Non-Hermitian Chern Bands

Abstract

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...

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