Non-Hermitian Boundary Modes and Topology Borgnia, Dan S; Kruchkov, Alex Jura; Slager, Robert-Jan
Physical review letters,
02/2020, Letnik:
124, Številka:
5
Journal Article
Recenzirano
Odprti dostop
We consider conditions for the existence of boundary modes in non-Hermitian systems with edges of arbitrary codimension. Through a universal formulation of formation criteria for boundary modes in ...terms of local Green's functions, we outline a generic perspective on the appearance of such modes and generate corresponding dispersion relations. In the process, we explain the skin effect in both topological and nontopological systems, exhaustively generalizing bulk-boundary correspondence to different types of non-Hermitian gap conditions, a prominent distinguishing feature of such systems. Indeed, we expose a direct relation between the presence of a point gap invariant and the appearance of skin modes when this gap is trivialized by an edge. This correspondence is established via a doubled Green's function, inspired by doubled Hamiltonian methods used to classify Floquet and, more recently, non-Hermitian topological phases. Our work constitutes a general tool, as well as a unifying perspective for this rapidly evolving field. Indeed, as a concrete application we find that our method can expose novel non-Hermitian topological regimes beyond the reach of previous methods.
The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked "non-Hermitian skin effect" necessitates redefinition of ...topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of the non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conventional bulk-boundary correspondence and introduces the non-Bloch bulk-boundary correspondence.
Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a non-Hermitian Hamiltonian are localized at ...the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodic-boundary spectrum of the Hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic non-Hermitian Hamiltonian, and, unlike in one dimension, is compatible with all point-group symmetries. We propose two new types of skin effect in two and higher dimensions: the corner-skin effect where all eigenstates are localized at corners of the system, and the geometry-dependent-skin effect where skin modes disappear for systems of a particular shape, but appear on generic polygons. An immediate corollary of our theorem is that any non-Hermitian system having exceptional points (lines) in two (three) dimensions exhibits skin effect, making this phenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.
The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly ...tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.
Higher-order phases are characterized by corner or hinge modes that arise due to the interesting interplay of localization mechanisms along two or more dimensions. In this work, we introduce and ...construct a novel class of "hybrid" higher-order skin-topological boundary modes in nonreciprocal systems with two or more open boundaries. Their existence crucially relies on nonreciprocal pumping in addition to topological localization. Unlike usual non-Hermitian "skin" modes, they can exist in lattices with vanishing net reciprocity due to the selective nature of nonreciprocal pumping: While the bulk modes remain extended due to the cancellation of nonreciprocity within each unit cell, boundary modes experience a curious spontaneous breaking of reciprocity in the presence of topological localization, thereby experiencing the non-Hermitian skin effect. The number of possible hybridization channels increases rapidly with dimensionality, leading to a proliferation of distinct phases. In addition, skin modes or hybrid skin-topological modes can restore unitarity and are hence stable, allowing for experimental observations and manipulations in non-Hermitian photonic and electrical metamaterials.
The discovery of topological phases in non-Hermitian open classical and quantum systems challenges our current understanding of topological order. Non-Hermitian systems exhibit unique features with ...no counterparts in topological Hermitian models, such as failure of the conventional bulk-boundary correspondence and non-Hermitian skin effect. Advances in the understanding of the topological properties of non-Hermitian lattices with translational invariance have been reported in several recent studies; however little is known about non-Hermitian quasicrystals. Here we disclose topological phases in a quasicrystal with parity-time (PT) symmetry, described by a non-Hermitian extension of the Aubry-André-Harper model. It is shown that the metal-insulating phase transition, observed at the PT symmetry breaking point, is of topological nature and can be expressed in terms of a winding number. A photonic realization of a non-Hermitian quasicrystal is also suggested.
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, and mechanical systems and genuine quantum systems such as electronic transport settings at material junctions and dissipative cold-atom setups are reviewed.
We propose a realistic cold-atom quantum setting where topological localization induces nonreciprocal pumping. This is an intriguing non-Hermitian phenomenon that illustrates how topology, when ...assisted with atom loss, can act as a "switch" for the non-Hermitian skin effect (NHSE), rather than as a passive property that is modified by the NHSE. In particular, we present a lattice-shaking scenario to realize a two-dimensional cold-atom platform, where nonreciprocity is switched on only in the presence of both atom loss and topological localization due to time-reversal symmetry breaking. The resultant nonreciprocal pumping is manifested by asymmetric dynamical evolution, detectable by atomic populations along the system edges. Our setup may trigger possible applications in nonreciprocal atomtronics, where loss and topological mechanisms conspire to control atomic transport. Its quantum nature will also facilitate future studies on the interplay between non-Hermiticity and many-body physics.
A unique feature of non-Hermitian systems is the skin effect, which is the extreme sensitivity to the boundary conditions. Here, we reveal that the skin effect originates from intrinsic non-Hermitian ...topology. Such a topological origin not merely explains the universal feature of the known skin effect, but also leads to new types of the skin effects-symmetry-protected skin effects. In particular, we discover the Z_{2} skin effect protected by time-reversal symmetry. On the basis of topological classification, we also discuss possible other skin effects in arbitrary dimensions. Our work provides a unified understanding about the bulk-boundary correspondence and the skin effects in 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.
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