Electrical circuit simulation of non-Hermitian lattice models

Xu Can-Hong; Xu Zhi-Cong; Zhou Zi-Yu; Cheng En-Hong; Lang Li-Jun

doi:10.7498/aps.72.20230914

Abstract

Quantum simulation serves as a significant tool for studying and understanding novel phenomena in the quantum world. In recent years, it has be realized that apart from quantum platforms, classical systems like photonic crystals, phononic crystals, and mechanical oscillators can also be used to simulate quantum models by analogizing the Schrödinger equation. Among these systems, electrical circuits have emerged as a promising simulation platform owing to their low cost, technological maturity, and ease of scalability, successfully simulating numerous important quantum phenomena. Meanwhile, non-Hermitian physics breaks the Hermiticity of systems’ Hamiltonians in traditional quantum mechanics, providing a fresh perspective for understanding the physics of quantum systems, particularly open quantum systems. Non-Hermitian systems, owing to their manifestation of unique phenomena absent in Hermitian systems, have become emerging research subjects in various fields of physics. However, many non-Hermitian phenomena require specialized configurations that pose relatively high technical thresholds on quantum platforms. For instance, the non-Hermitian skin effect typically requires systems to possess non-reciprocal hopping between lattice sites. Therefore, utilizing flexible electrical circuits to simulate non-Hermitian physics becomes a natural choice.This paper provides a short review of the current experimental progress in simulating non-Hermitian lattice models by using electrical circuits. It offers a brief introduction to the relevant knowledge of non-Hermitian physics, including mathematical concepts and novel phenomena, as well as the simulation theory of electrical circuits, including the mapping theory of the lattice models, the introduction of non-Hermiticity, and the measurement of physical quantities. The aim is to provide readers with a reference for better understanding or engagement in related researches, thus promoting further development in this field.

Keywords:

Authors and contacts

1.
School of Physics, South China Normal University, Guangzhou 510006, China
2.
Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China

Corresponding author: Cheng En-Hong, ehcheng@m.scnu.edu.cn ; Lang Li-Jun, ljlang@scnu.edu.cn

Funds: Project supported by the Guangdong Basic and Applied Basic Research Foundation, China (Grant No. 2019A1515111101) and the Startup Funding from South China Normal University, China.

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Cited By

图 1 能量E的实部(实线)和虚部(虚线)随非厄米强度γ的变化. 点线处为PT转变点, 其左侧为PT对称相(白色区域), 右侧为PT对称破缺相(灰色区域)

Figure 1. The real (solid lines) and imaginary (dashed lines) parts of the energy E versus the strength γ of the non-Hermiticity. The dotted line indicates the PT transition point, to the left side of which is the PT symmetric phase (white region) and to the right side of which is the PT-broken phase (gray region).

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图 2 (a)非厄米SSH模型^[59]分别在开边界(粉色)和周期边界(灰色)条件下的能谱E在复平面的示意图; (b)与(a)中能谱相对应的布里渊区(灰色)和广义布里渊区(粉色)的示意图, 其中β的定义见正文; (c)开边界条件下拓扑边缘态(红色)和趋肤态(灰色)的在实空间的几率分布$|\psi_i|^2$示意图, i为格点标记

Figure 2. (a) The sketch of the energy spectra in complex plane for the non-Hermitian SSH model in Ref. [59] respectively under open (pink) and periodic (gray) boundary conditions; (b) the sketch of the Brillouin zone (black) and the generalized Brillouin zone (pink) corresponding to the spectra with the same colors in (a), where the definition of β can be referred to in the main text; (c) the sketch of probability distribution $|\psi_i|^2$ of the topological end state (red) and the skin bulk states (gray) in real space under open boundary conditions, where i is the site index.

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图 3 (a)能谱ε随微扰z在二阶EP附近劈裂的示意图, 具有$\varepsilon \propto z^{1/2}$的形式^[110]; (b)能谱ε随微扰z在传统的二重简并点附近劈裂的示意图, 具有$\varepsilon \propto z$的形式^[110]

Figure 3. (a) The sketch of energy spectra ε versus the perturbation z around a two-order EP, satisfying $\varepsilon \propto z^{1/2}$ ^[110]; (b) the sketch of energy spectra ε versus the perturbation z around a traditional two-fold degenerate point, satisfying $\varepsilon \propto z$^[110].

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图 4 二聚体电路示意图^[132]. 由电感耦合的两个RLC回路构成, 其中M为互感, $-R$代表负电阻

Figure 4. The sketch of the dimer circuit consisting of two inductively coupled RLC tanks^[132], where M is the mutual inductance and $-R$ represents an negative resistance.

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图 5 (a)负阻抗的电路原理图^[133], 上下图分别表示对地端口和自由双端口的情况; (b) INIC的电路原理图^[137]; (c)电压跟随器的电路原理图^[136]. 以上所用运放的等效增益函数可用加法器和乘法器实现

Figure 5. (a) The schematic circuit for negative impedance^[133], where the upper and lower panels represent the one-port and two-port cases, respectively; (b) the schematic circuit for INIC^[136]; (c) the schematic circuit for a voltage follower^[136]. The equivalent gain functions of the operational amplifiers used above can be implemented using adders and multipliers.

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图 6 (a) PT对称的二聚体电路, 电路中的增益和损耗部分通过电容或互感耦合; (b) 实验测量的本征频率随着增益/损耗参数$\gamma/\gamma_{\rm{PT}}$的变换, 在$\gamma/\gamma_{\rm{PT}}=1$时发生PT对称破缺. 图来源于文献[113], 版权属于美国物理学会

Figure 6. (a) Circuit diagram of a PT-symmetric dimer, where the gain and loss parts are capacitively or inductively coupled; (b) experimentally measured eigenfrequencies versus the gain/loss parameter $\gamma/\gamma_{\rm{PT}}$, where the PT symmetry breaking occurs at $\gamma/\gamma_{\rm{PT}}=1$. All figures are adapted from Ref.[113] with the copyright © 2011 by the American Physical Society.

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图 7 (a)非布洛赫演化的拓扑电路; (b)实验测得的电压分布经拉普拉斯变换后在广义布里渊区中的等频分布; (c)四次方根非厄米SSH模型的电路图. 图(a)和(b)来源于文献[127], 图(c)来源于文献[145]. 版权属于美国物理学会

Figure 7. (a) Topolectrical circuit for the non-Bloch dynamics; (b) the isofrequency contour of the measured voltage distribution through the Laplace transform in the GBZ; (c) circuit diagram of a 4th-root non-Hermitian SSH model. Subfigures (a) and (b) are adapted from Ref. [127], and Figure (c) from Ref. [145]. Copyright © 2023 and 2022 by the American Physical Society.

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图 8 (a)单纯由增益/损耗诱导的非厄米拓扑模型; (b)实现(a)中模型的电路图. 图均来源于文献[74], 版权属于美国物理学会

Figure 8. (a) A non-Hermitian topological model whose topology is purely induced by gain/loss; (b) circuit diagram for the realization of the model in (a). All figures are adapted from Ref.[74] with the copyright © 2020 by the American Physical Society.

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图 9 (a)非厄米二阶拓扑电路的原胞示意图; (b)能隙间的一维边缘态, 彩色和圆圈分别表示实验测量和理论计算的结果; (c)实验测量的零维角态的电压分布; (d)实验测量的角态、边缘态和体态的阻抗响应随驱动频率的变化. 图均来源于文献[122], 版权属于美国物理学会

Figure 9. (a) Circuit diagram of a unit cell of the non-Hermitian second-order topological electric circuit; (b) one-dimensional gapped edge states, where the color map and the blue circles represent the data from the experiment and the theoretical calculation, respectively; (c) experimental voltage distributions of the zeroth dimensional corner states; (d) experimental impedance responses of corner states, edge states, and bulk states to the driving frequency. All figures are adapted from Ref. [122] with the copyright © 2022 by the American Physical Society.

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图 10 (a)传统的被动无线传感器与基于PT对称性的被动无线传感器的电路图; (b)基于PT对称性的被动无线传感器的反射谱的测量结果; (c)具有PT对称性的三阶无线传感器的电路图. 图(a)和(b)来源于文献[132], 图(c)来源于文献[152], 版权属于美国物理学会

Figure 10. (a) Circuit diagrams of a conventional passive wireless sensor and a PT-symmetry-based passive wireless sensor; (b) measured reflection spectra for the PT-symmetry-based passive wireless sensor; (c) circuit diagram of a PT symmetric third-order wireless sensing system. Subfigures (a) and (b) are adapted from Ref. [132] and subfigure (c) from Ref. [152], with copyright © 2020 and 2022 by the American Physical Society.

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