We study the nonlinear perturbation of a high-order exceptional point (EP) of the order equal to the system site number $L$ in a Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity. We find a class of discrete breathers that aggregate to one boundary, which we dubbed as $skin discrete breathers$ (SDBs). The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP. Specifically, the response of nonlinear energy to the perturbation is given by $E_m\propto \Gamma^{\alpha_{m}}$, where $\alpha_m=3^{m-1}$ is the power with $m=1,\cdots,L$ labeling the nonlinear energy bands. This is in sharp contrast to the $L$th root of a linear perturbation in general. These SDBs decay in a double-exponential manner, unlike the edge states or skin modes in linear systems, which decay exponentially. Furthermore, these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity. They are also stable, as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis. As nonreciprocal nonlinear models may be experimentally realized in various platforms, such as the classical platform of optical waveguides, where Kerr nonlinearity is naturally present, and the quantum platform of optical lattices with Bose-Einstein condensates, our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity, particularly on high-order EPs, and benchmark the relevant simulations.
The introduction of non-Hermiticity into traditional Hermitian quantum systems generalizes their basic notions and brings about many novel phenomena, e.g., the non-Hermitian skin effect that is exclusive to non-Hermitian systems, attracting enormous attention from almost all branches of physics. Contrary to the quantum platforms, classical systems have the advantages of low cost and mature techniques under room temperature. Among them, the classical electrical circuits are more flexible on simulating quantum tight-binding models in principle with any range of hopping under any boundary conditions in any dimension, and have become a powerful platform for the simulation of quantum matters. In this paper, by constructing an electrical circuit, we simulate by SPICE the static properties of a prototypical non-Hermitian model-the nonreciprocal Aubry-André (AA) model that has the nonreciprocal hopping and on-site quasiperiodic potentials.<br>The paper is organized as follows:Following the introduction, in Sec. II we review in detail the Laplacian formalism of electrical circuits and the mapping to the quantum tight-binding model. Then, in Sec. III, an electrical circuit is proposed with resistors, capacitors, inductors, and the negative impedance converters with current inversion (INICs), establishing a mapping between the circuit's Laplacian and the non-reciprocal AA model's Hamiltonian under periodic boundary conditions (PBCs) or open boundary conditions (OBCs). Especially, the nonreciprocity, the key of this model, is realized by INICs. In Sec IV, based on the mapping, for the proposed circuit under PBCs, we reconstruct the circuit's Laplacian via SPICE by measuring voltage responses of an AC current input at each node. The complex spectrum and its winding number <i>ν</i> can be calculated by the measured Laplacian, which are consistent with the theoretical prediction, showing <i>ν</i>=±1 for non-Hermitian topological regimes with complex eigenenergies and extended eigenstates, and ν=0 for topologically trivial regimes with real eigenenergies and localized eigenstates. In Sec V, for the circuit under OBCs, a similar method is used for measuring the node distribution of voltage response, which simulates the competition of non-Hermitian skin effects and the Anderson localization, depending on the strength of quasiperiodic potentials; the phase transition points also appear in the inverse participation ratios of voltage responses.<br>During the design process, the parameters of auxiliary resistors and capacitors are evaluated for obtaining stable responses, because the complex eigenfrequecies of the circuits are inevitable under PBCs. Our detailed scheme can directly instruct further potential experiments, and the designing method of the electrical circuit is universal and can in principle be applied to the simulation for other quantum tight-binding models.
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