As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the 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 non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.

正如陈数对量子霍尔效应的表征所首次证明的那样，拓扑学为实现凝聚态物质系统不受无序存在影响的稳健特性提供了指导原则。体-边界对应保证了在体表现出非零拓扑不变量的拓扑系统中出现无间隙边界模式。尽管最近的一些研究表明拓扑概念可能扩展到非线性系统，但拓扑不变量的非线性对应物尚未被理解。在这里，我们基于二维系统中的非线性特征值问题提出了陈数的非线性扩展，并证明了弱非线性范围之外体边界对应的存在。具体来说，我们发现了非线性引起的拓扑相变，其中拓扑边缘模式的存在取决于振荡模式的幅度。我们提出并分析了非线性陈绝缘体的最小模型，通过分析获得了其精确的体解。该模型表现出非线性陈数的振幅依赖性，为此我们确认了体-边界对应关系的非线性扩展。因此，我们的结果揭示了与线性状态绝热分离的真正非线性拓扑相的存在。