Active topolectrical circuits
Tejas Kotwal, Henrik Ronellenfitsch, Fischer Moseley, Alexander Stegmaier, Ronny Thomale, Jörn Dunkel
AActive topolectrical circuits
Tejas Kotwal,
1, 2
Henrik Ronellenfitsch, Fischer Moseley, Alexander Stegmaier, Ronny Thomale, and J¨orn Dunkel Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India Department of Mathematics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. Department of Physics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. Institut f¨ur Theoretische Physik und Astrophysik,Universit¨at W¨urzburg, D-97074 W¨urzburg, Germany (Dated: August 14, 2019)
The transfer of topological concepts from thequantum world to classical mechanical and elec-tronic systems has opened fundamentally new ap-proaches to protected information transmissionand wave guidance. A particularly promisingtechnology are recently discovered topolectricalcircuits that achieve robust electric signal trans-duction by mimicking edge currents in quantumHall systems. In parallel, modern active matterresearch has shown how autonomous units drivenby internal energy reservoirs can spontaneouslyself-organize into collective coherent dynamics.Here, we unify key ideas from these two previ-ously disparate fields to develop design principlesfor active topolectrical circuits (ATCs) that canself-excite topologically protected global signalpatterns. Building on a generic nonlinear oscilla-tor representation, we demonstrate both theoret-ically and experimentally the emergence of self-organized protected edge states in ATCs. Thegood agreement between theory, simulations andexperiment implies that ATCs can be realized inmany different ways. Beyond topological protec-tion, we also show how one can induce persis-tent localized bulk wave patterns by strategicallyplacing defects in 2D lattice ATCs. These resultslay the foundation for the practical implementa-tion of autonomous electrical circuits with robustfunctionality in arbitrarily high dimensions.
Information transfer and storage in natural and man-made active systems, from sensory organs [1–3] to theinternet, rely on the robust exchange of electrical sig-nals between a large number of autonomous units thatbalance local energy uptake and dissipation [4, 5]. Re-cent advances in the understanding of photonic [6–8],acoustic [9] and mechanical [10–13] metamaterials haveshown that topological protection [14–19] enables the sta-bilization and localization of signal propagation in pas-sive and active [20, 21] dynamical systems that violatetime-reversal and/or other symmetries. Important re-cent work has successfully applied these ideas to realizetopolectrical circuits [22] in the passive linear [23–28],passive nonlinear [29], and active linear regime [30]. De- spite substantial progress in the development of topolog-ical nonlinear optics [31–35] and transmission lines [36–39], however, the generalization to active [40, 41] non-linear circuits made from autonomously acting units stillposes an unsolved challenge.Exploiting a mathematical analogy with active Brow-nian particle systems [21], we develop and demonstrategeneral design principles for active topolectrical circuits(ATCs) that achieve self-organized, self-sustained, topo-logically protected current patterns. The main buildingblocks of ATCs are nonlinear dissipative elements thatexhibit an effectively negative resistance (NR) over a cer-tain voltage range. NRs can be realized using van der Pol(vdP) circuits [42], tunnel diodes, unijunction transis-tors, solid-state thyristors [43], or operational amplifiersset as negative impedance converters through current in-version [30], and the design principles described beloware applicable to all these systems. Indeed, we expectthem to apply to an even broader class of nonlinear sys-tems, as similar dynamics also describe electromagneticresonators with Kerr-type nonlinearities [34, 44, 45].Active electronic circuits with basic non-topological in-teractions have been studied previously as models of neu-ronal networks [42] and solitary signal transport [4, 5].To leading order, the NR elements in an active circuitcan be described [4, 5] by a Rayleigh-type resistance R ( V ) = − ( α − γV ), where α and γ are positive pa-rameters. A prototypical example is the vdP oscillatorcircuit with capacitance C and inductance L ii , as shownin Fig. 1(a). When expressed in terms of the rescaled di-mensionless voltage ˆ V i = x i = (cid:112) γ/α V i , the dynamicsof an isolated vdP-unit i is governed by [46]¨ x i − ε (cid:0) − x i (cid:1) ˙ x i + x i = 0 , (1)where ε = α (cid:112) L/C , t = √ LC ˆ t , and ˙ x i = dx i /d ˆ t . Equa-tion (1) is intimately related to that of a harmonicallytrapped active Brownian particle [47], with position co-ordinate y and velocity u = ˙ y described by the standardcubic friction model ˙ u = ε (1 − u / u − y . Upon takingthe time derivative and identifying u = ˙ y = x i , one re-covers Eq. (1). It was shown recently [21] that suitablycoupled mechanical chains of active Brownian particlescan autonomously actuate topological modes. This in- a r X i v : . [ c ond - m a t . o t h e r] A ug (a)(b) (c) FIG. 1. Schematic and dynamics of active topolectrical SSH circuits. (a) Circuit diagram of a basic vdP oscillator withcapacitance C , inductance L ii and nonlinear resistor R ( V ). (b,c) 1D and 2D SSH circuits where each node is a vdP oscillator.Thick lines indicate strong coupling s and thin lines weak coupling w (cid:28) s . (d) Oscillator dynamics in an undamped passive( ε = 0) 1D SSH circuit with 6 nodes. Applying a nonzero initial voltage at the first node, the oscillation remains exponentiallylocalized on that edge. (e–g) Dynamics of a 1D ATC ( ε >
0) for the same initial condition as in (d). The high-frequencytopological edge mode is activated first and induces slow synchronized bulk oscillations, which eventually actuate the secondtopological mode at the opposite end. Phase portraits of the boundary (f) and bulk (g) nodes show the limit cycles of the fastand slow oscillations. The edge oscillators show approximately circular limit cycles typical of weak nonlinearity while the bulkdynamics is of strong relaxation type despite ε being small. Simulation parameters in (d-e): g = 1, w = 0 . , s = g − w with ε = 0 in (d) and ε = 0 . x (0) = 2 and ˙ x (0) = x j (0) = ˙ x j (0) = 0 for j ≥ sight provides guidance for the design of ATCs.To design an ATC with desired topological properties,we generalize Eq. (1) by introducing suitably chosen cou-plings between vdP units i and j through inductances L ij ; see Fig. 1(b,c) for two examples. Assuming a lat-tice of vdP units and introducing the symmetric cou-pling matrix elements β ij = − L /L ij for i (cid:54) = j , and β ii = L (1 /L ii + (cid:80) k : k (cid:54) = i /L ik ), Eq. (1) generalizes to¨ x i − ε (cid:0) − x i (cid:1) ˙ x i + (cid:88) j β ij x j = 0 , (2)where L is the smallest inductance in the circuit (Sup-plementary Information).Interpreting each individual vdP unit as a node in thenetwork graph [Fig. 1(b,c)], the effective dimension d ofthe ATC can be tuned by increasing the number of cou-plings. In principle, arbitrary values d ≥ d -dimensional cubic lattices, rows of the couplingmatrix β = ( β ij ) corresponding to bulk nodes will have2 d off-diagonal entries. Below, we will focus on the cases d = 1 [Fig. 1(b)] and d = 2 [Fig. 1(c)] to demonstratethe implementation and key properties of ATCs. Gener-ally, through an appropriate choice of β , it is possible torealize a wide range of topological phases.As a first realization of an ATC, we consider the cou-pling matrix of a one-dimensional (1D) Su-Schrieffer-Heeger (SSH) model. Originally introduced to describepolyacetylene [48], the SSH model is known to supporttopologically protected boundary modes in a variety ofquantum [49] and classical [13, 21, 22] systems. The n × n SSH coupling matrix for the case of n = 6 unit vdPs reads β = g − w − w g − s − s g − w − w g − s
00 0 0 − s g − w − w g , (3)with the diagonal elements β ii = g > L ii are chosen such that g = L (1 /L ii + (cid:80) k : k (cid:54) = i /L ik ) forall i . Upon generalizing this coupling matrix by permit-ting non-uniform grounds g i , one can achieve additionalfrequency control (Supplementary Information). The off-diagonal entries w > s > w > ε = 0 and s (cid:29) w .Then, Eqs. (2) and (3) reduce to a linear dynamical sys-tem that exhibits topologically protected, exponentiallylocalized modes at both ends of the chain [49]. Theseedge modes have finite-frequency for g > g →
0. Their presence can be illustratedby considering an ideal passive ( ε = 0) circuit [22] in thealmost fully dimerized topological regime ( w (cid:28) s ) withall nodes initially at rest, x i ( t ) = ˙ x i ( t ) = 0 for t < x (0) >
0, the boundary node (a) (b)(c) (d) V i +15 V-15 V A D -15 VU1 R R A D A R A D A R U3 -15 VU2C L ii R R +15 V+15 V R FIG. 2. Experimental realization of a 1D active topolectric circuit with 4 nodes. (a) Implementation of a single activenode using a Chua diode circuit (Supplementary Information). (b) Four-node active topolectrical circuit using C = 42 nF, L = L = 10 . L = 525 µ H. To obtain the classical SSH coupling matrix Eq. (3), no inductors were used for thebulk nodes, such that L = L = 525 µ H. The corresponding dimensionless parameters are s = 1 . w = 0 . g = 1 . ε = 0 .
28. (c) Experimental setup on circuit board. (d) Experimental data (orange lines), simulations of the full circuitusing LTSpice (blue lines), and theoretical simulation using Eq. (2) match. Voltages V – V vary between ± will oscillate with a non-zero amplitude at frequency √ g ,while the amplitudes of the other nodes remain exponen-tially small [Fig. 1(d)]. In the next part, we will see thatATCs with ε > < ε (cid:28) √ g )significantly alters the dynamics of the edge and bulkoscillators [Fig. 1(e–g)]. In the topological regime, char-acterized by s (cid:29) w and g ≥ s + w , the boundary nodesare only weakly coupled to the bulk and behave similarlyto isolated vdP oscillators. The active local energy inputrenders the rest state ( x i = ˙ x i = 0) unstable, forcing theboundary nodes to approach a stable limit cycle corre-sponding to an oscillation at frequency √ g [Fig. 1(e,f)].By contrast, bulk nodes are strongly coupled to one ormore neighbors, resulting in a distinctly different bulkdynamics reminiscent of highly nonlinear relaxation os-cillations [Fig. 1(e,g)]. For fully decoupled pairs of vdPoscillators, one expects stable phase-locked solutions withrelative phase 0 or π [50, 51]. To obtain further analyticalinsight into the bulk dynamics, we linearize Eqs. (2) nearthe rest state. The resulting dynamical matrix has eigen-values λ k, ± = ( ε ± (cid:112) ε − µ k ) = ±√− µ k + O ( ε ), where µ k are the eigenvalues of the coupling matrix β (Sup-plementary Information). In the fully dimerized limit( w → µ k can be calculated explicitly and one findstwo degenerate eigenstates at µ k = g , corresponding tothe topologically protected edge modes, and n − µ k = g ± s (Supplementary In-formation). The bulk eigenvectors are pairwise localizedwith components (1 , ±
1) on one of the dimers, and zeroeverywhere else, corresponding to a low-energy in-phaseoscillation ( µ k = g − s ) and a high-energy anti-phase( µ k = g + s ) oscillation (Supplementary Information).This implies that the dimers’ dynamics near the reststate is approximately decoupled and governed by the eigenvalues √− g ∓ s . If g > s , all modes are oscillatory,while for g < s , the anti-phase mode becomes unstableand is not physically realizable. Armed with these an-alytical insights, we proceed to numerically characterizethe striking nonlinear effects that arise in ATCs. In thesimulations, we can fix g = 1 without loss of generality,which is equivalent to dividing Eq. (2) by g and rescal-ing t → t/ √ g , ε → ε/ √ g , s → s/g and w → w/g .In contrast to passive ( ε = 0) topolectrical circuits,which remain quiescent in the bulk when initiated atthe edge [Fig. 1(d)], ATCs with ε > t = 0, itrapidly settles into a limit-cycle oscillation [Fig. 1(e,f)],as predicted above. As it gets activated, the edgemode imparts a weak external forcing on the neighboringstrongly-coupled dimer. The combination of forcing andlocal energy input from the negative resistance drives thedimer into its low-energy state, characterized by a slownonlinear in-phase oscillation of the dimer nodes with pe-riod 2 π/ √ g − s [Fig. 1(e,g)]. The first bulk dimer then inturn activates the second dimer, and so on, resulting ina globally synchronized bulk state. The activation fronteventually reaches the last node, where it triggers thesecond topological edge mode [Fig. 1(f)]. The final stateof the chain is a robust attractor that is self-sustainedand could be used for active solitary signal transmission.As an experimental proof-of-concept demonstrationof the above general concepts, we built a 1D 4-nodeATC using a Chua diode circuit as the active ele-ment [Fig. 2(a,b)]. The measured time-series of the os-cillator voltages exhibit the theoretically predicted topo-logical edge modes and low-frequency bulk dynamics ++++ ++ + + + + p r o b a b ili t y (b) p r o b a b ili t y d e n s i t y g s gg + s (c) bulkboundary + ++ ++ + + + (a) + + FIG. 3. Attractor statistics of a 1D ATC. (a) An active SSHcircuit with n = 6 vdP nodes has four qualitatively distinctstable attractors. Circle size represents instantaneous abso-lute voltage | x i ( t ) | , symbol color the phase ϕ and backgroundcolor the time-averaged frequency ω = (cid:104) ˙ ϕ (cid:105) of the oscilla-tors. Two low-energy attractors exhibit in-phase dynamicsof the bulk dimer nodes, and two high-energy attractors showanti-phase dynamics on the bulk dimers. (b) Low-energy at-tractors are substantially more frequently realized when bulknodes are initialized with uniformly random x i (0) ∈ [ − , x i (0) ∈ [ − . , . ≈ .
5% of initial conditions. (c) Frequencies ofnonlinear bulk oscillations cluster around √ g ± s (solid/dash-dotted line), boundary oscillations around √ g (dashed line),as predicted by linearized theory. Simulation parameters: g = 1 , w = 0 . , s = g − w , and ε = 0 .
1. Histograms werecomputed from 50 ,
000 trajectories, integrated up to t = 1000. (Fig. 1). The quantitative agreement of the experimentswith theory as well as explicit circuit simulations (Sup-plementary Information) confirms that Eq. (2) providesa predictive framework for ATCs – and that it is straight-forward to realize topolectrical materials with off-the-shelf components. Indeed, the example in Fig. 2 is onlyone of many possible ATC implementations (Supplemen-tary Information).In practice, ATCs can be actuated with a wide range ofinitial conditions that can lead to different types of stableasymptotic behaviors (Fig. 3). To investigate the likeli-hood and characteristics of possible attractors, we usedEq. (2) to simulate an active SSH circuit with n = 6nodes with non-zero initial conditions on the bulk. Sincethe edge node initial conditions become negligible in thiscase due to the weak edge-bulk coupling, we fixed zero-initial conditions at the edges. Thus, the topological edgemodes are actuated by the bulk dynamics in these simu-lations. Examples of low-energy and high-energy attrac-tors with slow and fast bulk dimer oscillations, obtainedwith binary initial conditions in the bulk, x i (0) = ± . x i (0) = 0, are shown [Fig. 3(a)]. These attractors can be classified in terms of the relative signs of the n b = n − x i at a fixed time. Normaliz-ing by the sign of the first bulk oscillator, we numericallyfind that there exist four different stable attractors cor-responding to different possible combinations of in-phaseand anti-phase dimer oscillations [Fig. 3(a)].To estimate the likelihood of observing a specificattractor in experiments, we performed 50,000 simu-lations with randomly chosen bulk initial data. Toexplore a large neighborhood of the bulk limit cy-cle [Fig. 1(g)], initial conditions were sampled uni-formly from the 8-dimensional domain x i (0) ∈ [ − , x i (0) ∈ [ − . , . (cid:104) ˙ ϕ (cid:105) , where ϕ ( t ) =arctan( ˙ x ( t ) /x ( t )), agree remarkably well with the bulkoscillation frequencies √ g ± s and boundary oscilla-tion frequency √ g predicted by the linearized theoryabove [Fig. 3(c)]. Based on these observations, one ex-pects that low-energy attractors with in-phase bulk dimerdynamics will also be dominant in more complex ATCs,and that their bulk and edge frequencies can be esti-mated from spectra of the coupling matrix β in the weakcoupling limit.ATC implementations become particularly powerful in d ≥ w →
0. Then, the lowest-frequency mode is an in-phase state (1 ,
1; 1 ,
1) with pe-riod 2 π/ √ g − s (Supplementary Information). Similar (a) (b) (c) (d) (e) (f) / FIG. 4. Self-organized self-sustained nonlinear oscillations in 2D ATCs recapitulate topological boundary phenomena. (a) Con-nectivity of the 2D SSH grid in the topological T1-edge mode regime. Thick and thin lines indicate strong and weak couplings,respectively. (b) Snapshot of the vdP network dynamics on the grid (a), with nodes colored by the phase angle ϕ in the ( x i , ˙ x i )plane; background color shows time averaged node frequency ω = (cid:104) ˙ ϕ (cid:105) normalized by the maximum ˆ ω over all oscillators. Eventhough the initial voltage was non-zero only on the bottom left corner node, the circuit autonomously actuates two topologicaledge oscillations with mean frequency ˆ ω , while the bulk synchronizes in-phase (SI Movie 1). (c) Connectivity of the 2D SSHgrid in the topological T2-corner mode regime. (d) Snapshot of the vdP network dynamics on the grid (c), for the samecorner-localized initial conditions as in (b). This circuit self-organizes into a stable active state in which edges and cornersoscillate while the bulk synchronizes in-phase (SI Movie 2). (e,f) Snapshot of the vdP network dynamics on the grid (c) withedge (e) and bulk (f) defects. The vdP oscillator network remains polarized in the bulk, while active topological modes wraparound the defect (SI Movies 3,4). Simulation parameters: ε = 0 . , g = 1 , w = 5 × − , s = g/ − w . Initial conditions: x (0) = 2 and ˙ x (0) = x j (0) = ˙ x j (0) = 0 for j ≥ to the 1D case, in the limit ( g − s ) → + , the bulkquartets collectively synchronize to the low-frequency in-phase state. However, the dimers on the boundary nowoscillate at a frequency lower than that of the cornernodes, because ( g − s ) → s as ( g − s ) → + . Thisopens the intriguing possibility of using topological pro-tected modes to precisely control active oscillations in2D ATCs. In particular, by varying the ground induc-tances of each of the nodes one can control frequencies ofeach of the corner nodes, edge dimers, and bulk quartets(Supplementary Information).Initializing the 2D active SSH circuits with a nonzerovoltage at one of the corner nodes, one finds that essen-tial qualitative features of the dynamics seen in 1D ATCscarry over to the 2D case. In particular, the boundarynodes belonging to topologically protected corner modesand edge modes become activated one after the other andsettle down in their respective vdP limit cycles. Similarly,in the bulk, quartets of strongly coupled oscillators syn-chronize. Because the in-phase state is the lowest-energyattractor above the quiescent state, the bulk synchro-nizes in a global in-phase pattern [Fig. 4(b,d)]. Thus,in both 1D and 2D ATCs, topologically protected edgemodes become activated via self-sustained oscillations,while the bulk dynamics is almost decoupled, leading tosynchronization.Crucially, this ATC self-organization principle remainsvalid in the presence of lattice defects, demonstratingthat topological protection phenomena can survive inthe nonlinear regime ε > ε = 0) 2D SSH gridby removing a few unit cells does not affect the local-ized nature of the edge state, which now wraps around the defect due to topological protection from the linearcoupling. Self-sustained oscillations in nonlinear active( ε >
0) circuits inherit this topological protection glob-ally: in the presence of edge defects, all boundary nodescontinue to oscillate at a high non-zero frequency whilebulk quartets synchronize at low frequency [Fig. 4(e), SIMovie 3]. Similarly, bulk defects also lead to localizednonlinear edge oscillations [Fig. 4(f), SI Movie 4]. Theseresults show that the SSH network topology can be usedto precisely control the individual and collective behaviorof coupled nonlinear oscillators. Furthermore, the aboveideas can be extended to achieve control of active trav-eling wave patterns by means of non-topological defects.By strategically placing bulk defects, one can guide theself-organization of active wave patterns that can be ini- / FIG. 5. Control of active traveling wave patterns throughbulk defects. Activated in a single corner, the circuit self-organizes into a pre-programmed wave pattern (SI Movie 5).Simulation parameters: ε = 0 . , g = 1 , w = 5 × − , s = g/ − w . Initial conditions: x (0) = 2 and ˙ x (0) = x j (0) =˙ x j (0) = 0 for j ≥ tiated from a single corner node [Fig. 5, SI Movie 5].In conclusion, active topolectrical circuits promise awide range of applications, from active wave guides toautonomous electronic circuits with topologically pro-tected functionalities. The framework developed herecan be integrated with recently developed methods forthe inverse design of network-based metamaterial struc-tures [52, 53], to optimize and tailor the node couplingsand transmission properties. By designing the couplingmatrix β such that the associated dynamical matrix pos-sesses chiral [30] or other symmetries, it is possible torealize different topological phases by utilizing genericvdP-type nonlinearities. Another intriguing prospect isthe possibility of creating and studying electronic meta-materials with effective dimensions d > Acknowledgements.
This work was supported bya James S. McDonnell Foundation Complex SystemsScholar Award (J.D.) and the MIT Solomon BuchsbaumResearch Fund (J.D.). J.D. would like to thank the IsaacNewton Institute for Mathematical Sciences for supportand hospitality during the program ‘The MathematicalDesign of New Materials’ (supported by EPSRC grantEP/R014604/1) when work on this paper was under-taken. The work in W¨urzburg is funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) through Project-ID 258499086 - SFB 1170 andthrough the W¨urzburg-Dresden Cluster of Excellence onComplexity and Topology in Quantum Matter – ct.qmat
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NONDIMENSIONALIZATION OF VARIABLES AND PARAMETERS
Given a lattice circuit, we state the equations of its voltage dynamics using Kirchoff’s laws as follows: C ¨ V i − ( α − γV i ) ˙ V i + (cid:18) L ii + (cid:88) kk (cid:54) = i L ik (cid:19) V i − (cid:88) jj (cid:54) = i V j L ij = 0 , (S1)where V i is the voltage, C is the capacitance, L ii is the ground inductance, and α and γ are the vdP parameters ofthe oscillator at node i , while L ij are the coupling inductances between nodes i and j . The coupling inductances canbe either of L w or L s where the subscripts w and s stand for weak and strong coupling respectively (1 /L w < /L s ).Note that for nodes i and k that are not connected, we have 1 /L ik = 0.We introduce a voltage scale V i = V ˆ V i , a time scale t = τ ˆ t , and we scale all the inductances by the smallestinductance, say L , to get L ij = L ˆ L ij for all i and j . For τ , we use the natural time scale of an oscillator such that τ = √ L C , and the voltage scale is given by V = (cid:112) α/ (3 γ ). The dimensionless dynamics is then given by Eq. (2)where ˆ V i = x i , ε = α (cid:112) L /C , β ij = − L /L ij for i (cid:54) = j , and β ii = L (1 /L ii + (cid:80) kk (cid:54) = i /L ik ).In Eq. (3), the weak and strong couplings are given by w = L /L w and s = L /L s respectively, while L ii are chosensuch that g = L (1 /L ii + (cid:80) kk (cid:54) = i /L ik ) for each i . To generalize the coupling matrix, we replace g by g i to denote thateach of them need not be equal to the other as we shall see in a later section (Frequency control). EIGENVALUES OF THE COUPLING MATRIX
We now analyze the linear stability of the quiescent state ( x i = 0) of Eq. (2). We find for small deviations, bysetting ˙ x = y (cid:18) ˙ x ˙ y (cid:19) = (cid:18) − β ε (cid:19) (cid:18) xy (cid:19) . (S2)The eigenvalue equation immediately gives us λ − ελ = − µ, where µ is an eigenvalue of β and λ is an eigenvalue of the Jacobian. Thus, we obtain λ = 12 (cid:18) ε ± (cid:112) ε − µ (cid:19) , to conclude that the most unstable mode occurs for µ minimal.First, we compute the eigenvalues of the coupling matrix. We define B n = g − w . . . − w g − s ... − s . . . − w g − s − s g − w . . . − w g , (S3) C n − = g − s . . . − s . . .... − w g − s − s g − w . . . − w g , (S4)where B n is an n × n matrix and C n − is an ( n − × ( n −
1) matrix. Then the characteristic polynomial satisfiesthe recursion, χ ( B n , µ ) = ( g − µ ) χ ( C n − , µ ) − w χ ( B n − , µ ) , (S5a) χ ( C n − , µ ) = ( g − µ ) χ ( B n − , µ ) − s χ ( C n − , µ ) , (S5b)with initial conditions χ ( B , µ ) = ( g − µ ) − w and χ ( C , µ ) = g − µ . For simplicity we solve them in the fullydimerized limit, i.e., in the limiting case, w = 0, we find that χ ( B n , µ ) = ( g − µ ) (( g − µ ) − s ) n/ − , (S6)with two modes corresponding to eigenvalue g (the topological edge modes), and rest of the modes corresponding tothe symmetric eigenvalues g ± s .The eigenvectors x i belonging to the eigenvalue g are (cid:0) . . . (cid:1) (cid:124) and (cid:0) . . . (cid:1) (cid:124) which are precisely thetopological edge modes. Further, the eigenvectors corresponding to g − s are the modes (cid:0) . . . (cid:1) (cid:124) , (cid:0) . . . (cid:1) (cid:124) , ... (cid:0) . . . (cid:1) (cid:124) , and those belonging to the eigenvalue g + s are the modes (cid:0) − . . . (cid:1) (cid:124) , (cid:0) − . . . (cid:1) (cid:124) , ... (cid:0) . . . − (cid:1) (cid:124) . The most unstable modes are those with the minimum eigenvalue, which are the degenerate ones belonging to g − s .Thus, this explains why the in-phase state is the probabilistically more preferred state. This concludes the analysisnear the quiescent state.We have shown that the quiescent state is unstable by finding the ‘most preferred’ eigenvectors if perturbed. Wenow analyze the behavior of the system away from the quiescent state. To simplify analysis, it is often useful toconsider the system in the fully dimerized limit ( w = 0). This amounts to analyzing sub-circuits made up of nodesthat are only strongly coupled to each other. In essence, we are treating the ‘strong’ sub-circuit as independent to therest of the network since it is connected to the rest of the network only through weak bonds (in other words, assumethe weak bonds have zero strength). FULLY DIMERIZED LIMIT
For the 1D SSH chain, in the fully dimerized limit, the system decouples into two isolated oscillators from theboundary nodes, and the rest–oscillator pairs from the bulk, called dimers. The dynamics of an isolated oscillator isgiven by ¨ x − ε (1 − x ) ˙ x + gx = 0 , (S7)whose natural frequency is √ g/ (2 π ). In the topological state, the edge nodes of the SSH chain will oscillate with thisfrequency. The internal dynamics of a dimer however are given by the equations¨ x − ε (1 − x ) ˙ x + gx = sx , (S8a)¨ x − ε (1 − x ) ˙ x + gx = sx , (S8b)where x and x are the voltages of two adjacent nodes that are connected through a strong bond. It can beshown using eigenmode analysis, that the eigen-directions of the coupling matrix are (cid:0) (cid:1) (cid:124) (in-phase) and (cid:0) − (cid:1) (cid:124) (anti-phase) with frequencies √ g − s/ (2 π ) and √ g + s/ (2 π ) respectively. Strong bond strength s T i m e p e r i o d T Toward s=g = 1 T = : p g + s T = : p g ! s FIG. S1. Comparison of time periods of fully dimerized sub-circuits and bulk nodes in the SSH chain. Solid curves denotetheoretical time period for in-phase and anti-phase modes of the dimer. Diamonds and crosses denote time periods of bulknodes in the SSH chain at different values of s . t (ms) V V V V V V V V V V (a) t (ms) V V V V (b) / FIG. S2. (a) Theoretical time series of 10-node 1D SSH chain with localized frequency pattern. Darkening of backgroundcolor denotes frequency increase along length of the chain. (b) Experimental and theoretical time series comparison for thelow-energy in-phase attractor of 2D SSH sample in T2-corner mode regime. Since all ground inductances are equal, naturalfrequencies of all nodes are equal resulting in synchrony.
We see that the eigenmodes of the decoupled dimers and boundary nodes are consistent with the eigenmodes ofthe whole chain. In Fig. S1, there is a strong agreement between the time period of fully dimerized sub-circuits andthat of the bulk nodes in the SSH chain. Thus, we have validation that taking the fully dimerized limit is in factaccurate and hence, can prove to be advantageous as shown. It also explains why in the topological regime, we havecharacteristically different behavior for the boundary (isolated nodes) and the bulk (dimers).In the same way, we extend this theory for the 2D stacked SSH sample in which we analyze bulk quartets. Theinternal dynamics is given by a ring of van der Pol oscillators¨ x − ε (1 − x ) ˙ x + gx = sx + sx , (S9a)¨ x − ε (1 − x ) ˙ x + gx = sx + sx , (S9b)¨ x − ε (1 − x ) ˙ x + gx = sx + sx , (S9c)¨ x − ε (1 − x ) ˙ x + gx = sx + sx . (S9d)Once again, it can be shown that the eigenmodes of the coupling matrix include the in-phase state (denoted by (cid:0) (cid:1) ) and the anti-phase states (either (cid:0) − − (cid:1) or (cid:0) − − (cid:1) depending on which topologicalregime the network is in). It can be shown that the in-phase state is more stable than the anti-phase states, therebyfavoring synchronization in the bulk. The natural frequencies of the bulk are calculated to be √ g ± s ± s/ (2 π ). FREQUENCY CONTROL
We generalize the coupling matrix, Eq. (3) by replacing g by g i to get β = g − w − w g − s − s g − w − w g − s
00 0 0 − s g − w − w g , (S10)where g i = L (1 /L ii + (cid:80) kk (cid:54) = i /L ik ) such that L ii can be chosen independently for each i . Since the frequencies ofthe dimers and quartets are proportional to √ g i ± s and √ g i ± s ± s respectively, by varying L ii appropriately, wecan precisely control the frequency of each of the isolated nodes, dimers and quartets. We exploit this technique toconstruct two ATCs with specific frequency patterns.In Fig. S2 (a), we construct a 1D SSH chain of 10 nodes where the ground inductances are chosen such that thereis an increase in frequency by 10 kHz with each successive dimer/isolated node along the length of the chain. Weobtain a precise, localized frequency pattern as can be seen via the background color darkening along the length ofthe chain.In Fig. S2 (b), we consider a square 2D SSH sample in the T2-corner mode regime. Choosing the ground inductancesof all nodes to be equal, say L , we get the same in-phase frequency for all of the isolated nodes, dimers and quartets.This can be seen via the following computations: g c = L (cid:18) L + (cid:18) L w (cid:19)(cid:19) , (S11a) g e = L (cid:18) L + (cid:18) L w + 1 L s (cid:19)(cid:19) , (S11b) g b = L (cid:18) L + (cid:18) L w + 2 L s (cid:19)(cid:19) , (S11c) s = L L s , (S11d) g c = g e − s = g b − s = L (cid:18) L + 2 L w (cid:19) , (S11e)where the subscripts c , e and b stand for corner (isolated), edge (dimer) and bulk (quartet) respectively. This leadsto a probabilistically favored synchronized state where the frequency of all nodes are equal. For the experiment, wemake use of the same parameters used in Tab. II except that we include ground inductances for the bulk nodes thistime. Once again, due to its topological nature, such states are robust against lattice defects. ACTIVITY ONLY ON THE BOUNDARY
Global patterns can be realized in classical topological insulators by adding activity only to the boundary. Thisresults in patching up of the localized topological modes giving rise to a global, robust state. Moreover, these statescan be excited from the bulk as seen in the following simulations. We investigate by considering a 10 ×
10 2D SSHsample in the T1-edge mode regime, T2-corner mode regime and the non-topological regime [Fig. S3]. Upon enablingactivity only on the boundary, we excite the global topological state via an initial condition in the bulk. We observethat the samples in the topological regimes (a,b) display the same topological patterns with the added feature ofexponential decay in amplitude as seen in classical passive systems. However, for the non-topological regime (c), weobtain oscillations only on nodes adjacent to the corners, indicating that global patterns in ATCs are not a consequenceof adding activity alone; topology is also essential.
EXPERIMENTAL REALIZATION USING CHUA’S CIRCUIT
Our experimental realization of an active topolectrical circuit implements a 4-node version of the circuit in Fig. 1(b)for nodes V to V . The individual circuit elements are off-the-shelf components from Digi-Key Electronics (Digi-Key (a) (b) (c) / FIG. S3. Global topological states on 2D ATCs with activity only on the boundary (cf. Fig. 4). (a) Snapshot of the vdPnetwork dynamics in the T2-corner mode regime, with nodes colored by the phase angle ϕ in the ( x i , ˙ x i ) plane; backgroundcolor shows time averaged node frequency ω = (cid:104) ˙ ϕ (cid:105) normalized by the maximum ˆ ω over all oscillators. Even though the initialvoltage was non-zero only on one node in the bulk, the circuit autonomously actuates the entire boundary with exponentialdecay in amplitude along the bulk. (b) Snapshot of the vdP network dynamics in the T1-edge mode regime. The circuitautonomously actuates two topological edge oscillations with exponential decay in amplitude along the bulk. (d) Snapshot ofthe vdP network dynamics in the non-topological regime. The circuit autonomously actuates nodes adjacent to the corners,but this state is neither topological nor global. V V V V (a) experiment LTSpice theory c u rr e n t ( m A ) (b) measurementcubic fit FIG. S4. (a) Experimental and theoretical time series comparison for the high-energy anti-phase attractor. The initialconditions on V and V had opposite signs. Orange lines correspond to experimental data, blue lines to LTSpice simulationsand black lines to numerical simulations using Eq. (2) from the main paper. All voltages vary between ± I - V characteristic curve of the Chua circuit. We applied voltages between − I ( V ) = − . V +0 .
091 mS / V V . Electronics, Thief River Falls, MN, U.S.A). For the active nonlinear resistance, we constructed a realization of Chua’sdiode [54] [Fig. 2(a)], which features a cubic current-voltage relationship. We measured the current through the Chuacircuit when applying a voltage V i and found almost perfectly cubic behavior [Fig. S4(b)]. We fitted the curve toa cubic polynomial and discarded the small constant and quadratic terms, which resulted in the simplified model I ( V ) = − αV + γV , which is also shown in Fig. S4(b). The parameters of all circuit elements are collected in Tables Iand II.The circuits were powered using two Extech 382260 80W switching DC power supplies (FLIR Commerical Systems,Nashuah, NH, U.S.A.), and measurements were taken using a Rigol DS 1054Z 4-channel digital oscilloscope (RigolTechnologies U.S.A., Beaverton, OR, U.S.A.).Fig. S4 (a) shows an alternate dynamical attractor which is the high-energy bulk oscillation obtained using anti-phase initial conditions. This state is achieved using the same experimental setup in Fig. 2. TUNING WAVE-LIKE PATTERNS USING DEFECTS
Active traveling wave patterns can be induced in the bulk of an ATC by strategically placing bulk defects. Byreplacing one of the strong bonds by a weak bond in the desired bulk quartets, it is possible to construct arbitrary
Dimensional parameter Value Nondimensional parameter Value C
42 nF g . L e µ H w . L b Nil s . L w . ε . L s µ H α . Scaling factor Value γ .
091 mS / V V .
03 V R τ . µ s R L µ H R
400 Ω R R L w and L s denote the inductances in theweak and strong couplings respectively while L e and L b denote the ground inductances of the edge and bulk nodes. Note thatfor the bulk nodes, we do not use ground inductances. Component/Parameter Node 1 Node 2 Node 3 Node 4 α .
496 mS 2 .
494 mS 2 .
500 mS 2 .
500 mS γ . / V . / V . / V . / V L ii . µ H Nil Nil 486 . µ H L ii ESR 0 .
821 Ω Nil Nil 0 .
810 Ω C .
99 nF 41 .
63 nF 42 .
07 nF 42 .
06 nF C ESR 0 .
459 Ω 0 .
620 Ω 0 .
704 Ω 0 .
760 Ω L w . . L w ESR 2 .
520 Ω 2 .
528 Ω L s . µ H L s ESR 0 .
940 ΩTABLE II. Component values used for experiments in Figs. 2 and S4. ESR stands for equivalent series resistance, referring tothe parasitic resistance present in practical, non-ideal inductors and capacitors. For the values listed, ESR was measured at anAC frequency of 1 kHz. directional patterns. For instance, a circuit may be constructed to display text or images [Fig. 5]. The individualconstituent segments of this active pattern interact with each other to produce traveling wave-like dynamics.
ALTERNATIVE EXPERIMENTAL SETUP
Here we discuss an alternative way to experimentally implement the cubic nonlinearity analyzed in the main paper,and present LTSpice simulations of a 6-node active topolectric chain using this model.
Parameter scaling
For this realization, the scales for capacitance and inductance are set to as C = 100 nF and L = 1 . µ H. It followsthat the system’s frequency scale is ω = τ = √ L C = 2 π ·
411 kHz and that of resistance R = (cid:15) = (cid:113) L C = 3 .
87 Ω.These values are chosen to accommodate for what inductors and capacitors are easily available and to respect thelimitations of the following non-linear resistor implementation in terms of operating frequency and output current.
Implementation of the non-linear resistor
The vdP oscillator requires a non-linear resistor element that behaves according to R ( V ) − = − (cid:0) α − γV (cid:1) .Ref. [54] suggests such an implementation using a negative impedance converter to create the negative conductivityat low applied voltages in parallel to two diodes acting as large positive-valued conductivities at higher voltages. Theexact configuration used in our simulations is displayed in figure S5a. The diodes used are type 1N5817 Schottkydiodes, chosen for their low forward voltage drop to keep the oscillator chain’s operating voltage low. This reducesthe required current rating for other components. Figure S5b shows the non-linear resistor’s DC current responseand a third order polynomial fit to the region ±
200 mV. As can be seen, the element matches the U functionclosely up to about ±
250 mV. Beyond that, the current response is smaller than the ideal behavior. From the fitcurve, the parameters values of the non-linear resistor are determined as α = 0 . γ = 0 .
408 V − . In orderto test its usable frequency bandwidth, the circuit’s response to an AC voltage of 500 mV is simulated at variousfrequencies. Figure S6 shows the current response within one period plotted over the input voltage. At 10 kHz, thecurve approximately matches the DC behavior. Towards higher frequencies however, starting in the 100 kHz range,the current shows an increasing phase shift to the input voltage, visible as a loop forming in the current-voltage-plot.In simulations of the entire oscillator chain, this behavior has shown to cause problems at frequencies beyond 1 MHz,restricting the choice of other circuit parameters to the sub-megahertz range in ω π . Simulating the chain
Figure S7 shows the chain’s circuit diagram. Each node of the vdP oscillator chain is connected to ground via oneof the non-linear resistors and a 100 µ F capacitor in parallel, and to its neighboring nodes alternately by a 470 µ Hand a 1 . µ H inductor. The corresponding nondimensionalized parameter values are g = 1, w = 0 . s = 1 − w .The circuit has six nodes in total. To excite the left edge mode, a current I in can be fed into the first node. Idealized case
A simulation of the idealized case, neglecting parasitic resistances, was performed using LTSpice. The circuit isinitialized with all currents and voltages set to zero. The left edge mode at node 1 is briefly excited using a currentsource at resonance frequency to reach the steady-state oscillation. Figure S8a shows the result of the transientsimulation of all six nodes. In agreement with the numerical results of the main paper, the edges show a high-frequency oscillatory behavior, at about 410 kHz, while the bulk shows a slower oscillatory pattern at frequency25 kHz. Amplitudes are typically about 300mV. For the bulk nodes, an initial excitation at 575 kHz is observed thatdiminishes as the slow non-linear limit cycles settle. These oscillations are not visible in Fig. 1 of the main paper.This may be caused by the different means of initializating the system, where Fig. 1 uses nonzero initial conditionsfor x and ˙ x while here, we briefly resonantly excite node 1 by an input current. U in − + LT1363 (a) Circuit diagram of the alternative non-linear resistor for thevdP oscillator. (b) DC sweep of the non-linear resistor’s input current andpolynomial fit of third order. The resulting fit function is I = − (cid:0) . − · U − . − V − · U (cid:1) , corresponding toparameter values α = 0 . γ = 0 .
408 V − . FIG. S5. Alternative circuit implementing a cubic nonlinearity. (a) (b) (c)
FIG. S6. Current over voltage plot of the non-linear resistor’s response to an AC voltage of 500 mV at different frequencies. . µ H V R ( V ) 100nF470 µ H I in V R ( V ) 100nF1 . µ H . . . µ H V R ( V ) 100nF1 . µ H FIG. S7. Circuit diagram of the oscillator chain.
Including parasitic resistances
In order to obtain a more realistic simulation of the circuit, parasitic serial resistances are added to the inductorsand capacitors of the circuit and disorder in the form of a 10 % tolerance random deviation is applied to all passiveelements. The chosen serial resistances for the inductors are 24 . Q @ 411kHz = 50for 470 µ H one and 110 mΩ, corresponding to Q @ 411kHz = 35 for the 1 . µ H one. The capacitors at node 1 and6 have a serial resistance of 50 mΩ, those in the bulk 5 Ω. The different choice of the capacitor’s serial resistance isan attempt to reflect the frequency dependent resistance of capacitors and also demonstrates that the bulk cyclesare rather insensitive to a large serial resistance in the node’s capacitor. Figure S9 shows the results of the transientsimulation. Qualitatively, the circuit’s behavior does not change by much. The only obvious difference is that the bulknodes no longer show initial high frequency oscillations that now presumably are strongly damped by the parasiticresistances. Overall, the circuit’s voltage amplitudes have somewhat diminished with the edge nodes at about 175mVand the bulk nodes at around 230 mV. The frequency of the edge modes is slightly lower compared to the idealcase at about 400 kHz. Much more significant is the change in the bulk cycle frequency that is more than halved toabout 11 kHz. Re-running the simulation for various different parameter values shows that both the inductors andthe capacitor’s parasitic resistances contribute significantly to this effect, with the inductor’s serial resistance havingthe comparatively larger influence on the bulk cycle’s frequency. Figures S9b and S9c again show ˙ V over V of thecircuit’s fourth and sixth node. As for the ideal case, that of node four resembles a non-linear limit cycle and that ofnode six a linear mode. Compared to the ideal case, the amplitude at node six grows slower, caused by the additionaldamping of the parasitic resistances.When further increasing parasitic resistances, several modes of failure can occur that are listed in Table III. Themain concerns are the edge modes potentially being damped or the bulk oscillations stalling for large serial resistances.To a certain degree, this can be compensated by adjusting the non-linear resistor element for a stronger driving. Largerdisorder has relatively little effect on the qualitative behavior of the circuit, unless it locally induces a failure if someinductor or capacitor’s Q is too low. Element Threshold Effect R S of 470 µ H inductor >
30 Ω Bulk voltages stop oscillating, stall at constant value R S of 1 . µ H inductor >
200 mΩ Edge modes are damped R S of capacitors at edge >
150 mΩ Edge modes are damped R S of capacitors in bulk >
50 Ω No failure but bulk cycles take on trapezoid shapeTABLE III. Observed modes of failure induced by large parasitic resistances. (a) Transient voltage simulation of all six nodes for the ideal circuit. The horizontal axis is time in ms, verticalthe voltage in V.(b) ( V , ˙ V ) plot of node 4 for the ideal circuit. The first0 .
25 ms are cropped out due to the overlaid highfrequency oscillation as seen in S8a. The non-circularshape implies that it is a strongly non-linear limitcycle. (c) ( V , ˙ V ) plot of node 6 for the ideal circuit. Thecircular shape is close to that of a linear oscillation. FIG. S8. Simulation results for the idealized alternative active topolectric circuit using LTSpice. (a) Transient voltage simulation of all six nodes for the circuit with parasitic resistances. The horizontal axisis time in ms, vertical the voltage in V.(b) ( V , ˙ V ) plot of node 4 for the realistic circuit. Thefirst 0 .
25 ms are cropped out due to the overlaid highfrequency oscillation as seen in S8a. The non-circularshape implies that it is a strongly non-linear limit cycle. (c) ( V , ˙ V ) plot of node 6 for the realistic circuit. Thecircular shape is close to that of a linear oscillation.) plot of node 6 for the realistic circuit. Thecircular shape is close to that of a linear oscillation.