We review the current understanding of the role of topology in non-Hermitian (NH) systems, and its far-reaching physical consequences observable in a range of dissipative settings. Furthermore, the introduction of non-Hermiticity to topological systems offers a new degree of freedom to control wave propagation, such as concurrent existence of exceptional point and topological edge states, novel non-Hermiticity-induced topological . [] For dissipative systems, the associated eigenspectra are functions of the dissipation rates and an EP occurs at a critical dissipation rate c $\Gamma _c$ around which the real and imaginary part of two or more eigenvalues coalesce and bifurcate, respectively. 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. The band degeneracy, either the exceptional point of a non-Hermitian system or the Dirac point associated with a topological system, can feature distinct symmetry and topology. The robustness of these edge modes originates from yet another topological structure accompanying the branchpoint singularity . Understanding the topological properties of non-Hermitian systems has also been the focus of many research efforts [55-59]. Abstract: 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. This system is unique because we can create the topological insulating phase from a homogeneous resonator chain only by manipulating gain and loss with a certain order, leading to reconfigurable optical non-trivial topology. meaningful, adding another layer to the band topology, which is now called the spectral topology [12-16].
SPIE . This paper shows that non-hermitian quantum many-body systems, constructed as an ``analytic continuation'' of ergodic Hermitian systems, feature an exponential proliferation of exceptional points. SPIE . 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. Studies of non-Hermitian effects in quantum condensed matter systems, such as electronic materials, are less common. Among them, a unique feature emerges, known as the non-Hermitian skin effect. In particular, we elucidate how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm. The ANU Polariton BEC group has previously observed a non-Hermitian spectral degeneracy in this system and, in a separate study, detected the spectral winding around a pair of the exceptional points. 6. . Masaya Notomi and Kenta Takata "Non-Hermitian topology and exceptional points in coupled nanoresonators", Proc. As examples, non-Hermitian skin eects and exceptional points have been intensively studied. These boundary modes, also called skin modes, look quite similar to the boundary states in a topologically non-trivial insulator. FK Kunst, V Dwivedi. We propose a one-dimensional Floquet ladder that possesses two distinct topological transport channels with opposite directionality. 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. However, the selective excitation of the system in one among the infinitely many topological quasi-edge states . In this paper, we provide a topological classification of isolated EPs based on homotopy theory. Knot topology of exceptional point and non-Hermitian no-go theorem Haiping Hu, Shikang Sun, and . Third, the author reveals a certain relationship between the non-Bloch waves and non-Hermitian topology. The authors formulate a homotopy classification and knot theory of exceptional points and present a non-Hermitian no-go theorem governing the possible configurations of exceptional points and their splitting rules on a two-dimensional lattice. In this chapter, we review topological phases in Hermitian systems and explain non-Hermitian systems. The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. The direction of the EL can be identified by the corresponding Berry . Mapping between Non-Hermitian Quantum and Classical Models The non-Hermitian topology contained in the model of Eq. is a singularity in non-Hermitian systems which exhibits exotic functionalities such as high . Kunst, Flore K. Abstract. The appearance of the degenerate . Exceptional Points 1971 2004 1966 2015 In open systems, non-Hermiticity results from coupling with external bath. In particular, the classification indicates that an n-th order EP in two dimensions is fully characterized by the braid group Bn, with its . The team found that the topology of an energy surface in a non-Hermitian arrangement plays more of a role in how light behaves in a time evolving system than strict winding around an exceptional point. . 7-9. Masaya Notomi and Kenta Takata "Non-Hermitian topology and exceptional points in coupled nanoresonators", Proc. Most of the existing studies on the topology of non-Hermitian Hamiltonians con-
In interacting many-body systems, microscopic Hamiltonian is Hermitian, while one-body quasiparticle Hamiltonian is non-Hermitian due to damping. Exceptional points (EPs) are spectral degeneracies that emerge in open dynamical systems. Schematic diagram of the proposed non-Hermitian system based on coupled Fabry-Prot microcavities is illustrated in Fig. Abstract: 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. topological band theory in Hermitian systems. Knot topology of exceptional point and non-Hermitian no-go theorem Haiping Hu, Shikang Sun, and . Their topological structures called point-gap topology3-5 are unique to non-Hermitian systems. Exceptional points (EPs) are spectral degeneracies that emerge in open dynamical systems. The study of Non-Hermitian systems have gained an immense attention and importance in the recent times when it entered the area of topological systems 6,14,15,16 but the criticality in non . Special attentions are given to exceptional points - branch-point singularities on the complex eigenvalue manifolds that exhibit non-trivial topological properties. spectral topology that also emerges in non-Hermitian periodic systems, manifested as the winding of bands driven by crystal momentum. At this early stage of the field, several principles have been uncovered: (i) non-Hermitian systems have stable band degeneracies in two dimensions (2D), called exceptional points 15,16,17 (Fig. Thus, a natural question to ask is whether the finite non-Hermitian many-particle system has obvious topological properties. This system is unique because we can create the topological insulating phase from a homogeneous resonator chain only by manipulating gain and loss with a certain order, leading to reconfigurable optical non-trivial topology. "Our emulator is quite versatile in terms of the possibility of actually monitoring and digging into the dynamics of non-Hermitian systems . . Non-Hermitian Topology and Exceptional-Point Geometries. The signatures of this phase are two pairs of Kramers degenerate Floquet quasienergy bands that are separated by an imaginary gap. Exceptional non-Hermitian topological edge mode and its application to active matter: Authors: Kazuki Sone*, Yuto Ashida . Then we review topological classifications in terms of the ten-fold Altland-Zirnbauer symmetry class. 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. I will illustrate this physics through a concrete example: a honeycomb ferromagnet with Dzyaloshinskii-Moriya exchange, comparing interacting spin-wave calculations with an effective non-Hermitian model. Non-Hermitian theory is a theoretical framework that excels at describing open systems. The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. Subjects: Physics and Society, Mesoscale and Nanoscale Physics, Soft Condensed Matter, Statistical Mechanics loss mechanisms, there is an eminent need to reexamine topology within the context of non-Hermitian theories that describe open, lossy systems. In this paper, we provide a topological classification of isolated EPs based on homotopy theory. In this study, we give methods to theoretically detect skin effects and exceptional points by generalizing inversion symmetry. . . Physical Review B 99 (24), 245116, 2019. We systematically study the topology of the exceptional point (EP) in the finite non-Hermitian system. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. For the finite non-Hermitian many-particle systems, however, few studies have been done on the topological properties of EP. although the conventional notion of topological materials is based on hermitian hamiltonians, effective hamiltonians can become non-hermitian in nonconservative systems including both quantum and. The inclusion of non-Hermitian features in topological insulators has recently seen an explosion of activity. Initial interest revolved around exceptional points exhibiting unique topological features with no counterparts in Hermitian systems, such as Weyl exceptional rings [60], bulk Fermi arcs The authors formulate a homotopy classification and knot theory of exceptional points and present a non-Hermitian no-go theorem governing the possible configurations of exceptional points and their splitting rules on a two-dimensional lattice. The former type of topology exists both for Hermitian and non-Hermitian systems, while the latter is exclusive to non-Hermitian systems, has not been observed yet, and is the focus of the present work. Then we review topological classifications in terms of the ten-fold Altland-Zirnbauer symmetry class. 1, which consists of one pair of identical fiber Bragg gratings (FBGs) operating around 1550 n m with a bandwidth of 7 n m. E r 3 + - and C e 3 +-doped phosphosilicate sol-gel can be coated on the facets of each FBG to serve as active and lossy materials, respectively. In non-Hermitian systems, energy spectra . However, simple Hamiltonians without band . Quasiparticles in many-body systems generally have a finite lifetime due to electron-electron, electron-phonon and electron-impurity scatterings. Based on the concrete form of the Berry connection, we demonstrate that the exceptional line (EL), at which the eigenstates coalesce, can act as a vortex filament. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. The transport channels occur due to a Z 2 non-Hermitian Floquet topological phase that is protected by time-reversal symmetry. EJ Bergholtz, JC Budich, FK Kunst. 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. 1 Introduction. "This is the first direct measurement of a non-Hermitian topological invariant associated with an exceptional point in momentum space of a condensed matter system," says Dr Rui Su (Nanyang . 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. correspondence in the non-Hermitian version [48], and non-Hermitian skin effect [49]. February 24, 2021. Eq. However, a comprehensive theory of non-Hermitian topology for this system has not yet been developed. In contrast to the ingrained intuition that fre-quency levels are closed curves, each Fermi arc is an The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. . Exceptional points appear when both equations are satisfied simultaneously, i.e., when the two loops intersect. dI = 0 (dashed) [cf. We revisit the problem of classifying topological band structures in non-Hermitian systems. (12)] form closed loops in a two-dimensional parameter space. The band degeneracy, either the exceptional point of a non-Hermitian system or the Dirac point associated with a topological system, can feature distinct symmetry and topology. Properties . 1a). Exceptional topology of non-Hermitian systems. . The team found that the topology of an energy surface in a non-Hermitian arrangement plays more of a role in how light behaves in a time evolving system than strict winding around an exceptional . 10 (QGT), which includes the Berry curvature (the cornerstone of Hermi- The search for topological states in non-Hermitian systems, and more specifically in non-Hermitian lattice models, has become a newly emerging research front.Non-Hermitian systems are much more than a theoretical curiosity, they arise naturally in the description of the finite lifetime due to interactions, or more prominently, in photonic or acoustic systems. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. Non-Hermitian systems and topology: A transfer-matrix perspective. Based on the concrete form of the Berry connection, we demonstrate that the exceptional line (EL), at which the eigenstates coalesce, can act as a vortex filament. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. Publication. Abstract. Abstract: The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. We propose an anti-parity-time (anti-$\\mathcal{PT}$) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model, where the large non-Hermiticity constructively creates nontrivial topology and greatly expands the topological phase. which determines the topology in the non-Hermitian case . Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called "point-gap" and "line-gap" schemes. In the anti-$\\mathcal{PT}$-symmetric SSH model, the gain and loss are alternatively arranged in pairs under the inversion symmetry. The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. I will show that in small-gap systems, the decay of a quasiparticle can alter its energy-momentum dispersion significantly, for example, transform a two-dimensional Dirac point into a nodal arc that ends at topological exceptional points. Reviews of Modern Physics 93 (1), 015005, 2021. Their synergy will further produce more exotic topological effects in synthetic matter. The generalization of the Chern number to non-Hermitian Hamiltonians initiated this reexamination; however, there is no established connection between a non-Hermitian topological The multipartite non-Hermitian Su-Schrieffer-Heeger model is explored as a prototypical example of one-dimensional systems with several sublattice sites for unveiling intriguing insulating and metallic phases with no Hermitian counterparts. First, we show that various topological phases stem from a geometric phase. Second, a topological semimetal phase with exceptional points appears, The topological semimetal phase is unique to non-Hermitian systems because it is caused by the deformation of the generalized Brillouin zone by changes of system parameters. 54]. The generalization of inversion symmetry is unique to non-Hermitian systems. These phases are characterized by composite cyclic loops of multiple complex-energy bands encircling single or multiple exceptional points (EPs) on the . Exciting developments include tunable wave guides that are robust to disorder (1-3), structure-free systems (4, 5), and topological lasers and pumping (6-10).In these systems, active components are introduced to typically 1) break time-reversal symmetry to create topological .
