Exceptional points~(EPs) appear as degeneracies in the spectrum of nonHermitian matrices at which the eigenvectors coalesce. In general, an EP of order n may find room to emerge if 2(n−1) real constraints are imposed. Our results show that these constraints can be expressed in terms of the determinant and traces of the nonHermitian matrix. Our findings further reveal that the total number of constraints may reduce in the presence of unitary and antiunitary symmetries. Additionally, we draw generic conclusions for the lowenergy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudochiral symmetry enforces nth order EPs to disperse with the (n−1)th root. For two, three and fourband systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their lowenergy dispersion relations.
2021
Artificial Hawking Radiation in NonHermitian ParityTime Symmetric Systems
Marcus Stålhammar, Jorge LaranaAragon, Lukas Rødland, Flore Kunst
We show that the exceptional surfaces of linear threedimensional nonHermitian paritytimesymmetric twoband models attain the form of topologically stable tilted exceptional cones. By relating the exceptional cones to energy cones of twodimensional Hermitian paritytimesymmetric twoband models, we find a connection between the exceptional cone and the light cone of an observer in the vicinity of a Schwarzschild black hole. When the cone overtilts, lightlike particleantiparticle pairs are created resembling Hawking radiation. We also investigate dissipative features of the nonHermitian Hamiltonian related to the latter and comment on potential realizations in laboratory setups.
Exceptional topology of nonHermitian systems
Emil J. Bergholtz, Jan Carl Budich, Flore Kunst
Reviews of Modern Physics
93(1)
015005
(2021)

Journal
The current understanding of the role of topology in nonHermitian (NH) systems and its farreaching physical consequences observable in a range of dissipative settings are reviewed. 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. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open FermiSeifert surfaces, where conventional bandtouching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in singleband systems are detailed, and the manner in which a given physical context may affect the symmetrybased topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulkboundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative coldatom setups are reviewed.
2020
Phase transitions and generalized biorthogonal polarization in nonHermitian systems
Elisabet Edvardsson, Flore Kunst, Tsuneya Yoshida, Emil J. Bergholtz
Physical Review Research
2(4)
043046
(2020)

Journal
NonHermitian (NH) Hamiltonians can be used to describe dissipative systems, notably including systems with gain and loss, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulkboundary correspondence, invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a realspace invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we focus on the anisotropic SuSchriefferHeeger chain and study gap closings analytically. We also propose a generalization of a previously developed method with which to find all the bulk states of the system with open boundaries to NH models. Using the exact solutions for the bulk and boundary states, we elucidate genuinely NH aspects of the interplay between the bulk and boundary at the phase transitions.
Dissipative analog of fourdimensional quantum Hall physics
Fanny Terrier, Flore Kunst
Physical Review Research
2(2)
023364
(2020)

Journal
Fourdimensional quantum Hall (QH) models usually rely on synthetic dimensions for their simulation in experiment. Here, we study a QH system which features a nontrivial configuration of threedimensional Weyl cones on its boundaries. We propose a threedimensional analog of this model in the form of a dissipative Weyl semimetal (WSM) described by a nonHermitian (NH) Hamiltonian, which in the longtime limit manifests the anomalous boundary physics of the fourdimensional QH model in the bulk spectrum. The topology of the NH WSM is captured by a threedimensional winding number whose value is directly related to the total chirality of the surviving Weyl nodes. Upon taking open boundary conditions, instead of Fermi arcs, we find exceptional points with an order that scales with system size.
2019
Corner states of light in photonic waveguides
Ashraf El Hassan, Flore K. Kunst, Alexander Moritz, Guillermo Andler, Emil J. Bergholtz, Mohamed Bourennane
The recently established paradigm of higherorder topological states of matter has shown that not only edge and surface states(1,2) but also states localized to corners, can have robust and exotic properties(39). Here we report on the experimental realization of novel corner states made out of visible light in threedimensional photonic structures inscribed in glass samples using femtosecond laser technology(10,11). By creating and analysing waveguide arrays, which form twodimensional breathing kagome lattices in various sample geometries, we establish this as a platform for corner states exhibiting a remarkable degree of flexibility and control. In each sample geometry we measure eigenmodes that are localized at the corners in a finite frequency range, in complete analogy with a theoretical model of the breathing kagome(79,1214). Here, measurements reveal that light can be 'fractionalized,' corresponding to simultaneous localization to each corner of a triangular sample, even in the presence of defects.
NonHermitian systems and topology: A transfermatrix perspective
Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulkboundary correspondence. These features can change drastically for their nonHermitian generalizations, as exemplified by a general breakdown of bulkboundary correspondence and a localization of all states at the boundary, termed the nonHermitian skin effect. In this paper, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices, a realspace approach suitable for systems with periodic as well as open boundary conditions. We show that various qualitative properties of these systems can be easily deduced from the transfer matrix. For instance, the connection between the breakdown of the conventional bulkboundary correspondence and the existence of a nonHermitian skin effect, previously observed numerically, is traced back to the transfer matrix having a determinant not equal to unity. The vanishing of this determinant signals realspace exceptional points, whose order scales with the system size. We also derive previously proposed topological invariants such as the biorthogonal polarization and the Chern number computed on a complexified Brillouin zone. Finally, we define an invariant for and thereby clarify the meaning of topologically protected boundary modes for nonHermitian systems.
suggested by editors
NonHermitian extensions of higherorder topological phases and their biorthogonal bulkboundary correspondence
Elisabet Edvardsson, Flore K. Kunst, Emil J. Bergholtz
NonHermitian Hamiltonians, which describe a wide range of dissipative systems, and higherorder topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulkboundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface, and bulk degrees of freedom we establish that the nonHermitian extensions of higherorder topological phases exhibit an even richer phenomenology than their Hermitian counterparts and that this can be understood in a unifying way within our biorthogonal framework. Saliently this works in the presence of the nonHermitian skin effect, and also naturally encompasses genuinely nonHermitian phenomena in the absence thereof.
Extended Bloch theorem for topological lattice models with open boundaries
Flore K. Kunst, Guido van Miert, Emil J. Bergholtz
While the Bloch spectrum of translationally invariant noninteracting lattice models is trivially obtained by a Fourier transformation, diagonalizing the same problem in the presence of open boundary conditions is typically only possible numerically or in idealized limits. Here we present exact analytic solutions for the boundary states in a number of lattice models of current interest, including nodalline semimetals on a hyperhoneycomb lattice, spinorbit coupled graphene, and threedimensional topological insulators on a diamond lattice, for which no previous exact finitesize solutions are available in the literature. Furthermore, we identify spectral mirror symmetry as the key criterium for analytically obtaining the entire (bulk and boundary) spectrum as well as the concomitant eigenstates, and exemplify this for Chern and Z(2) insulators with open boundaries of codimension one. In the case of the twodimensional Lieb lattice, we extend this further and show how to analytically obtain the entire spectrum in the presence of open boundaries in both directions, where it has a clear interpretation in terms of bulk, edge, and corner states.
Boundaries of boundaries: A systematic approach to lattice models with solvable boundary states of arbitrary codimension
Flore K. Kunst, Guido van Miert, Emil J. Bergholtz
We present a generic and systematic approach for constructing Ddimensional lattice models with exactly solvable ddimensional boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of "sweet spots" in the space of possible tightbinding modelsthe exact solutions remain valid for any tightbinding parameters as long as they obey simple locality conditions that are manifest in the underlying lattice structure. Consequently, our models capture the physics of both (higher order) topological and nontopological phases as well as the transitions between them in a particularly illuminating and transparent manner.
Symmetryprotected nodal phases in nonHermitian systems
Jan Carl Budich, Johan Carlstrom, Flore K. Kunst, Emil J. Bergholtz
NonHermitian (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 nodal surfaces formed by EPs that are stable as long as a protecting symmetry is preserved, and that are connected by open Fermi volumes. We illustrate our findings with analytically solvable twoband lattice models in one and two spatial dimensions, and show how they are readily generalized to generic NH crystalline systems.
suggested by editors
2018
Biorthogonal BulkBoundary Correspondence in NonHermitian Systems
Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich, Emil J. Bergholtz
NonHermitian systems exhibit striking exceptions from the paradigmatic bulkboundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis) appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in nonHermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulkboundary correspondence and a quantized biorthogonal polarization that is formulated directly in systems with open boundaries. We illustrate our general insights by deriving the phase diagram for several microscopic open boundary models, including exactly solvable nonHermitian extensions of the SuSchriefferHeeger model and Chern insulators.
suggested by editors
Lattice models with exactly solvable topological hinge and corner states
Flore K. Kunst, Guido van Miert, Emil J. Bergholtz
We devise a generic recipe for constructing Ddimensional lattice models whose ddimensional boundary states, located on surfaces, hinges, corners, and so forth, can be obtained exactly. The solvability is rooted in the underlying lattice structure and as such does not depend on fine tuning, allowing us to track their evolution throughout various phases and across phase transitions. Most saliently, our models provide "boundary solvable" examples of the recently introduced higherorder topological phases. We apply our general approach to breathing and anisotropic kagome and pyrochlore lattices for which we obtain exact corner eigenstates, and to periodically driven twodimensional models as well as to threedimensional lattices where we present exact solutions corresponding to onedimensional chiral states at the hinges of the lattice. We relate the higherorder topological nature of these models to reflection symmetries in combination with their provenance from lowerdimensional conventional topological phases.
Transversal magnetotransport in Weyl semimetals: Exact numerical approach
Magnetotransport experiments on Weyl semimetals are essential for investigating the intriguing topological and lowenergy properties of Weyl nodes. If the transport direction is perpendicular to the applied magnetic field, experiments have shown a large positive magnetoresistance. In this work we present a theoretical scattering matrix approach to transversal magnetotransport in a Weyl node. Our numerical method confirms and goes beyond the existing perturbative analytical approach by treating disorder exactly. It is formulated in real space and is applicable to mesoscopic samples as well as in the bulk limit. In particular, we study the case of clean and strongly disordered samples.