non-Hermitian Skin Effect Research Articles

Geometric Response and Disclination-Induced Skin Effects in Non-Hermitian Systems.

We study the geometric response of three-dimensional non-Hermitian crystalline systems with nontrivial point-gap topology. For systems with fourfold rotation symmetry, we show that in the presence of disclination lines with a total Frank angle, which is an integer multiple of 2π, there can be nontrivial one-dimensional point-gap topology along the direction of the disclination lines. This results in disclination-induced non-Hermitian skin effects. By doubling a non-Hermitian Hamiltonian to a Hermitian three-dimensional chiral topological insulator, we show that the disclination-induced skin modes are zero modes of the effective surface Dirac fermion(s) in the presence of a pseudomagnetic flux induced by disclinations. Furthermore, we find that our results have a field theoretic description, and the corresponding geometric response actions (e.g., the Euclidean Wen-Zee action) enrich the topological field theory of non-Hermitian systems.

Guided accumulation of active particles by topological design of a second-order skin effect

Collective guidance of out-of-equilibrium systems without using external fields is a challenge of paramount importance in active matter, ranging from bacterial colonies to swarms of self-propelled particles. Designing strategies to guide active matter and exploiting enhanced diffusion associated to its motion will provide insights for application from sensing, drug delivery to water remediation. However, achieving directed motion without breaking detailed balance, for example by asymmetric topographical patterning, is challenging. Here we engineer a two-dimensional periodic topographical design with detailed balance in its unit cell where we observe spontaneous particle edge guidance and corner accumulation of self-propelled particles. This emergent behaviour is guaranteed by a second-order non-Hermitian skin effect, a topologically robust non-equilibrium phenomenon, that we use to dynamically break detailed balance. Our stochastic circuit model predicts, without fitting parameters, how guidance and accumulation can be controlled and enhanced by design: a device guides particles more efficiently if the topological invariant characterizing it is non-zero. Our work establishes a fruitful bridge between active and topological matter, and our design principles offer a blueprint to design devices that display spontaneous, robust and predictable guided motion and accumulation, guaranteed by out-of-equilibrium topology.

Exact solution of the single impurity problem in nonreciprocal lattices: Impurity-induced size-dependent non-Hermitian skin effect

Non-Hermitian non-reciprocal systems are known to be extremely sensitive to boundary conditions, exhibiting diverse localizing behaviors and spectrum structures when translational invariance is locally broken, either by tuning the boundary coupling strength, or by introducing an effective boundary using impurities or defects. In this work, we consider the single impurity problem in the Hatano-Nelson model and the Su-Schreieffer-Heeger model, which can be exactly solved with the single impurity being treated as an effective boundary of the system. From our exact solutions for finite-size systems, we unveil that increasing the impurity strength can lead to a transition of the bulk states from non-skin states to skin states, accompanied by the change of the spectrum structure from an ellipse in the complex plane to a segment along the real axis. These exact results indicate that the critical value of impurity strength is size-dependent, and increases exponentially with the lattice size when the impurity is strong or the system is large enough. OUr exact solutions are also useful for determining the point-gap topological transition in the concerned models.

Non-Hermitian topological end breathers

Nonlinearities in lattices with topologically nontrivial band structures can give rise to topological solitons, whose properties differ from both conventional lattice solitons and linear topological boundary states. We show that a Su-Schrieffer-Heeger-type lattice with both nonlinearity and nonreciprocal non-Hermiticity hosts a novel oscillatory soliton, which we call a topological end breather. The end breather is strongly localized to a self-induced topological domain near the end of the lattice, in sharp contrast to the extended topological solitons previously found in one-dimensional lattices. Its stable oscillatory dynamics can be interpreted as a Rabi oscillation between two self-induced topological boundary states, emerging from a combination of chiral lattice symmetry and the non-Hermitian skin effect. This demonstrates that non-Hermitian effects can give rise to a wider variety of topological solitons than was previously known to exist.

Steering non-Hermitian skin modes by synthetic gauge fields in optical ring resonators.

We show that the synthetic gauge fields for photons provide a versatile approach to generate and control the non-Hermitian skin effect. By utilizing indirectly coupled optical ring resonator arrays with long-range couplings and on-site gain and loss, we find that the skin effect appears once the gauge field is not an integer multiple of π. In addition to tunable localization direction, the skin modes display anisotropic behaviors with frequency-dependent decay length, which can be explained by the split subregion of the generalized Brillouin zone (GBZ) and an effective model under adiabatic elimination. Through numerical simulation, we can also demonstrate exotic features in propagation effects enabled by the skin effect, including asymmetric transmission and reconfigurable accumulation interface. Our study paves the way to dynamically steer skin modes, which may find applications in laser, optical switch, and signal processing.

Unsupervised Learning of Non-Hermitian Topological Phases.

Non-Hermitian topological phases bear a number of exotic properties, such as the non-Hermitian skin effect and the breakdown of conventional bulk-boundary correspondence. In this Letter, we introduce an unsupervised machine learning approach to classify non-Hermitian topological phases based on diffusion maps, which are widely used in manifold learning. We find that the non-Hermitian skin effect will pose a notable obstacle, rendering the straightforward extension of unsupervised learning approaches to topological phases for Hermitian systems ineffective in clustering non-Hermitian topological phases. Through theoretical analysis and numerical simulations of two prototypical models, we show that this difficulty can be circumvented by choosing the "on-site" elements of the projective matrix as the input data. Our results provide a valuable guidance for future studies on learning non-Hermitian topological phases in an unsupervised fashion, both in theory and experiment.

Observation of Non-Bloch Parity-Time Symmetry and Exceptional Points.

Parity-time (PT)-symmetric Hamiltonians have widespread significance in non-Hermitian physics. A PT-symmetric Hamiltonian can exhibit distinct phases with either real or complex eigenspectrum, while the transition points in between, the so-called exceptional points, give rise to a host of critical behaviors that holds great promise for applications. For spatially periodic non-Hermitian systems, PT symmetries are commonly characterized and observed in line with the Bloch band theory, with exceptional points dwelling in the Brillouin zone. Here, in nonunitary quantum walks of single photons, we uncover a novel family of exceptional points beyond this common wisdom. These "non-Bloch exceptional points" originate from the accumulation of bulk eigenstates near boundaries, known as the non-Hermitian skin effect, and inhabit a generalized Brillouin zone. Our finding opens the avenue toward a generalized PT-symmetry framework, and reveals the intriguing interplay between PT symmetry and non-Hermitian skin effect.

Interaction-induced double-sided skin effect in an exciton-polariton system

The non-Hermitian skin effect can be realized through asymmetric hopping between forward and backward directions, where all the modes of the system are localized at one edge of a finite 1D lattice. Here we show theoretically that in a finite chain of 1D exciton-polariton micropillars with symmetric hopping, the inherent nonlinearity of the system can exhibit a double-sided skin (bi-skin) effect based of the fluctuations of the system, where the modes of the system are localized at the two edges of the system. To show the topological origin of such modes, we calculate the winding number. The bi-skin effect can be detected experimentally as an intensity drop at one edge of the chain and an increase at the opposite edge upon an increase of the polariton density under continuous wave excitation.

Nonlinear non-Hermitian skin effect

Distant boundaries in linear non-Hermitian lattices can dramatically change energy eigenvalues and corresponding eigenstates in a nonlocal way. This effect is known as non-Hermitian skin effect (NHSE). Combining non-Hermitian skin effect with nonlinear effects can give rise to a host of novel phenomenas, which may be used for nonlinear structure designs. Here we study nonlinear non-Hermitian skin effect and explore nonlocal and substantial effects of edges on stationary nonlinear solutions. We show that fractal and continuum bands arise in a long lattice governed by a nonreciprocal discrete nonlinear Schrodinger equation. We show that stationary solutions are localized at the edge in the continuum band. We consider a non-Hermitian Ablowitz-Ladik model and show that nonlinear exceptional point disappears if the lattice is infinitely long.

Real-space topological invariant and higher-order topological Anderson insulator in two-dimensional non-Hermitian systems

We study the characterization and realization of higher-order topological Anderson insulator (HOTAI) in non-Hermitian systems, where the non-Hermitian mechanism ensures extra symmetries as well as gain and loss disorder. We illuminate that the quadrupole moment ${Q}_{xy}$ can be used as the real space topological invariant of non-Hermitian higher-order topological insulator (HOTI). Based on the biorthogonal bases and non-Hermitian symmetries, we prove that ${Q}_{xy}$ can be quantized to 0 or 0.5. Considering the disorder effect, we find the disorder-induced phase transition from normal insulator to non-Hermitian HOTAI. Furthermore, to clarify the universality of real-space topological invariant in non-Hermitian systems, we elucidate that ${Q}_{xy}$ is also applicable for samples with the non-Hermitian skin effect. Our work enlightens the study of the combination of disorder and non-Hermitian HOTI.

Topological Field Theory of Non-Hermitian Systems.

Non-Hermiticity gives rise to unique topological phases without Hermitian analogs. However, the effective field theory has yet to be established. Here, we develop a field-theoretical description of the intrinsic non-Hermitian topological phases. Because of the dissipative and nonequilibrium nature of non-Hermiticity, our theory is formulated solely in terms of spatial degrees of freedom, which contrasts with the conventional theory defined in spacetime. Our theory provides a universal understanding of non-Hermitian topological phenomena such as the unidirectional transport in one dimension and the chiral magnetic skin effect in three dimensions. Furthermore, it systematically predicts new physics; we illustrate this by revealing transport phenomena and skin effects in two dimensions induced by a perpendicular spatial texture. From the field-theoretical perspective, the non-Hermitian skin effect, i.e., the anomalous localization due to non-Hermiticity, is shown to be a signature of an anomaly.

Non-Hermitian band topology with generalized inversion symmetry

Non-Hermitian skin effects and exceptional points are topological phenomena characterized by integer winding numbers. In this study, we give methods to theoretically detect skin effects and exceptional points by generalizing inversion symmetry. The generalization of inversion symmetry is unique to non-Hermitian systems. We show that parities of the winding numbers can be determined from energy eigenvalues on the inversion-invariant momenta when generalized inversion symmetry is present. The simple expressions for the winding numbers allow us to easily analyze skin effects and exceptional points in non-Hermitian bands. We also demonstrate the methods for (second-order) skin effects and exceptional points by using lattice models.

Delocalization of topological edge states

The non-Hermitian skin effect (NHSE) in non-Hermitian lattice systems depicts the exponential localization of eigenstates at the system's boundaries. It has led to a number of counterintuitive phenomena and challenged our understanding of bulk-boundary correspondence in topological systems. This work aims to investigate how the NHSE localization and topological localization of in-gap edge states compete with each other, with several representative static and periodically driven 1D models, whose topological properties are protected by different symmetries. The emerging insight is that at critical system parameters, even topologically protected edge states can be perfectly delocalized. In particular, it is discovered that this intriguing delocalization occurs if the real spectrum of the system's edge states falls on the same system's complex spectral loop obtained under the periodic boundary condition. We have also performed sample numerical simulation to show that such delocalized topological edge states can be safely reconstructed from time-evolving states. One particular application of delocalized topological edge states in assisting adiabatic edge state pumping is also computationally demonstrated.

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.

Non-Hermitian Skin Effects in Hermitian Correlated or Disordered Systems: Quantities Sensitive or Insensitive to Boundary Effects and Pseudo-Quantum-Number

There is a common belief in the condensed matter community that bulk quantities become insensitive to the boundary condition in the infinite-volume limit. Here we reconsider this statement in terms of recent arguments of non-Hermitian skin effects-strong dependence of spectra on boundary conditions for the non-Hermitian Hamiltonians-in the traditional Green's function formalism. We find the criterion for quantities to be sensitive or insensitive against the boundary condition in Hermitian correlated or disordered systems, which is characterized by the residue theorem. We also discuss the uncertainty of the quasiparticle energy under the skin effects in terms of non-normal pseudospectra, which can be tested via the sharp optical absorption from the bulk-surface coupling. Our result indicates that pseudo quantum number emerges as a consequence of large nonnormality.

Quasi-stationary solutions in 1D non-Hermitian systems

Eigenstates exhibit localization at an open edge in a non-Hermitian lattice due to non-Hermitian skin effect. We here explore another interesting feature of non-Hermitian skin effect and predict quasi-stationary solutions, which are approximately time-independent. We show that the transition from such states to eigenstates is dramatically non-perturbative. We consider a non-Hermitian variant of the Su-Schrieffer-Heeger (SSH) model and predict non-topological but robust quasi-stationary zero energy modes.

Non-Bloch band theory in bosonic Bogoliubov–de Gennes systems

In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the non-Hermitian skin effect, such as instability against infinitesimal perturbations and reentrant behavior, in terms of the non-Bloch band theory. This result indicates that our theory is powerful tool for studying non-Hermitian nature in bosonic BdG systems.

Majorana polarization in non-Hermitian topological superconductors

Topological invariants play an important role in characterizing topological phases. However, the topological invariants of Hermitian systems usually fail to characterize non-Hermitian topological systems due to non-Hermiticity. In this work, we generalize the Majorana polarization, which is initially defined to describe Hermitian topological superconductors, as a topological edge invariant to characterize the non-Hermitian topological superconductors. The definition of the generalized Majorana polarization depends upon two inequivalent particle-hole symmetries in non-Hermitian systems. The spinless Kitaev chain model and topological superconductor model on the honeycomb lattice are considered to examine the reliability and validity of the generalized Majorana polarization. We find that the phase transitions obtained by using Majorana polarization are consistent with the commonly used complex-energy point gap descriptions, which indicates the Majorana polarization is a reliable topological invariant to characterize the topological phase transition in non-Hermitian topological superconductors. In addition, the non-Hermitian skin effect on Majorana bound states in non-Hermitian topological superconductors is also discussed.

Spectral deformations in non-Hermitian lattices with disorder and skin effect: A solvable model

We derive analytical results on energy spectral phase transitions and deformations in the simplest model of one-dimensional lattice displaying the non-Hermitian skin effect, namely the Hatano-Nelson model with unidirectional hopping, under on-site potential uncorrelated disorder in complex energy plane. While the energy spectrum under open boundary conditions (OBC) exactly reproduces the distribution of on-site potential disorder, the energy spectrum under periodic boundary conditions (PBC) undergoes spectral deformations, from one or more closed loops in the fully delocalized phase, with no overlap with the OBC spectrum, to a mixed spectrum (closed loops and some OBC energies) in the mobility edge phase, to a complete collapse toward the OBC spectrum in the bulk localized phase. Such transitions are observed as the strength of disorder is increased. Depending on the kind of disorder, different interesting behaviors are found. In particular, for continuous disorder with a radial distribution in complex energy plane it is shown that in the delocalized phase the energy spectrum under PBC is locked and fully insensitive to disorder, while transition to the bulk localized phase is signaled by the change of a topological winding number. When the disorder is described by a discrete distribution, the bulk localization transition never occurs, while topological phase transitions associated to PBC energy spectral splittings can be observed.

Non-Bloch quench dynamics

We study the quench dynamics of non-Hermitian topological models with non-Hermitian skin effects. Adopting the non-Bloch band theory and projecting quench dynamics onto the generalized Brillouin zone, we find that emergent topological structures, in the form of dynamic skyrmions, exist in the generalized momentum-time domain, and are correlated with the non-Bloch topological invariants of the static Hamiltonians. The skyrmion structures anchor on the fixed points of dynamics whose existence are conditional on the coincidence of generalized Brillouin zones of the pre- and post-quench Hamiltonians. Global signatures of dynamic skyrmions, however, persist well beyond such a condition, thus offering a general dynamic detection scheme for non-Bloch topology in the presence of non-Hermitian skin effects. Applying our theory to an experimentally relevant, non-unitary quantum walk, we explicitly demonstrate how the non-Bloch topological invariants can be revealed through the non-Bloch quench dynamics.