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...

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.