Introduction

Non-Hermitian physics has emerged as a groundbreaking paradigm for manipulating phases of matter, fundamentally transforming our understanding of both quantum and classical systems. This revolution has been marked by the discovery of various exotic phenomena, including exceptional points, where eigenvalues and eigenvectors coalesce1,2,3,4,5,6, the braiding topology of complex eigen-spectra7,8,9,10,11, and the non-Hermitian skin effect, characterized by the anomalous localization of eigenstates at system boundaries12,13,14,15,16. These phenomena, rooted in the intrinsic non-Hermiticity of systems, have unlocked new dimensions of topological physics. In recent years, the synergy between non-Hermitian effects and topological physics has further broadened this field, leading to the discovery of unique non-Hermitian topological phenomena. These include the generalized bulk-edge correspondence in one dimension, which redefines the relationship between bulk and boundary states17,18,19, geometric-dependent non-Hermitian skin effects in high dimensions20,21,22,23,24, skin-topological modes25,26,27, the non-Hermitian morphing of midgap topological modes28, and the development of non-Hermitian topological sensors, which offer unprecedented sensitivity due to the amplification of boundary responses29,30. Inspired by these theoretical advances, experimental explorations of non-Hermitian topological physics have been undertaken across a diverse array of platforms31,32,33,34,35,36,37, heralding a new era of applications in science and technology.

Recently, the investigation of non-Hermitian topological physics has extended beyond linear systems into the nonlinear domain, unveiling a plethora of novel nonlinear non-Hermitian topological effects. These include the discovery of non-Hermitian topological phase transitions controlled by nonlinearity38, the observation of non-reciprocal topological solitons39, and nonlinear exceptional points40,41. These findings not only deepen our understanding of non-Hermitian physics but also open new avenues for exploring the rich interplay between topology, nonlinearity, and non-Hermiticity. Among the various nonlinear phenomena, nonlinear synchronization42 has been of particular interest. This phenomenon, which refers to the collective oscillatory behavior of interacting units that evolve synchronously, has been extensively studied since Huygens’ discovery of two coupled pendulum clocks exhibiting identical oscillation frequencies. Over the centuries, synchronization has been recognized as a fundamental process, playing crucial roles in diverse fields such as electrical engineering, radio technology, biology, and statistical mechanics43,44,45,46,47,48,49. In recent years, significant progress has been made in exploring the interaction between topological physics and nonlinear synchronization50,51,52,53,54,55,56. It has been demonstrated that topological boundary states in one-dimensional (1D) Su-Schrieffer-Heeger models and higher-order topological corner states in two-dimensional (2D) systems can effectively support the synchronized dynamics of nonlinear oscillators localized at boundaries and corners51. Moreover, the presence of topological lasing edge states in 2D lattices has been shown to induce synchronization among coupled nonlinear oscillators at boundaries52. These studies provide compelling evidence of the ability of topological states to control nonlinear synchronization, offering valuable insights into the construction of topologically protected synchronization mechanisms. Despite these theoretical achievements, the interaction between synchronization and the myriad of novel effects in non-Hermitian topological systems remains largely unexplored. Moreover, to the best of our knowledge, the experimental validation of topological synchronization still remains elusive. Therefore, further investigation into the interplay between non-Hermitian topology and nonlinear synchronization, alongside the development of new experimental platforms to realize these novel effects, is of critical importance.

In this work, we theoretically investigate and experimentally demonstrate the realization of non-Hermitian topological synchronization. Our study reveals that as the lattice size increases, a 2D non-Hermitian nonlinear model undergoes three distinct types of topological synchronization transitions. The critical lattice sizes at which these synchronization transitions occur can be precisely controlled by tuning the topological bandgap. Moreover, we uncover that the boundary site geometry serves as an additional degree of freedom, enabling the manipulation of topological boundary synchronization configurations. Experimentally, we leverage the correspondence between electric circuit networks and tight-binding lattice models to design and fabricate nonlinear topolectrical circuits, thereby achieving the observation of non-Hermitian topological synchronization. Our results not only deepen the understanding of the connection between non-Hermitian topology and nonlinear synchronization but also hold potential for future applications in synchronization-based technologies.

Results

Non-Hermitian topological synchronization transitions with varied lattice sizes

We consider a 2D non-Hermitian nonlinear lattice model composed of two parts, including the linear coupling part with four sublattices in a unit cell, and the nonlinear part of onsite Stuart-Landau oscillators. The intra-cell couplings along x- and y-axis are denoted by \(\pm iu\) and \(ib\) (or \(-ib\)), and the corresponding inter-cell couplings are represented by \(\pm i{t}_{1}\) and \(\pm i{t}_{2}\), as shown in the top chart of Fig. 1a. In this case, the linear part of the Hamiltonian in momentum space (kx,ky) is written as

$$H({k}_{x},{k}_{y})= [u+cos({k}_{x})+\,\cos ({k}_{y})]{I}_{2}\otimes {\sigma }_{y}+sin({k}_{y}){I}_{2}\otimes {\sigma }_{x}+\,\sin ({k}_{x}){\sigma }_{z}\otimes {\sigma }_{z}\\ +ib{\sigma }_{x}\otimes {\sigma }_{z},$$
(1)

which can be unitarily transformed into a non-Hermitian topological model, representing the coupling between Qi-Wu-Zhang model and its time-reversal counterpart (see Supplementary Note 1). Hence, the linear part of our Hamiltonian possesses non-Hermitian band topology. By tunning system parameters, it exhibits midgap topological edge modes at zero energy, where the corresponding imaginary part of the eigenenergy is positive, leading to a significant gain effect known as topological lasing modes (see Supplementary Fig. 1)57. As for the nonlinear part, each lattice site possesses a Stuart-Landau oscillator, whose dynamical equation is described by \(\dot{Z}=(i{\omega }_{0}+\alpha -\beta |Z{|}^{2})Z\) with \(Z\), \(\alpha\) and \(\beta\) being the complex-valued amplitude, gain coefficient, and nonlinear coefficient, respectively. In this case, each isolated oscillator can exhibit the self-excited oscillation with the frequency and absolute value of the amplitude being \({\omega }_{0}\) and \(\sqrt{\alpha /\beta }\). Thus, the nonlinear dynamical equations at four sublattices can be written as

$$\dot{{Z}_{a}}(x,y)= \, (i{\omega }_{0}+\alpha -\beta |{Z}_{a}(x,y){|}^{2}){Z}_{a}(x,y)-i[ib{Z}_{c}(x,y)-iu{Z}_{b}(x,y)-i{t}_{1}{Z}_{a}(x+1,y)\\ -i{t}_{1}{Z}_{b}(x+1,y)+i{t}_{1}{Z}_{a}(x-1,y)-i{t}_{1}{Z}_{b}(x-1,y)-i{t}_{2}{Z}_{b}(x,y+1)],$$
$$\dot{{Z}_{b}}(x,y) = \, (i{\omega}_{0}+\alpha -\beta |{Z}_{b}(x,y){|}^{2}){Z}_{b}(x,y)-i[-ib{Z}_{d}(x,y)+iu{Z}_{a}(x,y) +i{t}_{1}{Z}_{a}(x+1,y) \,\\ +i{t}_{1}{Z}_{b}(x+1,y)+i{t}_{1}{Z}_{a}(x-1,y)-i{t}_{1}{Z}_{b}(x-1,y)+i{t}_{2}{Z}_{a}(x,y-1)],$$
$$\dot{{Z}_{c}}(x,y) = \, (i{\omega }_{0}+\alpha -\beta |{Z}_{c}(x,y){|}^{2}){Z}_{c}(x,y)-i[ib{Z}_{a}(x,y)-iu{Z}_{d}(x,y)+i{t}_{1}{Z}_{c}(x+1,y) \\ \,-i{t}_{1}{Z}_{d}(x+1,y)-i{t}_{1}{Z}_{c}(x-1,y)-i{t}_{1}{Z}_{d}(x-1,y)-i{t}_{2}{Z}_{d}(x,y+1)],$$
$$\dot{{Z}_{d}}(x,y) = \, (i{\omega }_{0}+\alpha -\beta |{Z}_{d}(x,y){|}^{2}){Z}_{d}(x,y)-i[-ib{Z}_{b}(x,y)+iu{Z}_{c}(x,y)+i{t}_{1}{Z}_{c}(x+1,y) \\ \,-i{t}_{1}{Z}_{d}(x+1,y)+i{t}_{1}{Z}_{c}(x-1,y)+i{t}_{1}{Z}_{d}(x-1,y)+i{t}_{2}{Z}_{c}(x,y-1)],$$
(2)

where \({Z}_{\gamma }(x,y)\) represents the complex amplitude of the γth sublattice (\(\gamma =a,b,c,d\)) in the unit cell located at (x, y) with \(x\in [1,{N}_{x}]\) and \(y\in [1,{N}_{y}]\). It is noted that our model integrates non-Hermitian couplings, nonlinearity, and topological states, as illustrated in the bottom chart of Fig. 1a, offering a multi-degree-of-freedom platform to manipulate non-Hermitian topological synchronization.

Fig. 1: Numerical results of lattice size-dependent non-Hermitian topological synchronization.
figure 1

a The top chart illustrates a schematic of a two-dimensional lattice model featuring onsite nonlinear oscillators. The intra-cell and inter-cell couplings correspond to the site couplings within the same and different units. The bottom chart illustrates the creation of non-Hermitian topological synchronization induced by the interplay among non-Hermitian physics, topological states, and nonlinearity. b1, b2 Numerical results of time-domain dynamics of \({S}_{e}(t)={{\mathrm{Re}}}\{{\sum }_{i\in {N}_{e}}{{\mathrm{Z}}}_{{\mathrm{i}}}\left({{\mathrm{t}}}\right)/{{\mathrm{N}}}_{{\mathrm{e}}}\}\) and \({S}_{b}(t)={{\mathrm{Re}}}\{{\sum }_{i\in {N}_{b}}{{\mathrm{Z}}}_{{\mathrm{i}}}\left({{\mathrm{t}}}\right)/{{\mathrm{N}}}_{{\mathrm{b}}}\}\) for the system with \({N}_{x}={N}_{y}=5\). b3 Numerical results of frequency spectra obtained from the discrete Fourier transform on \({S}_{e}(t)\) and \({S}_{b}(t)\) in b1, b2. Numerical results of \({S}_{e}\left(t\right),\,{S}_{b}(t)\) and the corresponding frequency spectra are shown in c1–c3 for \({N}_{x}={N}_{y}=8\) and (d1–d3) for \({N}_{x}={N}_{y}=21\).

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Then, we investigate the wave dynamics of the above 2D lattice model by sovling Eq. (1). Here, the random amplitudes of \({Z}_{i}(x,y)\in\)[−0.1, 0.1] are used as the initial condition, and other parameters are set as \({\omega }_{0}=0\), \(\alpha =0.02\), \(\beta =8\), \(u=-1\), \(b=0.5\), \({t}_{1}=0.5\) and \({t}_{2}=1\). Two quantities defined by \({S}_{e}(t)=Re\{{\sum}_{i\in {N}_{e}}\frac{{Z}_{i}(t)}{{N}_{e}}\}\) and \({S}_{b}(t)=Re\{{\sum}_{i\in {N}_{b}}\frac{{Z}_{i}(t)}{{N}_{b}}\}\) are used to manifest the collective behaviors of boundary and bulk oscillators, where \({N}_{e}\) and \({N}_{b}\) are the amounts of boundary and bulk units. We start to focus on the lattice structure with \({N}_{x}={N}_{y}=5\). Figure 1(b1, b2) displays numerical results of \({S}_{e}(t)\) and \({S}_{b}(t)\) in the asymptotic regime for t > 6300, that is sufficient for the system to reach a steady state. The associated frequency spectra obtained from the discrete Fourier transform (FT) of \({S}_{e}(t)\) and \({S}_{b}(t)\) are plotted in Fig. 1b3 with red and blue lines. It is evident that \({S}_{e}(t)\) and \({S}_{b}(t)\) exhibit two constant amplitudes in the steady state, being consistent with the conditions that all boundary and bulk oscillators exhibit zero-frequency oscillations. The appearance of zero-frequency global synchronization is verified by a thousand of random initial states. It is worth noting that, except for the synchronization displayed in Fig. 1b, there are other two synchronized states with different spatial profiles of \({Z}_{i}(x,y)\) (see Supplementary Note 2). The spatial distribution of these three synchronized states aligns with the eigen-profiles of the three nonlinear topological zero-energy modes. These eigen-profiles are obtained by solving the nonlinear eigen-equation using the Newton-gradient method, derived by substituting the harmonic wave solution \({Z}_{a,b,c,d}({{{\boldsymbol{x}}}},{{{\boldsymbol{y}}}},{{{\boldsymbol{t}}}})={Z}_{a,b,c,d}({{{\boldsymbol{x}}}},{{{\boldsymbol{y}}}}){{{{\boldsymbol{e}}}}}^{-{{{\boldsymbol{i}}}}{{{\boldsymbol{\varepsilon }}}}{{{\boldsymbol{t}}}}}\,\) into Eq. (2) (see Supplementary Note 3). This consistency confirms that the frequency synchronization is governed by the zero-energy nonlinear topological boundary states. To further emphasize the critical role of topology on the nonlinear synchronization, we analyze the wave dynamics of a trivial nonlinear square-shape lattice model without topological boundary states (see Fig. S4 in Supplementary Note 4). The results exhibit that frequency synchronization is absent, with the spatial profile at the long-time limit becoming randomly distributed rather than localized at the boundaries. Moreover, random multi-frequency oscillations emerge across both boundary and bulk sites. These results confirm that topology is essential for the emergence of size-dependent topological frequency synchronization. Furthermore, we also perform the linear stability analysis on three nonlinear topological zero-energy modes, and find that any perturbations around these three states are attenuated, showing that these three synchronized states correspond to stable fixed points of the nonlinear dynamics. We call these stable fixed points as non-Hermitian topological global frequency synchronization. Moreover, we clarify that throughout the analysis, we have set \({\omega }_{0}=0\). A nonzero \({\omega }_{0}\) would lead to a limit cycle instead of stable fixed points.

Then, we study the nonlinear dynamics of our system with \({N}_{x}={N}_{y}=8\). Figure 1(c1, c2) display the evolutions of \({S}_{e}(t)\) and \({S}_{b}(t)\). The corresponding FT frequency spectra are plotted in Fig. 1c3. Different from the above revealed stable fixed points, we find that the boundary and bulk oscillators show distinct dynamical behaviors. All boundary sites still exhibit the zero-frequency synchronization. While, the bulk sites exhibit the double-frequency oscillation with \(\omega =0\) and \(0.8796\), as shown in the FT frequency spectra of bulk sites. By analyzing FT frequency spectra of all bulk oscillators, we find that bulk sites, which locate at relatively large distances from boundary units, exclusively exhibit single-frequency oscillation at \(\omega =0.8796\). Conversly, the remaining bulk sites in close to boundary units show double-frequency dynamics. These results show that our system can transform from fixed points to a new synchronized state, called the separated bulk/boundary non-Hermitian topological synchronization, with increased lattice size.

By further enlarging the lattice size to \({N}_{x}={N}_{y}=21\), the zero-frequency synchronization of boundary oscillators still exists, while bulk sites show multi-frequency dynamics with \(\omega\) in the range from 0.85 to 0.9, as shown in Fig. 1d1, d2. We call this state as the topological boundary synchronization, which still exists with \({N}_{x,y}\) being larger than 21. From the above results, we find that the configuration of nonlinear dynamics in our model is closely deepened on the system size, and exhibits two types of representative synchronization transitions from non-Hermitian topological global frequency synchronization to separated bulk-boundary non-Hermitian synchronization and from separated bulk/boundary non-Hermitian synchronization to non-Hermitian topological boundary synchronization. Furthermore, similar to topological boundary states, we note that topological frequency synchronization also exhibits robustness against disorder. However, there is a notable difference: the robustness of non-Hermitian topological frequency synchronization is inherently weaker than that of the topological boundary states. Specifically, when a certain level of frequency disorder is introduced, the existence of topological boundary states merely requires the bandgap to remain open. In contrast, achieving topological frequency synchronization not only necessitates the presence of topological boundary states but also requires sufficiently strong coupling between them to facilitate their collapse into a single topological boundary state (see Supplementary Note 5).

The origin of the appearance of size-dependent configurations of synchronization stems from the effective coupling between nonlinear topological boundary states and bulk states. When the system size is relatively small (\({N}_{x}={N}_{y}=5\)), the localization length of nonlinear topological boundary state is comparable to the lattice size. It is worth noting that linear topological boundary states exhibit a shorter localization length compared to their nonlinear counterparts. Furthermore, the wave amplitudes of nonlinear topological zero modes are significantly larger than those of other nonlinear bulk or boundary modes, causing them to dominate the nonlinear evolution. This phenomenon can be intuitively understood by considering the linear regime. Specifically, due to the largest imaginary eigenenergy exhibited by the linear topological lasing mode at zero energy, this mode experiences a significant gain effect, where \({{{\rm{Imag}}}}({\varepsilon }_{0})+\alpha \gg \beta |{Z}_{t=0}{|}^{2}\). In contrast, other bulk and edge linear modes exhibit either lossy effects or a much weaker gain effect compared to the topological lasing modes. Consequently, the topological lasing mode undergoes significantly amplified wave amplitudes compared to other modes and dominates the system’s dynamics. As the wave amplitude of the topological lasing mode increases, the nonlinear loss effect also intensifies. The nonlinear topological zero mode emerges when a balance is achieved between the nonlinear loss term \(-\beta |{Z}_{final}{|}^{2}\) and the linear gain effect. In this case, the nonlinear zero-energy topological mode can dominate the wave dynamics of all boundary and bulk oscillators. When the lattice size exceeds the localization length of nonlinear topological edge states, the central bulk sites are no longer influenced by the zero-energy nonlinear topological boundary state, but are solely governed by a nonlinear bulk eigenmode with the largest amplitude. In addition, bulk sites situated in the intermediate region between boundary and central bulk units can exhibit dual-frequency dynamics governed by two modes. The steady-state distribution is consistent with the superposition of a nonlinear topological zero-energy state and a bulk state at \(\omega =0.8796\). Lastly, if the lattice length is further increased, a greater number of nonlinear bulk modes can significantly influence the wave dynamics of bulk sites, resulting in the emergence of a multi-frequency bulk oscillation. In this scenario, the steady-state profile corresponds to a superposition of the nonlinear topological zero-energy state and multiple nonlinear bulk states (see Supplementary Note 3).

To further find transition points of the lattice length for different synchronized configurations, we introduce two order-parameters to distinguish these nonlinear dynamics. The first one is an extended version of Kuramoto order parameter, defined by \({O}_{1}={\{\max [R(t)]-\min [R(t)]\}}_{t > {t}_{0}}\) with \(R(t)=|{\sum}_{l=[1,4{N}_{x}{N}_{y}]}{Z}_{l}(t)|\) and \({t}_{0}\) being sufficiently large to reach a steady state. The order-parameter \({O}_{1}\) can distinguish the global synchronized state with \({O}_{1}=0\) from other dynamical states with \({O}_{1}\ne 0\). The second order-parameter, which is expressed as \({O}_{2}={\{\max [F(t)]-\min [F(t)]\}}_{t > {t}_{0}}\) with \(F(t)=|\{R(t)-\frac{\max [R(t)]+\,\min [R(t)]}{2}\}+i\{{\rm H}[R(t)]\}|\) and \({\rm H}[R(t)]\) being the Hilbert transformation on R(t), is used to distinguish the dual-frequency oscillation with multi-frequency oscillation (see Supplementary Note 6 for details). Figure 2(a1, a2) present numerical results on the variation of two order-parameters as a function of the lattice length with \({N}_{x}={N}_{y}=L\). It is clearly shown the topological stable point with \({O}_{1}={O}_{2}=0\) exists in the region from \(L=4\) to \(L=6\) (the blue domain). The separated bulk/boundary synchronization with \({O}_{1}\ne 0\), \({O}_{2}=0\) appears when the lattice length is within the range from \(L=7\) to \(L=9\) (the green domain). When the lattice length is larger than \(L=9\) (the orange domain), topological boundary synchronization appears accompanied with random bulk oscillations. It is noted that, by increasing the absolute value of \(u\) without closing the topological band gap, the localization length of the topological boundary states also increases. This results in the increase of synchronization transition points, without leading to the disappearance of the second phase. Figure 2(b1, b2, c1, c2) display the variations of two order parameters as a function of the lattice length with \(u=-1.2\) and \(u=-1.3\), respectively. It is clearly shown that two synchronization transition points of the lattice length are both increased with the absolute value of \(u\) being increased, which can reduce the topological bandgap of the linear Hamiltonian and enlarge the localization length of the nonlinear topological zero-energy modes (see Supplementary Note 7 for details). The enlarged localization length of the zero-energy topological boundary state makes topological frequency global synchronization appear in the model with a larger size.

Fig. 2: Numerical results of two order parameters for characterizing size-controlled non-Hermitian topological synchronization.
figure 2

a1, a2 The numerical result of the first order-parameter \({O}_{1}\) and the second order-parameter O2 as a function of the lattice length with \(u=-1\). The blue, green, and orange domains represent regions of non-Hermitian topological global frequency synchronization, separated boundary/bulk non-Hermitian topological synchronization, and non-Hermitian topological boundary synchronization. b1, b2, c1, c2. Numerical results of O1 and O2 as a function of the lattice length with u = −1.2 and u = −1.3.

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Boundary site geometry-dependent non-Hermitian topological synchronization

Except for the lattice size-dependent non-Hermitian topological synchronization, in the following, we investigate the influence of the boundary site geometry on non-Hermitian topological synchronization. Here, we consider a lattice model with the contour profile in the form of a triangle, as shown in Fig. 3a. The lattice length along x- and y-axis are \({N}_{x}=40\) and \({N}_{y}=20\), respectively. In Fig. 3(b1, b5), we calculate the eigenprofiles of five represented nonlinear topological boundary states using Newton-gradient method. Different from nonlinear topological boundary states in the square-shaped lattice model, which show the extended feature along boundaries, nonlinear topological boundary states of the triangle-shaped model exhibit different spatial profiles at three boundaries. Specifically, the nonlinear topological boundary states with eigenenergies 0, 0.5917, and 0.2618 are extendedly distributed on bottom, left, and right boundaries, respectively. Additionally, the right boundary exhibits two nonlinear topological boundary states at 0.5236 and 0.7854, showing localized features at opposite ends along this boundary. It is noted that these corner-localized nonlinear topological boundary states are absent in the square lattice model, which results from the boundary site geometry-dependent non-Hermitian skin effects of nonlinear topological boundary states (see Supplementary Note 8 for the detailed explanation). Moreover, it is worth noting that the triangular lattice model, without removing the lattice sites on the left and right boundaries, exhibits the eigen-spectrum as that of the square lattice, where the topological boundary states are uniformly extended along all boundaries (see Supplementary Note 8). These findings confirm that our triangular lattice model exhibits eigen-spectra and spatial profiles for topological boundary states that are dependent on the boundary site geometry. Furthermore, it is worth noting that the boundary site geometry-dependent non-Hermitian skin effect of topological boundary modes is fundamentally distinct from the geometry-dependent non-Hermitian skin effect of bulk modes20,21,22,23. Specifically, changes in the site geometry at the boundary modify the eigen-spectrum of boundary states, resulting in a nonzero spectral area that signifies the presence of the non-Hermitian skin effect of topological boundary states25,26,27. Under fully open boundary conditions, this leads to the appearance of skin-topological modes at the ends of these boundaries, while the bulk modes remain extended.

Fig. 3: Numerical results of boundary site geometry-dependent non-Hermitian topological synchronization.
figure 3

a The lattice model has a triangular contour profile with lattice length along the x- and y-axes being \({N}_{x}=40\) and \({N}_{y}=20\). b1–b5. Numerical results of five nonlinear topological boundary eigenstates with different eigenvalues using the Newton-gradient method. c1–g1, c2–g2. The numerical results of time-domain dynamics Z(t) and the corresponding frequency spectra of several represented lattice sites. The time-domain dynamics, Z(t), are shown with distinct colors corresponding to the five stars with matched colors in a.

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Next, we show that the boundary site geometry-dependent spatial distribution of nonlinear topological boundary states can significantly affect non-Hermitian topological synchronization at boundaries along different directions. Figure 3(c1, c2) display numerical results of the wave dynamics and FT frequency spectrum of a bottom-boundary lattice site (marked by the purple pentagram in Fig. 3a). Here, all systematic parameters are the same as that used in Fig. 2, and the random initial condition is applied. It is shown that the zero-energy topological boundary state governs the evolution of lattice sites at the bottom boundary. The little fluctuations result from the weak excitation of bulk states due to the finite size effect, which can be eliminated by increasing the size of the system. Figure 3d presents the calculated waveform of a left-boundary lattice site (the green pentagram in Fig. 3a) and its corresponding FT frequency spectrum. We find that the single-frequency oscillation can also occur at left-boundary lattice sites. But, it is distinguished from the topological zero-energy synchronization observed at bottom-boundary sites and instead is governed by another nonlinear topological boundary state at \({{{\rm{\varepsilon }}}}=\)0.5917. Figure 3e–g show waveforms and FT spectra of other three right-boundary lattice sites. It is found that the site marked by the blue (black) pentagram exhibits the double-frequency oscillation at \({{{\rm{\varepsilon }}}}=0.2618\) and \(0.5236\) (\(0.7854\)). The central site marked by the red pentagram shows the three-frequency oscillation with \({{{\rm{\varepsilon }}}}=0.2618\), \(0.5236\) and \(0.7854\). The completely different dynamical behaviors of three right-boundary lattice sites result from the coexistence of an extended topological state at \({{{\rm{\varepsilon }}}}=0.2618\) and two end-localized topological states at \({{{\rm{\varepsilon }}}}=0.5236\) and \(0.7854\) along the right boundary. In Supplementary Note 9, we also calculate the nonlinear dynamics of lattice sites at three corners of the triangle-shape lattice model. It is shown that these three corner sites manifest the superposition of nonlinear dynamics of two intersecting boundaries. From the above results, we can see that the boundary site geometry can be used as a new degree of freedom to manipulate non-Hermitian topological synchronization. This arises from the varying dispersions of topological boundary states across different boundary site geometries and the presence of non-Hermitian skin-topological modes. To further demonstrate the importance of topology in boundary site geometry-dependent nonlinear synchronization, we analyze the dynamical behavior of a trivial triangle-shaped lattice model. Similar to the behavior observed in the trivial square-shaped lattice model, nonlinear synchronization is absent, and instead, random multi-frequency oscillations occur on both boundary and bulk sites.

Experimental observation of non-Hermitian topological synchronization by nonlinear circuits

Motivated by recent experimental breakthroughs in realizing various topological states using electric circuits58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76, in this part, we design non-Hermitian nonlinear topological circuits to observe the non-Hermitian topological synchronization. Figure 4a illustrates the scheme for implementing inter-cell and intra-cell couplings in the circuit. An effective unit cell contains four circuit nodes, where voltages at these four sub-nodes within the unit cell at (x, y) are denoted as \({V}_{x,y}^{\gamma }\) \((\gamma =a,b,c,d)\), representing the \({Z}_{\gamma }(x,y)\) in the lattice model. The positive resistance \({R}_{b}\) (\(ib\)), negative resistance \(-{R}_{b}\) (\(-ib\)), and two negative impedance converters (INIC) \(\pm {R}_{u}\) (\(\pm iu\)) are employed for intra-cell couplings. Circuit nodes within two adjacent units are connected through an INIC, representing non-reciprocal inter-cell coupling \(\pm i{t}_{1}\) (or \(\pm i{t}_{2}\)) along x-axis (y-axis). The internal structures of the negative resistance and INIC are depicted in two insets. In addition, each node is also grounded with a capacitor C, a Chua diode \({R}_{C}\) (see Supplementary Note 10) and a non-reciprocal resistance \(\pm {R}_{g}\) (or positive resistance \({R}_{g}\)). In this case, the voltage dynamical equation possesses the identical form as Eq. (2) (see Supplementary Note 11 for the detailed derivation), indicating that our circuit simulator can implement non-Hermitian topological synchronization. The photo of the fabricated circuit sample with \({N}_{x}={N}_{y}=5\) is shown in Fig. 4b, with the enlarged unit cell being plotted in the right inset. Here, circuit parameters are set as \({R}_{b}=200\varOmega\), \({R}_{u}=100\varOmega\), \({R}_{1}=200\varOmega\), \({R}_{2}=100\varOmega ,\) \(C=1000nF,\) \({R}_{3}=5k\varOmega\), \({R}_{4}=4k\varOmega\), and \({R}_{5}=796k\varOmega\).

Fig. 4: Experimental results of geometric-size-dependent non-Hermitian topological synchronization.
figure 4

a The schematic diagram of the designed electric circuit to simulate the geometric-size-dependent topological synchronization. Two subcharts present the structure of a negative resistance and the non-reciprocal coupling. b The photograph image of twenty-five circuit cells in the fabricated circuit sample for simulating geometric-size-dependent synchronization. The right chart plots the enlarged photo of a single circuit cell. c1–c4 Colored lines show the measured voltage signals from four circuit nodes, corresponding to the four dots with matched colors in b, with \({N}_{x}={N}_{y}=5\). d1, d2 Measured results of time-domain voltage signals \({V}_{e}(t)={\sum }_{i\in {N}_{e}}{V}_{i}(t)/{N}_{e}\) and \({V}_{b}(t)={\sum }_{i\in {N}_{b}}{V}_{i}(t)/{N}_{b}\) of the steady of the system with \({N}_{x}={N}_{y}=5\). e1, e2 Measured voltage signals of a boundary site 68 and a bulk site 77 with \({N}_{x}=8,\,{{N}}_{y}=5\). The purple and orange dots in the inset indicate the locations of the measured voltage signals in the circuit lattice model. f1, f2 Measured results of time-domain voltage signals \({S}_{e}(t)\) and \({S}_{b}(t)\) of the steady of the system with \({N}_{x}=8,{N}_{y}=5\). g1, g2 The corresponding frequency spectra of the voltage signals \({S}_{e}(t)\) and \({S}_{b}(t)\). Here, other circuit parameters are set as Rb = 200Ω, Ru = 100Ω, R1 = 200Ω, R2 = 100Ω, C = 1000 nF, R3 = 5kΩ, R4 = 4 kΩ, and R5 = 796 kΩ.

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To explore the lattice size-dependent non-Hermitian topological synchronization, we firstly measure voltage dynamics of the circuit sample with \({N}_{x}={N}_{y}=5\). Here, the initial voltage signal is set as \({V}_{1,1}^{a}=1V\), while other nodes possess zero-valued initial voltages. Figure 4(c1, c2) presents the measured voltage signals of \({V}_{1,1}^{a}(t)\) and \({V}_{1,3}^{b}(t)\) at two boundary nodes (green and orange dots in Fig. 4b). Voltage signals of two bulk nodes \({V}_{3,3}^{a}(t)\) and \({V}_{3,2}^{b}(t)\) (pink and blue dots in Fig. 4b) are shown in Fig. 4(c3, c4). We can see that the voltages of both boundary nodes and bulk nodes rapidly converge to a steady state without any oscillation. It is worth noting that the measured voltage signals exhibit a slight fluctuation compared to simulation results (see Supplementary Note 12), owing to the inherent instability of active circuit components. To further illustrate the collective behavior of boundary and bulk circuit nodes, the experimentally measured \({V}_{e}(t)={\sum}_{i\in {N}_{e}}\frac{{V}_{i}(t)}{{N}_{e}}\) and \({V}_{b}(t)={\sum}_{i\in {N}_{b}}\frac{{V}_{i}(t)}{{N}_{b}}\) in the long-time limit (t > 90 ms) are presented in Fig. 4(d1, d2). It is shown that both the summed boundary and bulk voltages exhibit zero-frequency dynamics, manifesting the observation of topological global frequency synchronization.

Then, we turn to a larger circuit sample with \({N}_{x}=8\) and \({N}_{y}=5.\) Figure 4(e1, e2) display the measured voltage signals of a boundary node and a bulk node depicted in the inset. Figure 4(f1, f2) present the summed voltages of boundary \({V}_{e}(t)\) and bulk \({V}_{b}(t)\) nodes, respectively, with their FT frequency spectra being illustrated in Fig. 4(g1, g2). Different from the topological global frequency synchronization, it is shown that boundary circuit nodes can still exhibit the synchronized oscillation at 0 kHz, while bulk circuit nodes show a double-frequency oscillation at 0 kHz and 1.428 kHz, which are matched to the corresponding eigenenergy’s of the lattice mode at \(\varepsilon =0\) and 0.897. These compelling evidences clearly show the realization of the separated bulk-boundary topological synchronization, being consistent with the theoretical prediction.

To further observe the boundary site geometry-dependent non-Hermition topological synchronization, we fabricate the circuit network with a triangular shape, as shown in Fig. 5a. Details on the left and right boundaries of the triangular circuit are presented in Supplementary Note 13. Figure 5(b1) depicts the measured voltage signal \({V}_{8,1}^{d}(t)\) of a bottom-boundary circuit node (the purple pentagram), with the corresponding FT frequency spectrum showing in Fig. 5(b2). It is clearly shown that the bottom-boundary circuit node can exhibit the zero-frequency oscillation, being consistent with the square-shape circuit. We further measure the voltage signal \({V}_{4,4}^{b}(t)\) of a circuit node at the left boundary (the green pentagram), as shown in Fig. 5(c1). The corresponding FT frequency spectrum is presented in Fig. 5(c2). We can see that the oscillation frequency of the left-boundary circuit node is equal to 1.052 kHz, matching the eigenenergy of a midgap nonlinear topological boundary state localized on the left boundary. Finally, we measure the voltage signals \({V}_{14,3}^{a}(t)\) and \({V}_{10,7}^{c}(t)\) of two circuit nodes on the right boundary (the blue and black pentagrams) of the triangular circuit, as illustrated in Fig. 5(e1, f1), and the FT frequency spectra are displayed in Fig. 5(e2, f2). It is shown that circuit nodes at the right boundary can exhibit multi-frequency oscillations with frequencies being 0.451 kHz, 0.902 kHz, and 1.353 kHz, corresponding to the simultaneous excitation of an extended topological boundary state and two corner-localized topological boundary states. This phenomenon is consistent with the theoretical prediction in Fig. 3e–g. Above experimental results clearly demonstrate the experimental observation of boundary site geometry-dependent non-Hermitian topological synchronization.

Fig. 5: Experimental results of the boundary site geometry-dependent non-Hermitian topological synchronization.
figure 5

a The schematic diagram for realizing a lattice model with the contour profile in the form of a triangle in the electric circuit to simulate boundary site geometry-dependent synchronization. The dashed red, blue, and black circles enclose the left, right, and bottom boundaries of the circuit lattice model, respectively. b1, b2. Measured voltage signal marked by the purple pentagram at the bottom boundary and the corresponding frequency spectrum. c1, c2. Measured voltage signal marked by the green pentagram at the left boundary and the corresponding frequency spectrum. d1–d2, e1–e2. Measured voltage signals and corresponding frequency spectra of circuit nodes marked by the blue and black pentagrams at the right boundary.

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Conclusions

In conclusion, we have theoretically proposed and experimentally demonstrated the realization of geometry-dependent non-Hermitian topological synchronization. Specifically, we have presented three types of nonlinear dynamical behaviors in our model with different geometric sizes, including topological global frequency synchronization, separated bulk-boundary topological frequency synchronization, and boundary topological synchronization accompanied by random oscillations in the bulk. In addition, the boundary site geometry controlled topological boundary synchronization has also been demonstrated. By leveraging the perfect correspondence between electric circuits and lattice models, we have designed and fabricated nonlinear top electrical circuits to experimentally observe geometry-dependent non-Hermitian topological synchronization. Our results deepen the understanding of the connection between non-Hermitian topology and nonlinear synchronization, offering potential values for future applications.

Methods

Sample fabrications and circuit measurements

We exploit electric circuits by using LCEDA program software, where the PCB composition, stack-up layout, internal layer, and grounding design are suitably engineered. The designed PCBs have six layers. Except for the top and bottom two layers, there are two power layers ( ± 2.5 V and ±20 V) and two grounding layers. Moreover, all the internal layers are connected through blind buried holes. In order to achieve the splicing function, male pins (female sockets) are positioned along both the upper boundary (lower boundary) and right boundary (left boundary) of each circuit board. For the realization of the INIC, we use the operational amplifier (OpAmp) of LT1363, whose performance is good enough in experiments. In addition, two surface-mounted resistors are used as the auxiliary resistors in the positive and negative feedback loops of the OpAmp. The OpAmps and multiplier (AD633JNZs) are supplied by external voltages of \(\pm 2.5\)V and ±20 V, respectively. All capacitors and resistors are packaged in the form of 0603. Besides, to ensure the tolerance of circuit elements, we use a WK6500B impedance analyzer to select circuit elements with a high accuracy (the disorder strength is only 1%). The values of all circuit elements are large enough to ignore the influence of effective resistances and parasitic capacitances in the circuit sample.

As for the time-domain voltage measurement, we will initially establish the system’s initial state by assigning an initial voltage to each lattice node. In other words, at the initial moment, each lattice node is connected to an externally excited DC voltage source (+\({V}_{0}\) or -\({V}_{0}\)), or grounded (0 V). In this case, we use the relay model G6K (Omron) to connect the circuit nodes and the external voltage source (or GND layer). The relays are controlled by a signal of 5 V through a mechanical switch. With this setting, the external signals can be removed at the same time. Thus, we connect the nodes to an oscilloscope by coaxial cables and measure the voltage signal after the switch is turned off. Additionally, a 4-channel oscilloscope DSO7104B (Agilent Technologies) is used in experiments to collect the time-domain voltage signals. For each circuit, we made three rounds of measurements at least 20 times per round to verify the reproducibility of the obtained results.