Nonlinear perturbation of a high-order exceptional point: skin discrete breathers and the hierarchical power-law scaling

Abstract We study the nonlinear perturbation of a high-order exceptional point (EP) order equal to system site number $L$ in Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity. find class discrete breathers that aggregate one boundary, which we dubbed as $skin breathers$ (SDBs). The spectrum these SDBs shows hierarchical power-law scaling near EP. Specifically, response energy is given by $E_m\propto \Gamma^{\alpha_{m}}$, where $\alpha_m=3^{m-1}$ power $m=1,\cdots,L$ labeling bands. This sharp contrast $L$th root linear general. These decay double-exponential manner, unlike edge states or skin modes systems, exponentially. Furthermore, can survive over full range nonlinearity strength are continuously connected self-trapped limit large They also stable, confirmed defined fidelity an adiabatic evolution from stability analysis. As nonreciprocal models may be experimentally realized various platforms, such classical platform optical waveguides, naturally present, quantum lattices Bose-Einstein condensates, our analytical results inspire further exploration interplay between non-Hermiticity, particularly on EPs, benchmark relevant simulations.