non-Hermitian Skin Effect Research Articles

Nonreciprocal exciton-polariton ring lattices

Recent experiments have shown the transfer of orbital angular momentum (OAM) from a non-resonant laser onto an exciton-polariton condensate, despite earlier views that the phase information of such a laser should be lost during the process of polariton condensation. We study with a phenomenological theory the interplay of a usual angular momentum independent gain and an angular momentum preserving gain. We find that even when the latter is much smaller, it is enough to favour condensation into a given orbital angular momentum state. This further allows a breaking of symmetry in the system, which further manifests in non-reciprocal one-way propagation in a lattice of coupled rings. Even though we consider only Hermitian reciprocal coupling between rings, the local non-Hermiticity generates an effective non-reciprocal coupling and supports a non-Hermitian topological invariant (winding number) associated to a non-Hermitian skin effect.

Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk–boundary correspondence

We investigate novel features of three-dimensional non-Hermitian Weyl semimetals, paying special attention to the unconventional bulk–boundary correspondence. We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram, which is also confirmed by the energy spectra from our numerical results. It is shown that, in sharp contrast to Hermitian systems, the conventional (Bloch) bulk–boundary correspondence breaks down in non-Hermitian topological semimetals, which is caused by the non-Hermitian skin effect. We establish the non-Bloch bulk–boundary correspondence for non-Hermitian Weyl semimetals: the topological edge modes are determined by the non-Bloch Chern number of the bulk bands. Moreover, these topological edge modes can manifest as the unidirectional edge motion, and their signatures are consistent with the non-Bloch bulk–boundary correspondence. Our work establishes the non-Bloch bulk–boundary correspondence for non-Hermitian topological semimetals.

Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric

Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and the absence of non-Hermitian skin effects. Here we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach, we present and study the pseudo-Hermitian version of well-known Hermitian topological phases, ranging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to three-dimensional Weyl semimetals and chiral topological insulators. We analyze their topological bulk states through non-Hermitian generalizations of Abelian and non-Abelian tensor Berry connections and quantum metrics. Although our pseudo-Hermitian models and their Hermitian counterparts share the same topological invariants, their band geometries are different. We indeed show that some of our pseudo-Hermitian phases naturally support nearly flat topological bands, opening the route to the study of pseudo-Hermitian strongly interacting systems. Finally, we provide an experimental protocol to realize our models and measure the full non-Hermitian quantum geometric tensor in synthetic matter.

Acoustic non-Hermitian skin effect from twisted winding topology

The recently discovered non-Hermitian skin effect (NHSE) manifests the breakdown of current classification of topological phases in energy-nonconservative systems, and necessitates the introduction of non-Hermitian band topology. So far, all NHSE observations are based on one type of non-Hermitian band topology, in which the complex energy spectrum winds along a closed loop. As recently characterized along a synthetic dimension on a photonic platform, non-Hermitian band topology can exhibit almost arbitrary windings in momentum space, but their actual phenomena in real physical systems remain unclear. Here, we report the experimental realization of NHSE in a one-dimensional (1D) non-reciprocal acoustic crystal. With direct acoustic measurement, we demonstrate that a twisted winding, whose topology consists of two oppositely oriented loops in contact rather than a single loop, will dramatically change the NHSE, following previous predictions of unique features such as the bipolar localization and the Bloch point for a Bloch-wave-like extended state. This work reveals previously unnoticed features of NHSE, and provides the observation of physical phenomena originating from complex non-Hermitian winding topology.

Probing phase transitions in non-Hermitian systems with multiple quantum coherences

Understanding the interplay between quantum coherence and non-Hermitian features would enable the devising of quantum technologies based on dissipative systems. In turn, quantum coherence can be characterized in terms of the language of multiple quantum coherences (MQCs) originally developed in solid-state nuclear magnetic resonance (NMR), nowadays applied to the detection of quantum chaos, and to the study of criticality in many-body quantum systems. Here we show the usefulness of MQCs for probing equilibrium phase transitions in non-Hermitian systems. To do so, we investigate the connection of quantum coherences and critical points for several paradigmatic non-Hermitian Hamiltonians. For a non-Hermitian two-level system, MQCs witness the parity-symmetry breaking phase transition from the unbroken to the broken phase. Furthermore, for the non-Hermitian transverse field Ising model, MQCs capture the Yang-Lee phase transition in which the ground state energy acquires a nonzero imaginary component. For the disordered Hatano-Nelson (HN) model with periodic boundary conditions, MQCs testify the emergence of mobility edges in the spectrum of this model. In addition, MQCs signal the topological phase transition exhibited by the complex energy spectra of the disorder-free HN model. Finally, we comment on experimentally probing phase transitions in NMR systems realizing non-Hermitian Hamiltonians. Our results have applications to non-Hermitian quantum sensing, quantum thermodynamics, and in the study of the non-Hermitian skin effect.

Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions

We study the one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore how to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as those of the periodical system when the hopping parameters fulfill certain conditions, whereas the eigenstates display the non-Hermitian skin effect.

Data and code for figures of "Acoustic non-Hermitian skin effect from twisted winding topology" by Li Zhang et al.

Data and code for figures of "Acoustic non-Hermitian skin effect from twisted winding topology" by Li Zhang et al.

Data and code for figures of "Acoustic non-Hermitian skin effect from twisted winding topology" by Li Zhang et al.

Data and code for figures of "Acoustic non-Hermitian skin effect from twisted winding topology" by Li Zhang et al.

Dislocation non-Hermitian skin effect

We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response.

Scaling rule for the critical non-Hermitian skin effect

Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at a boundary of the systems with open boundary conditions. In this paper, we study dependence of the localization length of the eigenstates on a system size in a specific non-Hermitian model with a critical non-Hermitian skin effect, where the energy spectrum undergoes discontinuous transition in the thermodynamic limit. We analytically show that the eigenstates exhibit remarkable localization, known as scale-free localization, where the localization length is proportional to a system size. Our result gives a theoretical support for the scale-free localization, which has been proposed only numerically in previous works.

Supermetal-insulator transition in a non-Hermitian network model

We study a non-Hermitian and non-unitary version of the two-dimensional Chalker-Coddington network model with balanced gain and loss. This model belongs to the class D^dagger with particle-hole symmetry^dagger and hosts both the non-Hermitian skin effect as well as exceptional points. By calculating its two-terminal transmission, we find a novel contact effect induced by the skin effect, which results in a non-quantized transmission for chiral edge states. In addition, the model exhibits an insulator to 'supermetal' transition, across which the transmission changes from exponentially decaying with system size to exponentially growing with system size. In the clean system, the critical point separating insulator from supermetal is characterized by a non-Hermitian Dirac point that produces a quantized critical transmission of 4, instead of the value of 1 expected in Hermitian systems. This change in critical transmission is a consequence of the balanced gain and loss. When adding disorder to the system, we find a critical exponent for the divergence of the localization length \nu \approx 1, which is the same as that characterizing the universality class of two-dimensional Hermitian systems in class D. Our work provides a novel way of exploring the localization behavior of non-Hermitian systems, by using network models, which in the past proved versatile tools to describe Hermitian physics.

Boundary condition independence of non-Hermitian Hamiltonian dynamics

Non-Hermitian skin effect, namely that the eigenvalues and eigenstates of a non-Hermitian tight-binding Hamiltonian have significant differences under open or periodic boundary conditions, is a remarkable phenomenon of non-Hermitian systems. Inspired by the presence of the non-Hermitian skin effect, we study the evolution of wave-packets in non-Hermitian systems, which can be determined using the single-particle Green's function. Surprisingly, we find that in the thermodynamical limit, the Green's function does not depend on boundary conditions, despite the presence of skin effect. We proffer a general proof for this statement in arbitrary dimension with finite hopping range, with an explicit illustration in the non-Hermitian Su-Schrieffer-Heeger model. We also explore its applications in non-interacting open quantum systems described by the master equation, where we demonstrate that the evolution of the density matrix is independent of the boundary condition.

Reconstructed conventional bulk-boundary correspondence in the non-Hermitian s -wave nodal-ring superconductor

Recently, the interplay between the non-Hermitian skin effect (NHSE) and topological phases has attracted a lot of attention. When the NHSE occurs in a topological system, the conventional bulk-boundary correspondence (BBC) based on the Bloch band theory is destroyed. In this work, we investigate a non-Hermitian nodal-ring semimetal with the $s$-wave superconducting (SC) term. We find that the SC term introduces the intrinsic coupling between the different general Brillouin zones (GBZs) of the particle and hole subbands in our model, which drives the total GBZ to the BZ. Furthermore, when the SC term is finite, we show a possibility that the GBZ is the same as the BZ. We also find that the Bloch winding number of the quasi-one-dimensional model can predict the number of Majorana zero states faithfully in this case, which implies that the conventional BBC is reconstructed in our model. The localization strengths of the Majorana zero states are also calculated to show the reconstruction of conventional BBC. The finite-size effect, related experimental regimes in the electric circuit, and the fragility of this reconstructed conventional BBC are also discussed in this work.

Observation of higher-order non-Hermitian skin effect

Beyond the scope of Hermitian physics, non-Hermiticity fundamentally changes the topological band theory, leading to interesting phenomena, e.g., non-Hermitian skin effect, as confirmed in one-dimensional systems. However, in higher dimensions, these effects remain elusive. Here, we demonstrate the spin-polarized, higher-order non-Hermitian skin effect in two-dimensional acoustic higher-order topological insulators. We find that non-Hermiticity drives wave localizations toward opposite edges upon different spin polarizations. More interestingly, for finite systems with both edges and corners, the higher-order non-Hermitian skin effect leads to wave localizations toward two opposite corners for all the bulk, edge and corner states in a spin-dependent manner. We further show that such a skin effect enables rich wave manipulation by configuring the non-Hermiticity. Our study reveals the intriguing interplay between higher-order topology and non-Hermiticity, which is further enriched by the pseudospin degree of freedom, unveiling a horizon in the study of non-Hermitian physics.

Exact Solution of Non-Hermitian Systems with Generalized Boundary Conditions: Size-Dependent Boundary Effect and Fragility of the Skin Effect.

Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditions and lattice size, and thus an important question is whether non-Hermitian skin effects can survive when deviating from the open boundary condition. To unveil the origin of boundary sensitivity, we present exact solutions for one-dimensional non-Hermitian models with generalized boundary conditions and study rigorously the interplay effect of lattice size and boundary terms. Besides the open boundary condition, we identify the existence of non-Hermitian skin effect when one of the boundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, we find that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit, although it can survive in a finite size system. Moreover, we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which is different from either open or periodical boundary case.

Nontrivial point-gap topology and non-Hermitian skin effect in photonic crystals

We show that two-dimensional non-Hermitian photonic crystals made of lossy material can exhibit non-trivial point gap topology in terms of topological winding in its complex frequency band structure. Such crystals can be either made of lossy isotropic media which is reciprocal, or lossy magneto-optical media which is non-reciprocal. We discuss the effects of reciprocity on the properties of the point gap topology. We also show that such a non-trivial point gap topology leads to non-Hermitian skin effect when the photonic crystal is truncated. In contrast to most previous studies on point gap topology which used tight-binding models, our work indicates that such non-Hermitian topology can studied in photonic crystals, which represent a more realistic system that has technological significance.

Non-Hermitian skin effect beyond the tight-binding models

The energy bands of non-Hermitian systems exhibit nontrivial topological features that arise from the complex nature of the energy spectrum. Under periodic boundary conditions (PBCs), the energy spectrum describes rather generally closed loops in complex plane, characterized by integer nonzero winding numbers. Such nontrivial winding provides the topological signature of the non-Hermitian skin effect (NHSE), i.e., the macroscopic condensation of bulk states at the lattice edges under open boundary conditions (OBCs). In spite of the great relevance of band winding in the non-Hermitian topological band theory and the related NHSE, most of current results rely on tight-binding models of non-Hermitian systems, while exact Bloch wave function analysis of the NHSE and related topological band theory is still lacking. While tight-binding models can correctly describe narrow-band electronic states with a relatively weak degree non-Hermiticity, they are not suited to describe high-energy wide-band electronic states and/or regimes corresponding to strong non-Hermiticity. Here we consider the single-particle continuous Schr\"odinger equation in a periodic potential, in which non-Hermiticity is introduced by an imaginary vector potential in the equation, and show that the NHSE is ubiquitous under OBC and characterized by a nonvanishing integer winding number, even though the energy spectrum under PBC always comprises an open curve, corresponding to high-energy electronic states. We also show that the interior of the PBC energy spectrum corresponds to the complex eigenenergies sustaining localized (edge) states under semi-infinite boundary conditions.

Disorder-driven phase transition in the second-order non-Hermitian skin effect

The non-Hermitian skin effect exhibits a collapse of extended bulk modes into an extensive number of localized boundary states in the open boundary condition. Here, we demonstrate a disorder-driven phase transition of a trivial non-Hermitian system to the higher-order non-Hermitian skin effect phase. In contrast to clean systems, the disorder-induced boundary modes form an arc in the complex energy plane, which is a manifestation of the disorder-driven energy-dependent phase transition. At the phase transition, localized corner modes and bulk modes characterized by trivial Hamiltonians coexist within a single band, while they remain separated in the complex energy plane. This behavior is analogous to the mobility edge phenomena in disordered Hermitian systems. Using effective medium theory and numerical diagonalizations, we provide a systematic characterization of the disorder-driven phase transition.

Non-Hermitian skin effect as an impurity problem

A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian Su-Schrieffer-Heeger models and finally we contrast the edge states localization with that of bulk states.

Compatibility of transport effects in non-Hermitian nonreciprocal systems

Based on a general transport theory for nonreciprocal non-Hermitian systems and a topological model that encompasses a wide range of previously studied examples, we (i) provide conditions for effects such as reflectionless and transparent transport, lasing, and coherent perfect absorption, (ii) identify which effects are compatible and linked with each other, and (iii) determine by which levers they can be tuned independently. For instance, the directed amplification inherent in the non-Hermitian skin effect does not enter the spectral conditions for reflectionless transport, lasing, or coherent perfect absorption, but allows to adjust the transparency of the system. In addition, in the topological model the conditions for reflectionless transport depend on the topological phase, but those for coherent perfect absorption do not. This then allows us to establish a number of distinct transport signatures of non-Hermitian, nonreciprocal, and topological behavior, in particular (1) reflectionless transport in a direction that depends on the topological phase, (2) invisibility coinciding with the skin-effect phase transition of topological edge states, and (3) coherent perfect absorption in a system that is transparent when probed from one side.