  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.
