non-Hermitian Bulk-boundary Correspondence Research Articles

Restoration of non-Hermitian bulk-boundary correspondence by counterbalancing skin effect

In systems exhibiting the non-Hermitian skin effect (NHSE), the bulk spectrum under open boundary conditions (OBC) significantly differs from that of its periodic counterpart. This disparity renders the conventional bulk-boundary correspondence (BBC) inapplicable. Here we propose an intuitive approach called doubling and swapping to restore the BBC, using the non-Hermitian Su-Schrieffer-Heeger model as an example. Explicitly, we construct a modified system free of NHSE by swapping the asymmetric intracell hoppings in every second primitive unit cell. Importantly, this change does not alter the OBC spectrum. As a result, the modified periodic system can serve as the bulk for defining topological invariants that accurately predict edge states and topological phase transitions. The basic principle is applicable to many other systems. By extending the study to disordered systems in which the asymmetric hoppings are randomly swapped, we show that two types of winding numbers can also be defined to account for the NHSE and topological edge states, respectively.

Non-Hermitian extended midgap states and bound states in the continuum

We investigate anomalous localization phenomena in non-Hermitian systems by solving a class of generalized Su–Schrieffer–Heeger/Rice–Mele models and by relating their provenance to fundamental notions of topology, symmetry-breaking, and biorthogonality. We find two types of bound states in the continuum, both stable even in the absence of chiral symmetry: the first being skin bulk states, which are protected by the spectral winding number. The second type is constituted by boundary modes associated with a quantized biorthogonal polarization. Furthermore, we find an extended state stemming from the boundary state that delocalizes while remaining in the gap at bulk critical points. This state may also delocalize within a continuum of localized (skin) states. These results clarify fundamental aspects of topology and symmetry in light of different approaches to the anomalous non-Hermitian bulk-boundary correspondence and are of direct experimental relevance for mechanical, electrical, and photonic systems.

Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulk-boundary correspondence is lost. Here, we restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and without symmetries. We obtain the desired NH Hamiltonian from the mean-field evolution of driven-dissipative cavity arrays, in which NH terms-in the form of non-reciprocal hopping amplitudes, gain and loss-are explicitly modeled via coupling to (engineered and non-engineered) reservoirs. This approach removes the arbitrariness in the definition of the topological invariant, as point-gapped spectra differing by a complex-energy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically non-trivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamiltonians so obtained via the singular value decomposition, which allows to express the NH bulk-boundary correspondence in the following simple form: an integer value \nuν of the topological invariant defined in the bulk corresponds to \vert \nu\vert|ν| singular vectors exponentially localized at the system edge under open boundary conditions, in which the sign of \nuν determines which edge. Non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.

Non-Abelian effects in dissipative photonic topological lattices

Topology is central to phenomena that arise in a variety of fields, ranging from quantum field theory to quantum information science to condensed matter physics. Recently, the study of topology has been extended to open systems, leading to a plethora of intriguing effects such as topological lasing, exceptional surfaces, as well as non-Hermitian bulk-boundary correspondence. Here, we show that Bloch eigenstates associated with lattices with dissipatively coupled elements exhibit geometric properties that cannot be described via scalar Berry phases, in sharp contrast to conservative Hamiltonians with non-degenerate energy levels. This unusual behavior can be attributed to the significant population exchanges among the corresponding dissipation bands of such lattices. Using a one-dimensional example, we show both theoretically and experimentally that such population exchanges can manifest themselves via matrix-valued operators in the corresponding Bloch dynamics. In two-dimensional lattices, such matrix-valued operators can form non-commuting pairs and lead to non-Abelian dynamics, as confirmed by our numerical simulations. Our results point to new ways in which the combined effect of topology and engineered dissipation can lead to non-Abelian topological phenomena.

Universal characteristics of one-dimensional non-Hermitian superconductors

We establish a non-Bloch band theory for one-dimensional(1D) non-Hermitian topological superconductors. The universal physical properties of non-Hermitian topological superconductors are revealed based on the theory. According to the particle-hole symmetry, there exist reciprocal particle and hole loops of generalized Brillouin zone. The critical point of quantum phase transition, where the energy gap closes, appears when the particle and hole loops intersect at Bloch points. If the non-Hermitian system has non-Hermitian skin effects, the non-Hermitian skin effect should be the Z 2 skin effect: the corresponding eigenstates of particle and hole localize at opposite ends of an open chain, respectively. The non-Bloch band theory is applied to two examples, non-Hermitian p- and s-wave topological superconductors. In terms of Majorana Pfaffian, a Z 2 non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.

Designing non-Hermitian real spectra through electrostatics

Non-hermiticity presents a vast newly opened territory that harbors new physics and applications such as lasing and sensing. However, only non-Hermitian systems with real eigenenergies are stable, and great efforts have been devoted in designing them through enforcing parity-time (PT) symmetry. In this work, we exploit a lesser-known dynamical mechanism for enforcing real-spectra, and develop a comprehensive and versatile approach for designing new classes of parent Hamiltonians with real spectra. Our design approach is based on a new electrostatics analogy for modified non-Hermitian bulk-boundary correspondence, where electrostatic charge corresponds to density of states and electric fields correspond to complex spectral flow. As such, Hamiltonians of any desired spectra and state localization profile can be reverse-engineered, particularly those without any guiding symmetry principles. By recasting the diagonalization of non-Hermitian Hamiltonians as a Poisson boundary value problem, our electrostatics analogy also transcends the gain/loss-induced compounding of floating-point errors in traditional numerical methods, thereby allowing access to far larger system sizes.

Phase Diagram of a Non-Hermitian Chern Insulator: Destabilization of Chiral Edge States and Bulk–Boundary Correspondence

A non-Hermitian Chern insulator with gain/loss-type non-Hermiticity shows a peculiar gap closing; when a nontrivial Chern insulator phase changes to a gapless phase, conduction and valence bands are combined into one band owing to destabilization of chiral edge states. A previous recipe of non-Hermitian bulk–boundary correspondence is insufficient for this system since such a gap closing is beyond its scope. Here, we revise the recipe by taking the peculiar gap closing into account and apply it to the non-Hermitian Chern insulator. We demonstrate that the bulk–boundary correspondence holds in the system including a destabilization of chiral edge states. A phase diagram derived from the bulk–boundary correspondence is shown to be consistent with spectra of the system.

Non-Hermitian bulk-boundary correspondence and singular behaviors of generalized Brillouin zone

The bulk boundary correspondence, which connects the topological invariant, the continuum band and energies under different boundary conditions, is the core concept in the non-Bloch band theory, in which the generalized Brillouin zone (GBZ), appearing as a closed loop generally, is a fundamental tool to rebuild it. In this work, it can be shown that the recovery of the open boundary energy spectrum by the continuum band remains unchanged even if the GBZ itself shrinks into a point. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of the energy, but not from the topological invariant. Meanwhile, regardless of the fact that the GBZ comes out with the closed loop, the bulk boundary correspondence cannot be well characterized yet because of the ill-definition of the topological number. Here, the results obtained may be useful for improving the existing non-Bloch band theory.

Bulk–Boundary Correspondence in a Non-Hermitian Chern Insulator

A scenario of non-Hermitian bulk–boundary correspondence proposed for one-dimensional topological insulators is adapted to a non-Hermitian Chern insulator to examine its applicability to two-dimens...

Two-Dimensional Quantum Walk with Non-Hermitian Skin EffectsSupported by the National Natural Science Foundation of China (Grant Nos. 11974331, 11674306, and 61590932), and the National Key R&D Program (Grant Nos. 2016YFA0301700 and 2017YFA0304100).

We construct a two-dimensional, discrete-time quantum walk, exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states is consistent with the Floquet winding number, calculated using a non-Bloch band theory, invoking time-dependent generalized Brillouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined via the local biorthogonal eigenwave functions of a non-unitary Floquet operator. Our work aims to stimulate further studies of non-Hermitian Floquet topological phases where skin effects play a key role.

Exceptional topology of non-Hermitian systems

The current understanding of the role of topology in non-Hermitian (NH) systems and its far-reaching physical consequences observable in a range of dissipative settings are reviewed. In particular, how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm is discussed. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. Furthermore, new notions of gapped phases including topological phases in single-band systems are detailed, and the manner in which a given physical context may affect the symmetry-based topological classification is clarified. A unique property of NH systems with relevance beyond the field of topological phases consists of the anomalous relation between bulk and boundary physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, a picture of intriguing phenomena such as the NH bulk-boundary correspondence and the NH skin effect is put together. Finally, applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits,s and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.

Non-Hermitian bulk-boundary correspondence in a periodically driven system

Bulk-boundary correspondence, connecting the bulk topology and the edge states, is an essential principle of the topological phases. However, the bulk-boundary correspondence is broken down in general non-Hermitian systems. In this paper, we construct one-dimensional non-Hermitian Su-Schrieffer-Heeger model with periodic driving that exhibits non-Hermitian skin effect: all the eigenstates are localized at the boundary of the systems, whether the bulk states or the zero and the $\pi$ modes. To capture the topological properties, the non-Bloch winding numbers are defined by the non-Bloch periodized evolution operators based on the generalized Brillouin zone. Furthermore, the non-Hermitian bulk-boundary correspondence is established: the non-Bloch winding numbers ($W_{0,\pi}$) characterize the edge states with quasienergies $\epsilon=0, \pi$. In our non-Hermitian system, a novel phenomenon can emerge that the robust edge states can appear even when the Floquet bands are topological trivial with zero non-Bloch band invariant, which is defined in terms of the non-Bloch effective Hamiltonian. We also show that the relation between the non-Bloch winding numbers ($W_{0,\pi}$) and the non-Bloch band invariant ($\mathcal{W}$): $\mathcal{W}= W_{0}- W_{\pi}$.

Generalized Brillouin zone and non-Hermitian band theory

The energy band theory is one of the cornerstones of condensed matter physics. It also has wide applications in other branches of physics. Recently, a number of questions from non-Hermitian physics call for a generalization of energy band theory to non-Hermitian systems. In the study of non-Hermitian topological states, it has been found that such a generalization necessitates redefinitions of certain fundamental concepts of band theory. In particular, the non-Hermitian skin effect (NHSE) causes the breakdown of Bloch-band picture and conventional bulk-boundary correspondence. To calculate the energy spectra and define topological invariants, the standard Brillouin zone gives way to the generalized Brillouin zone (GBZ). Many intriguing non-Hermitian phenomena, including the non-Hermitian skin effect, can be precisely characterized in terms of the generalized Brillouin zone. The non-Hermitian band theory based on the concept of generalized Brillouin zone, now generally known as the non-Bloch band theory, has successfully described and predicted a number of novel non-Hermitian phenomena. The present article provides a brief introduction to the main concepts of non-Bloch band theory, and its applications in the non-Hermitian bulk-boundary correspondence, Green’s functions, wave dynamics, chiral damping, and non-Bloch parity-time symmetry.

Generalized Bloch band theory for non-Hermitian bulk–boundary correspondence

Abstract Bulk–boundary correspondence is the cornerstone of topological physics. In some non-Hermitian topological systems this fundamental relation is broken in the sense that the topological number calculated for the Bloch energy band under the periodic boundary condition fails to reproduce the boundary properties under the open boundary. To restore the bulk–boundary correspondence in such non-Hermitian systems a framework beyond the Bloch band theory is needed. We develop a non-Hermitian Bloch band theory based on a modified periodic boundary condition that allows a proper description of the bulk of a non-Hermitian topological insulator in a manner consistent with its boundary properties. Taking a non-Hermitian version of the Su–Schrieffer–Heeger model as an example, we demonstrate our scenario, in which the concept of bulk–boundary correspondence is naturally generalized to non-Hermitian topological systems.

Non-Hermitian Bulk-Boundary Correspondence and Auxiliary Generalized Brillouin Zone Theory.

We provide a systematic and self-consistent method to calculate the generalized Brillouin zone (GBZ) analytically in one-dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a n-band non-Hermitian Hamiltonian is constituted by n distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit.

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.

Topological Phase Transition Driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics.

Quantum phase transitions are intriguing and fundamental cooperative phenomena in physics. Analyzing a superconducting nanowire with spin-dependent non-Hermitian hopping, we discover a topological quantum phase transition driven by infinitesimal cascade instability. The anomalous phase transition is complementary to the universal non-Bloch wave behavior of non-Hermitian systems. We show that an infinite small magnetic field drastically suppresses the non-Hermitian skin effect, inducing a topological phase with Majorana boundary states. Furthermore, by identifying the bulk topological invariant, we establish the non-Hermitian bulk-boundary correspondence that does not have a Hermitian counterpart. We also discuss an experimental realization of the system by using the spin-current injection to a quantum wire.