Topological circuit: a playground for exotic topological physics
doi:10.37188/CO.2021-0095
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摘要:在过去的十年里，材料的拓扑相变以及它们的奇异物理现象在固态电子学领域掀起了研究热潮。近些年来，人们开始在其他系统中重现和模拟电子体系中的各种拓扑现象，包括冷原子气体、离子阱、光子学、声子学、机械波和电路体系。在这些体系平台中，由电感电容所组成的拓扑电路因具有高度灵活的设计自由度、高性价比、易加工和易集成的独特优势而备受关注。除此之外，在拓扑电路中可以方便地设计具有任意长程耦合、非线性、非互易、增益等效应的拓扑模型，从而实现很多在电子体系和光学体系中难以实现的非线性、非阿贝尔和非厄米的拓扑相变材料。本文作为拓扑电路领域的第一篇综述，系统性地回顾了过去六年拓扑电路领域的主要进展，重点关注其理论建模、电路设计与实验测量，并对拓扑电路与电子和光学体系中的拓扑绝缘体着重进行讨论和区别。本综述涵盖了多种不同类型的拓扑电路，包括含有非平庸边界态、高阶拓扑角模式以及外尔粒子的厄米拓扑电路，拥有拓扑节线和节点态的高维拓扑电路，具有趋肤效应和由增益/衰减导致的非厄米拓扑电路，自感应拓扑边界态的非线性拓扑电路，以及具有非阿贝尔规范场效应的拓扑电路。Abstract:Exploring topological phases of matter and their exotic physics appeared as a rapidly growing field of study in solid-state electron systems in the past decade. In recent years, there has been a trend on the emulation of topological insulators/semimetals in many other systems, including ultracold quantum gases, trapped ions, photonic, acoustic, mechanical, and electrical circuit systems. Among these platforms, topological circuits made of simple capacitive and inductive circuit elements emerged as a very competitive platform because of its highly controllable degrees of freedom, lowercost, easy implementation, and great flexibility for integration. Owing to the unique advantages of electrical circuits such as arbitrary engineering of long-range hopping, convenient realization of nonlinear, nonreciprocal, and gain effects, highly flexible measurement, many of the nonlinear, non-abelian, and non-Hermitian physics can be potentially realized and investigated using the electrical circuit platform. In this review, we provide the first short overview of the main achievements of topological circuits developed in the past six years, primarily focusing on their theoretical modeling, circuit construction, experimental characterization, and their distinction from their counterparts in quantum electronics and photonics. The scope of this review covers a wide variety of topological circuits, including Hermitian topological circuits hosting nontrivial edge state, higher-order corner state, Weyl particles; higher dimensional topological circuits exhibiting nodal link and nodal knot states; non-Hermitian topological circuits showing skin effects, gain and loss induced nontrivial edge state; self-induced topological edge state in nonlinear topological circuit; topological circuit having non-Abelian gauge potential.
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图 1（a-b）具有π/2感应强度的Hofstadter电路模型中的自旋依赖的拓扑边界态[9]。（c）三维拓扑电路中受电路参数调控的拓扑节线态和外尔态[16]。（d-f）二维形式的Su–Schrieffer–Heeger拓扑电路[17]。
Figure 1.(a-b) Spin-dependent topological edge state in an electrical circuit mimicking π/2 flux Hofstadter model[9]. (c) 3D topological circuit with topological nodal line state and Weyl state controlled by the circuit parameters[16]. (d-f) Two-dimensional version of the Su–Schrieffer–Heeger model circuit[17].
图 3（a-c）二维高阶拓扑电路中由四极子动量导致的拓扑角模式的实验观测[33]。（d-e）三维高阶拓扑电路中由八极子动量所保护的零维拓扑角模式[34]。
Figure 3.(a-c) Experimental observation of a quadrupole moment induced corner state in a 2D electrical circuit[33]. (d-e) 3D topological circuit exhibiting 0D corner state topologically protected by octupole moment of the bulk circuit[34].
图 4（a-c）具有二阶陈数的四维拓扑电路中的量子霍尔效应的实验验证[40]。（d,e）四维拓扑电路中的Seifert表面，即三维动量空间中二维表面引出的节线或者节点[41]。
Figure 4.(a-c) Experimental implementation of the first 4D quantum Hall model in electrical circuit characterized by a nonzero second Chern number[40]. (d,e) 4D circuit topological circuit exhibiting a Seifert surface, which is 2D surface with its boundary tracing out a link or knot in 3D momentum space[41].
图 5（a-c）由增益和衰减导致的拓扑相变的实验验证[48]。（d,e）具有非互易效应的非厄米拓扑电路中的趋肤效应的实验验证[13]。
Figure 5.(a-c) Experimental realization of a non-Hermitian electrical circuit whose nontrivial topology is induced by the introduction of gain and loss[48]. (d,e) Experimental observation of non-Hermitian skin effect in a variation of SSH topological circuit with nonreciprocity[13].
图 6（a-c）具有编织耦合的传输线网络中的非阿贝尔拓扑电荷的实验观测[63]。（d,e）具有非阿贝尔规范场效应的拓扑电路中的Rashba-Dresselhaus自旋-轨道耦合现象的实验验证[65]。
Figure 6.(a-c) Experimental observation of non-Abelian topological charges in a transmission line network with braiding connectivity[63]. (d,e) Experimental demonstration of the Rashba-Dresselhaus spin-orbit interaction (SOI) in a passive electrical circuit with non-Abelian gauge fields[65].
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