The breakdown of the bulk-boundary correspondence in non-Hermitian (NH) topological systems is an open, controversial issue. In this paper, to resolve this issue, we ask the following question: Can a (global) topological invariant completely describe the topological properties of a NH system as its Hermitian counterpart? Our answer is no. One cannot use a global topological invariant (including non-Bloch topological invariant) to accurately characterize the topological properties of the NH systems. Instead, there exist a new type of topological invariants that are absence in its Hermitian counterpart -- the state dependent topological invariants. With the help of the state-dependent topological invariants, we develop a new topological theory for NH topological system beyond the general knowledge for usual Hermitian systems and obtain an exact formulation of the bulk-boundary correspondence, including state-dependent phase diagram, state-dependent phase transition and anomalous transport properties (spontaneous topological current). Therefore, these results will help people to understand the exotic topological properties of various non-Hermitian systems.
We investigate novel features of three dimensional non-Hermitian Weyl semimetals, paying special attention to its 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-…
Bulk-boundary correspondence is the cornerstone of topological physics. In some non-Hermitian topological system 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 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 generalize…
In two-dimensional anomalous Floquet insulators, non-Hermitian boundary state engineering can be used to completely separate chiral boundary states from bulk bands in the quasienergy spectrum. The topological properties of such spectrally separated boundary states are no longer restricted by the strict bulk-boundary correspondence of Hermitian systems. We show that this additional topological freedom enables one to faithfully transfer the topological properties of a boundary attached to a Floquet insulator to a non-Hermitian Floquet chain obtained by physically cutting off the boundary from th…
In this paper, we review our non-Bloch band theory in one-dimensional non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su-Schrieffer-Heeger model, the bulk-edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge stat…
We address the breakdown of the bulk-boundary correspondence observed in non-Hermitian systems, where open and periodic systems can have distinct phase diagrams. The correspondence can be completely restored by considering the Hamiltonian's singular value decomposition instead of its eigendecomposition. This leads to a natural topological description in terms of a flattened singular decomposition. This description is equivalent to the usual approach for Hermitian systems and coincides with a recent proposal for the classification of non-Hermitian systems. We generalize the notion of the entang…
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 …
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-dimensional systems. This scenario employs bulk geometry under a modified periodic boundary condition and boundary geometry under an open boundary condition. The bulk geometry is used to define a topological number, whereas the boundary geometry is used to observe the presence or absence of a topological boundary state. It is demonstrated that the bulk--boundary correspondence holds in a two-dimensional C…
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 $π$ modes. To capture the topological properties, the non-Bloch winding numbers are defined by the non-Bloch periodize…
Non-Hermitian skin effect and critical skin effect are unique features of non-Hermitian systems. In this Letter, we study an open system with its dynamics of single-particle correlation function effectively dominated by a non-Hermitian damping matrix, which exhibits $\mathbb{Z}_2$ skin effect, and uncover the existence of a novel phenomenon of helical damping. When adding perturbations that break anomalous time reversal symmetry to the system, the critical skin effect occurs, which causes the disappearance of the helical damping in the thermodynamic limit although it can exist in small size sy…
Recently, topological quantum states of non-Hermitian systems, exhibiting rich new exotic states, have attracted great attention in condensed-matter physics. As for the demonstration, most of non-Hermitian topological phenomena previously focused on are in one- and two-dimensional systems. Here, we investigate three-dimensional non-Hermitian nodal-line semimetals in the presence of a particle gain-and-loss perturbation. It is found that this perturbation will split the original nodal ring into two exceptional rings (ERs). The topological nature of the bulk electronic structure is characterized…
In previous studies, the topological invariants of 1D non-Hermitian systems have been defined in open boundary condition (OBC) to satisfy the bulk-boundary correspondence. The extreme sensitivity of bulk energy spectra to boundary conditions has been attributed to the breakdown of the conventional bulk-boundary correspondence based on the topological invariants defined under periodic boundary condition (PBC). Here we propose non-Hermitian many-body polarization as a topological invariant for 1D non-Hermitian systems defined in PBC, which satisfies the bulk-boundary correspondence. Employing ma…
We establish exact relations between the winding of "energy" (eigenvalue of Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with periodic boundary condition, and the presence of "skin modes" with open boundary condition in non-hermitian systems. We show that the nonzero winding with respect to any complex reference energy leads to the presence of skin modes, and vice versa. We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the non-vanishing current through the system.
Capital to topological insulators, the bulk-boundary correspondence ties a topological invariant computed from the bulk (extended) states with those at the boundary, which are hence robust to disorder. Here we put forward an ordering unique to non-Hermitian lattices, whereby a pristine system becomes devoid of extended states, a property which turns out to be robust to disorder. This is enabled by a peculiar type of non-Hermitian degeneracy where a macroscopic fraction of the states coalesce at a single point with geometrical multiplicity of $1$, that we call a phenomenal point.
Non-Hermitian Topology from Edge Transport in Hermitian Quantum Anomalous Hall Systems
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；School of Physical Science and Technology, Soochow University（苏州大学物理科学与技术学院）；Institute for Advanced Study, Soochow University（苏州大学前沿科学高等研究院）；Institute for Nanoelectronic Devices and Quantum Computing, Fudan University（复旦大学纳米电子器件与量子计算研究所）；Hefei National Laboratory（合肥国家实验室）
Non-Hermitian materials can not only exhibit exotic energy band structures but also an anomalous velocity induced by non-Hermitian anomalous Berry connection as predicted by the semiclassical equations of motion for Bloch electrons. However, it is not clear how the modified semiclassical dynamics modifies transport phenomena. Here, we theoretically demonstrate the emergence of anomalous oscillations driven by either an external dc or ac electric field, which arise from non-Hermitian anomalous Berry connection. Moreover, it is a well-known fact that geometric structures of electric wave functio…
Recent work has extended topological band theory to open, non-Hermitian Hamiltonians, yet little is understood about how non-Hermiticity alters the topological quantization of associated observables. We address this problem by studying the quantum anomalous Hall effect (QAHE) generated in the Dirac surface states of a 3D time-reversal-invariant topological insulator (TI) that is proximity-coupled to a metallic ferromagnet. By constructing a contact self-energy for the ferromagnet, we show that in addition to generating a mass gap in the surface spectrum, the ferromagnet can introduce a non-Her…
Non-Hermitian topological systems show quite different properties as their Hermitian counterparts. An important, puzzled issue on non-Hermitian topological systems is the existence of defective edge states beyond usual bulk-boundary correspondence (BBC) that localize either on the left edge or the right edge of the one-dimensional system. In this paper, to understand the existence of the defective edge states, the theory of anomalous bulk-boundary correspondence (A-BBC) is developed that distinguishes the non-Bloch bulk-boundary correspondence (NB-BBC) from non-Hermitian skin effect. By using …
It was known that for non-Hermitian topological systems due to the non-Hermitian skin effect, the bulk-edge correspondence is broken down. In this paper, by using one-dimensional Su-SchriefferHeeger model and two-dimensional (deformed) Qi-Wu-Zhang model as examples, we focus on a special type of non-Hermitian topological system without non-Hermitian skin effect-topological systems under non-Hermitian similarity transformation. In these non-Hermitian systems, the defective edge states and the breakdown of bulk-edge correspondence are discovered. To characterize the topological properties, we in…
The Comment by Xiong et al. (arXiv:1610.06275) criticizing my Letter [Phys. Rev. Lett. 116, 133903 (2016)] was rejected by Physical Review Letters. In this Reply, I show that all their claims are wrong.
Topological states in non-Hermitian systems are known to exhibit some anomalous features. Here, we find two new anomalous features of non-Hermitian topological states. We consider a one dimensional nonreciprocal Hamiltonian and show that topological robustness can be practically lost for a linear combination of topological eigenstates in non-Hermitian systems due to the non-Hermitian skin effect. We consider a two dimensional non-Hermitian Chern insulator and show that chirality of topological states can be broken at some parameters of the Hamiltonan. This implies that the topological states a…
The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk topological invariants defined for infinite periodic systems and the occurrence of protected zero-energy surface states in finite samples, is a ubiquitous and widely observed phenomenon in topological matter. In non-Hermitian generalizations of topological systems, however, this fundamental correspondence has recently been found to be qualitatively altered, largely owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary conditions. In this work, we report on two contributions towards comprehen…
In this paper, with the help of Hermitian/non-Hermitian dynamic strings, the theory of non-Hermitian topological order is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations (e-particle and m-particle) are Hermitian tight-binding lattice model; the effective model for fermionic topological excitation (f-particle) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudo-spin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topolog…
The hallmark of symmetry-protected topological (SPT) phases is the existence of anomalous boundary states, which can only be realized with the corresponding bulk system. In this work, we show that for every Hermitian anomalous boundary mode of the ten Altland-Zirnbauer classes, a non-Hermitian counterpart can be constructed, whose long time dynamics provides a realization of the anomalous boundary state. We prove that the non-Hermitian counterpart is characterized by a point-gap topological invariant, and furthermore, that the invariant exactly matches that of the corresponding Hermitian anoma…
We consider a 3-dimensional (3D) non-Hermitian exceptional line semimetal model and take open boundary conditions in x, y, and z directions separately. In each case, we calculate the parameter regions where the bulk-boundary correspondence is broken. The breakdown of the bulk-boundary correspondence is manifested by the deviation from unit circles of generalized Brillouin zones (GBZ) and the discrepancy between spectra calculated with open boundary conditions (OBC) and periodic boundary conditions (PBC). The consistency between OBC and PBC spectra can be recovered if the PBC spectra are calcul…
A non-Hermitian extension of a Chern insulator and its bulk-boundary correspondence are investigated. It is shown that in addition to the robust chiral edge states that reflect the nontrivial topology of the bulk (nonzero Chern number), anomalous helical edge states localized only at one edge can appear, which are unique to the non-Hermitian Chern insulator.
The conventional bulk-boundary correspondence breaks down in non-Hermitian systems. In this paper, we reestablish the bulk-boundary correspondence in one-dimensional non-Hermitian systems by applying the scattering theory, which is a systematical way in various symmetry classes. Based on the scattering theory, it is discovered that the topological invariant can be obtained by solving a generalized eigenproblem without calculating the generalized Brillouin zone. As a direct consequence, we unveil a new type of topological phase transition without typical bulk enengy gap closing and an unstable …
Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge sta…
Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-H…
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 $n$-th 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 boun…
In Hermitian topological systems, the bulk-boundary correspondence strictly constraints boundary transport to values determined by the topological properties of the bulk. We demonstrate that this constraint can be lifted in non-Hermitian Floquet insulators. Provided that the insulator supports an anomalous topological phase, non-Hermiticity allows us to modify the boundary states independently of the bulk, without sacrificing their topological nature. We explore the ensuing possibilities for a Floquet topological insulator with non-Hermitian time-reversal symmetry, where the helical transport …
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 conf…
Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological pha…
A non-Hermitian topological insulator is fundamentally different from conventional topological insulators. The non-Hermitian skin effect arises in a nonreciprocal tight binding lattice with open edges. In this case, not only topological states but also bulk states are localized around the edges of the nonreciprocal system. We discuss that controllable switching from topological edge states into topological extended states in a chiral symmetric non-Hermitian system is possible. We show that the skin depth decreases with non-reciprocity for bulk states but increases with it for topological zero …
The generalized Brillouin zone (GBZ), which is the core concept of the non-Bloch band theory to rebuild the bulk boundary correspondence in the non-Hermitian topology, appears as a closed loop generally. In this work, we find that even if the GBZ itself collapses into a point, the recovery of the open boundary energy spectrum by the continuum bands remains unchanged. 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 …
Both non-Hermitian systems and the behaviour of emitters coupled to structured baths have been studied intensely in recent years. Here we study the interplay of these paradigmatic settings. In a series of examples, we show that a single quantum emitter coupled to a non-Hermitian bath displays a number of unconventional behaviours, many without Hermitian counterpart. We first consider a unidirectional hopping lattice whose complex dispersion forms a loop. We identify peculiar bound states inside the loop as a manifestation of the non-Hermitian skin effect. In the same setting, emitted photons m…
Topological edge modes are excitations that are localized at the materials' edges and yet are characterized by a topological invariant defined in the bulk. Such bulk-edge correspondence has enabled the creation of robust electronic, electromagnetic and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter and geophysical flows. Recently, the advent of non-Hermitian topological systems---wherein energy is not conserved---has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Herm…
We classify topological phases of non-Hermitian systems in the Altland-Zirnbauer classes with an additional reflection symmetry in all dimensions. By mapping the non-Hermitian system into an enlarged Hermitian Hamiltonian with an enforced chiral symmetry, our topological classification is thus equivalent to classifying Hermitian systems with both chiral and reflection symmetries, which effectively change the classifying space and shift the periodical table of topological phases. According to our classification tables, we provide concrete examples for all topologically nontrivial non-Hermitian …
Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and $π$ quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, arising from the winding of the time evolution unitary rather than the bulk Floquet bands at the end of the driving period. In this paper, we systematically derive the first and inversion-symmetric second-order AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D, …
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 provid…
Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees …
The chiral anomaly underlies a broad number of phenomena, from enhanced electronic transport in topological metals to anomalous currents in the quark-gluon plasma. The discovery of topological states of matter in non-Hermitian systems -- effective descriptions of dissipative systems -- raises the question of whether there are anomalous conservation laws that remain unaccounted for. To answer this question, we consider both $1+1$ and $3+1$ dimensions, presenting a unified formulation to calculate anomalous responses in Hermitianized, anti-Hermitianized and non-Hermitian systems of massless elec…
We present a one-dimensional coupled ring resonator lattice exhibiting a variant of the non- Hermitian skin effect (NHSE) that we call the anomalous Floquet NHSE. Unlike existing approaches to achieving the NHSE by engineering gain and loss on different ring segments, our design uses fixed on-site gain or loss in each ring. The anomalous Floquet NHSE is marked by the existence of skin modes at every value of the Floquet quasienergy, allowing for broadband asymmetric transmission. Varying the gain/loss induces a non-Hermitian topological phase transition, reversing the localization direction of…
Bulk boundary correspondence is crucial to topological insulator as it associates the boundary states (with zero energy, chiral or helical) to topological numbers defined in bulk. The application of this correspondence needs a prerequisite condition which is usually not mentioned explicitly: the boundaries themselves cannot alter the bulk states, so as to the topological numbers defined on them. In non-hermitian models with fractional winding number, we prove that such precondition fails and the bulk boundary correspondence is cut out. We show that, as eliminating the hopping between the bound…
Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where o…
The chiral hinge modes are the key feature of a second order topological insulator in three dimensions. Here we propose a quadrupole index in combination of a slab Chern number in the bulk to characterize the flowing pattern of chiral hinge modes along the hinges at the intersection of the surfaces of a sample. We further utilize the topological field theory to demonstrate the correspondent connection of the chiral hinge modes to the quadrupole index and the slab Chern number, and present a picture of three-dimensional quantum anomalous Hall effect as a consequence of chiral hinge modes. The t…
Berry phases strongly affect the properties of crystalline materials, giving rise to modifications of the semiclassical equations of motion that govern wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry connection have been analyzed to characterize the topology of these systems. While the topological classification of non-Hermitian systems is being developed, little attention has been paid to the impact of the new geometric phases on dynamics and transport. In this work, we derive the full set of semiclassical equations of motion for wave-packet dynamics in a system g…
