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.

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.Hamiltonians are Hermitian in the standard quantum mechanics. Nevertheless, non-Hermitian Hamiltonians [1, 2] are highly useful in describing many phenomena such as various open systems [3][4][5][6][7][8][9][10][11][12] and waves propagations with gain and loss . Recently, topological phenomena in non-Hermitian systems have attracted considerable attention. For example, an electron's non-Hermitian self energy stemming from disorder scatterings or electron-electron interactions [40][41][42] can generate novel topological effects such as bulk Fermi arcs connecting exceptional points [40,41] (a photonic counterpart has been observed experimentally [43]). The interplay between non-Hermiticity and topology has been a growing field with a host of interesting theoretical and experimental [76][77][78][79][80][81][82] progresses witnessed in recent years.A central principle of topological states is the bulkboundary (or bulk-edge) correspondence, which asserts that the robust boundary states are tied to the bulk topological invariants. Within the band theory, the bulk topological invariants are defined using the Bloch Hamiltonian [83][84][85][86]. This has been well understood in the usual context of Hermitian Hamiltonians; nevertheless, it is a subtle issue to generalize this correspondence to non-Hermitian systems [44][45][46][47][48][53][54][55][56]. As demonstrated numerically [46,53,54,56], the bulk spectra of one-dimensional (1D) open-boundary systems dramatically differ from those with periodic boundary condition, suggesting a breakdown of bulk-boundary correspondence. This issue has been resolved[56] in 1D non-Hermitian Su-Schrieffer-Heeger (SSH) model: The topological end modes are determined by the non-Bloch winding number[56] instead of topological invariants defined by Bloch Hamiltonian [45][46][47][48][49][50][51][52], which suggests a generalized bulk-boundary correspondence [56].However, the general implications of these results based solely on a simple 1D model remain to be understood (e.g., Is the physics specific to 1D?). Moreover, the topology of this 1D model requires a chiral symmetry [85], which is often fragile in real systems. Thus, we are motivated to study 2D non-Hermitian...

Topology of non-Hermitian systems is drastically shaped by the non-Hermitian skin effect, which leads to the generalized bulk-boundary correspondence that replaces the conventional one. The essential part in formulations of bulk-boundary correspondence is a general and computable definition of topological invariants. In this paper, we introduce a construction of non-Hermitian topological invariants based directly on real-space wavefunctions, which provides a general and straightforward approach for determining non-Hermitian topology. As an illustration, we apply this formulation to several representative models of non-Hermitian systems, efficiently obtaining their topological invariants in the presence of non-Hermitian skin effect. Our formulation also provides a surprisingly simple dual picture of the generalized Brillouin zone of non-Hermitian systems, leading to a unique perspective of bulk-boundary correspondence.Non-Hermitian Hamiltonians have widespread applications in physics. For example, when a quantum-mechanical system is open, meaning that its interaction with the environment is nonnegligible, its effective Hamiltonian is non-Hermitian [1][2][3][4][5]. The ubiquitous loss and engineered gain in classical wave phenomena [6][7][8][9], the finite quasiparticle lifetimes [10][11][12][13][14][15], and certain statistical-mechanical models [16], etc, are naturally described in terms of non-Hermitian Hamiltonians. Recently, growing efforts have been invested in uncovering novel topological phases in non-Hermitian systems. Among other observations, we mention that non-Hermiticity calls for revised bulk-boundary correspondence [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and novel topological invariants [19-21, 23, 33-40], introduces new symmetries that enrich the topological classifications of bands [41][42][43], brings in non-Hermitian topological semimetals exhibiting exceptional band degeneracies without Hermitian counterparts [44][45][46][47][48][49][50][51][52][53][54], and many other interesting phenomena .Crucial to understanding the band topology is the non-Hermitian skin effect (NHSE) [19,24], namely the exponential localization of (nominally bulk) continuum-spectrum eigenstates to boundaries. Its meanings and consequences are under active studies [20, 25, 28-30, 34, 48, 85-96]. In particular, NHSE underlies the breakdown of conventional bulkboundary correspondence and suggests the non-Bloch bulkboundary correspondence as its generalization [19,22], and leads to the concepts of generalized Brillouin zone (GBZ) and non-Bloch topological invariants [19,20,23,33,34].It is the purpose of this paper to offer a dual picture of non-Hermitian bands. For the usual Hermitian bands, Fourier transformation precisely connects the Brillouin zone and real space. Such a simple picture cannot, however, be straightforwardly generalized to non-Hermitian bands, because the eigenstates lose the extendedness of Bloch waves by the NHSE. Somewhat surprisingly, we find that non-Hermitian topological invari...

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary condition but exponential under open boundary condition. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wavefront. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.Model.-The system is illustrated in Fig.1(a). Our Hamiltonian H = ij h ij c † i c j , where c † i , c i are fermion creation and annihilation operators at site i (including