Electrical circuit simulation of nonreciprocal Aubry-André models
Authors: Cheng En-Hong , Lang Li-Jun ,   Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China,   Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China
Published: May 2022

<sec>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. </sec><sec>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 <inline-formula><tex-math id="M1">\begin{document}$\nu$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M1.png"/></alternatives></inline-formula> can be calculated by the measured Laplacian, which are consistent with the theoretical prediction, showing <inline-formula><tex-math id="M2">\begin{document}$\nu=\pm 1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M2.png"/></alternatives></inline-formula> for non-Hermitian topological regimes with complex eigenenergies and extended eigenstates, and <inline-formula><tex-math id="M3">\begin{document}$\nu=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="16-20220219_M3.png"/></alternatives></inline-formula> 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. </sec><sec>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. </sec>