  As first demonstrated by the characterization of the quantum Hall effect by
the Chern number, topology provides a guiding principle to realize robust
properties of condensed matter systems immune to the existence of disorder. The
bulk-boundary correspondence guarantees the emergence of gapless boundary modes
in a topological system whose bulk exhibits nonzero topological invariants.
Although some recent studies have suggested a possible extension of the notion
of topology to nonlinear systems such as photonics and electrical circuits, the
nonlinear counterpart of topological invariant has not yet been understood.
Here, we propose the nonlinear extension of the Chern number based on the
nonlinear eigenvalue problems in two-dimensional systems and reveal the
bulk-boundary correspondence beyond the weakly nonlinear regime. Specifically,
we find the nonlinearity-induced topological phase transitions, where the
existence of topological edge modes depends on the amplitude of oscillatory
modes. We propose and analyze a minimal model of a nonlinear Chern insulator
whose exact bulk solutions are analytically obtained and indicate the amplitude
dependence of the nonlinear Chern number, for which we confirm the nonlinear
counterpart of the bulk-boundary correspondence in the continuum limit. Thus,
our result reveals the existence of genuinely nonlinear topological phases that
are adiabatically disconnected from the linear regime, showing the promise for
expanding the scope of topological classification of matter towards the
nonlinear regime.
