Recently a type of robust exceptional points was found that is insensitive to the coupling disorder in the bulk. Here we show that a disparity emerges when the number of coupled cavities in this one-dimensional array changes from even to odd. The robust exceptional point only exists in the former case, whereas the location of the exceptional point in the latter depends inversely on the size of the cavity array and is subjected to coupling disorder in the bulk. We further show that the exceptional points in these two cases are second and third order, respectively. We elucidate the origin of the robust EP as a restricted bulk zero mode, which shares the same robustness against coupling disorder and has a finite amplitude adjacent to the boundary. This finding enables us to identify robust EPs in higher dimensional systems reliably, which can exist in the presence of either sublattice symmetry or the non-Hermitian particle-hole symmetry evolved from it.
Weyl points are the simplest topologically-protected degeneracy in a three-dimensional dispersion relation. The realization of Weyl semimetals in photonic crystals has allowed these singularities and their consequences to be explored with electromagnetic waves. However, it is difficult to achieve nonlinearities in such systems. One promising approach is to use the strong-coupling of photons and excitons, because the resulting polaritons interact through their exciton component. Yet topological polaritons have only been realized in two dimensions. Here, we predict that the dispersion relation f…
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…
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.
We consider a 3-dimensional (3D) non-Hermitian exceptional line semimetal model and take open boundary conditions in x, y, and z directions separately. In each case, we calculate the parameter regions where the bulk-boundary correspondence is broken. The breakdown of the bulk-boundary correspondence is manifested by the deviation from unit circles of generalized Brillouin zones (GBZ) and the discrepancy between spectra calculated with open boundary conditions (OBC) and periodic boundary conditions (PBC). The consistency between OBC and PBC spectra can be recovered if the PBC spectra are calcul…
We introduce a topological theory to study quasiparticles in interacting and/or disordered many-body systems, which have a finite lifetime due to inelastic and/or elastic scattering. The one-body quasiparticle Hamiltonian includes both the Bloch Hamiltonian of band theory and the self-energy due to interactions, which is non-Hermitian when quasiparticle lifetime is finite. We study the topology of non-Hermitian quasiparticle Hamiltonians in momentum space, whose energy spectrum is complex. The interplay of band structure and quasiparticle lifetime is found to have remarkable consequences in ze…
Topological semimetals feature a diversity of nodal manifolds including nodal points, various nodal lines and surfaces, and recently novel quantum states in non-Hermitian systems have been arousing widespread research interests. In contrast to Hermitian systems whose bulk nodal points must form closed manifolds, it is fascinating to find that for non-Hermitian systems exotic nodal manifolds can be bounded by exceptional points in the bulk band structure. Such exceptional points, at which energy bands coalesce with band conservation violated, are iconic for non-Hermitian systems. In this work, …
Topological heavy-fermion systems in three dimensions are usually classified as topological insulators or semimetals. Here, we theoretically predict a different type of heavy-fermion system (dubbed exceptional heavy-fermion semimetal) by studying a three-dimensional periodic Anderson model consisting of strongly correlated localized $f$ electrons and itinerant conduction $c$ electrons in a zincblende lattice. Due to the breaking of inversion symmetry, the quasiparticle lifetimes at different sublattices are distinct, leading to the emergence of Weyl exceptional rings in the complex pole of the…
We consider the impact of disorder on the spectrum of three-dimensional nodal-line semimetals. We show that the combination of disorder and a tilted spectrum naturally leads to a non-Hermitian self-energy contribution that can split a nodal line into a pair of exceptional lines. These exceptional lines form the boundary of an open and orientable bulk Fermi ribbon in reciprocal space on which the energy gap vanishes. We find that the orientation and shape of such a disorder-induced bulk Fermi ribbon is controlled by the tilt direction and the disorder properties, which can also be exploited to …
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 e…
Capital to topological insulators, the bulk-boundary correspondence ties a topological invariant computed from the bulk (extended) states with those at the boundary, which are hence robust to disorder. Here we put forward an ordering unique to non-Hermitian lattices, whereby a pristine system becomes devoid of extended states, a property which turns out to be robust to disorder. This is enabled by a peculiar type of non-Hermitian degeneracy where a macroscopic fraction of the states coalesce at a single point with geometrical multiplicity of $1$, that we call a phenomenal point.
We demonstrate that the non-Hermitian parity-time (PT) symmetric interfaces formed between amplifying and lossy crystals support dissipationless edge states. These PT edge states exhibit gapless spectra in the complex band structure interconnecting complex-valued bulk bands as long as exceptional points (EPs) of edge states exist. As a result, regimes exist where the edge states can spectrally overlap with the bulk continuum without hybridization, and leakage into the bulk states is suppressed due to the PT symmetry. Two exemplary PT symmetric systems, based on valley and quantum hall topologi…
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…
We consider a non-Hermitian superconducting system by coupling a conventional superconductor to a ferromagnet lead and demonstrate the emergence of exceptional points when an external Zeeman field is applied. We discover that, depending on the non-Hermiticity and the Zeeman field, the exceptional points mark the ends of lines with zero real energy, thus giving rise to topologically protected and highly tunable bulk Fermi arcs, which we coin bulk Bogoliubov Fermi arcs due to their superconducting nature. We show that these bulk Bogoliubov Fermi arcs are the non-Hermitian counterparts of the Her…
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 explore the topological properties of non-Hermitian nodal-link semimetals with dissipative cold atoms in a three-dimensional optical lattice. We construct a two-band continuum model in three dimensions with a spin-dependent gain and loss, where the exceptional points in the energy spectrum can comprise a double Hopf link. The topology of the bulk band is characterized by a winding number defined for a one-dimensional loop in the momentum space and a topological transition of the nodal structures emerges as the change of the non-Hermiticity strength. A non-Bloch theory is built to describe t…
We study the vibrational spectrum of a constrained classical ring. Due to the presence of 2-order exceptional points, a topologically trivial band at the infinity can make the vibrational band topologically nontrivial. The symmetry, which is believed to be indispensable in topological models, is absent in this model. The fractional boundary states can be found in such classical system. Furthermore, the other aspect of the bulk boundary correspondence is revealed: an extra fractional exceptional point is topologically protected in bulk by the boundary states.
Periodically driven non-Hermitian systems can exhibit rich topological band structure and non-Hermitian skin effect, without analogs in their static or Hermitian counterparts. In this work we investigate the exceptional band-touching points in the Floquet quasi-energy bands, the topological characterization of such exceptions points and the Floquet non-Hermitian skin effect (FNHSE). Specifically, we exploit the simplicity of periodically quenched two-band systems in one dimension or two dimensions to analytically obtain the Floquet effective Hamiltonian as well as locations of the many excepti…
The traditional bulk-boundary correspondence assuring robust gapless modes at the edges and surfaces of insulating and nodal topological materials gets masked in non-Hermitian (NH) systems by the skin effect, manifesting an accumulation of a macroscopic number of states near such interfaces. Here we show that dislocation lattice defects are immune to such skin effect or at most display a \emph{weak} skin effect (depending on its relative orientation with the Burgers vector), and as such they support robust topological modes in the bulk of a NH system, specifically when the parent Hermitian pha…
The existence of magnetostriction in bulk BiFeO3 is still a matter of investigation and it is also an issue to investigate the magnetostriction effect in nano BiFeO3. Present work demonstrates the existence of magnetostrictive strain in superparamagnetic BiFeO3 nanoparticles at room temperature and the magnetoelectric coupling properties in composite form with P(VDFTrFE). Despite few reports on the magnetostriction effect in bulk BiFeO3 evidenced by the indirect method, the direct method (strain gauge) was employed in this work to examine the magnetostriction of superparamagnetic BiFeO3. In ad…
Exceptional Topology Survives Strong Hermitian Fields in Radiative Atomic Arrays
Institute of Atomic and Molecular Sciences, Academia Sinica（中央研究院原子与分子科学研究所）；Department of Physics, National Taiwan University（台湾大学物理系）；Department of Physics, University of California, Berkeley（加州大学伯克利分校物理系）；Department of Physics and Astronomy, Purdue University（普渡大学物理与天文系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）；Department of Physics, National Taiwan Normal University（台湾师范大学物理系）；Department of Physics, National Central University（中央大学物理系）
We analyze chiral topological edge modes in a non-Hermitian variant of the 2D Dirac equation. Such modes appear at interfaces between media with different "masses" and/or signs of the "non-Hermitian charge". The existence of these edge modes is intimately related to exceptional points of the bulk Hamiltonians, i.e., degeneracies in the bulk spectra of the media. We find that the topological edge modes can be divided into three families ("Hermitian-like", "non-Hermitian", and "mixed"), these are characterized by two winding numbers, describing two distinct kinds of half-integer charges carried …
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.…
We investigate the potential of the surface states of 3D nodal-line semimetals to produce surface and 3D quantized Hall effects in which the electronic currents flow unidirectionally along the surface of the material. This phenomenology is based on the fact that, irrespective of its orientation, a strong magnetic field gives rise to a zero Landau level with huge 2D degeneracy of states in the nodal-line semimetal. When the magnetic field is perpendicular to the plane of the nodal line, we show that the evanescent states within the nodal ring are transmuted into Landau surface states which corr…
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…
Multimode bulk acoustic systems show promise for use in superconducting quantum computation. They can serve as a medium term memory storage, with exceptional coherence times demonstrated, and they exhibit a mode density that is physically highly compact. Here we experimentally demonstrate accessing individual acoustic modes without being hindered by the uniform frequency spacing of the modes. We use sideband control where a low-frequency modulation is applied to the transmon qubit energy. The amplitude of the modulation defines the qubit-acoustic mode coupling, and its frequency detunes the ac…
We discuss the topology of Bogoliubov excitation bands from a Bose-Einstein condensate in an optical lattice. Since the Bogoliubov equation for a bosonic system is non-Hermitian, complex eigenvalues often appear and induce dynamical instability. As a function of momentum, the onset of appearance and disappearance of complex eigenvalues is an exceptional point (EP), which is a point where the Hamiltonian is not diagonalizable and hence the Berry connection and curvature are ill-defined, preventing defining topological invariants. In this paper, we propose a systematic procedure to remove EPs fr…
Recently, topological quantum states of non-Hermitian systems, exhibiting rich new exotic states, have attracted great attention in condensed-matter physics. As for the demonstration, most of non-Hermitian topological phenomena previously focused on are in one- and two-dimensional systems. Here, we investigate three-dimensional non-Hermitian nodal-line semimetals in the presence of a particle gain-and-loss perturbation. It is found that this perturbation will split the original nodal ring into two exceptional rings (ERs). The topological nature of the bulk electronic structure is characterized…
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…
We consider the role of non-triviality resulting from a non-Hermitian Hamiltonian that conserves twofold PT-symmetry assembled by interconnections between a PT-symmetric lattice and its time reversal partner. Twofold PT-symmetry in the lattice produces additional surface exceptional points that play the role of new critical points, along with the bulk exceptional point. We show that there are two distinct regimes possessing symmetry-protected localized states, of which localization lengths are robust against external gain and loss. The states are demonstrated by numerical calculation of a quas…
A single unit cell contains all the information about the bulk system, including the topological feature. The topological invariant can be extracted from a finite system, which consists of several unit cells under certain environment, such as a non-Hermitian external field. We investigate a non- Hermitian finite-size Kitaev chain with PT-symmetric chemical potentials. Exact solution at the symmetric point shows that Majorana edge modes can emerge as the coalescing states at exceptional points and PT symmetry breaking states. The coalescing zero mode is the finite-size projection of the convent…
