Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 10, No. 1, 2024 330 Observation of Multiple Topological edge States in Square‐Root electric circuits Jingxiang Gao*, Ming Zhang, Xiefei Cheng, Xingcheng Tang, Runjia Guo School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China * Corresponding author Abstract: Square-root topological states are new topological phases, whose intriguing topological properties are inherited from the parent lattice Hamiltonian. Because of the square-root procedure, the bulk gap of the parent Hamiltonian is doubled. In this letter, we report the observation of the square-root topological insulators (SRTIs) in topological LC circuits, whose squared Hamiltonian includes a Su-Schrieffer-Heeger (SSH) model, a well-known example of topological insulators (TI). The multiple localized edge states falling in different bandgaps are observed with characteristic phase structures, in sharp contrast to the discrete diffraction in a topologically trivial structure. These edge states with zero energy are manifested by a prominent impedance peak at the midgap frequency and directly observed by impedance measurements. Our work opens up an alternative gateway towards actively controllable topological systems and may bring about new possibilities in topology-driven electronic devices. Keywords: Topological insulator, square-root operation, edge states, electrical circuit. 1. Introduction Topological band theory provides a theoretical framework for the abundance of topological states of quantum matter, such as insulators1,2, (semi-)metals3, and superconductors4,5. The system described by Hamiltonians has been the main focus of research on topological band theory. This theory can be used to classify the structures of periodic energy bands by using topological invariants related to the eigenstates of energy in the momentum space. In the past few years, the intriguing topological physical effects have been extended to artificial periodic structures including photonic crystals6-9, phononic crystals (PCs)10-16, electric circuits13,17-19, etc. Due to the high precision of sample fabrication, the electric circuit system provides an excellent platform to explore intriguing topological properties. To date, searching and realizing new topological states in circuits continues to attract a great deal of attention20-31. Recently, a square-root TI (SRTI) is proposed32. Its topological properties are demonstrated to be inherited from the parent Hamiltonian 33-36. The square-root procedure had played an important role in deriving the Dirac equation in relativistic quantum mechanics from the quadratic Klein- Gordon equation, which revealed the chirality for electrons. This procedure provides an antidiagonal block matrix form and leads to symmetric spectra for square-root Hamiltonian 37-40. For lattice models, SRTI can be generated by inserting additional sites and breaking up the coupling between the original sites. So far, only few experiments for SRTI have been realized in quantum matter. Very recently, electric circuit experiment confirmed the square-root topological insulator33,41,42. In order to directly observe the finite-energy topological states, scientists introduced the extra grounded inductors to each node, which shifts topological states to zero- energy without affecting their spatial distributions. So obtaining “nonzero-energy” topological modes with zero energy is the point in all kinds of square-root electric circuits. In this letter, we demonstrate the multiple topological edge States in square-root electric circuits, by inserting a set of additional nodes in the LC SSH model. Because of the square- root procedure, the bulk gap of the parent Hamiltonian is doubled. We find that the multiple localized edge states fall in different bandgaps with characteristic phase structures. These edge states with zero energy are directly observed by impedance measurements. Our work substantiates the emerging square-root TI and may bring about new possibilities in actively controllable topology-driven electronic devices. 2. Results and Discussion Su-Schrieffer-Heeger Electrical Circuit. We start with the non-Hermitian SSH chain, which is shown in the upper panel of Figure 1a. The SSH chain consists of two nonequivalent sites (A and B) in each unit cell. The intracell and intercell coupling strengths are denoted as ν and μ . Following the general recipe for the construction of SRTI developed by Ezawa,43 by inserting another two sites (C and D) which separate the original ones and taking the square root of coupling strengths, as shown in the lower panel of Figure 1a, the square-root SSH chain has been designed. Now, there are four sites labeled by A, B, C, and D in each unit cell, with symmetrical intracell coupling ν, symmetrical intracell coupling μ and symmetrical intercell coupling μ . According to Kirchhoff's law and Ohm's law, we can formulate the current and voltage equations for the 11 nodes. After simplification of the resulting current equation, we obtained the following: 𝐼 𝑖𝜔 𝐶 𝐶 𝐶 𝐼 𝐻 𝑘 𝑉 (1) with 𝐻 𝑘 ⎣ ⎢ ⎢ ⎡ 0 0 𝐶 𝐶 𝑒 0 0 𝐶 𝐶 𝐶 𝐶 0 0 𝐶 𝑒 𝐶 0 0 ⎦ ⎥ ⎥ ⎤ 𝐻 0 0 𝐻 (2) with matrix elements 𝐻 𝐶 𝐶 𝐼 𝐻 with 331 𝐻 0 𝐶 𝐶 𝑒 𝐶 𝐶 𝑒 0 (3) Where I represents the 4 × 4 unit matrix, 𝐼 and 𝑉 denote the input currents and response voltages of four nodes, respectively. The residual Hamiltonian, denoted as 𝐻 , consistently represents a topologically mundane system, while 𝐻 stands for the Hamiltonian of the SSH chain. The square of 𝐻 𝑘 equals the direct sum of the parent Hamiltonian 𝐻 and a residual Hamiltonian 𝐻 . As 𝐻 𝑘 corresponds to the square-root of 𝐻 , we refer to this new composite lattice model as the square-root SSH chain. Figure 1. Lattice models for SRTIs. (a) Top panel: the schematic for the SSH model with squared hopping parameters. Bottom panel: the schematic for the square-root SSH model with intracell coupling ν and intercell coupling μ. (b) Circuit implementation of the square-root SSH model, and the gray solid box marks one unit cell. (c)–(e) Band structure for different parameter: 𝐶 1.5 nF, 𝐶 1 nF for (c), 𝐶 𝐶 1 nF for (d), 𝐶 1 nF, 𝐶 1.5 nF for (e). The parameters are set to 𝐿 10 μH. When analyzing the eigenvalues of the circuit system, assuming 𝐼 0, Eq. (1) can be reformulated as 𝐻 𝑘 𝑉 𝐶 𝐶 𝑉. By solving for the eigenvalue 𝜀 of 𝐻 𝑘 , we ultimately derive the dispersion relation of the circuit as 𝑓 1 2𝜋 2𝐶 2𝐶 𝜀⁄ . It is important to note that we mitigate the impact of the main diagonal elements on the system during resonance by judiciously selecting 𝐶 𝐶 𝐶 , 𝐶 2𝐶 and 𝐶 2𝐶 . The band structures for different capacitance coupling are shown in Figure 1c-e, which is gapped at 𝑘 𝜋 point for 𝐶 𝐶 1⁄ . It is therefore possible to realize the topological insulator phase due to the gap opening. The band structures in Figure 1c and Figure 1e show the same gap opening conditions. However, we suggest that there is a non-trivial bandgap in the former case, while the latter was trivial. In Figure 1d, when 𝐶 𝐶 the bandgap closes at 𝑘 𝜋 , representing topological phase transition points. For each band, we can evaluate a 1D topological invariant, Zak’s winding phase 𝜃 𝑑𝑘A 𝑘 , where 𝐴 𝑘 𝑖⟨𝑣 𝑘 |𝜕 |𝑣 𝑘 ⟩ is the Berry connection of the 𝑖 th band and |𝑣 𝑘 ⟩ is the corresponding eigenstate44. For a standard 1D topological insulator, the winding phase takes quantized values of 𝜋 (or 0) corresponding to encircling (or not encircling) a singularity in quasi-momentum phase space. When 𝐶 1 nF, 𝐶 1.5 nF, there are four bands with the quantized Zak’s winding phases (𝜃 𝜋 2⁄ ). Edge States. We consider a finite-size SSH chains circuit with 𝑁 40 nodes. The Laplacian of the circuit can be written as 𝐽 𝜔 ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ 𝑀 0 𝐽 0 0 ⋯ 0 𝑀 𝐽 𝐽 0 ⋯ 𝐽 𝐽 𝑀 0 0 ⋯ 0 𝐽 0 𝑀 𝐽 ⋯ 0 0 0 𝐽 𝑀 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ 𝑁 𝑁 (4) with 𝑀 𝑖𝜔 2𝐶 2𝐶 1 𝜔 𝐿⁄ and 𝐽 𝑖𝜔𝐶 𝑚 1,2 . 332 Figure 2. The admittance spectrums and eigenfrequencies spectrums of the circuit model. (a), (b) The admittance spectrums and eigenfrequencies spectrums of circuit model for 𝐶 𝐶 3 2⁄⁄ . (c)The theoretical spectrum of J(ω) depending on the frequency with grounded inductors L, where the red segments denote the edge states. (d) Admittances for 𝐶 𝐶 2 3⁄⁄ with 𝐿 10 μH. The red and gray dots represent the edge and bulk, respectively. Subsequently, we computed the eigenfrequencies and eigenfunctions of 𝐽 𝜔 . Figures 2a and 2b display the admittance spectrum eigenfrequencies spectrum of the square-root SSH chain when 𝐶 𝐶 3 2⁄⁄ . It can be observed that there are no isolated states, indicating that the system is trivial in this case. Figure 2c displays the admittance spectrum of the square-root SSH chain, wherein the edge states are plotted as red curves. Figure. 2d shows the eigenvalue spectrum and edge states in the bandgap, where red circles and grey circles represent edge states and bulk states, respectively. The three edge degenerate states are numerically identified at three frequencies, 𝑓 610 kHz , 𝑓 890 kHz and 𝑓 910 kHz ,with their density distributions localized at the left and right end of the model. In the theoretical calculations, we adopt 𝐿 10 μH. The nontrivial topological feature of the 1D topological insulator is manifested by its distinct edge state, which appears at the boundaries of a finite chain. However, it should be noted that the observables of topological circuits are different from those in the quantum and photonics systems. Topological circuits are commonly studied through a two- point impedance measured between two adjacent nodes, a and b, subject to an external current excitation, 𝐼 , flowing through them, which is expressed as24,45,46 𝑍 ∑ , , , (5) in which 𝜓 , 𝑖 𝑎 𝑜𝑟 𝑏 and 𝑗 𝜔 represent the eigenstates and eigenvalues of 𝐽 𝜔 , respectively. As the root of 𝑗 𝜔 corresponds to the eigenfrequency of the circuit, 𝑍 𝜔 diverges when the denominator 𝑗 𝜔 crosses zero. Consequently, each pole in 𝑍 𝜔 signifies a mode (block or edge) in a finite circuit. Therefore, strong resonant peaks on the 𝜔 impedance spectra can be used to identify topological states at circuit boundaries. Impedance observation. We choose electric elements 𝐶 1 nF, 𝐶 1.5 nF, 𝐿 10 μH. Figure 3a shows tight-binding model of ten unit cells. We determine the ground impedance between two representative nodes and other bulk nodes as a function of excitation frequency, as shown in Figure 3b, where we select the A, B, and other bulk nodes to characterize the properties of the edge and bulk states, respectively. In the theoretical calculation, we have averaged the result after 103 realizations of uniformly distributed disorder (we assume a 2% tolerance of each electric element). At the resonant frequency 𝑓 , 𝑓 and 𝑓 , theoretical curves(red) display a strong peak, which confirms the very existence of the edge states. We demonstrate the distribution of the theoretical impedance in this circuit (see Figure 3c-e). Robustness of the edge states. Topological states show resilience and robustness against disorder and defects in general. In order to examine the robustness of the BIC, we introduce defects in the fourth and seventh unit cells of the circuit structure. The specific circuit diagrams are shown in Figure 4a. Figure 4b and Figure 4c depict the impedance map at node 𝐴 and B, the impedance at the frequency 𝑓 , 𝑓 and 𝑓 still shows the strong localization around the original edge. The frequency of the edge states remains stable. 333 Figure 3. Realization of the SRTI in LC circuit. (a) Tight-binding model of ten unit cells. (b) Theoretical (frequency domain solver) impedance spectra of the finite circuit chain. (c)-(e) Distribution of the measured impedance for the edge states at 𝑓 , 𝑓 and 𝑓 , respectively. Figure 4. Robustness verification of the square-root LC circuit. (a) Circuit diagrams of the defects introduced in the fourth and seventh unit cells, respectively. (b) (c) Impedance simulation diagrams at points A and B. 3. 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