Realizing exceptional points of any order in the presence of symmetry

Sharareh Sayyad, Flore K. Kunst

Exceptional points~(EPs) appear as degeneracies in the spectrum of non-Hermitian 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 non-Hermitian 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 low-energy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudo-chiral symmetry enforces nth order EPs to disperse with the (n−1)th root. For two-, three- and four-band systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their low-energy dispersion relations.

Anomalous Behaviors of Quantum Emitters in Non-Hermitian Baths

Zongping Gong, Miguel Bello, Daniel Malz, Flore K. Kunst

Both non-Hermitian systems and the behaviour of emitters coupled to structured baths have been studied intensely in recent years. Here we study the interplay of these paradigmatic settings. In a series of examples, we show that a single quantum emitter coupled to a non-Hermitian bath displays a number of unconventional behaviours, many without Hermitian counterpart. We first consider a unidirectional hopping lattice whose complex dispersion forms a loop. We identify peculiar bound states inside the loop as a manifestation of the non-Hermitian skin effect. In the same setting, emitted photons may display spatial amplification markedly distinct from free propagation, which can be understood with the help of the generalized Brillouin zone. We then consider a nearest-neighbor lattice with alternating loss. We find that the long-time emitter decay always follows a power law, which is usually invisible for Hermitian baths. Our work points toward a rich landscape of anomalous quantum emitter dynamics induced by non-Hermitian baths.

Bound states and photon emission in non-Hermitian nanophotonics

Zongping Gong, Miguel Bello, Daniel Malz, Flore K. Kunst

We establish a general framework for studying the bound states and the photon-emission dynamics of quantum emitters coupled to structured nanophotonic lattices with engineered dissipation (loss). In the single-excitation sector, the system can be described exactly by a non-Hermitian formalism. We have pointed out in the accompanying letter [Gong \emph{et al}., arXiv:2205.05479] that a single emitter coupled to a one-dimensional non-Hermitian lattice may already exhibit anomalous behaviors without Hermitian counterparts. Here we provide further detail on these observations. We also present several additional examples on the cases with multiple quantum emitters or in higher dimensions. Our work unveils the tip of the iceberg of the rich non-Hermitian phenomena in dissipative nanophotonic systems.

Topological interface of light

Flore K. Kunst

Upon combining dissipative and nonlinear effects in a bipartite lattice of cavity polaritons, dissipatively stabilized bulk gap solitons emerge, which create a topological interface.

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Symmetry-protected exceptional and nodal points in non-Hermitian systems

Sharareh Sayyad, Marcus Stålhammar, Lukas Rødland, Flore K. Kunst

One of the unique features of non-Hermitian (NH) systems is the appearance of non-Hermitian degeneracies known as exceptional points~(EPs). The occurrence of EPs in NH systems requires satisfying constraints whose number can be reduced in the presence of some symmetries. This results in stabilizing the appearance of EPs. Even though two different types of EPs, namely defective and non-defective EPs, may emerge in NH systems, exploring the possibilities of stabilizing EPs has been only addressed for defective EPs, at which the Hamiltonian becomes non-diagonalizable. In this letter, we show that certain discrete symmetries, namely parity-time, parity-particle-hole, and pseudo-Hermitian symmetry, may guarantee the occurrence of both defective and non-defective EPs. We extend this list of symmetries by including the non-Hermitian time-reversal symmetry in the two-band systems. <br>We further show that the non-defective EPs manifest themselves by i) the diagonalizability of non-Hermitian Hamiltonian at these points and ii) the non-diagonalizability of the Hamiltonian along certain intersections of non-defective EPs. Two-band and four-band models exemplify our findings. Through an example, we further reveal that ordinary (Hermitian) nodal points may coexist with defective EPs in non-Hermitian models when the above symmetries are relaxed.

PT symmetry-protected exceptional cones and analogue Hawking radiation

Marcus Stålhammar, Jorge Larana-Aragon, Lukas Rødland, Flore Kunst

We show that the exceptional surfaces of linear three-dimensional non-Hermitian parity-time-symmetric two-band models attain the form of topologically stable tilted exceptional cones. By relating the exceptional cones to energy cones of two-dimensional Hermitian parity-time-symmetric two-band 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, light-like particle-antiparticle pairs are created resembling Hawking radiation. We also investigate dissipative features of the non-Hermitian Hamiltonian related to the latter and comment on potential realizations in laboratory setups.

Phase transitions and generalized biorthogonal polarization in non-Hermitian systems

Elisabet Edvardsson, Flore Kunst, Tsuneya Yoshida, Emil J. Bergholtz

Non-Hermitian (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 bulk-boundary 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 real-space 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 Su-Schrieffer-Heeger 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 four-dimensional quantum Hall physics

Fanny Terrier, Flore Kunst

Four-dimensional 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 three-dimensional Weyl cones on its boundaries. We propose a three-dimensional analog of this model in the form of a dissipative Weyl semimetal (WSM) described by a non-Hermitian (NH) Hamiltonian, which in the long-time limit manifests the anomalous boundary physics of the four-dimensional QH model in the bulk spectrum. The topology of the NH WSM is captured by a three-dimensional 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.

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 higher-order 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(3-9). Here we report on the experimental realization of novel corner states made out of visible light in three-dimensional photonic structures inscribed in glass samples using femtosecond laser technology(10,11). By creating and analysing waveguide arrays, which form two-dimensional 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(7-9,12-14). 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.

Non-Hermitian systems and topology: A transfer-matrix perspective

Flore K. Kunst, Vatsal Dwivedi

Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulk-boundary correspondence. These features can change drastically for their non-Hermitian generalizations, as exemplified by a general breakdown of bulk-boundary correspondence and a localization of all states at the boundary, termed the non-Hermitian skin effect. In this paper, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices, a real-space 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 bulk-boundary correspondence and the existence of a non-Hermitian 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 real-space 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 non-Hermitian systems.

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Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence

Elisabet Edvardsson, Flore K. Kunst, Emil J. Bergholtz

Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order 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 bulk-boundary 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 non-Hermitian extensions of higher-order 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 non-Hermitian skin effect, and also naturally encompasses genuinely non-Hermitian 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 nodal-line semimetals on a hyperhoneycomb lattice, spin-orbit coupled graphene, and three-dimensional topological insulators on a diamond lattice, for which no previous exact finite-size 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 two-dimensional 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 D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges, and surfaces. These solvable models represent a class of "sweet spots" in the space of possible tight-binding models-the exact solutions remain valid for any tight-binding 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.

Symmetry-protected nodal phases in non-Hermitian systems

Jan Carl Budich, Johan Carlstrom, Flore K. Kunst, Emil J. Bergholtz

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 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 two-band lattice models in one and two spatial dimensions, and show how they are readily generalized to generic NH crystalline systems.

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Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems

Flore K. Kunst, Elisabet Edvardsson, Jan Carl Budich, Emil J. Bergholtz

Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary 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 non-Hermitian systems, their combined biorthogonal density penetrates the bulk precisely when phase transitions occur. This leads to generalized bulk-boundary 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 non-Hermitian extensions of the Su-Schrieffer-Heeger model and Chern insulators.

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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 D-dimensional lattice models whose d-dimensional 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 higher-order 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 two-dimensional models as well as to three-dimensional lattices where we present exact solutions corresponding to one-dimensional chiral states at the hinges of the lattice. We relate the higher-order topological nature of these models to reflection symmetries in combination with their provenance from lower-dimensional conventional topological phases.

Transversal magnetotransport in Weyl semimetals: Exact numerical approach

Jan Behrends, Flore Kunst, Bjoern Sbierski

Magnetotransport experiments on Weyl semimetals are essential for investigating the intriguing topological and low-energy 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.