non-Hermitian Topological Systems Research Articles

Hall conductance of a non-Hermitian Weyl semimetal

In recent years, non-Hermitian (NH) topological semimetals have garnered significant attention due to their unconventional properties. In this work, we explore one of the transport properties, namely the Hall conductance of a three-dimensional dissipative Weyl semi-metal formed as a result of the stacking of two-dimensional Chern insulators. We find that unlike Hermitian systems where the Hall conductance is quantized, in presence of non-Hermiticity, the quantized Hall conductance starts to deviate from its usual nature. We show that the non-quantized nature of the Hall conductance in such NH topological systems is intimately connected to the presence of exceptional points. We find that in the case of open boundary conditions, the transition from a topologically trivial regime to a non-trivial topological regime takes place at a different value of the momentum than that of the periodic boundary spectra. This discrepancy is solved by considering the non-Bloch case and the generalized Brillouin zone (GBZ). Finally, we present the Hall conductance evaluated over the GBZ and connect it to the separation between the Weyl nodes, within the non-Bloch theory.

Topological phase transitions of generalized Brillouin zone

It has been known that the bulk-boundary correspondence (BBC) of the non-Hermitian skin effect is characterized by the topology of the complex eigenvalue spectra, while the topology of the wave function gives rise to Hermitian BBC with conventional boundary modes. In this work, we go beyond the known description of the non-Hermitian topological phase and find a different type of BBC that appears in generalized boundary conditions. The generalized Brillouin zone (GBZ) possesses non-trivial topological structures in the intermediate boundary condition between open and periodic boundary conditions. Unlike the conventional BBC, the topological phase transition is characterized by the generalized momentum touching of GBZ, which manifests as exceptional points. As a realization of our proposal, we suggest the non-reciprocal Kuramoto oscillator lattice, where phase slips accompany exceptional points as a signature of such topological phase transition. Our work establishes an understanding of non-Hermitian topological matter by complementing the non-Hermitian BBC as a general foundation of the non-Hermitian topological systems.

Parity–time-symmetric photonic topological insulator

Topological insulators are a concept that originally stems from condensed matter physics. As a corollary to their hallmark protected edge transport, the conventional understanding of such systems holds that they are intrinsically closed, that is, that they are assumed to be entirely isolated from the surrounding world. Here, by demonstrating a parity–time-symmetric topological insulator, we show that topological transport exists beyond these constraints. Implemented on a photonic platform, our non-Hermitian topological system harnesses the complex interplay between a discrete coupling protocol and judiciously placed losses and, as such, inherently constitutes an open system. Nevertheless, even though energy conservation is violated, our system exhibits an entirely real eigenvalue spectrum as well as chiral edge transport. Along these lines, this work enables the study of the dynamical properties of topological matter in open systems without the instability arising from complex spectra. Thus, it may inspire the development of compact active devices that harness topological features on-demand.

A scheme for realizing nonreciprocal interlayer coupling in bilayer topological systems

Nonreciprocal interlayer coupling is difficult to practically implement in bilayer non-Hermitian topological photonic systems. In this work, we identify a similarity transformation between the Hamiltonians of systems with nonreciprocal interlayer coupling and on-site gain/loss. The similarity transformation is widely applicable, and we show its application in one- and two-dimensional bilayer topological systems as examples. The bilayer non-Hermitian system with nonreciprocal interlayer coupling, whose topological number can be defined using the gauge-smoothed Wilson loop, is topologically equivalent to the bilayer system with on-site gain/loss. We also show that the topological number of bilayer non-Hermitian C6v-typed domain-induced topological interface states can be defined in the same way as in the case of the bilayer non-Hermitian Su–Schrieffer–Heeger model. Our results show the relations between two microscopic provenances of the non-Hermiticity and provide a universal and convenient scheme for constructing and studying nonreciprocal interlayer coupling in bilayer non-Hermitian topological systems. This scheme is useful for observation of non-Hermitian skin effect in three-dimensional systems.Graphical

Topological invariant for multiband non-Hermitian systems with chiral symmetry

Topology plays an important role in non-Hermitian systems. How to characterize a non-Hermitian topological system under open-boundary conditions (OBCs) is a challenging problem. A one-dimensional (1D) topological invariant defined on a generalized Brillion zone (GBZ) was recently found to successfully describe the topological property of the two-band Su-Schrieffer-Heeger model. But for a 1D multiband chiral symmetric system under OBCs, it is still controversial how to define the topological invariant. We show in this paper by exact proof and detailed demonstration that to acquire the topological invariant for multiband non-Hermitian models with chiral symmetry, the GBZ as the integral domain should be replaced by a more generalized closed loop. Our work thus establishes the non-Bloch bulk-boundary correspondence for 1D multiband chiral symmetric non-Hermitian systems.

Entanglement Phase Transition Induced by the Non-Hermitian Skin Effect

Recent years have seen remarkable development in open quantum systems effectively described by non-Hermitian Hamiltonians. A unique feature of non-Hermitian topological systems is the skin effect, anomalous localization of an extensive number of eigenstates driven by nonreciprocal dissipation. Despite its significance for non-Hermitian topological phases, the relevance of the skin effect to quantum entanglement and critical phenomena has remained unclear. Here, we find that the skin effect induces a nonequilibrium quantum phase transition in the entanglement dynamics. We show that the skin effect gives rise to a macroscopic flow of particles and suppresses the entanglement propagation and thermalization, leading to the area law of the entanglement entropy in the nonequilibrium steady state. Moreover, we reveal an entanglement phase transition induced by the competition between the unitary dynamics and the skin effect even without disorder or interactions. This entanglement phase transition accompanies nonequilibrium quantum criticality characterized by a nonunitary conformal field theory whose effective central charge is extremely sensitive to the boundary conditions. We also demonstrate that it originates from an exceptional point of the non-Hermitian Hamiltonian and the concomitant scale invariance of the skin modes localized according to the power law. Furthermore, we show that the skin effect leads to the purification and the reduction of von Neumann entropy even in Markovian open quantum systems described by the Lindblad master equation. Our work opens a way to control the entanglement growth and establishes a fundamental understanding of phase transitions and critical phenomena in open quantum systems far from thermal equilibrium.

Topological laser with higher-order corner states in the 2-dimensional Su-Schrieffer-Heeger model.

A nonlinear non-Hermitian topological laser system based on the higher-order corner states of the 2-dimensional (2D) Su-Schrieffer-Heeger (SSH) model is investigated. The topological property of this nonlinear non-Hermitian system described by the quench dynamics is in accordance with that of a normal 2D SSH model. In the topological phase, all sites belonging to the topological corner states begin to emit stable laser light when a pulse is given to any one site of the lattice, while no laser light is emitted when the lattice is in the trivial phase. Furthermore, the next-nearest-neighbor (NNN) couplings are introduced into the strong-coupling unit cells of the 2D SSH model, which open a band gap in the continuous band structure. In the topological phase, similar to the case of 2D SSH model without NNN couplings, the corner sites can emit stable laser light due to the robustness of the higher-order corner states when the NNN couplings are regarded as the perturbation. However, amplitude of the stimulated site does not decay to zero in the trivial phase, because the existence of the NNN couplings in the strong-coupling unit cells make the lattice like one in the tetramer limit, and a weaker laser light is emitted by each corner.

Robustness versus sensitivity in non-Hermitian topological lattices probed by pseudospectra

Non-Hermitian topological systems simultaneously possess two antagonistic features: ultrasensitivity 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 pseudospectrum theory on the prototypical non-Hermitian Su-Schrieffer-Heeger lattice. Topological modes around the underlying third-order exceptional point (EP3) are robust with respect to chiral perturbations but sensitive to diagonal perturbations. In fact, exactly at the EP3 the chiral symmetry leads to a suppressed sensitivity, that corresponds to an EP2. Finally, for nonlinearly induced perturbations we provide a connection between the pseudospectrum approach and a nonlinear phase shift, which is relevant for experiments.

Non-Hermitian total-loss high-order topological insulator based on 1D Su–Schrieffer–Heeger (SSH)

Non-Hermiticity alters topology with the presence of non-Hermitian factors in topological systems. Most existing non-Hermitian topological systems derive their topological phases from Hermitian components, that is, the gain and loss of the system are considered simultaneously. In this work, we reveal two-dimensional non-Hermitian high-order topological insulator based on one-dimensional SSH chain, the nontrivial topology of which induced by total-loss. By introducing the loss of a specific configuration, we get a bandgap with corner and edge states whose topology is characterized by the gapped wannier band. In addition, we demonstrate the existence of ’real-energy’ edge states in pseudo-PT symmetric domain wall system. These results can be easily implemented in experiments, and promote the research of topological transmission of lossy systems in the real world.

Experimental Identification of the Second-Order Non-Hermitian Skin Effect with Physics-Graph-Informed Machine Learning.

Topological phases of matter are conventionally characterized by the bulk-boundary correspondence in Hermitian systems. The topological invariant of the bulk in d dimensions corresponds to the number of (d - 1)-dimensional boundary states. By extension, higher-order topological insulators reveal a bulk-edge-corner correspondence, such that nth order topological phases feature (d - n)-dimensional boundary states. The advent of non-Hermitian topological systems sheds new light on the emergence of the non-Hermitian skin effect (NHSE) with an extensive number of boundary modes under open boundary conditions. Still, the higher-order NHSE remains largely unexplored, particularly in the experiment. An unsupervised approach-physics-graph-informed machine learning (PGIML)-to enhance the data mining ability of machine learning with limited domain knowledge is introduced. Through PGIML, the second-order NHSE in a 2D non-Hermitian topoelectrical circuit is experimentally demonstrated. The admittance spectra of the circuit exhibit an extensive number of corner skin modes and extreme sensitivity of the spectral flow to the boundary conditions. The violation of the conventional bulk-boundary correspondence in the second-order NHSE implies that modification of the topological band theory is inevitable in higher dimensional non-Hermitiansystems.

Different phases in non-Hermitian topological semiconductor stripe laser arrays.

As a novel branch of topology, non-Hermitian topological systems have been extensively studied in theory and experiments recently. Topological parity-time (PT)-symmetric semiconductor stripe laser arrays based on the Su-Schreiffer-Heeger model are proposed. The degree of non-Hermicity can be tuned by altering the length of the cavities, and PT symmetry can be realized by patterned electrode. Three laser arrays working in different non-Hermitian phases are analyzed and fabricated. With the increasing degree of non-Hermicity, the peaks of output intensities move from the edge to the bulk. The proposed semiconductor stripe laser array can function as an active, flexible, and feasible platform to investigate and explore non-Hermitian topology for further developments in this field.

Non-Hermitian topological systems with eigenvalues that are always real

The effect of non-Hermiticity in band topology has sparked many discussions on non-Hermitian topological physics. It has long been known that non-Hermitian Hamiltonians can exhibit real energy spectra under the condition of parity-time ($PT$) symmetry---commonly implemented with balanced loss and gain---but only when non-Hermiticity is relatively weak. Sufficiently strong non-Hermiticity, on the other hand, will destroy the reality of energy spectra, a situation known as spontaneous $PT$-symmetry breaking. Here, based on nonreciprocal coupling, we show a systematic strategy to construct non-Hermitian topological systems exhibiting bulk and boundary energy spectra that are always real, regardless of weak or strong non-Hermiticity. Such nonreciprocal-coupling-based non-Hermiticity can directly drive a topological phase transition and determine the band topology, as demonstrated in a few non-Hermitian systems from one dimensional to two dimensional. Our work develops a theory that can guarantee the reality of energy spectra for non-Hermitian Hamiltonians, and offers an avenue to explore non-Hermitian topological physics.

Critical hybridization of skin modes in coupled non-Hermitian chains

Non-Hermitian topological systems exhibit a plethora of unusual topological phenomena including the extreme localization of eigenstates, known as the non-Hermitian skin effect (NHSE), which occurs in open chains. However, the non-Hermitian characteristics of coupled non-Hermitian chains remain largely unexplored. Here, we report on the eigenstate localization in coupled non-Hermitian chains with dissimilar inverse skin lengths in which the NHSE can be switched on and off by modulating the interchain coupling strength. In the limit of small interchain strength, the NHSE is present at both ends of the coupled system because of the weak hybridization of the eigenstates of the individual chains. The eigenspectrum under open boundary conditions (OBC) exhibits a discontinuous jump known as the critical NHSE (CNHSE) as the chain length increases. However, when the interchain coupling strength and hence, the hybridization between eigenstates become significant, the NHSE and CNHSE vanish. Instead, a peculiar ``half-half skin localization'' occurs in composite chains with opposite signs of inverse skin lengths, where half of the eigenstates are exponentially localized at one chain and the remainder of the eigenstates on the other chain. Our results provide new insights into the non-Hermitian phenomena in coupled systems.

Competition between band topology and non-Hermiticity

The introduction of non-Hermiticity to topological systems profoundly modifies their topological properties, leading to unprecedented phenomena beyond the descriptions of Bloch band theory, e.g., the breakdown of the conventional bulk-boundary correspondence. However, the comprehensive interplay between band topology and non-Hermiticity remains elusive. Here, based on a non-Hermitian Su-Schrieffer-Heeger model, we demonstrate that in the non-Hermitian topological systems, band topology and non-Hermiticity interplay by competing with each other, exhibiting a transition from symmetry dominance to non-Hermitian dominance. The former is featured with two edge states separately distributed at the two ends of the Su-Schrieffer-Heeger chain, similar to their Hermitian counterparts. In the latter, however, driven by non-Hermiticity, the two edge states become localized toward the same chain end, exhibiting the non-Hermitian skin effect. We find that such a transition is a universal behavior in the non-Hermitian systems, which is responsible for the breakdown of the conventional bulk-boundary correspondence and is the key factor to understanding the non-Hermitian topological properties. Furthermore, it is shown that this transition can be modified by the mode coupling between the edge states. As a comparison, we propose a non-Hermitian square-root Su-Schrieffer-Heeger model, where the edge states do not couple with each other and the mode coupling effect can be deactivated. Our work explicitly reveals the interplay between band topology and non-Hermiticity, which lays the foundation for the studies of non-Hermitian topological physics.

Non-Hermitian CNH=2 Chern insulator protected by generalized rotational symmetry

We propose a non-Hermitian topological system protected by the generalized rotational symmetry which invokes rotation in space and Hermitian conjugation. The system, described by the tight-binding model with reciprocal imaginary next-nearest-neighbor hopping, is found to host two pairs of in-gap edge modes in the gapped topological phase, and is characterized by the non-Hermitian (NH) Chern number ${C}_{\mathrm{NH}}=2$. The quantization of the NH Chern number is shown to be protected by the generalized rotational symmetry ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}^{+}(\mathrm{g}\stackrel{P\vec}{k})={\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}}_{g}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}(\stackrel{P\vec}{k}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}}_{g}^{+}$ of the system. Our finding paves the way towards non-Hermitian topological systems characterized by large values of topological invariants and hosting multiple in-gap edge states, which can be used for topologically resilient multiplexing.

Detecting Non-Bloch Topological Invariants in Quantum Dynamics.

Non-Bloch topological invariants preserve the bulk-boundary correspondence in non-Hermitian topological systems, and are a key concept in the contemporary study of non-Hermitian topology. Here we report the dynamic detection of non-Bloch topological invariants in single-photon quantum walks, revealed through the biorthogonal chiral displacement, and crosschecked with the dynamic spin textures in the generalized quasimomentum-time domain following a quantum quench. Both detection schemes are robust against symmetry-preserving disorders, and yield consistent results with theoretical predictions. Our experiments are performed far away from any boundaries, and therefore underline non-Bloch topological invariants as intrinsic properties of the system that persist in the thermodynamic limit. Our work sheds new light on the experimental investigation of non-Hermitian topology.

Anomalous non-Abelian statistics for non-Hermitian generalization of Majorana zero modes

In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this paper we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, due to the particle-hole-symmetry breaking, NH Majorana zero modes have anomalous non-Abelian statistics with continuously tunable braiding Berry phase from $\ensuremath{\pi}/8$ to $3\ensuremath{\pi}/8$. This is quite different from the usual non-Abelian statistics with fixed braiding Berry phase $\ensuremath{\pi}/4$. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the $\ensuremath{\pi}/8$ gate can be reached and thus universal topological quantum computation becomes possible.

Observation of topological properties of non-Hermitian crystal systems with diversified coupled resonators chains

Non-Hermiticity extends the topological phase beyond the given Hermitian structure. Whereas the phases of non-Hermitian topological systems derived from Hermitian components have been extensively explored, the topological properties of an acoustic crystal that occur purely due to non-Hermiticity require further investigation. In this letter, we describe the development of an acoustic crystal with an adjustable loss that is composed of a chain of one-dimensional, coupled acoustic resonators. Each unit cell can contain three or six resonators, which are equivalent to 3 × 3 or 6 × 6 non-Hermitian Hamiltonian matrices, respectively. The topological properties of the crystal were verified by calculating the defined topological invariant, and the states of the edge and interface of the acoustic crystal were obtained by using a practical model. We obtained the states of the edges and the interface for both odd and even numbers of resonators in each unit cell and found that the location of the inductive loss had an important effect on the topological properties. This results here can guide research on advanced wave control for sensing and communication applications.

Light-driven Lifshitz transitions in non-Hermitian multi-Weyl semimetals

Non-Hermitian topological systems are the newest additions to the growing field of topological matter. In this work, we report the light-driven exceptional physics in a multi-Weyl semimetal. The driving is not only a key ingredient to control the position of the exceptional contours (ECs); light also has the ability to generate new ECs. Interestingly, we also demonstrate topological charge distribution and Lifshitz transition, which are controllable by the driving field in such generated ECs. Our findings present a promising platform for the manipulation and control over exceptional physics in non-Hermitian topological matter.

Non-Hermitian semi-Dirac semi-metals

Recently, many novel and exotic phases have been proposed by considering the role of topology in non-Hermitian systems, and their emergent properties are of wide current interest. In this work we propose the non-Hermitian generalization of semi-Dirac semimetals, which feature a linear dispersion along one momentum direction and a quadratic one along the other. We study the topological phase transitions in such two-dimensional semi-Dirac semimetals in the presence of a particle gain-and-loss term. We show that such a non-Hermitian term creates exceptional points (EPs) originating out of each semi-Dirac point. We map out the topological phase diagram of our model, using winding number and vorticity as topological invariants of the system. By means of numerical and analytical calculations, we examine the nature of edge states for different types of semi-Dirac models and establish bulk-boundary correspondence and absence of the non-Hermitian skin effect, in one class. On the other hand, for other classes of semi-Dirac models with asymmetric hopping, we restore the non-Hermitian skin effect, an anomalous feature usually present in non-Hermitian topological systems.