Majorana-like end states in one-dimensional dimerized Kitaev topolectrical circuit

Majorana zero modes (MZMs) have attracted tremendous attention in condensed matter and materials physics communities due to the implications in topological quantum computation. One-dimensional (1D) dimerized Kitaev chain is a prototype model for MZMs, but its realization remains a challenge in material systems. Here, we develop a distinctive approach to achieve Majorana-like end states (MESs) by implementing practical dimerized Kitaev topolectrical circuits. Specifically, two arrays of inductors are arranged to simulate particles and antiparticles, while intra- and inter-array capacitive connections are used to model hopping and superconducting pairing. Three topological phases can be achieved by tuning the capacitance, i.e. the trivial phase, Su–Schrieffer–Heeger topological phase and Kitaev phase, with distinct field strength distributions in real space. Majorana splitting is observed around a domain wall in the circuit, and we propose an efficient experimental observable-edge distance-to characterize the process as premonition of topological phase transition. Remarkably, dynamics of the Gaussian wave packet in time domain provide an excellent signal to detect MESs in experiments, as only MESs allow nonlocal propagation in circuit network. Our results not only manifest the superiorities of topolectrical circuits for exotic topological states, but also pave the way for possible applications in electrical engineering and signal processing.


Introduction
During the past decade, tremendous efforts have been made in the search of various exotic topological states of matter, including but not limited to topological insulators, Dirac/Weyl semimetals, topological nodal line semimetals, and topological superconductors (TSCs) [1][2][3][4][5][6]. Realization of TSCs that support Majorana zero modes (MZMs) is an exciting topic of both fundamental scientific importance and technological potential applications, as MZMs are regarded as promising candidates for fault-tolerant quantum computing [7,8]. MZMs correspond to neutral excitations comprised of an equal superposition of electrons and holes. The unique properties of Majorana quasiparticles lie in the fact that they are their own antiparticles, closely related to Majorana fermions that exhibit an extensive ground-state degeneracy and non-abelian exchange statistics. So far, a few material systems have been put forward for observing MZMs, such as the interface between topological insulators and superconductors [9][10][11], semiconductors with strong spin-orbit coupling (SOC) [12][13][14][15], phase-biased Josephson junctions [16] and magnetic atomic chains [17,18].
In this contribution, we theoretically introduce a topolectrical circuit and demonstrate the realization of Majorana-like end states (MESs) in circuit systems. It should be noted that electrical circuits are classical systems and cannot simulate TSC due to the lack of particle-hole symmetry. Nevertheless, MESs exhibit intriguing physical properties that can simulate the braiding behavior of Majorana fermions [31]. Recent reports showed that electrical circuits provide a powerful platform to simulate a plethora of exotic topological quantum states [32][33][34][35], even of higher-orders/dimensions [36][37][38]. By using simple inductors and capacitors, we construct a specific circuit network to simulate the 1D dimerized Kitaev chain. The dimerized Kitaev chain is the prototype model for spinless p-wave TSCs, in which the competition between Su-Schrieffer-Heeger (SSH) order and Kitaev order gives rise to a rich phase diagram [39]. However, so far there are no practical systems that can host this model due to the complexity of the dimerized Kitaev structure. We arrange two arrays of inductors to simulate particles and antiparticles, while intra-and inter-array connections can be used to simulate hopping and superconducting pairing. By tuning the capacitance, three distinct topological phases emerge in the circuit, including the trivial phase, SSH topological phase and Kitaev phase, which manifest different band structures and field strength distributions. Interestingly, Majorana splitting can be induced by increasing the superconducting pairing strength that fosters the topological phase transition from trivial to Kitaev phase, where the field strength peak at the domain wall splits into two peaks located at two ends of the circuit chain, and an experimental observable-edge distance-is proposed to facilitate the detection. We further investigate the dynamics of Gaussian wave packet in time domain, and demonstrate that MESs realized in the topolectrical circuit allow nonlocal propagation of wave packet throughout the circuit, which may greatly benefit the experimental observation.

Model and methodology
We begin with the tight-binding model of the 1D dimerized Kitaev chain as schematically depicted in figure 1(a). The Hamiltonian can be written as [39], where i is the index of unit cell with A/B denoting the sublattice coordinate. μ is the chemical potential and t is the electron hopping term. Δ and η represent the p-wave superconducting pairing amplitude and the dimerization parameter, respectively. As such, the Hamiltonian harbors characteristics of both SSH and Kitaev models [40,41]: with μ = 0 and Δ = 0, it reduces to SSH model, and with η = 0 it is Kitaev type.
In the dimerized Kitaev chain, TSCs may appear and the MZMs at two ends can be determined by solving the Bogoliubov-de Gennes (BdG) equation. By diagonalizing the BdG Hamiltonian matrix, the eigenstates is obtained. With the four-component operator with where where s denotes the distance between adjacent sites. We then consider the practical circuit implementation of the dimerized Kitaev chain. As shown in figure 1(b), the designed circuit involves two arrays of inductors within a LC network. Each unit cell contains four sublattices (A, B, A * and B * ) and each sublattice is simulated by an inductor. We define U a = U a+ − U a− as the voltage across an inductor, with a representing the four inductors, and +/− the two ends of an inductor. Parallel (+ to + and − to −) or antiparallel (+ to − and − to +) connection can be used to realize positive or negative coupling. It is able to define the top array as particles and the bottom array as antiparticles, so that the hopping terms can be simulated by connections between inductors within the same array using capacitors with capacitance t(1 ± η), while superconducting pairing is simulated by inter-array connections using capacitors with capacitance Δ(1 ± η). Therefore, different from previous proposal in using node voltage as the basis of wave function and circuit Laplacian to solve the impedance distribution [31], here we employ the voltage difference across an inductor as the basis of wave function and the dynamic equation to describe the dynamics of field strength distribution.
The dynamic properties of this circuit can be analyzed by using Kirchhoff's law. The detailed derivation can be found in note in supplementary material (https://stacks.iop.org/NJP/24/043032/mmedia). Here we only present the dynamic matrix of the circuit, and the equation of motion in k space is written as, where T with T the transpose symbol and ω the excitation frequency. Clearly, the circuit dynamic matrix is similar to the tight-binding matrix, except that the on-site potential is described by m 0 , and there is a constant coefficient L/2 that is attributed to inductors. The opposite chemical potentials of particles and antiparticles are realized by using different grounding capacitors between the top and bottom arrays. It should be noted that the eigenvalue of M k is 1/ω 2 because the magnitude of matrix elements (expressed by the product of inductance and capacitance) corresponds to resonant frequency in LC circuits, similar to the energy in electronic systems. Without losing generality, L and t are set to 1.

Topological band structures for the infinite and finite LC chain
The calculated band structures for an infinite LC chain with different capacitance are displayed in figures 2(a)-(d), with the detailed band evolution and phase diagram presented in figure S1 in note in the supporting information. There are four bands with an energy gap at X point, and the band shape looks similar with different capacitance. To explore the topological properties, we computed the parities of the wave functions for these energy bands. As the band gap closes and reopens at X point, we focus on the parities at this point, labeled by '+' and '−' in the figure. When η is positive with η > Δ, the parities of the top two bands are '−' and the bottom two bands are '+', corresponding to a SSH-like trivial phase ( figure 2(a)). With decreased η, the band gap reduces until a Dirac point is formed when η = Δ. Further decrease of η reopens the gap. This gap closing and reopening process is associated with band inversion, as the parities between the middle two bands exchange signs. When η < Δ, the system enters the Kitaev phase and MESs emerge ( figure 2(b)). When η becomes negative and |η| < Δ, the parities of the top two bands as well as the bottom two bands exchange signs, and the system remains in the Kitaev phase ( figure 2(c)). When η is further reduced with |η| > Δ, the parities of the top two bands become '+' and the bottom two bands are '−', so that the system becomes the SSH-like topological phase ( figure 2(d)). Consequently, three distinct topological phases can be achieved by tuning the capacitance in the circuit, including the trivial phase, Kitaev phase and SSH topological phase.
To further confirm the distinct topology of the three phases, we constructed a finite LC circuit consisting of 200 unit cells and calculated the discrete energy levels, which are presented in figures 2(e)-(h). For the trivial phase, there are no gap states inside the bulk gap, indicating trivial topology (figure 2(e)). When the circuit is in the Kitaev phase, two degenerate in-gap states can be clearly identified, which can be assigned to MESs (figures 2(f) and (g)). For the SSH topological phase, four degenerate edge states exist within the gap (figure 2(h)), as each MES consists of two edge states.

MESs and topological phase transition
In electrical circuits, it is convenient to measure the field strength distribution of topological states in real space. We calculated field strength distribution of the finite LC circuit, as defined by U n = q |U n,q |, with q = A, B, A * and B * , and present the results in figure 3. In the trivial phase, no field strength can be observed throughout the circuit chain ( figure 3(a)). In the SSH topological phase, obvious edge states appear at the two ends of the chain ( figure 3(b)). The Kitaev phase exhibits MESs at the two ends (figures 3(c) and (d)), and the difference between SSH edge states and MESs lies in their drastically different response towards grounding capacitors (analog to chemical potential), for which the SSH edge states are sensitive to the grounding capacitors, while MESs maintain robust (see figures S2 and S3 in note III). We further introduced a domain wall to the finite chain to explore the possible existence of solitons. Here, the domain wall refers to a point defect in the dimerized pattern, where we have added a pair of extra B-type inductors in the circuit chain. As shown in figures 3(e) and (f), solitons appear around the domain wall in SSH-like trivial and nontrivial phases, while there are no solitons in the Kitaev phase besides the MESs at two ends (figures 3(g) and (h)). Therefore, the field strength distribution provides valuable information on the topological states in real space, which should greatly benefit experimental characterization.
Topological phase transition has been extensively studied in condensed matter physics [42][43][44]. However, given a fixed structure, it is rather difficult to actively control and measure the phase transition. Topolectrical circuits hold advantages in a sense that the physical properties can be readily controlled by tuning the inductance and capacitance of the electrical components [31][32][33][34][35]. In this regard, we investigated the topological phase transition of the designed circuit by varying the capacitance (so that the superconducting pairing Δ can be tuned). Figure 4(a) shows the field strength distribution with respect to Δ and site index. As discussed above, when Δ is smaller than the hopping term η, the system is in the SSH trivial phase, where a soliton with a peak of field strength is observed around the domain wall. Decrease of Δ narrows the band gap and at Δ = η, the gap closes with field strength spreading all over the chain. When Δ > η, the system enters Kitaev phase, so that the soliton disappears and MESs emerge with two peaks localized at the two ends of chain. Tracing the whole process, it is easy to identify the Majorana splitting, where the field strength peak around the domain wall splits into two peaks at the ends. This provides an excellent signal of the topological phase transition.
To facilitate the observation of Majorana splitting in electrical circuit, we proposed an experimental observable, i.e., edge distance, as defined by, . d edge represents the field strength weighted average distance from the nearest edge (d i ), so it is closely related to the field strength distribution. For a finite circuit chain with 200 unit cells, the peak around the domain wall corresponds to d edge ∼ 100 and the two peaks at the ends correspond to d edge ∼ 0. We plot the edge distance as a function of Δ in figure 4(b). It can be seen that the edge distance reduces monotonously with the increase of Δ, which provides an efficient observable to characterize the premonition of Majorana splitting in real experiments.

Dynamics in time domain
We further investigated the dynamics of the topological states in time domain to understand the different behaviors of SSH edge states and MESs under external excitation. For this purpose, we introduced a Gaussian wave packet in the finite circuit chain as well as in a constructed circuit heterostructure consisting of a finite circuit chain and a normal metal. The normal metal is simulated by an array of inductors connected by capacitors with the same capacitance without dimerization, and this metallic chain is connected to the upper array of the dimerized chain, as indicated in the insert of figure 5. The fourth Runge-Kutta method was used to explore the dynamics in time domain (more details in note IV).
The simulated time evolution of the Gaussian wave packet are shown in figure 5 and the supplementary videos. For the circuit chain in SSH topological phase, an excitation source at left end confines the Gaussian wave packet at the left, with no excitations extending into the bulk region ( figure 5(a)), corresponding to metallic topological edge states and insulating bulk states. When the circuit chain is in the Kitaev phase, the wave packet propagates throughout the chain ( figure 5(b)), consistent with the fact that MZMs behave non-locally [6]. For the constructed heterostructure with a finite circuit chain and a normal metal, an excitation source at the interface prevents the wave packet travelling into the circuit chain in the SSH topological phase, so that the wave packet propagates only in the metal region ( figure 5(c)). However, when the circuit chain is in Kitaev phase, the wave packet splits into two subpackets, which propagate in opposite directions simultaneously without influencing each other ( figure 5(d)). This demonstrates that MESs in this circuit allow non-local propagation, exhibiting the non-local properties of real MZMs, while the edge states of SSH topological phases are strongly localized at the ends. In sharp contrast, the dynamics for non-topological scenarios behave as a normal insulator ( figure S4). Besides, we have also calculated the dynamics in the presence of capacitance/inductance randomness and parasitic/contact resistances, and found that the topological modes can still be observed in practical experiments (see figure S5 in the supporting information). We believe that this signature is significant for facilitating the detection of MESs in topolectrical circuit.

Conclusion
In summary, we have demonstrated the implementation of dimerized Kitaev topolectrical circuit for the realization of MESs by using inductors and capacitors. The 1D dimerized Kitaev chain is successfully modeled by arranging two arrays of inductors for particles and antiparticles, while hopping and superconducting pairing are simulated by intra-and inter-array capacitive connections. Three distinct phases can be obtained by tuning the capacitance, including the trivial phase, SSH topological phase and Kitaev phase, which exhibit different field strength distributions in real space. Majorana splitting is observed during the topological phase transition from trivial to Kitaev phase, where the field strength peak at the domain wall splits into two peaks at two ends, and can be detected by an experimental observable-edge distance. We highlight the dynamics of Gaussian wave packets in time domain, as only MESs allow nonlocal propagation along the circuit. With high flexibility and tunability, our proposed topolectrical circuit not only provides an ideal platform for MESs, but also holds significant potential for future electrical engineering and signal processing applications. Meanwhile, the presented model is general, and can be applicable to other physical systems like photonic [45,46], acoustic [47,48], polaritonic [49] and mechanical systems [50]. Considering the rapid developments of this discipline [51][52][53][54][55][56], we believe that our work will stimulate considerable attention in a broad range of scientific and engineering communities.