Topological phases of matter transcend the paradigm of Ginzburg–Landau theory in condensed matter physics, with the absence of any symmetry breaking but derived from geometry, and have attracted extensive investigation in various fields over the past few decades [1–9]. Topological phases are defined by the global wavefunctions of the dispersion bands that pervade the entire system rather than local orbitals, so that they are particularly robust to local perturbations such as defects and impurities. In essence, band structure is the sufficient condition for the existence of topological phases. Since the first discovery of topological phases in quantum electronic systems [1,2], novel and exotic topological properties have been developed in diverse platforms with their own unique advantages such as optics [10,11], acoustics [12,13], mechanics [14], and electric circuits [15–17] in classical regimes and ultra-cold atoms [18,19], trapped ions [20,21], and Fock-state lattices [22,23] in quantum regimes.

One-dimensional (1D) topological phases bring some new insights because of their manipulability and experimental accessibility. The Su–Schrieffer–Heeger (SSH) model of polyacetilene [24,25], as a starting point for 1D topological models based on tight-binding approximation, is a dimerized chain by having two different alternating hopping amplitudes between nearest-neighboring lattice site hosts. Recently, in the context of SSH chains, a variety of extended configurations with new physics and phenomena have been proposed [9,26–30]. On the one hand, special inconsistent inter-site interactions, such as periodically modulated hopping, nonreciprocal hopping, environment-induced coupling, and multisite coupling, have been introduced to raise a plethora of distinct topological phenomena including but not limited to the non-Hermitian skin effect [31–35], non-Hermitian real spectra [36], dissipative and Floquet topological phase transition [37–40], and trimer topological phases [41–44]. On the other hand, with respect to on-site potentials, the introduction of on-site gain and loss not only provides a pointcut to combine the non-Hermitianity and topological phases for widening topological family [45–47], but also can drive topologically trivial systems and induce topological phase transitions solely by deliberate design [48–52].

Herein, we present a quasi-one-dimensional (quasi-1D) tight-binding configuration without any staggered hopping and on-site potentials. We consider unit cells of multiple identical resonators with uniform coupling between every two sites and the same strength as the inter-cell coupling, i.e., only one kind of coupling strength and resonators in the whole chain. The system then forms complete multimer due to the zero effective intracell hopping induced by special coupling paths. Conceivably, considering the bulk-boundary correspondence (BBC), degenerate topological edge states with full localization exist in finite systems. Interestingly, with the increase of resonators in the unit cell, pairs of nontrivial flat bands appear at two fixed frequencies, which correspond to fully dimerized subspaces derived from the Hilbert spaces. Moreover, for even multimer chains, topological bound states in the continuum (BICs) [53] naturally form via the bandgap of nontrivial flat bands just covered by a trivial band. We experimentally implement the idea by using AC circuits consisting of uniform capacitors and inductors.

We start by considering a tight-binding system consisting of n(n≥3) identical resonators coupled to each other with the same hopping amplitude κ, as shown in Fig. 1(a). Here, we consider a Hermitian system in which the intrinsic and coupling losses of all the resonators are ignored and κ is sufficiently small compared to the frequency of resonators ω0. The systems can be represented by the Hamiltonian Hn=(0κκ⋯κ0κ⋯κκ0⋯⋮⋮⋮⋱)n×n,(1)characterized by the diagonal elements of the matrix being zero and all the others being κ. Interestingly, there are always degenerate states with a fixed frequency independent of n in the system. Specifically, with reference to ω0, one of its eigenvalues is λn=(n−1)κ with the normalized eigenvectors |ψn⟩=(1/n,1/n,…,1/n)′ while the others are λi=−κ where i=1,2,…,n−1 with the corresponding eigenvectors |ψi⟩=(0,…,0,1/2)′ where the ith element |ψi⟩i=1/2. In terms of the splitting of eigenvalues, the local effective coupling between the degenerate modes can be seen as zero to some extent.

With this supposition, as shown in Figs. 1(b)–1(d), we design a class of quasi-1D lattices with the above coupling multiple resonators as their unit cell. The unit cells are coupled to their nearest neighbor through [n/2] independent coupling channels with the same hopping amplitudes κ. Considering the zero effective intracell hopping, we can expect that our chains are topologically nontrivial with complete multimerization. In bulk momentum space, the Bloch Hamiltonian of the chain can be written as Hn(k)=(0κ⋯κ+κe−ikaκ0⋯κ⋮⋮⋱⋮κ+κeikaκ⋯0)n×n,(2)where a is the lattice constant between the units and k is the Bloch wave number. The Bloch Hamiltonian shows that the Bloch term κe±ika only exists in all anti-diagonal elements, provided that the diagonal term remains zero. The multichannel inter-cell coupling term κ appears in all elements of the matrix except the diagonal entries. Obviously, the Hamiltonian displays inversion (I) symmetry, i.e., IHn(k)I−1=Hn(−k). For clarity, the system is classified into two patterns via n that is odd or even in the following analysis. We find the analytical solutions of its energy spectra expressed as ωn,o(k)=(−2⋮n/3−1−(A++i3A−)/2n/3−1−(A+−i3A−)/20⋮n/3−1+A+)κ,(3)when n is odd where A±=C/B±B, B=(D2−C3+D)1/3, C=(n−1)coska/3+(n2+3)/9, and D=(n2−n)coska/6+(n/3)3+(n−3)/6. Surprisingly, there are n−3 flat bands equally divided at ωn=0 and ωn=−2κ. And only two bandgaps with the width G1=(n−2)2+1−(n−1)2−1+1 and G2=(n−2)2+1+n−2 exist in the energy spectra. For the case where n is even, the eigenfrequency is given by ωn,e(k)=(−2⋮n/2−1−n2/4+1+ncoska0⋮n/2−1+(n2/4+1+ncoska)κ,(4)characterized by having n/2−1 flat bands at ωn=0 and ωn=−2κ, respectively. The (n/2)th band and the top (nth) band are symmetric with respect to (n/2−1)κ. It is noteworthy that as the parameter n varies, there consistently exists one bandgap with a width of G=n−2 due to the lower nonflat band precisely overlapping the bandgap of the flat bands. In both cases, the band structure of the bulk Hamiltonian is not symmetric around zero, indicating that our chain is chiral symmetry broken. In detail, except for the top band that exceeds zero, the other bands are always distributed between −2κ and zero.

Figures 2(a)–2(c) show the band structures in first Brillouin zone for different n. Here, in the presence of inversion symmetry, we introduce the Zak phase, defined as Zj=−i∫−π/aπ/a⟨ψk,j|∂k|ψk,j⟩dk, to characterize the topology of our 1D multimer system where j specifies the occupied band index with corresponding Bloch wavefunctions |ψk,j⟩ [54]. We can obtain nonzero quantized Zak phases of bands for various n, indicating the topological nontriviality of our chains. Particularly, the top band possesses a Zak phase of zero while the flat bands always have Zak phases of −π. We can block-diagonalize Hn(k) by unitary transformation Un−1Hn(k)Un=HnBD(k) to show the separation between flat and nonflat bands clearly. For the simplest case with n=4, the unitary matrix and the block-diagonal Hamiltonian are U4=12(101010−10010−10101),(5)H4BD(k)=(12+e−ika002+eika10000−1e−ika00eika−1)κ,(6)respectively. As expected, the 2×2 blocks have the same form as the SSH Hamiltonian where the upper one is topologically trivial corresponding to the blue bands and the lower block is topologically nontrivial with complete dimerization corresponding to the flat bands. More generally, the bulk Hamiltonians for larger n can be divided into a topological trivial dimerized subspace and n/2−1 nontrivial fully dimerized subspaces by the unitary transformation when n is even. Moreover, the lower trivial band for the even chain always spans the bandgap between the flat bands by meeting the upper and lower flat bands at the boundary and center of the first Brillouin zone, respectively. Similarly, for odd n, we can get the block-diagonal Hamiltonian composed of a 3×3 block and (n−3)/2 same nontrivial 2×2 blocks (see Appendix C for details) where the 3×3 subspace owns two nontrivial lower bands in contact with the upper and lower flat bands independently.

Considering the BBC, under the open boundary condition, we show the normalized eigenvalue spectra ωi/κ of finite multimer chains with 60 resonators in Figs. 2(d)–2(f) and the corresponding wavefunctions |ψi⟩ in Figs. 2(g)–2(i) for n=3, 4, and 5. Mathematically, the wavefunction solution is not unique for the finite-size chains with n>3, owing to the fact that the rank of the Hamiltonian matrix is smaller than the matrix dimension. The wavefunction distributions depicted in Figs. 2(h) and 2(i) exemplify potential numerical solutions for n=4 and n=5, respectively. Correspondingly, there are pairs of degenerate edge states marked by the colored dots at the exact detuning −κ with the same number as the nontrivial bands. The topological edge modes of the odd multimer chain sit in the lower bandgap G1. Remarkably, the topological edge states are clearly embedded in the continuous spectrum of the lower nontrival band and are the so-called topological BICs [53] when n is even. Because of the complete multimerization, the wavefunctions of exact −1 energy edge modes are absolutely localized at the two boundary cells without any distribution in the bulk, while the bulk wavefunctions are diffused throughout the whole chains. Therefore, the edge states are also completely robust to the local perturbations of frequency and coupling strength of bulk resonators.

We employ periodic radio-frequency inductor-capacitor (LC) circuits featuring flexible hopping channels to experimentally observe the tight-binding modes. Here, the lattice nodes are capacitively coupled to ground and inductively coupled to each other. The multimer chains can be represented by the admittance matrix Jω (also termed circuit Laplacian) [17,47,55]. The voltage response V(ω) of the nodes to an input current I(ω) at frequency ω follows Kirchhoff’s law: I(ω)=J(ω)V(ω), where the vectors I(ω)=[I1,I2,…,Is]′ and V(ω)=[V1,V2,…,Vs]′ for s nodes circuit. For our uniform hopping chains, using the same size capacitors C and inductors L, we have the circuit Laplacian J(ω)=1iω[(nL−ω2C)I+H](7)with H=(0−1/L−1/L⋯−1/L0−1/L⋯−1/L−1/L0⋯⋮⋮⋮⋱)s×s,(8)where I is the s×s unit matrix and H can represent our theoretical model accurately with the hopping amplitude −1/L. We construct periodic LC circuits with 24 nodes for trimer and tetramer configurations as shown in Figs. 3(a) and 4(a), respectively. By solving the eigenvalues Ei of H numerically, we can obtain the general admittance eigenspectral dispersion as fi=(n/L+Ei)/C/(2π) with degenerate edge states labeled by red shown in Figs. 3(b) and 4(b). It is obvious that the edge states shown in Fig. 4(b) are located in the middle of the continuous energy band for the tetrameric circuit. Note that the inverted nonlinear spectra are due to the negative frequency-dependent hopping amplitude of inductive coupling.

In the experiment implementation, we choose the circuit components: C=1nF with ±1% tolerance and L=1.1μH with ±5% deviation. Details of the sample fabrication and impedance measurements are provided in the Appendix A. Measured impedances of nodes 1, 13, and 17 to ground (|Z1|, |Z13|, and |Z17|) versus the frequency of input circuit are shown in Figs. 3(c) and 4(c) for trimer and tetramer chain, respectively. The peak frequencies of the impedances are in good agreement with the calculated eigenvalues, despite some slight frequency shift of the measured impedance peaks due to component tolerances. In Fig. 3(c), the highest impedance peak near 9.57 MHz of the edge node inside the band gap (about 9.25–10.1 MHz) with impedance valleys for bulk nodes denotes the topological modes unambiguously. More intuitively, we measure the impedance distribution of the degenerate topological edge modes with strong locality at both ends at 9.57 MHz shown in Fig. 3(d). Differently, corresponding to the calculated eigenspectrum, the impedance peak of the edge node is accompanied by the impedance peaks of the bulk nodes near the frequency of edge state 10.52 MHz in Fig. 4(c), representing the existence of a topological bound state in a nontopological continuum where the bound edge states are shown in Fig. 4(d).

In summary, we theoretically and experimentally demonstrated degenerate topological edge states in a class of topological multimer chains consisting of identical resonators with uniform hopping. By designing deliberate coupling paths, the systems exhibit full multimerization with fully dimerized SSH subspaces corresponding to flat bands that can be separated from their Hilbert spaces. We also show topological BICs by embedding the degenerate topological bound states into a continuous band in even chains naturally. These phenomena indicate some potential applications in various fields. Our scheme is experimentally accessible and can also be implemented in coupled waveguide arrays [56], optical and acoustic coupled cavity arrays [53,57], cold atom lattices [58], and three-dimensional circuit quantum electrodynamics [59]. Our work sheds new light on the construction of topological phases.

Acknowledgment. C.-M. Hu acknowledgments support from NSERC Discovery Grants and NSERC Discovery Accelerator Supplements.