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Topological circuits, an exciting field just emerged over the last two years, have become a very accessible platform for realizing and exploring topological physics, with many of their physical phenomena and potential applications as yet to be discovered. In this work, we design and experimentally demonstrate a topologically nontrivial band structure and the associated topologically protected edge states in an RF circuit, which is composed of a collection of grounded capacitors connected by alternating inductors in the x and y directions, in analogy to the Su–Schrieffer–Heeger model. We take full control of the topological invariant (i.e., Zak phase) as well as the gap width of the band structure by simply tuning the circuit parameters. Excellent agreement is found between the experimental and simulation results, both showing obvious nontrivial edge state that is tightly bound to the circuit boundaries with extreme robustness against various types of defects. The demonstration of topological properties in circuits provides a convenient and flexible platform for studying topological materials and the possibility for developing flexible circuits with highly robust circuit performance.

Topological insulators (TI), which insulate in the bulk but conduct on the surface, have been the subject of many recent studies in physics aimed at achieving topologically protected nontrivial band structures and the associated exotic phenomena, with possible implementation in diverse fields ranging from solid in electronics [

The previously demonstrated Chern and Z2 topological insulators are enabled by the presence of Berry curvatures or non-Abelian Berry curvatures. It is expected that many of the topological features found in condensed-matter physics can find their analogues in RF circuits. Many topological phenomena and properties, including the robust edge state, can be readily reproduced with electrical circuits. Importantly, topological circuits represent a highly flexible platform for investigating topological phenomena due to the convenient connections between nodes at arbitrarily long distances. This may lead to realization of 3D topological systems without introducing extra synthetic dimensions [

The SSH model has attracted increasing research interests in the past decades, due to its rich physical phenomena, including topologically protected edge states, fractional charge, PT symmetry, and topological soliton excitation [

We start with a brief review of the 1D SSH model, which was originally developed to describe the 1D polyacetylene and is one of the simplest models to host topological properties [_{
a} and_{
b} in both the

_{
a} and_{
b} indicate, respectively, the coupling strength between capacitors inside the unit cell (intracoupling) and between two adjacent unit cells (intercoupling)._{
a}=39nH,_{
b}=220nH,

Having described the unit cell of 2D SSH circuit, we proceed to provide the mathematical tool for analyzing its topological properties. We apply the Kirchhoff's Law to the circuit unit cell by assuming the voltages _{
x} and_{
y} denoting the phase of Block wave vector propagating in the

Substituting_{
a}=39nH,_{
b}=220nH,_{
x} and_{
y}, producing the 2D band structure as shown in Figure _{
a} added to the right and bottom edges, which accounts for a quarter of unit cell (Figure

_{
a}._{
a} and_{
b}, respectively, while the black sphere represents the grounded capacitors. The red wavelike curve indicates the nontrivial edge states.

Although topologically protected edge states appear at the edge of topological circuit, they are closely related to the bulk states through the bulk-edge correspondence, which can predict, from the bulk circuit Laplacian, the number of topologically protected edge modes present in a finite-sized TI. To demonstrate the origin of these edge states, we calculate the topological invariant, Zak phase, from the grounded circuit Laplacian matrix_{
a}
_{
b}, where a nontrivial edge mode appears at the four boundaries, whereas the edge mode disappears as we exchange the value of inductors (_{
a}
_{
b}, see Supplementary Figure

We can obtain the spectrum of eigenmodes of the finite-sized circuit for different choices of_{
a} and_{
b} by calculating the eigenvalues (^{
2}) of the dynamical matrix _{
a} and_{
b} are exchanged (Supplementary Figure _{
a}=_{
b} (Supplementary Figure

It can be expected, according to the bulk-boundary correspondence, that our 2D SSH circuit supports an edge mode localized at the four edges in the nontrivial regime. To support this expectation, we design and fabricate a circuit board which incorporates 7.25×7.25 unit cells, as shown by the photographs in Figures

Reflectance measurement was firstly carried out to experimentally characterize the 2D SSH circuit. The absorption spectrum, which represents the amount of RF energy pumped into the circuit, can be simply obtained from _{
11} is measured with a 50 Ohm coaxial cable, it reaches zero (linear scale) when the input impedance of a certain node equals 50 Ω, leading to a maximum absorptance of unity. The expected edge mode distribution is illustrated in Figure

Because such edge modes are protected by the topological nature of the SSH circuit, they are thus robust to certain types of defects and disorder. We demonstrate the robustness of the 2D SSH circuit by removing a square-shaped patch which consists of 3×3 unit cells, as illustrated in Figure _{
a} connected to the ground, such that the topology nature is still protected by the inversion symmetry of the 2D SSH circuit. As shown by Figures

The insulating bulk and conducting edge nature of topological insulator circuit can also be revealed by inspecting the transmission coefficients (_{
21}) between two nodes on the edge and bulk. We note, in the following tests, that the RF signal is pumped into the 2D SSH circuit from the second resonator at the bottom edge with a stimulating port (port 1, indicated by the yellow star in Figure

To further demonstrate the robustness of the topologically protected edge state, we measure the transmission coefficient of the edge and bulk states under the presence of different number of small defects. As illustrated in Figure

In this work, we presented the design and experimental realization of a 2D SSH circuit exhibiting a nontrivial band structure and topologically protected edge state. We experimentally identified the topologically protected edge modes in a sample with 7.25×7.25 unit cells, which were located on all the edges and decayed rapidly into the bulk sites. The circuit performance is robust against component tolerance of ~5% and component Q factor of ~10 (see Supplementary Figures

The proposed topological circuit may find potential applications in flexible electronics, a technology for assembling electronic circuits on flexible substrates, such as polyimide and polyester films [

The software Agilent Design System is employed for the numerical simulation of the finite-sized 2D SSH circuit having 7.25×7.25 unit cells, which is built with the exact value as the real components selected for the fabricated sample. The simulated absorption spectra given in Figure

To minimize the loss effect to the topological properties of circuits, two inductors with 220nH±2% (Murata, LQW2BAN39NG00#) and 39nH±2% (Murata, LQW2UASR22G00#) inductances are selected, with Q-factors reaching ~38 and ~35 at 40 MHz, respectively. The self-resonance frequency of both inductors is over 500MHz, far beyond the operational frequency of our topological circuit. Chip multilayer ceramic capacitors of 1000 pF ±5% (GRM1882C1H102JA01-01A) are selected for the grounded capacitors. Keysight N5230C VNA was employed to measure the reflection and transmission coefficient, which had been calibrated using a 50 Ω calibration unit (E5052D) before measurements. The transmission coefficient between two nodes was performed by means of two-port transmission measurement using a pair of microwave cables, with one serving as the excitation and the other probing the response.

The authors declare that there are no conflicts of interest regarding the publication of this article.

Shuo Liu, Wenlong Gao, and Qian Zhang contributed equally to this work and are co-first authors. Shuo Liu, Wenlong Gao, and Shaojie Ma carried out the analytical modelling and numerical simulations. Shuo Liu, Qian Zhang, and Lei Zhang completed the sample fabrication and circuit measurements. As the principal investigators of the projects, Shuang Zhang, Tie Jun Cui, and Yuan Jiang Xiang conceived the idea, suggested the designs, and planned, coordinated, and supervised the work. Shuo Liu and Shuang Zhang contributed to the writing of the manuscript. All authors discussed the theoretical and numerical aspects and interpreted the results.

This work was supported by the European Research Council Consolidator Grant (TOPOLOGICAL), the Royal Society, the Wolfson Foundation, Horizon 2020 Action Project no. 734578 (D-SPA), and the National Key Research and Development Program of China (Grant no. 2017YFA0700201), in part by the National Natural Science Foundation of China (Grants nos. 61631007, 61571117, 61875133, and 11874269) and the 111 Project (Grant no. 111-2-05), and in part by the China Postdoctoral Science Foundation (Grant no. 2018M633129).

Figure S1: circuit schematic with the node voltages_{
1}
_{
4} on the four nodes and current_{
1}
_{
8} in the corresponding branches. Figure S2: bandwidth of the lower and higher bandgaps with respect to inductances_{
a}
_{
b}. Figure S3: numerically simulated absorption spectra of the 2D SSH circuit for the edge site when_{
a}=220 nH and_{
b}=39nH. Figure S4: the spectrum of eigenmodes of the circuit for different choices of_{
a} and_{
b}, obtained by calculating the eigenvalues of the dynamical matrix of the finite-sized circuit. Figure S5: results of the absorptance distribution at the higher bandgap for the case without and with defect. Figure S6: distribution of average absorptance of the three bulk bands. Figure S7: experimentally measured and numerically simulated absorption spectra of the 2D SSH circuit for the bulk site and edge site. Figure S8: statistical data of the absorptance distribution in Figure

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