Circuit Implementation of a Four-Dimensional Topological Insulator
CCircuit Implementation of a Four-Dimensional Topological Insulator
You Wang, Hannah M. Price, Baile Zhang,
1, 3 and Y. D. Chong
1, 3 Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore 637371, Singapore School of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, United Kingdom Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore
The classification of topological insulators predicts the existence of high-dimensional topologicalphases that cannot occur in real materials, as these are limited to three or fewer spatial dimensions.We use electric circuits to experimentally implement a four-dimensional (4D) topological lattice.The lattice dimensionality is established by circuit connections, and not by mapping to a lower-dimensional system. On the lattice’s three-dimensional surface, we observe topological surface statesthat are associated with a nonzero second Chern number but vanishing first Chern numbers. The4D lattice belongs to symmetry class AI, which refers to time-reversal-invariant and spinless systemswith no special spatial symmetry. Class AI is topologically trivial in one to three spatial dimensions,so 4D is the lowest possible dimension for achieving a topological insulator in this class. This workpaves the way to the use of electric circuits for exploring high-dimensional topological models.
Introduction.—
Topological insulators are materialsthat are insulating in the bulk but host surface statesprotected by nontrivial topological features of their bulkbandstructures [1, 2]. They are classified according tosymmetry and dimensionality [3–7], with each class hav-ing distinct and interesting properties. The celebratedtwo-dimensional Quantum Hall (2DQH) phase [8], forinstance, has topological edge states that travel unidi-rectionally on the one-dimensional (1D) edge, whereasthree-dimensional (3D) topological insulators based onspin-orbit coupling have surface states that act like mass-less 2D Dirac particles. The classification of topologicalinsulators contains hypothetical high-dimensional phases[3] that cannot be realized with real materials, since elec-trons only move in one, two, or three spatial dimensions.These include several types of four-dimensional Quan-tum Hall (4DQH) phases, which are characterised by a4D topological invariant called the second Chern num-ber and exhibit a much richer phenomenology than the2DQH phase [9–12]. In recent years, topological phaseshave been implemented in a range of engineered systemsincluding cold atom lattices [13], photonic structures [14],acoustic and mechanical resonators [15, 16], and elec-tric circuits [17–28]. Some of these platforms can re-alise lattices that are hard to achieve in real materials,raising the intriguing prospect of using them to createhigh-dimensional topological insulators. Although therehave been demonstrations of “topological pumps” thatmap 4D topological lattice states onto lower-dimensionalsystems [29–32], there has been no experimental realisa-tion of a 4D topological insulator with protected surfacestates on a 3D surface.Here, we describe the implementation of a 4DQH phaseusing electric circuits to access higher dimensions. Sinceelectric circuits are defined in terms of lumped (discrete)elements and their interconnections, lattices with genuinehigh-dimensional structure can be explicitly constructed by applying the appropriate connections [33–35]. In thisway, we experimentally implement a 4D lattice hostingthe first realisation of a Class AI topological insulator[5, 6], which has no counterpart in three or fewer spatialdimensions.In the symmetry-based classification of topologicalphases [3–7], Class AI includes time-reversal (T) sym-metric, spinless systems that are not protected by anyspecial spatial symmetries. Whereas the 2DQH phase istied to nontrivial values of the first Chern number, whichrequires T-breaking [36], 4DQH phases rely on the sec-ond Chern number, which does not [9–12]. Even thoughthe Class AI conditions are ubiquitous [13, 14], the classis topologically trivial in one to three dimensions [3–7].Hence, realising a Class AI topological insulator requiresgoing to at least 4D. We focus on a theoretical 4D lat-tice model recently developed by one of the authors [37],which exhibits a nonzero second Chern number with van-ishing first Chern numbers. Hence, we obtain the firstobservations of topological surface states that are intrin-sically tied to 4D band topology, with no connection tolower-dimensional topological invariants.The present approach, based on circuit connections,is distinct from other recently-investigated methods foraccessing higher-dimensional models. One of the alter-natives involves manipulating internal degrees of free-dom, such as oscillator modes, to act as synthetic di-mensions [38–52]. Although there have been theoret-ical proposals for using synthetic dimensions to build4D topological lattices [40, 43], all experiments so farhave been limited to 1D and 2D [51]. Another approachinvolves adiabatic topological pumping schemes, whichmap high-dimensional models onto lower-dimensional se-tups by replacing spatial degrees of freedom with tunableparameters [29–32]. As mentioned above, 2D topologicalpumps based on cold atoms and photonics have recentlybeen used to explore Class A (T-broken) 4DQH systems a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n [30, 53, 54]. However, topological pumps have the draw-back of being inherently limited to probing specific quasi-static solutions of a high-dimensional system, withoutrealising a genuinely high-dimensional lattice. Moreover,in those experiments the second Chern number in 4D isnot truly independent of the first Chern numbers in 2D,which are nonzero.Our 4D lattice is implemented using electric circuitswith carefully chosen capacitive and inductive connec-tions. The lattice model has two topologically distinctphases: a 4DQH phase and a conventional insulator,with the choice of phase governed by a parameter m thatmaps to certain combinations of capacitances and induc-tances. Using impedance measurements that are equiv-alent to finding the local density of states (LDOS), weshow that the 4DQH phase hosts surface states on the3D surface, while the conventional insulator phase hasonly bulk states. By varying the driving frequency, weshow that the topological surface states span a frequencyrange corresponding to a bulk bandgap, as predicted bytheory. Our experimental results also agree well with cir-cuit simulations. This work demonstrates that electriccircuits are a flexible and practical way to realise higher-dimensional lattices, paving the way for the explorationof other previously-inaccessible topological phases. The 4D latticemodel is shown schematically in Fig. 1(a). The spatial co-ordinates are denoted x , y , z , and w . The lattice containsfour sublattices labelled A, B, C and D, with sites con-nected by real nearest neighbour hoppings ± J . The fourbands host two pairs of Dirac points in the Brillouin zone;each pair is the time-reversed counterpart of the other.To control the pairs separately, long-range hoppings withamplitudes ± J (cid:48) and ± J (cid:48)(cid:48) are added within the x - z plane[these long-range hoppings are omitted from Fig. 1(a) forclarity, but are shown in Fig. 1(c)]. Upon adding mass+ m to the A and B sites, and − m to the C and D sites,the Dirac masses for the different Dirac point pairs closeat m = J (cid:48) − J (cid:48)(cid:48) and m = J (cid:48)(cid:48) − J (cid:48) . These gap closings aretopological transitions, such that, for J (cid:48)(cid:48) = − J (cid:48) , the sec-ond Chern number of the lower bands is -2 (nontrivial)if | m | < | J (cid:48) | . Since T is unbroken, the first Chern num-ber is always zero, so the model exhibits QH behaviourstemming purely from the second Chern number [37].For the experiment, we set J = 1 and J (cid:48) = − J (cid:48)(cid:48) = 2, sothat the topological transition of the bulk lattice occursat m = ±
6. We take a finite 4D lattice with three unitcells (6 sites) in the x and z directions, and one unit cell(2 sites) in y and w . Periodic boundary conditions areapplied along y and w to mitigate finite-size effects, andare implemented using nearest neighbor type connectionsbetween opposite ends of the lattice. The lattice has atotal of 144 sites, of which we consider 16 to be bulk sites(defined as being more than 2 sites away from a surface)and 128 to be surface sites. Circuit realization.—
The finite 4D lattice is imple-
A CD B a bc
A BCD
C B A D zxwy xz C = 1 nF L = 2 mH C' = 2 nF L' = 1 mH A C
A D
FIG. 1: Model of the 4D Quantum Hall lattice and its circuitimplementation. (a) Schematic of the 4D tight-binding model.Each unit cell consists of four sites labelled A-D. Hollow andfilled circles respectively denote positive ( m ) and negative( − m ) on-site masses, while yellow solid lines and blue dashesrespectively denote positive ( J ) and negative ( − J ) hoppings.(c) Long-range hoppings of the tight-binding lattice. (b) Pho-tographs of the circuit. (c) Schematic of the circuit; positive(negative) masses are realised by capacitors (inductors) con-necting the sites to ground, and hoppings are realised usingcapacitors or inductors connecting different sites. mented with a set of connected printed circuit boards,shown in Fig. 1(b). Each site i of the tight-binding modelmaps to a node on the circuit, and the mass term maps toa circuit component of conductance − D ii connecting thenode to ground. Each hopping J ij between sites i and j maps to a circuit element of conductance D ij connect-ing the nodes. We add extra grounding components withconductance D (cid:48) ii in parallel with − D ii . If an external ACcurrent I i flows into each node i at frequency f , and V i is the complex AC voltage on that node, Kirchhoff’s lawstates that I i = ( − D ii + D (cid:48) ii ) V i + (cid:88) j D ij ( V i − V j ) . (1)We define D ij ( f ) = iαH ij ( f ) , (2)where α is a positive real constant. Then capacitances(inductances) correspond to positive (negative) real H ij .We require that at a reference working frequency f = f ,the values of H ij ( f ) match the desired tight-binding lat-tice Hamiltonian. We map the positive nearest neigh-bor hopping J = 1 to capacitance C = 1 nF by taking α = 2 πf C . The long-range hopping J (cid:48) then maps tocapacitance C (cid:48) = 2 nF. By setting f = 1 / (2 π √ L C ) ≈
113 kHz, the negative nearest neighbor hopping maps toinductance L = 2 mH, and the negative long-range hop-ping J (cid:48)(cid:48) = − L (cid:48) = 1 mH.The grounding conductance of node i is parameterisedas − D ii + D (cid:48) ii . We tune D (cid:48) ii so that for f = f and D ii obeying Eq. (2), D (cid:48) ii + (cid:80) j (cid:54) = i D ij = iαE for a target energy E . The required D (cid:48) ii is dependent on the m parameter.Eq. (1) now becomes [55] I i ( f ) ≡ (cid:88) j L ij V j = − iα (cid:88) j (cid:104) H ij ( f ) − E δ ij (cid:105) V j ( f ) , (3)where L ij are the components of the circuit Laplacian L .In our experiments, we measure the impedance be-tween a given node r and the common ground by applyinga 1 V sine wave of frequency f on that node, and mea-suring the voltage V r and the current I r . The impedancebetween node r and the ground is the r th diagonal termof the inverse of the circuit Laplacian L : V r = (cid:88) j ( L − ) rj I j = Z r I r . (4)Using Eq. (3), one can show that [20] Z r = iα lim (cid:15) → (cid:88) n | ψ n ( r ) | E n − E + i(cid:15) , (5)where ψ n ( r ) is the n -th energy eigenstate’s amplitude onsite r , and E n is the corresponding eigenenergy. There-fore Re[ Z r ] = (1 /πα ) (cid:80) n δ ( E − E n ) | ψ n ( r ) | is, up to ascale factor, the LDOS of the target lattice at energy E when measured at f = f . Experimental results.—
Fig. 2(a) shows the band dia-gram of the infinite bulk tight-binding model as a func-tion of the mass detuning parameter m . For | m | <
6, thesystem is in a 4DQH phase, with a topologically nontriv-ial bandgap centered at E = 0, which hosts topologicalsurface states. The band diagram for the 144-site tight-binding model is shown in Fig. 2(b). The colors of the ac de f b Bulk bandsBulk bandsSurface states S u r f a c e c o n c e n t r a t i o n E m mm = 0, E = 0 m = 0, E = 1 m = 4, E = 0 m = 8, E = 0 wy wywy wy z x z xz xz x FIG. 2: (a) Calculated band diagram for the infinite 4D lat-tice. The bulk bands are shown in gray. For | m | <
6, there isa bandgap associated with nontrivial second Chern number,accompanied by topological surface states (shaded green). For | m | >
6, the bandgap is trivial. (b) Calculated band diagramfor 144-site lattice with periodic boundary conditions along y and w . Colors indicate the degree of surface concentrationof the energy states, as defined in the main text. Due tofinite-size effects, surface states occur at | m | (cid:46) | m | ≈
4. The parameters correspond-ing to subplots (c)–(f) are indicated with pink dots. (c)–(f)Experimentally obtained LDOS maps for different m and E ,measured at working frequency f = f . Surface states are ob-served in (c) and (d), consistent with theoretical predictions. curves indicate the degree to which each eigenstate isconcentrated on the surface, as defined byln [ (cid:104)| ψ ( r ) |(cid:105) surf . / (cid:104)| ψ ( r ) |(cid:105) bulk ] , (6)where ψ ( r ) denotes the energy eigenfunction, whose mag-nitudes are averaged over either surface or bulk sites. Dueto the finite lattice size, both the bulk and surface spec-trum are split into sub-bands. The closing of the bulkgap is shifted to | m | ≈
4, and the surface states occurmost prominently at small values of E and | m | . In theSupplementary Information, we plot band diagrams forincreasing lattice sizes, showing that the spectra comeinto close agreement with Fig. 2(a) as finite-size effectsbecome negligible [55].We now fabricate a set of circuits with parameters m ∈ { , , . . . , } and E ∈ { , } . Figure 2(c)–(f) showsthe measured LDOS (at f = f ) for four representa-tive samples. From the experimental data, which agreeswell with circuit simulations [55], we see that the sur-face LDOS is high and the bulk LDOS is low when inthe topologically nontrivial bandgap [Fig. 2(c) and (d)].For E = 0, m = 4, which corresponds roughly to thegap-closing point, there is no significant difference be-tween the surface and bulk LDOS. For E = 0, m = 8,the LDOS on all sites is low, consistent with being in atopologically trivial bandgap.To further quantify the differences between the 4DQHand conventional insulator phases, Fig. 3(a) plots theexperimentally-determined ratio of the mean LDOS onsurface sites to the mean LDOS on bulk sites, for dif-ferent values of the mass detuning parameter m . Thesemeasurements are performed at f = f , corresponding to E = 0. With increasing m , the ratio decreases sharplyfrom around 4 . Z r ], averaged over surface or bulksites. As explained above, our impedance measurementsprobe the response at fixed energy E of an effectiveHamiltonian H ( f ) that depends parametrically on thefrequency f [Eq. (5)]. At the reference frequency, H ( f )matches the target tight-binding model; at other frequen-cies, H ( f ) deviates from the form of the target model(e.g., the positive and negative hoppings become unequalin magnitude, lifting the band degeneracy). However, solong as E lies in the bulk bandgap of H ( f ), the secondChern number is unchanged [55]. Our experimental re-sults at small values of m indeed show a strong surfaceresponse over a range of frequencies spanning a bulk gap[Fig. 3(b) and (c)]. Increasing m closes the bulk gap, andthereafter the surface and bulk LDOS measures exhibitno notable frequency dependent features. These resultsalso agree well with simulations [Fig. 3(g)–(j)]. Discussion.—
We have used electric circuits to imple-ment a 4D lattice hosting a 4D Quantum Hall phase.This is the first experimental demonstration of a 4D topo-logical lattice, and of a Class AI topological insulator.This is also the first experimental exploration of a 4DQHmodel with nontrivial second Chern number but trivialfirst Chern numbers. Using impedance measurements,we have demonstrated that the LDOS on the 3D sur- face is enhanced in the 4DQH phase, due to the presenceof topological surface states; the enhanced surface re-sponse is shown to span the frequency range of the bulkbandgap. The gap closing is also clearly observed, de-spite being shifted by finite-size effects. In future work,it is desirable to find ways to probe the detailed fea-tures of the 3D surface states, which are predicted to betwo robust isolated Weyl points of the same chirality, asituation that does not occur in lower-dimensional topo-logical models [37]. This successful implementation of4D lattices of very substantial size (144 sites) shows thatelectronic circuits are an excellent platform for exploringexotic band topological effects, and a promising alterna-tive to the “synthetic dimensions” approach [38, 39, 45]to realizing higher-dimensional lattices.While this work was being done, we became aware ofa related theoretical proposal to use circuits for realizingsimilar Class AI topological insulators [56].We are grateful to C. H. Lee and T. Ozawa for helpfuldiscussions. This work was supported by the SingaporeMOE Academic Research Fund Tier 3 Grant MOE2016-T3-1-006, Tier 1 Grants RG187/18 and RG174/16(S),and Tier 2 Grant MOE2018-T2-1-022(S). HMP is sup-ported by the Royal Society via grants UF160112,RGF/EA/180121 and RGF/R1/180071. [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011).[3] A. Altland and M. R. Zirnbauer, Phys. Rev. B , 1142(1997).[4] A. Kitaev, V. Lebedev, and M. Feigel’man, in AIP Con-ference Proceedings (AIP, 2009).[5] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Phys. Rev. B , 195125 (2008).[6] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Lud-wig, New J. Phys. , 065010 (2010).[7] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,Rev. Mod. Phys. , 035005 (2016).[8] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980).[9] J. Avron, L. Sadun, J. Segert, and B. Simon, Phys. Rev.Lett. , 1329 (1988).[10] J. Fr¨ohlich and B. Pedrini, in Mathematical Physics 2000 (World Scientific, 2000), pp. 9–47.[11] S.-C. Zhang, Science , 823 (2001).[12] S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, andI. Spielman, Science , 1429 (2018).[13] N. R. Cooper, J. Dalibard, and I. B. Spielman, Rev. Mod.Phys. , 015005 (2019).[14] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi,L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil-berberg, et al., Rev. Mod. Phys. , 015006 (2019).[15] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, andB. Zhang, Phys. Rev. Lett. , 114301 (2015).[16] S. D. Huber, Nat. Phys. , 621 (2016). af bg ch di ej S u r f a c e / bu l k r a t i o S u r f a c e / bu l k r a t i o E x pe r i m en t s S i m u l a t i on s Frequency (kHz)
BulkSurface R e [ ⟨⟩ ] ( a . u . ) Z r R e [ ⟨⟩ ] ( a . u . ) Z r m = 0 m = 1 m = 4 m = 8 m = 0 m = 1 m = 4 m = 8 m FIG. 3: Comparison of bulk and surface contributions to the LDOS. (a) Ratio of surface to bulk LDOS, measured at f = f ,versus m . (b)–(e) Mean values of the LDOS measure Re[ Z r ] on surface and bulk sites, versus working frequency f . Forthese subplots, measurements were only taken over sites in the 2D plane ( y, w ) = (1 , f (corresponding to E = 0) is indicated by the vertical dotted line. For small m , we observe an elevated surface LDOSmeasure over a range of frequencies coincident with a bulk gap. Upon increasing m , the gap closes. (f)–(j) Simulation resultscorresponding to (a)–(e), computed using the same lattice size and circuit parameters.[17] J. Ningyuan, C. Owens, A. Sommer, D. Schuster, andJ. Simon, Phys. Rev. X , 021031 (2015).[18] V. V. Albert, L. I. Glazman, and L. Jiang, Phys. Rev.Lett. , 173902 (2015).[19] Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Al, Na-ture Electronics , 178 (2018).[20] C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W.Molenkamp, T. Kiessling, and R. Thomale, Comm. Phys. , 39 (2018).[21] S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W.Molenkamp, T. Kiessling, F. Schindler, C. H. Lee,M. Greiter, T. Neupert, et al., Nat. Phys. , 925 (2018).[22] K. Luo, R. Yu, and H. Weng, Research , 1 (2018).[23] M. Ezawa, Phys. Rev. B , 201402 (2018).[24] Y. Wang, L.-J. Lang, C. H. Lee, B. Zhang, and Y. D.Chong, Nat. Comm. , 1102 (2019).[25] Y. Lu, N. Jia, L. Su, C. Owens, G. Juzeli¯unas, D. I.Schuster, and J. Simon, Phys. Rev. B , 020302 (2019).[26] M. Serra-Garcia, R. S¨usstrunk, and S. D. Huber, Phys.Rev. B , 020304 (2019).[27] T. Helbig, T. Hofmann, C. H. Lee, R. Thomale, S. Imhof,L. W. Molenkamp, and T. Kiessling, Phys. Rev. B ,161114 (2019).[28] T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, andR. Thomale, Phys. Rev. Lett. , 247702 (2019).[29] D. Thouless, Phys. Rev. B , 6083 (1983).[30] Y. E. Kraus, Z. Ringel, and O. Zilberberg, Phys. Rev.Lett. , 226401 (2013).[31] I. Petrides, H. M. Price, and O. Zilberberg, Phys. Rev.B , 125431 (2018).[32] C. H. Lee, Y. Wang, Y. Chen, and X. Zhang, Phys. Rev.B , 094434 (2018).[33] D. Juki´c and H. Buljan, Phys. Rev. A , 013814 (2013).[34] M. Ezawa, Phys. Rev. B , 081401 (2019).[35] L. Li, C. H. Lee, and J. Gong, Comm. Phys. , 135(2019).[36] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. , 405 (1982).[37] H. M. Price, Four-dimensional topological lattices withoutgauge fields (2018), arXiv:1806.05263.[38] O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein,Phys. Rev. Lett. , 133001 (2012).[39] A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B.Spielman, G. Juzeli¯unas, and M. Lewenstein, Phys. Rev.Lett. , 043001 (2014).[40] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, andN. Goldman, Phys. Rev. Lett. , 195303 (2015).[41] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider,J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte,et al., Science , 1510 (2015).[42] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, andI. B. Spielman, Science , 1514 (2015).[43] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, andI. Carusotto, Phys. Rev. A , 043827 (2016).[44] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, andN. Goldman, Phys. Rev. B , 245113 (2016).[45] L. Yuan, Y. Shi, and S. Fan, Optics Letters , 741(2016).[46] T. Ozawa and I. Carusotto, Phys. Rev. Lett. , 013601(2017).[47] H. M. Price, T. Ozawa, and N. Goldman, Phys. Rev. A , 023607 (2017).[48] L. Yuan, Q. Lin, M. Xiao, and S. Fan, Optica , 1396(2018).[49] H. M. Price, T. Ozawa, and H. Schomerus, arXiv preprintarXiv:1907.04231 (2019).[50] E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A.Bandres, A. Szameit, and M. Segev, Nature , 356(2019).[51] T. Ozawa and H. M. Price, Nature Reviews Physics ,349 (2019).[52] A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao, andS. Fan, Science , 59 (2020).[53] M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, Nature , 55 (2018).[54] O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P.Chen, Y. E. Kraus, and M. C. Rechtsman, Nature ,59 (2018). [55] See online Supplemental Information.[56] R. Yu, Y. X. Zhao, and A. P. Schnyder, arXiv preprintarXiv:1906.00883 (2019).
Supplementary Information
SUPPLEMENTARY NOTE 1: CIRCUIT DESIGN DETAILS
The circuit is divided into several printed circuit boards (PCBs), stacked on top of each other. As shown in Fig. S1,each PCB is divided into 6 × x - z plane; seeFig. 1(c) of the main text. Each x - z lattice plane actually consists of several PCBs stacked with vertical electricalinterconnects, in order to fit all the necessary circuit components.Let I i be the external current injected into node i , V j the voltage (relative to ground) on node j , and D ij theconductance between nodes i and j for i (cid:54) = j . Moreover, let the conductance between node i and ground be D ( g ) ii = − D ii + D (cid:48) ii . (S1)By Kirchhoff’s laws, I i = D ( g ) ii V i + (cid:88) j D ij ( V i − V j ) (S2)= (cid:88) j (cid:34) − D ij + (cid:32) − D ii + D (cid:48) ii + (cid:88) k D ik (cid:33) δ ij (cid:35) V j (S3)= (cid:88) j − D ij + D (cid:48) ii + (cid:88) k (cid:54) = i D ik δ ij V j . (S4)Note that in Eq. (S2), the sum can be taken either over all j , or equivalently over j (cid:54) = i . We now adjust D (cid:48) ii so that,at a reference working frequency f , D (cid:48) ii ( f ) + (cid:88) j (cid:54) = i D ij ( f ) = iαE (S5)for each node i , with some constant α and target energy E . At f = f , Eq. (S4) then becomes I i ( f ) = − iα (cid:88) j (cid:104) H ij ( f ) − E δ ij (cid:105) V j ( f ) , (S6) D ij ( f ) ≡ iαH ij ( f ) . (S7)We require H ij ( f ) to match the target tight-binding Hamiltonian, which has parameters J = 1, J (cid:48) = − J (cid:48)(cid:48) = 2. Forreal α , positive (negative) real values of H ij correspond to capacitances (inductances). As described in the main text, FIG. S1: Photographs showing the topmost PCB (left) and the stack of PCBs (right). by choosing α and f we can assign the following circuit elements to the lattice model’s hopping terms: C ↔ J = 1 (positive NN hopping) C (cid:48) = 2 C ↔ J (cid:48) = 2 (positive long range hopping) L ↔ − J = − L (cid:48) = L / ↔ J (cid:48)(cid:48) = − πf = 1 / (cid:112) L C , α = 2 πf C . (S9)For each node, we determine the grounding conductance required to satisfy Eq. (S5). Suppose node i is connectedto other nodes by p i type- C capacitors, q i type- L inductors, p (cid:48) i type- C (cid:48) capacitors, and q (cid:48) i type- L (cid:48) inductors (theseconnections depend on which sublattice the node lies on, and whether it lies in the bulk or on the surface). Then (cid:88) j (cid:54) = i D ij ( f ) = 2 πip i f C + q i πif L + 2 πip (cid:48) i f C (cid:48) + q (cid:48) i πif L (cid:48) = 2 πif C (cid:18) p i + 2 p (cid:48) i − ( q i + 2 q (cid:48) i ) f f (cid:19) . (S10)Taking f = f and plugging into Eq. (S5) gives D (cid:48) ii ( f ) = iαE − (cid:88) j (cid:54) = i D ij ( f )= 2 πif C (cid:0) E − p i − p (cid:48) i + q i + 2 q (cid:48) i (cid:1) . (S11)The on-site mass term is H ii ( f ) = ± m , depending on whether the node is on the A,B or C,D sublattices. Hence, thegrounding conductance must satisfy D ( g ) ii ( f ) = − D ii ( f ) + D (cid:48) ii ( f )= 2 πif C (cid:0) E ∓ m − p i − p (cid:48) i + q i + 2 q (cid:48) i (cid:1) . (S12)To achieve this in the experiment, we connect each node i to ground with 6 − p i type- C capacitors, 3 − q i type- L inductors, 4 − p (cid:48) i type- C (cid:48) capacitors, and 4 − q (cid:48) i type- L (cid:48) inductors. Additionally, (i) we connect each node to groundby an extra inductor L g , and (ii) if node i belongs to sublattice C or D, we connect it to ground by an extra capacitor C m = 2 mC . As a result, the grounding conductance of node i at an arbitrary frequency f is D ( g ) ii ( f ) = 2 πi (6 − p i ) f C + (3 − q i )2 πif L + 2 πi (4 − p (cid:48) i ) f C (cid:48) + (4 − q (cid:48) i )2 πif L (cid:48) + 12 πif L g + 2 πi ( m ∓ m ) f C (S13)where ∓ refers to sublattice A,B or C,D respectively. At f = f , this satisfies Eq. (S12) if we pick L L g = 3 + m − E. (S14)Hence, D ( g ) ii ( f ) = 2 πif C (cid:20)
14 + m ∓ m − p i − p (cid:48) i + (cid:16) E − − m + q i + 2 q (cid:48) i (cid:17) f f (cid:21) . (S15)Returning to Eq. (S4), define the quantity in the parentheses—which gives rise to the E term in Eq. (S6)—as iα E i ( f ) = D (cid:48) ii ( f ) + (cid:88) k (cid:54) = i D ik ( f ) (S16)= D ( g ) ii ( f ) + D ii ( f ) + (cid:88) j (cid:54) = i D ij ( f ) f → f −→ iαE. (S17)Eq. (S6) then generalises to I i ( f ) = − iα (cid:88) j (cid:104) H ij ( f ) − E i ( f ) δ ij (cid:105) V j ( f ) . (S18)Now observe that in Eq. (S17), the first term D ( g ) ii ( f ) is defined by Eq. (S15) for any f , and the third term is likewisedefined by Eq. (S10) for any f . However, D ii ( f ) is defined only at f = f . This turns out not to be a problem forour system of equations, since this term is exactly cancelled by the Hamiltonian term in Eq. (S18), which possessesthe same ambiguity. We are therefore free to give D ii ( f ) any frequency dependence, consistent with its value at f (i.e., D ii ( f ) = iαH ii ( f ) = ± iαm ). A convenient choice is D ii ( f ) = iαE − D ( g ) ii ( f ) − (cid:88) j (cid:54) = i D ij ( f ) f → f −→ iαH ii ( f ) (S19) ⇒ iα E i ( f ) = iαE for all i, f. (S20)With this choice, E i ( f ) becomes i -independent, and Eq. (S18) simplifies to I i ( f ) = − iα (cid:88) j (cid:104) H ij ( f ) − E δ ij (cid:105) V j ( f ) . (S21)This can be interpreted as a family of response equations with an f -dependent Hamiltonian and fixed energy E . Forgeneral f , the Hamiltonian’s hopping terms are determined by the circuit elements summarised in Eq. (S8), and itson-site mass term is determined by Eq. (S19); for f = f , it reduces to the target Hamiltonian.Suppose E is in a topological gap of the target Hamiltonian, so that topological edge states exist at frequency f . Aswe vary f away from f , the Hamiltonian varies smoothly, deviating from the form of the target Hamiltonian (e.g., thepositive and negative nearest neighbor hoppings become unequal in magnitude). Throughout this variation, so longas E lies in a gap, the topological properties are unchanged and the topological edge states continue to exist. Thus,the f -dependent response of the circuit behaves like a bandstructure. For small m , the circuit exhibits a finite-widthtopological bandgap in f -space; tuning m closes this bandgap, and causes the topological edge states to disappear. SUPPLEMENTARY NOTE 2: FINITE-SIZE EFFECTS IN THE TIGHT-BINDING MODEL
As shown in Fig. 1(a) of the main text, the bandgap of the infinite tight-binding model closes at m = 6. However,as shown in Fig. 1(b), for a finite lattice of the same size as our experimental sample the gap closing occurs around m = 4; moreover, the edge states appear at small values of m . E m -3-2-10123 S u r f a c e c o n c e n t r a t i o n FIG. S2: Eigenvalue pairs closest to E = 0 for lattices with { , , , , , } unit cells along x , z (lengths labelled in the upperright corner), and one unit cell along y , w (with periodic boundary conditions). Black curves show the bulk band edges. E = 0) for a series of lattices with { , , , , , } unit cells along both x and z . Along y and w , theselattices remain one unit cell wide with periodic boundary conditions (equivalent to k y = k w = 0). The colors indicatewhether the eigenstate is concentrated on the surface (red) or in the bulk (blue). As the size of the lattice increases,the eigenvalues at large m (in the conventional insulator regime) approach the predicted bulk band edges, while theeigenvalues in the topological insulator regime spread over a larger range of m corresponding to the topologicallynontrivial gap. SUPLEMENTARY NOTE 3: FREQUENCY RESPONSE FOR DIFFERENT MASS PARAMETERS
Fig. S3 plots the frequency dependence of the LDOS measure Re[ Z r ], averaged over on surface and bulk sites, for m = 0 , , . . . ,
8. In Fig. 3(b)–(e) and (g)–(j), only a few values of m were shown for brevity. The experimental resultsare shown in Fig. S3(a), and the corresponding simulation results are shown in Fig. S3(b). The discrepancies betweenexperimental and simulation results can be explained by the disorder in the experimental samples: according to the m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 0 m = 1 m = 2 b Simulations a Experiments m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 Frequency (kHz)Frequency (kHz) R e [ ⟨⟩ ] ( a . u . ) Z r R e [ ⟨⟩ ] ( a . u . ) Z r FIG. S3: (a) Measured frequency dependence of Re[ Z r ] averaged over surface and bulk sites, for m = 0 , , . . . ,
8. The measure-ments are taken over sites in the 2D plane ( y, w ) = (1 , f is indicated by the vertical dotted lines. (b) The correspondingcircuit simulation results. SUPPLEMENTARY NOTE 4: DETAILS OF CIRCUIT SIMULATIONS
All circuit simulations are performed with ngspice , a free software circuit simulator. To model circuit losses, wetreat each capacitor and inductor as having a 5Ω resistance in series with the purely capacitive or inductive element.This has the same order of magnitude as the resistances stated in the data sheets for the individual circuit components;we pick a uniform value of 5Ω to represent the various hard-to-characterise resistances in the PCBs.The simulations are performed like the experiments: i.e., we apply a sine wave voltage source to each node, usethe steady-state voltage and current to determine the complex impedance, and hence obtain the LDOS on each site.In Fig. S4, we show how the simulated frequency-dependent LDOS measure is affected by circuit resistance. Theupper row shows the outcomes for 5Ω resistances, identical to the results shown in Fig. S3 and Fig. 3 of the main text(which are a good match for experimental results). The lower row shows the results with an order of magnitude lowerresistance (0 . E = 0 , m = 0 , , E = 1 , m = 0 (using 5Ω resistances),corresponding to the experimental results shown in Fig. 2(c)–(f) of the main text. R e [ ⟨⟩ ] ( a . u . ) Z r R e [ ⟨⟩ ] ( a . u . ) Z r Frequency (kHz) m = 0 m = 1 m = 2 m = 0 m = 1 m = 25 Ω Ω FIG. S4: Simulated frequency-dependent LDOS measure for different circuit component resistances: 5Ω (upper row), corre-sponding to the first row of Fig. S3(b), and 0 .
5Ω (lower row). a b c d m = 0, E = 0 m = 0, E = 1 m = 4, E = 0 m = 8, E = 0 wywywy wy z x z x z x z x FIG. S5: Simulated LDOS distribution for E = 1 , m = 0 and E = 0 , m = 0 ,,
5Ω (lower row). a b c d m = 0, E = 0 m = 0, E = 1 m = 4, E = 0 m = 8, E = 0 wywywy wy z x z x z x z x FIG. S5: Simulated LDOS distribution for E = 1 , m = 0 and E = 0 , m = 0 ,, ,,