Topological-to-topological transition induced by on-site nonlinearity in a one-dimensional topological insulator

Recent studies have extended the notion of band topology to nonlinear systems by defining nonlinear counterparts of eigenvalue problems. They have found the nonlinearity-induced topological transition regarding edge-localized eigenvectors, while it has required complicated nonlinearity such as off-diagonal one. Thus, the existence of nonlinearity-induced edge modes has been unclear under homogeneous on-site nonlinearity, which is ubiquitously found in nature. We here reveal that such on-site nonlinearity can induce transitions of topological modes, where topological modes converging to zero begin to converge to nonzero values. Since such nonlinearity-induced transition keeps the bulk band topology unchanged, we can regard it as a transition from a conventional topological mode to one unique to nonlinear systems. We analyze a nonlinear eigenvalue problem by rewriting it to a dynamical system in the spatial direction and clarify that the nonlinearity-induced transition is a result of the bifurcation in the spatial dynamics. We also propose a possible setup to observe the nonlinearity-induced transition that uses a gradual amplification of nonlinear waves. These results provide a general designing principle of topological insulators controlled by nonlinearity.