Strongly nonlinear topological phases of cascaded topoelectrical circuits

doi:10.1007/s11467-023-1292-4

Front. Phys.

2023, Vol. 18

Issue (3) : 33311 https://doi.org/10.1007/s11467-023-1292-4

RESEARCH ARTICLE

Strongly nonlinear topological phases of cascaded topoelectrical circuits

Jijie Tang¹, Fangyuan Ma², Feng Li²(

), Honglian Guo¹(

), Di Zhou²(

)

¹. College of Science, Minzu University of China, Beijing 100081, China
². Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurement (MOE), School of Physics, Beijing Institute of Technology, Beijing 100081, China

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Abstract

Circuits provide ideal platforms of topological phases and matter, yet the study of topological circuits in the strongly nonlinear regime, has been lacking. We propose and experimentally demonstrate strongly nonlinear topological phases and transitions in one-dimensional electrical circuits composed of nonlinear capacitors. Nonlinear topological interface modes arise on domain walls of the circuit lattices, whose topological phases are controlled by the amplitudes of nonlinear voltage waves. Experimentally measured topological transition amplitudes are in good agreement with those derived from nonlinear topological band theory. Our prototype paves the way towards flexible metamaterials with amplitude-controlled rich topological phases and is readily extendable to two and three-dimensional systems that allow novel applications.

Keywords strongly nonlinear Berry phase topological electrical

Corresponding Author(s): Feng Li,Honglian Guo,Di Zhou

About author:

*These authors equally shared correspondence to this manuscript.

Issue Date: 09 May 2023

Cite this article:

Jijie Tang,Fangyuan Ma,Feng Li, et al. Strongly nonlinear topological phases of cascaded topoelectrical circuits[J]. Front. Phys. , 2023, 18(3): 33311.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1292-4
https://academic.hep.com.cn/fop/EN/Y2023/V18/I3/33311

Fig.1 (a) The unit cell of the underlying nonlinear topoelectrical circuit. (b) The unit cell of the linear circuit that helps to construct an interface between two semi-lattices in Fig.3. (c) The voltage-dependence of the nonlinear capacitors, where the dots represent the data measured by the network analyzer, and the fitting Gaussian curve is adopted for numerical computations. (d) The numerical calculation of the nonlinear band gap and topological phase transitions of the circuit unit cell in (a) for growing nonlinearity as voltage amplitude increases. At the transition point marked by the vertical red line, the degree of nonlinearity (horizontal axis) reads 0.332, which grants the considered topoelectrical circuit the strongly nonlinear regime.

Fig.2 Topological phase diagram of the nonlinear electrical circuit in Fig.1(a). In the horizontal axis, we vary the parameter of the linear capacitor

C 2

from

14.0 p F

37.0 p F

. The vertical axis represents the amplitude of responding voltage fields. The blue curve denotes the numerical result of the topological transition voltage amplitudes for varying linear capacitor

C 2

. The blue area depicts the numerical result of the topological transition voltage under the influence of fluctuations in the nonlinear capacitors

C 1

with a range of

± 10 %

. The experimentally measured transition voltages are depicted by the square marks with error bars. The inset illustrates the transition of the topological index when

C 2 = 23 p F

Fig.3 (a) Schematic illustration of the first prototype of the nonlinear topoelectrical circuit, whose interface connects the left and right half-lattices. Encircled by the red dashed box, the right-sided unit cells are composed of linear elements, as shown in Fig.1(b). The left-sided semi-lattice, as enclosed by the green dashed box, is composed of unit cells whose intra-cell and inter-cell couplings are switched comparing with Fig.1(a). (b) Photograph of the experimentally constructed circuit board from the design in (a), whose right and left sides contain 4 unit cells each. (c) The absence of topological interface mode in the small amplitude regime. (d) Topological interface mode on the verge of nonlinear topological phase transition. The mode amplitude reads

3.09 V

, which approaches the transition point

A c = 2.97 V

. (e) Nonlinear topological interface mode becomes clearer for amplitude at

7.84 V

. (f?h) Impedance diagram of variable capacitor replaced by linear capacitor

37 p F

in (f),

25 p F

(g), and

15 p F

in (h).

Fig.4 (a) Schematic illustration of the second prototype of the nonlinear topoelectrical circuits. As encircled by the green dashed boxes, the unit cells of the two semi-lattices are designed from Fig.1(a). (b) Experimental setup of (a). (c) The interface hosts a topological mode with the excitation power of

0.065 V

. (d) Nonlinear topological interface mode on the verge of disappearance, as the external power

1.91 V

approaches the topological transition amplitude

A c = 2.97 V

. (e) For the power that further rises to

6.17 V

, trivial localized modes take place on the interface. (f?h) Impedance diagram of variable capacitor replaced by linear capacitor

37 p F

in (f),

25 p F

(g), and

15 p F

in (h).

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