modes, which enables the out-coupling of topological 'audio lasing' modes with the desired handedness. We foresee that these findings will stimulate progress in non-destructive testing and acoustic sensing.

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.

Non-Hermitian topological phase transitions in superlattices and the optical Dirac equation

Optical superlattices with sublattice symmetry subjected to a synthetic imaginary gauge field undergo a topological phase transition in the Bloch energy spectrum, characterized by the change of a spectral winding number. For a narrow gap, the phase transition is of universal form and described by a non-Hermitian Dirac equation with Lorentz-symmetry violation. A simple photonic system displaying such a phase transition is discussed, which is based on light coupling in co-propagating gratings.

Exact Solution of Non-Hermitian Systems with Generalized Boundary Conditions: Size-Dependent Boundary Effect and Fragility of the Skin Effect

Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditions and lattice size, and thus an important question is whether non-Hermitian skin effects can survive when deviating from the open boundary condition. To unveil the origin of boundary sensitivity, we present exact solutions for one-dimensional non-Hermitian models with generalized boundary conditions and study rigorously the interplay effect of lattice size and boundary terms. Besides the open boundary condition, we identify the existence of non-Hermitian skin effect when one of the boundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, we find that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit, although it can survive in a finite size system. Moreover, we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which is different from either open or periodical boundary case.

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.

Quantized classical response from spectral winding topology

Topologically quantized response is one of the focal points of contemporary condensed matter physics. While it directly results in quantized response coefficients in quantum systems, there has been no notion of quantized response in classical systems thus far. This is because quantized response has always been connected to topology via linear response theory that assumes a quantum mechanical ground state. Yet, classical systems can carry arbitrarily amounts of energy in each mode, even while possessing the same number of measurable edge states as their topological winding. In this work, we discover the totally new paradigm of quantized classical response, which is based on the spectral winding number in the complex spectral plane, rather than the winding of eigenstates in momentum space. Such quantized response is classical insofar as it applies to phenomenological non-Hermitian setting, arises from fundamental mathematical properties of the Green's function, and shows up in steady-state response, without invoking a conventional linear response theory. Specifically, the ratio of the change in one quantity depicting signal amplification to the variation in one imaginary flux-like parameter is found to display fascinating plateaus, with their quantized values given by the spectral winding numbers as the topological invariants.

Observation of Non-Hermitian Topology with Nonunitary Dynamics of Solid-State Spins

Non-Hermitian topological phases exhibit a number of exotic features that have no Hermitian counterparts, including the skin effect and breakdown of the conventional bulk-boundary correspondence. Here, we implement the non-Hermitian Su-Schrieffer-Heeger Hamiltonian, which is a prototypical model for studying non-Hermitian topological phases, with a solid-state quantum simulator consisting of an electron spin and a ^{13}C nuclear spin in a nitrogen-vacancy center in a diamond. By employing a dilation method, we realize the desired nonunitary dynamics for the electron spin and map out its spin texture in the momentum space, from which the corresponding topological invariant can be obtained directly. From the measured spin textures with varying parameters, we observe both integer and fractional winding numbers. The non-Hermitian topological phase with fractional winding number cannot be continuously deformed to any Hermitian topological phase and is intrinsic to non-Hermitian systems. Our result paves the way for further exploiting and understanding the intriguing properties of non-Hermitian topological phases with solid-state spins or other quantum simulation platforms.

Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing

It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap Δ, but also on the localization length ξ of the eigenmodes. In particular, we show that the longest relaxation time τ that is maximized over initial states and local observables is given by τ∼Δ^{-1}(1+L/ξ) with L being the system size. This implies that the longest relaxation time can diverge for L→∞ without gap closing.

Geometric Response and Disclination-Induced Skin Effects in Non-Hermitian Systems

We study the geometric response of three-dimensional non-Hermitian crystalline systems with nontrivial point-gap topology. For systems with fourfold rotation symmetry, we show that in the presence of disclination lines with a total Frank angle, which is an integer multiple of 2π, there can be nontrivial one-dimensional point-gap topology along the direction of the disclination lines. This results in disclination-induced non-Hermitian skin effects. By doubling a non-Hermitian Hamiltonian to a Hermitian three-dimensional chiral topological insulator, we show that the disclination-induced skin modes are zero modes of the effective surface Dirac fermion(s) in the presence of a pseudomagnetic flux induced by disclinations. Furthermore, we find that our results have a field theoretic description, and the corresponding geometric response actions (e.g., the Euclidean Wen-Zee action) enrich the topological field theory of non-Hermitian systems.

Guided accumulation of active particles by topological design of a second-order skin effect

Collective guidance of out-of-equilibrium systems without using external fields is a challenge of paramount importance in active matter, ranging from bacterial colonies to swarms of self-propelled particles. Designing strategies to guide active matter and exploiting enhanced diffusion associated to its motion will provide insights for application from sensing, drug delivery to water remediation. However, achieving directed motion without breaking detailed balance, for example by asymmetric topographical patterning, is challenging. Here we engineer a two-dimensional periodic topographical design with detailed balance in its unit cell where we observe spontaneous particle edge guidance and corner accumulation of self-propelled particles. This emergent behaviour is guaranteed by a second-order non-Hermitian skin effect, a topologically robust non-equilibrium phenomenon, that we use to dynamically break detailed balance. Our stochastic circuit model predicts, without fitting parameters, how guidance and accumulation can be controlled and enhanced by design: a device guides particles more efficiently if the topological invariant characterizing it is non-zero. Our work establishes a fruitful bridge between active and topological matter, and our design principles offer a blueprint to design devices that display spontaneous, robust and predictable guided motion and accumulation, guaranteed by out-of-equilibrium topology.

Steering non-Hermitian skin modes by synthetic gauge fields in optical ring resonators

We show that the synthetic gauge fields for photons provide a versatile approach to generate and control the non-Hermitian skin effect. By utilizing indirectly coupled optical ring resonator arrays with long-range couplings and on-site gain and loss, we find that the skin effect appears once the gauge field is not an integer multiple of π. In addition to tunable localization direction, the skin modes display anisotropic behaviors with frequency-dependent decay length, which can be explained by the split subregion of the generalized Brillouin zone (GBZ) and an effective model under adiabatic elimination. Through numerical simulation, we can also demonstrate exotic features in propagation effects enabled by the skin effect, including asymmetric transmission and reconfigurable accumulation interface. Our study paves the way to dynamically steer skin modes, which may find applications in laser, optical switch, and signal processing.

Direct Measurement of Topological Properties of an Exceptional Parabola

Non-Hermitian systems can produce branch singularities known as exceptional points (EPs). Different from singularities in Hermitian systems, the topological properties of an EP can involve either the winding of eigenvalues that produces a discriminant number (DN) or the eigenvector holonomy that generates a Berry phase. The multiplicity of topological invariants also makes non-Hermitian topology richer than its Hermitian counterpart. Here, we study a parabola-shaped trajectory formed by EPs with both theory and acoustic experiments. By obtaining both the DNs and Berry phases through the measurement of eigenvalues and eigenfunctions, we show that the EP trajectory endows the parameter space with a nontrivial fundamental group. Our findings not only shed light on exotic non-Hermitian topology but also provide a route for the experimental characterization of non-Hermitian topological invariants.

Simulating Exceptional Non-Hermitian Metals with Single-Photon Interferometry

We experimentally simulate in a photonic setting non-Hermitian (NH) metals characterized by the topological properties of their nodal band structures. Implementing nonunitary time evolution in reciprocal space followed by interferometric measurements, we probe the complex eigenenergies of the corresponding NH Bloch Hamiltonians, and study in detail the topology of their exceptional lines (ELs), the NH counterpart of nodal lines in Hermitian systems. We focus on two distinct types of NH metals: two-dimensional systems with symmetry-protected ELs, and three-dimensional systems possessing symmetry-independent topological ELs in the form of knots. While both types feature open Fermi surfaces, we experimentally observe their distinctions by analyzing the impact of symmetry-breaking perturbations on the topology of ELs.

Phase diagram and quantum criticality of a non-HermitianXYmodel with a complex transverse field

We study a one dimensional non-Hermitian quantumXYmodel with a complex transverse field, where the imaginary transverse field can be generated by three-level atoms with spontaneous decay. Many-body spectrum can be obtained analytically. The phase diagram in thermodynamical limit indicates that the imaginary transverse field induces non-Hermitian degeneracy and shrinks the ferromagnetic phase for anisotropic interaction. We also analyze the effect of imaginary field on quantum criticality through ground state geometry phase. It causes a shift of transition point but does not change the order of phase transition in anisotropic interaction region. However, the phase transition is related to the original Hermitian degeneracy in anisotropic interaction region, which indicates that the imaginary transverse field cannot change phase boundary and critical behavior.

Unsupervised Learning of Non-Hermitian Topological Phases

Non-Hermitian topological phases bear a number of exotic properties, such as the non-Hermitian skin effect and the breakdown of conventional bulk-boundary correspondence. In this Letter, we introduce an unsupervised machine learning approach to classify non-Hermitian topological phases based on diffusion maps, which are widely used in manifold learning. We find that the non-Hermitian skin effect will pose a notable obstacle, rendering the straightforward extension of unsupervised learning approaches to topological phases for Hermitian systems ineffective in clustering non-Hermitian topological phases. Through theoretical analysis and numerical simulations of two prototypical models, we show that this difficulty can be circumvented by choosing the "on-site" elements of the projective matrix as the input data. Our results provide a valuable guidance for future studies on learning non-Hermitian topological phases in an unsupervised fashion, both in theory and experiment.

Observation of Non-Bloch Parity-Time Symmetry and Exceptional Points

Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay between PT symmetry and non-Hermitian skin effect.

Topological Field Theory of Non-Hermitian Systems

Non-Hermiticity gives rise to unique topological phases without Hermitian analogs. However, the effective field theory has yet to be established. Here, we develop a field-theoretical description of the intrinsic non-Hermitian topological phases. Because of the dissipative and nonequilibrium nature of non-Hermiticity, our theory is formulated solely in terms of spatial degrees of freedom, which contrasts with the conventional theory defined in spacetime. Our theory provides a universal understanding of non-Hermitian topological phenomena such as the unidirectional transport in one dimension and the chiral magnetic skin effect in three dimensions. Furthermore, it systematically predicts new physics; we illustrate this by revealing transport phenomena and skin effects in two dimensions induced by a perpendicular spatial texture. From the field-theoretical perspective, the non-Hermitian skin effect, i.e., the anomalous localization due to non-Hermiticity, is shown to be a signature of an anomaly.

Bulk-Boundary Correspondence for Non-Hermitian Hamiltonians via Green Functions

Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function.

Non-Hermitian Skin Effects in Hermitian Correlated or Disordered Systems: Quantities Sensitive or Insensitive to Boundary Effects and Pseudo-Quantum-Number

There is a common belief in the condensed matter community that bulk quantities become insensitive to the boundary condition in the infinite-volume limit. Here we reconsider this statement in terms of recent arguments of non-Hermitian skin effects-strong dependence of spectra on boundary conditions for the non-Hermitian Hamiltonians-in the traditional Green's function formalism. We find the criterion for quantities to be sensitive or insensitive against the boundary condition in Hermitian correlated or disordered systems, which is characterized by the residue theorem. We also discuss the uncertainty of the quasiparticle energy under the skin effects in terms of non-normal pseudospectra, which can be tested via the sharp optical absorption from the bulk-surface coupling. Our result indicates that pseudo quantum number emerges as a consequence of large nonnormality.

Nonunitary Scaling Theory of Non-Hermitian Localization

Non-Hermiticity can destroy Anderson localization and lead to delocalization even in one dimension. However, a unified understanding of non-Hermitian delocalization has yet to be established. Here, we develop a scaling theory of localization in non-Hermitian systems. We reveal that non-Hermiticity introduces a new scale and breaks down the one-parameter scaling, which is the central assumption of the conventional scaling theory of localization. Instead, we identify the origin of unconventional non-Hermitian delocalization as the two-parameter scaling. Furthermore, we establish the threefold universality of non-Hermitian localization based on reciprocity; reciprocity forbids delocalization without internal degrees of freedom, whereas symplectic reciprocity results in a new type of symmetry-protected delocalization.

Non-reciprocal phase transitions

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter^1-6, non-equilibrium systems^7-9, networks of neurons^10,11, social groups with conformist and contrarian members¹², directional interface growth phenomena^13-15 and metamaterials^16-20. Although wave propagation in non-reciprocal media has recently been closely studied^1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points²¹. We describe the emergence of these phases using insights from bifurcation theory^22,23 and non-Hermitian quantum mechanics^24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

Generating arbitrary topological windings of a non-Hermitian band

The nontrivial topological features in the energy band of non-Hermitian systems provide promising pathways to achieve robust physical behaviors in classical or quantum open systems. A key topological feature of non-Hermitian systems is the nontrivial winding of the energy band in the complex energy plane. We provide experimental demonstrations of such nontrivial winding by implementing non-Hermitian lattice Hamiltonians along a frequency synthetic dimension formed in a ring resonator undergoing simultaneous phase and amplitude modulations, and by directly characterizing the complex band structures. Moreover, we show that the topological winding can be controlled by changing the modulation waveform. Our results allow for the synthesis and characterization of topologically nontrivial phases in nonconservative systems.

Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the "discriminant number." Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.

Non-Hermitian generalizations of extended Su-Schrieffer-Heeger models

Non-Hermitian generalizations of the Su-Schrieffer-Heeger (SSH) models with higher periods of the hopping coefficients, called the SSH3 and SSH4 models, are analyzed. The conventional construction of the winding number fails for the Hermitian SSH3 model, but the non-Hermitian generalization leads to a topological system due to a point gap on the complex plane. The non-Hermitian SSH3 model thus has a winding number and exhibits the non-Hermitian skin effect. Moreover, the SSH3 model has two types of localized states and a zero-energy state associated with special symmetries. The total Zak phase of the SSH3 model exhibits quantization, and its finite value indicates coexistence of the two types of localized states. Meanwhile, the SSH4 model resembles the SSH model, and its non-Hermitian generalization also exhibits the non-Hermitian skin effect. A careful analysis of the non-Hermitian SSH4 model with different boundary conditions shows the bulk-boundary correspondence is restored with the help of the generalized Brillouin zone or the real-space winding number. The physics of the non-Hermitian SSH3 and SSH4 models may be tested in various simulators.

Knots and Non-Hermitian Bloch Bands

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A Z_{2} knot invariant, the global biorthogonal Berry phase Q as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature, for example, the Hopf link, the trefoil knot, the figure-8 knot, and the Whitehead link.

Non-Hermitian Bulk-Boundary Correspondence and Auxiliary Generalized Brillouin Zone Theory

We provide a systematic and self-consistent method to calculate the generalized Brillouin zone (GBZ) analytically in one-dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a n-band non-Hermitian Hamiltonian is constituted by n distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit.

Localized photonic states and dynamic process in nonreciprocal coupled Su-Schrieffer-Heeger chain

We investigate the localized photonic states and dynamic process in one-dimensional nonreciprocal coupled Su-Schrieffer-Heeger chain. Through numerical calculation of energy eigenvalue spectrum and state distributions of the system, we find that different localized photonic states with special energy eigenvalues can be induced by the nonreciprocal coupling, such as zero-energy edge states, interface states and bound states with pure imaginary energy eigenvalues. Moreover, we analyze the dynamic process of photonic states in such non-Hermitian system. Interestingly, it is shown that the nonreciprocal coupling has an evident gathering effect on the photons, which also break the trapping effect of topologically protected edge states. In addition, we consider the impacts of on-site defect potentials on the dynamic process of photonic states for the system. It is indicated that the photons go around the defect lattice site and still present the gathering effect, and different forms of laser pulses can be induced with the on-site defect potentials in different lattice sites. Furthermore, we present the method for the quantum simulation of current model based on the circuit quantum electrodynamic lattice.

Anomalous Topological Edge States in Non-Hermitian Piezophononic Media

The bulk-boundary or bulk-edge correspondence is a principle relating surface confined states to the topological classification of the bulk. By marrying non-Hermitian ingredients in terms of gain or loss with media that violate reciprocity, an unconventional non-Bloch bulk-boundary correspondence leads to unusual localization of bulk states at boundaries-a phenomenon coined non-Hermitian skin effect. Here, we numerically employ the acoustoelectric effect in electrically biased and layered piezophononic media as a solid framework for non-Hermitian and nonreciprocal topological mechanics in the MHz regime. Thanks to a non-Hermitian skin effect for mechanical vibrations, we find that the bulk bands of finite systems are highly sensitive to the type of crystal termination, which indicates a failure of using traditional Bloch bands to predict the wave characteristics. More surprisingly, when reversing the electrical bias, we unveil how topological edge and bulk vibrations can be harnessed either at the same or opposite interfaces. Yet, while bulk states are found to display this unconventional skin effect, we further discuss how in-gap edge states in the same instant, counterintuitively are able to delocalize along the entire layered medium. We foresee that our predictions will stimulate new avenues in echo-less ultrasonics based on exotic wave physics.