non-Hermitian Topological Systems Research Articles

Non-Hermitian bulk–boundary correspondence in quantum dynamics

Bulk-boundary correspondence, a central principle in topological matter relating bulk topological invariants to edge states, breaks down in a generic class of non-Hermitian systems that have so far eluded experimental effort. Here we theoretically predict and experimentally observe non-Hermitian bulk-boundary correspondence, a fundamental generalization of the conventional bulk-boundary correspondence, in discrete-time non-unitary quantum-walk dynamics of single photons. We experimentally demonstrate photon localizations near boundaries even in the absence of topological edge states, thus confirming the non-Hermitian skin effect. Facilitated by our experimental scheme of edge-state reconstruction, we directly measure topological edge states, which match excellently with non-Bloch topological invariants calculated from localized bulk-state wave functions. Our work unequivocally establishes the non-Hermitian bulk-boundary correspondence as a general principle underlying non-Hermitian topological systems, and paves the way for a complete understanding of topological matter in open systems.

Generalized bulk-edge correspondence for non-Hermitian topological systems

A modified periodic boundary condition adequate for non-hermitian topological systems is proposed. Under this boundary condition a topological number characterizing the system is defined in the same way as in the corresponding hermitian system and hence, at the cost of introducing an additional parameter that characterizes the non-hermitian skin effect, the idea of bulk-edge correspondence in the hermitian limit can be applied almost as it is. We develop this framework through the analysis of a non-hermitian SSH model with chiral symmetry, and prove the bulk-edge correspondence in a generalized parameter space. A finite region in this parameter space with a nontrivial pair of chiral winding numbers is identified as topologically nontrivial, indicating the existence of a topologically protected edge state under open boundary.

Geometric characterization of non-Hermitian topological systems through the singularity ring in pseudospin vector space

This work unveils how geometric features of two-band non-Hermitian Hamiltonians can completely classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, where a singularity ring (SR) captures the degeneracy structure of generic exceptional points. The SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our explicit study of chiral, sublattice, particle-hole and conjugated particle-hole symmetries. In 1D, the winding number about the SR corresponds to the band vorticity measurable through the Berry phase. In 2D, more complicated winding behavior leads to a variety of phases that illustrate the richness of the interplay between SR topology and geometry beyond mere Chern number classification. Through a normalization procedure that puts generic 2-band non-Hermitian Hamiltonians on equal footing, our SR approach also allows for vivid visualization of the non-Hermitian skin effect.

Non-Bloch topological invariants in a non-Hermitian domain wall system

We study non-Bloch bulk-boundary correspondence in a non-Hermitian Su-Schieffer-Heeger model in a domain-wall configuration where the left and right bulks have different parameters. Focusing on the case where chiral symmetry is still conserved, we show that non-Hermitian skin effects of bulk states persist in the system, while the definition of the non-Bloch winding number of either bulk depends on parameters on both sides of the boundary. Under these redefined non-Bloch topological invariants, we confirm non-Bloch bulk-boundary correspondence under the domain-wall configuration, which exemplifies the impact of boundary conditions in non-Hermitian topological systems.

Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization

A prominent feature of some one-dimensional non-Hermitian systems is that all right-eigenstates of the non-Hermitian Hamiltonian are localized in one end of the chain. The topological and trivial phases are distinguished by the emergence of zero-energy modes within the skin states in the presence of the chiral symmetry. Skin states are formed when the system is nonreciprocal, where it is said nonreciprocal if the absolute values of the right- and left-going hoppings amplitudes are different. Indeed, the zero-energy edge modes emerge at both edges in the topological phase of the reciprocal non-Hermitian system. Then, analyzing higher-order topological insulators in nonreciprocal systems, we find the emergence of topological zero-energy modes within the skin states formed in the vicinity of one corner. Explicitly we explore the anisotropic honeycomb model in two dimensions and the diamond lattice model in three dimensions. We also study an electric-circuit realization of these systems. Electrical circuits with (without) diodes realize the nonreciprocal (reciprocal) non-Hermitian topological systems. Topological phase transitions are observable by measuring the impedance resonance due to zero-admittance topological corner modes.

Finite-size effects in non-Hermitian topological systems

We systematically investigate the finite-size effects in non-Hermitian one-dimensional (1D) Su-Schrieffer-Heeger (SSH) and two-dimensional (2D) Chern insulator models. Using a combination of analytical and numerical calculations, we show that the non-Hermitian intra-cell hoppings in the SSH model can modify the localization lengths of bulk and end states, giving rise to a complex finite-size energy gap that exhibits an oscillating exponential decay as the chain length grows. However, the imaginary staggered on-site potentials in the SSH model only change the end-state energy, leaving the localization lengths of the system unchanged. In this case, the finite-size energy gap can undergo a transition from real values to imaginary values. We observed similar phenomena for the finite-size effect in 2D Chern insulator systems.

Open and closed boundaries in non-Hermitian topological systems

We investigate the boundary condition dependence of non-Hermitian topological systems. We consider a complex extension of the Su-Schrieffer-Heeger model. We discuss that topological zero-energy modes exist if the system has open edges. We show that robust zero-energy states appear for the closed lattice too. We also consider a complex extension of the Bernevig-Hughes-Zhang Hamiltonian. We show that the translationally invariant system has a complex-valued band structure. We discuss that $\mathcal{PT}$ symmetry is restored for helical edge states if the translational invariance is lost in one direction.

Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics

Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequence of this symmetry unification is the emergence of unique non-equilibrium topological phases that have no counterparts in Hermitian systems. We illustrate this by presenting a non-Hermitian counterpart of the Majorana chain in an insulator with time-reversal symmetry and that of the quantum spin Hall insulator in a superconductor with particle-hole symmetry. Our work establishes a fundamental symmetry principle in non-Hermitian physics and paves the way towards a unified framework for non-equilibrium topological phases.

Topological edge states with skin effect in a trimerized non-Hermitian lattice

In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.

Hall conductance of a non-Hermitian Chern insulator

In this letter we study the Hall conductance for a non-Hermitian Chern insulator and quantitatively describe how the Hall conductance deviates from a quantized value. We show the effects of the non-Hermitian terms on the Hall conductance are two folds. On one hand, it broadens the density-of-state of each band, because of which there always exists a non-universal bulk contribution. On the other hand, it adds decay term to the edge state, because of which the topological contribution also deviates from the quantized Chern number. We provides a simple formula for the topological contribution for a general two-band non-Hermitian Chern insulator, as a non-Hermitian version of the Thouless-Kohmoto-Nightingale-de Nijs formula. It shows that the derivation from quantized value increases either when the strength of the non-Hermitian term increases, or when the momentum dependence of the non-Hermitian term increases. Our results can be directly verified in synthetic non-Hermitian topological systems where the strength of the non-Hermitian terms can be controlled.

PT symmetry protected non-Hermitian topological systems

We study PT symmetry protected topological phase in non-Hermitian 1D and 2D systems. We show that topological phase exist in non-Hermitian PT symmetric systems for both gapped and gapless systems and discuss appearance of exceptional points. We apply our formalism to a complex extension of the SSH model, topological semimetals and nodal superconductors.

Non-Hermitian Topological Systems

Non-Hermitian Topological Systems