In non-Hermitian systems, the defective band degeneracies, so-called exceptional points (EPs), can form robust exceptional lines (ELs) in 3D momentum space in the absence of any symmetries. Here, we show that a natural orientation can be assigned to every EL according to the eigenenergy braiding around it, and prove the source-free principle of ELs as a corollary of the generalized Fermion doubling theorem for EPs on an arbitrary closed oriented surface, which indicates that if several ELs flow into a junction, the same number of outflow ELs from the junction must exist. Based on this principle, we discover three different mechanisms that can stabilize the junction of ELs and therefore guarantee the formation of various types of exceptional chains (ECs) under the protection of mirror, mirror-adjoint, or ${C}_2\mathcal{T}$ symmetries. Furthermore, we analyze the thresholdless perturbations to a Hermitian nodal line and map out all possible EC configurations that can be evolved. By strategically designing the structure and materials, we further exhibit that these exotic ECs can be readily observed in non-Hermitian photonic crystals. Our results directly manifest the combined effect of spatial symmetry and topology on the non-Hermitian singularities and pave the way for manipulating the morphology of ELs in non-Hermitian crystalline systems.
We show that topologically protected defect states can exist in open (leaky or lossy) systems even when these systems are topologically trivial in the closed limit. The states appear from within the continuum, thus in absence of a band gap, and are generated via exceptional points (a spectral transition that occurs in open wave and quantum systems with a generalized time-reversal symmetry), or via a degeneracy induced by charge-conjugation-symmetry (which is related to the pole transition of Majorana zero modes). We demonstrate these findings for a leaking passive coupled-resonator optical wav…
So far, topological band theory is discussed mainly for systems described by eigenvalue problems. Here, we develop a topological band theory described by a generalized eigenvalue problem (GEVP). Our analysis elucidates that non-Hermitian topological band structures may emerge for systems described by a GEVP with Hermitian matrices. The above result is verified by analyzing a two-dimensional toy model where symmetry-protected exceptional rings (SPERs) emerge although the matrices involved are Hermitian. Remarkably, these SPERs are protected by emergent symmetry, which is unique to the systems d…
Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and parity-time ($\mathcal{PT}$) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in non-Hermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by $\mathcal{PT}$ symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as second-order knotted surfaces of arbitrary genus, third-order knots and fourth-order points.
Non-Hermitian skin effects and exceptional points are topological phenomena characterized by integer winding numbers. In this study, we give methods to theoretically detect skin effects and exceptional points by generalizing inversion symmetry. The generalization of inversion symmetry is unique to non-Hermitian systems. We show that parities of the winding numbers can be determined from energy eigenvalues on the inversion-invariant momenta when generalized inversion symmetry is present. The simple expressions for the winding numbers allow us to easily analyze skin effects and exceptional point…
The ideas of topology have found tremendous success in Hermitian physical systems, but even richer properties exist in the more general non-Hermitian framework. Here, we theoretically propose and experimentally demonstrate a new topologically-protected bulk Fermi arc which---unlike the well-known surface Fermi arcs arising from Weyl points in Hermitian systems---develops from non-Hermitian radiative losses in photonic crystal slabs. Moreover, we discover half-integer topological charges in the polarization of far-field radiation around the Fermi arc. We show that both phenomena are direct cons…
The generic nature of band touching points in three-dimensional band structures is at heart of the rich phenomenology, topological stability and novel Fermi arc surface states associated with Weyl semimetals. Here we report on the corresponding scenario emerging in systems effectively described by non-Hermitian Hamiltonians. Remarkably, three-dimensional non-Hermitian systems have generic band touchings along one-dimensional closed contours forming exceptional rings and links in reciprocal space. The associated Seifert surfaces support open "Fermi ribbons" where the real part of the energy gap…
We study a topological band degeneracy in non-Hermitian systems with parity-time ($PT$) and parity-particle-hole ($CP$) symmetries. In $d$-dimensional non-Hermitian systems, it is shown that $(d-1)$-dimensional exceptional surfaces can appear from band touching thanks to $PT$ or $CP$ symmetry. We investigate the topological stability and zero-gap quasiparticles for the exceptional surfaces due to the band degeneracy. We also demonstrate the band degeneracy by using lattice models of a topological semimetal and a superconductor.
By exploring the extraordinary property of exceptional points (EPs) in non-Hermitian systems, we here demonstrate that losses can play constructive roles in controlling bistable states. We experimentally realize the EP in a non-Hermitian meta-molecule of coupled resonators in microwave regime. By increasing the loss, we first observe the bistable state suppression at the weak-dissipative regime, but then the bistable state recovery in strong-dissipative regime. Both the experimental and theoretical analysis demonstrate that the revival of bistable states results from the revival of the field i…
The usual concepts of topological physics, such as the Berry curvature, cannot be applied directly to non-Hermitian systems. We show that another object, the quantum metric, which often plays a secondary role in Hermitian systems, becomes a crucial quantity near exceptional points in non-Hermitian systems, where it diverges in a way that fully controls the description of wavepacket trajectories. The quantum metric behaviour is responsible for a constant acceleration with a fixed direction, and for a non-vanishing constant velocity with a controllable direction. Both contributions are independe…
The Hermitian systems possess unitary scattering; however, the Hermiticity is unnecessary for a unitary scattering although the scattering under the influence of non-Hermiticity is mostly non-unitary. Here we prove that the unitary scattering is protected by certain type of pseudo-Hermiticity and unaffected by the degree of non-Hermiticity. The energy conservation is violated in the scattering process and recovers after scattering. The subsystem of the pseudo-Hermitian scattering center including only the connection sites is Hermitian. These findings provide fundamental insights on the unitary…
Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic topologically stable exceptional points according to two types of complex-energy gaps and fundamental symmetries of charge conjugation, parity, and time reversal. This classification reveals unique non-Hermitian gapless structures with no Hermitian analogs and systematically predicts unknown non-Hermitian semimetals and nodal superconductors; a topological dumbbe…
Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal (C and K) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity (Q symmetry) or the inversion (P) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symm…
Understanding how local potentials affect system eigenmodes is crucial for experimental studies of nontrivial bulk topology. Recent studies have discovered many exotic and highly non-trivial topological states in non-Hermitian systems. As such, it would be interesting to see how non-Hermitian systems respond to local perturbations. In this work, we consider chiral and particle-hole -symmetric non-Hermitian systems on a bipartite lattice, including SSH model and photonic graphene, and find that a disordered local potential could induce bound states evolving from the bulk. When the local potenti…
Unconventional Weyl points with topological charges higher than 1 can transform into various complex unconventional Weyl exceptional contours under non-Hermitian perturbations. However, theoretical studies of these exceptional contours have been limited to tight-binding models. Here, we propose to realize unconventional Weyl exceptional contours in photonic continua -- non-Hermitian anisotropic chiral plasma, based on ab initio calculation by Maxwell's equations. By perturbing in-plane permittivity, an unconventional Weyl point can transform into a quadratic Weyl exceptional circle, a Type-I W…
Non-Hermitian (NH) Hamiltonians have become an important asset for the effective description of various physical systems that are subject to dissipation. Motivated by recent experimental progress on realizing the NH counterparts of gapless phases such as Weyl semimetals, here we investigate how NH symmetries affect the occurrence of exceptional points (EPs), that generalize the notion of nodal points in the spectrum beyond the Hermitian realm. Remarkably, we find that the dimension of the manifold of EPs is generically increased by one as compared to the case without symmetry. This leads to no…
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 symmetr…
Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional…
Engineered non-Hermitian systems featuring exceptional points can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, electronics, to atomic physics. Here we introduce and present non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sen…
We provide a systematic study of non-Hermitian topologically charged systems. Starting from a Hermitian Hamiltonian supporting Weyl points with arbitrary topological charge, adding a non-Hermitian perturbation transforms the Weyl points to one-dimensional exceptional contours. We analytical prove that the topological charge is preserved on the exceptional contours. In contrast to Hermitian systems, the addition of gain and loss allows for a new class of topological phase transition: when two oppositely charged exceptional contours touch, the topological charge can dissipate without opening a g…
Exceptional points play a pivotal role in the topology of non-Hermitian systems, and significant advances have been made in classifying exceptional points and exploring the associated phenomena. Exceptional surfaces, which are hypersurfaces of exceptional degeneracies in parameter space, can support hypersurface singularities, such as cusps, intersections and swallowtail catastrophes. Here we topologically classify the intersection singularity of exceptional surfaces for a generic pseudo-Hermitian system with parity-time symmetry. By constructing the quotient space under equivalence relations …
We provide classification of gapless phases in non-Hermitian systems according to two types of complex-energy gaps: point gap and line gap. We show that exceptional points, at which not only eigenenergies but also eigenstates coalesce, are characterized by gap closing of point gaps with nontrivial topological charges. Moreover, we find that bulk Fermi arcs accompanying exceptional points are robust because of topological charges for real line gaps. On the basis of the classification, some examples are also discussed.
The fidelity susceptibility has been used to detect quantum phase transitions in the Hermitian quantum many-body systems over a decade, where the fidelity susceptibility density approaches $+\infty$ in the thermodynamic limits. Here the fidelity susceptibility $χ$ is generalized to non-Hermitian quantum systems by taking the geometric structure of the Hilbert space into consideration. Instead of solving the metric equation of motion from scratch, we chose a gauge where the fidelities are composed of biorthogonal eigenstates and can be worked out algebraically or numerically when not on the exc…
The efficient transfer of excitations between different levels of a quantum system is a task with many applications. Among the various protocols to carry out such a state transfer in driven systems, rapid adiabatic passage (RAP) is one of the most widely used. Here we show both theoretically and experimentally that adding a suitable amount of loss to the driven Hamiltonian turns a RAP protocol into a scheme for encircling an exceptional point including the chiral state transfer associated with it. Our work thus discloses an intimate connection between a whole body of literature on RAP and rece…
Non-Hermitian topological systems simultaneously posses two antagonistic features: ultra sensitivity due to exceptional points and robustness of topological zero energy modes, and it is unclear which one prevails under different perturbations. We study that question by applying the pseudospectra theory on the prototypical non-Hermitian SSH (NHSSH) lattice. Topological modes are robust with respect to chiral perturbations and sensitive to parity-time (PT) symmetry preserving perturbations. In fact, the chiral symmetry exactly at the exceptional point leads to the suppression of its sensitivity,…
We propose an application of a parity-time symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model by embedded it in a two-dimensional square lattice tube. The coalescence state at the exceptional point of non-Hermitian SSH model is chiral and selectively controls helical transport and amplification. Two typical helicity-dependent scattering dynamics are observed. If the incidence has an identical helicity with the embedded non-Hermitian SSH model, we observe a perfect transmission without reflection. However, if the incidence has an opposite helicity with the embedded non-Hermitian SSH model…
We construct localized beams that are located at the edge of a non-Hermitian Glauber Fock (NGF) lattice made of coupled waveguides and can propagate for a long distance without almost no diffraction. Specifically, we calculate the closed-form of the eigenstates at the exceptional point of a semi-infinite NGF lattice composed of waveguides which are coupled in a unidirectional manner. We use the closed-form solution to construct the non-diffracting beams in finite NGF lattices. We provide a numerical approach to find other lattices that are capable of supporting non-diffracting beams at an exce…
Exceptional rings in nonlinear non-Hermitian planar optical microcavities: implementation, signal enhancement, and topology
Paderborn University（帕德博恩大学）；Wyant College of Optical Sciences, University of Arizona（亚利桑那大学怀特光学科学学院）；Department of Physics, University of Arizona（亚利桑那大学物理系）
Exceptional points (EPs) are degeneracies in open wave systems with coalescence of at least two energy levels and their corresponding eigenstates. In higher dimensions, more complex EP physics not found in two-state systems is observed. We consider the emergence and interaction of multiple EPs in a four coupled optical waveguides system by non-Hermitian coupling showing a unique EP formation pattern in a phase diagram. In addition, absolute phase rigidities are computed to show the mixing of the different states in definite parameter regimes. Our results could be potentially important for deve…
Exceptional points are non-Hermitian degeneracies in open quantum and wave systems at which not only eigenenergies but also the corresponding eigenstates coalesce. This is in strong contrast to degeneracies known from conservative systems, so-called diabolic points, at which only eigenenergies degenerate. Here we connect these two kinds of degeneracies by introducing the concept of the distance of a given exceptional point in matrix space to the set of diabolic points. We prove that this distance determines an upper bound for the response strength of a non-Hermitian system with this exceptiona…
This paper theoretically and numerically studies the response characteristics of non-Hermitian resonant photonic systems operating near an exceptional point (EP), where two resonant eigenmodes coalesce. It is shown that a system near an EP can exhibit a non-Lorentzian frequency response, whose line shape and intensity strongly depend on the modal decay rate and coupling parameters for the input waves, unlike a normal Lorentzian response around a single resonance. In particular, it is shown that the peak intensity of the frequency response is inversely proportional to the fourth power of the mo…
Topological materials exhibit edge-localized scattering-free modes protected by their nontrivial bulk topology through the bulk-edge correspondence in Hermitian systems. While topological phenomena have recently been much investigated in non-Hermitian systems with dissipations and injections, the fundamental principle of their edge modes has not fully been established. Here, we reveal that in non-Hermitian systems robust gapless edge modes can ubiquitously appear owing to a mechanism that is distinct from bulk topology, thus indicating the breakdown of the bulk-edge correspondence. The robustn…
Recently, a Dirac exceptional point (EP) was reported in a non-Hermitian system. Unlike a Dirac point in Hermitian systems, this Dirac EP has coalesced eigenstates in addition to the degenerate energy. Also different from a typical EP, the two energy levels connected at this Dirac EP remain real in its vicinity and display a linear instead of square root dispersion, forming a tilted Dirac cone in the hybrid space consisting of a momentum dimension and a synthetic dimension for the strength of non-Hermiticity. In this report, we first present simple three-band and two-band matrix models with a …
Exceptional-point-constrained locking of boundary-sensitive topological transitions in chiral non-Hermitian SSH-type lattices
Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所与量子光学技术与器件国家重点实验室）；Department of Physics, Fudan University（复旦大学物理系）
Exceptional points (EPs) are peculiar band singularities and play a vital role in a rich array of unusual optical phenomena and non-Hermitian band theory. In this paper, we provide a topological classification of isolated EPs based on homotopy theory. In particular, the classification indicates that an $n$-th order EP in two dimensions is fully characterized by the braid group B$_n$, with its eigenenergies tied up into a geometric knot along a closed path enclosing the EP. The quantized discriminant invariant of the EP is the writhe of the knot. The knot crossing number gives the number of bul…
The spectral, dynamical and topological properties of physical systems described by non-Hermitian (including $\mathcal{PT}$-symmetric) Hamiltonians are deeply modified by the appearance of exceptional points and spectral singularities. Here we show that exceptional points in the continuum can arise in non-Hermitian (yet admitting and entirely real-valued energy spectrum) optical lattices with engineered defects. At an exceptional point, the lattice sustains a bound state with an energy embedded in the spectrum of scattered states, similar to the von-Neumann Wigner bound states in the continuum…
Exceptional points, at which two or more eigenfunctions of a Hamiltonian coalesce, occur in non-Hermitian systems and lead to surprising physical effects. In particular, the behaviour of a system under parameter variation can differ significantly from the familiar Hermitian case in the presence of exceptional points. Here we analytically derive the probability of a non-adiabatic transition in a two-level system driven through two consecutive exceptional points at finite speed. The system is Hermitian far away from the exceptional points. In the adiabatic limit an equal redistribution between t…
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 …
