Experimental observation of edge-dependent quantum pseudospin Hall effect
Huanhuan Yang, Lingling Song, Yunshan Cao, X. R. Wang, Peng Yan
EExperimental observation of edge-dependent quantum pseudospin Hall e ff ect Huanhuan Yang , Lingling Song , Yunshan Cao , X. R. Wang , ∗ and Peng Yan † School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices,University of Electronic Science and Technology of China, Chengdu 610054, China and Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
It is a conventional wisdom that the helical edge states of quantum spin Hall (QSH) insulator are particularlystable due to the topological protection of time-reversal symmetry. Here, we report the first experimental ob-servation of an edge-dependent quantum (pseudo-)spin Hall e ff ect by employing two Kekul´e electric circuitswith molecule-zigzag and partially-bearded edges, where the chirality of the circulating current in the unit cellmimics the electron spin. We observe a helicity flipping of the topological in-gap modes emerging in oppositeparameter regions for the two edge geometries. Experimental findings are interpreted in terms of the mirrorwinding number defined in the unit cell, the choice of which exclusively depends on the edge shape. Our worko ff ers a deeper understanding of the boundary e ff ect on the QSH phase, and pave the way for studying thespin-dependent topological physics in electric circuits. A paradigm in the topological band insulator family [1, 2] isthe quantum spin Hall (QSH) insulator, which has an insulat-ing gap in the bulk, but supports gapless helical states on theboundary [3–6]. QSH insulators are characterized by the topo-logical Z invariant, defined in the presence of time-reversalsymmetry. Because of the symmetry protection, the helicaledge states are robust against the electronic backscattering [5–9], ushering in a new era in spintronics and quantum comput-ing [10–13]. Counterintuitively, Freeney et al. recently re-ported an edge-dependent topology in artificial Kekul´e lattices[14]. The mechanism is that the edge geometries of samplesdetermine the choice of the unit cell, and further dictate thevalue of topological invariants [15–19]. However, the experi-mental evidence of an edge-dependent quantum (pseduo-)spinHall e ff ect is still lacking.Recently, the topolectrical circuit springs up as a powerfulplatform to study the fundamental topological physics [20–29], since simple inductor-capacitor (LC) networks can fullysimulate the tight-binding model in condensed matter physics.In this Letter, we fabricate two kinds of Kekul´e LC circuitswith molecule-zigzag and partially-bearded edges (see Fig.1). By measuring the node-ground impedance and monitor-ing the spatiotemporal voltage signal propagation, we observethe quantum pseudospin Hall e ff ect emerging in the oppositeparameter regions with flipped helicities for the two di ff erentedge terminations, where the chirality of the circulating cur-rent in the unit cell mimics the spin. Quantized mirror windingnumber is proposed to explain our experimental findings.We consider two finite-size artificial Kekul´e circuits withmolecule-zigzag and partially-bearded edge terminations, asshown in Figs. 1 (a) and 1(b), respectively. The circuits con-sist of two types of capacitors C A , C B and inductor L . Theresponse of the circuit at frequency ω is given by Kirchho ff ’slaw: I a ( ω ) = (cid:88) b J ab ( ω ) V b ( ω ) , (1)where I a is the external current flowing into node a , V b is thevoltage of node b , and J ab ( ω ) = i ω (cid:104) C ab + δ ab ( (cid:80) n C an − ω L a ) (cid:105) is the circuit Laplacian, with C ab the capacitance between nodes a and b . Based on Eq. (1), one can explicitly ex-press the circuit Laplacian J I ( ω ) and J II ( ω ) of the two cir-cuits in Figs. 1(a) and 1(b) [30]. At the resonant frequency ω = / √ (2 C A + C B ) L , the diagonal elements of circuitLaplacians vanish, and the circuit model is equivalent to thetight-binding model with − ω C A and − ω C B being two hop-ping coe ffi cients.We fabricate two printed circuit boards with di ff erent edgegeometries displayed in Figs. 2(a) and 2(b), respectively. Inexperiments, we adopt C A = C B =
10 nF or 0 . L = µ H (all circuit elements have a 2% tolerance), with theresonant frequency being ω / π = / [2 π √ (2 C A + C B ) L ] = .
65 kHz or 556 .
13 kHz, respectively.We measure the distributions of impedance between eachnode and the ground by an analyser (Keysight E4990A),with the results plotted in Figs. 2(c)-2(f). For deviceswith molecule-zigzag edge at C A / C B = . C A / C B =
10 [Fig. 2(f)], we observethat the impedance concentrates on the sample edge, the valueof which is larger than one thousand Ohms, indicating the ex-istence of edge states. Theoretically, the impedance betweennode a and b is given by [26]: Z ab = V a − V b I ab = (cid:88) n | ψ n , a − ψ n , b | j n , (2)where | ψ n , a − ψ n , b | is the amplitude di ff erence between a and b nodes of the n th eigenstate, and j n is the n -th eigenvalue.We plot the numerical results in the insets of Figs. 2(c)-2(f),showing an excellent agreement with the experimental mea-surements.It’s known that the QSH insulator allows bidirectional prop-agation states along the boundary. However, we cannot di-rectly observe the time-resolved wave dynamics by measuringthe impedance. To solve this problem, we monitor and recordthe spatiotemporal voltage signal in the circuits. Specifically,we impose a sinusoidal voltage signal v ( t ) = v sin( ω t ) withthe amplitude v = a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n
14 5 6 2 37 8 910 11 12 13 14 15 16 17 18 1920 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4042 43 44 45 46 47 48 49 50 5152 53 54 55 56 57 58 59 6085 86 87 88 89 90 91 9294 95 96 97 98 99 100 101 61 4162 63 64 65 66 67 68 69 70 71 7273 7484 75 76 77 78 79 80 81 82 8393102 103 104 105 106 107 108132 133 134 135 136 137139 140 141 142 143152 153 154163 164166 168 167162 165155 156157 158 159 160 161144 145 146 147148 149 150 151 138109 110 111 112 113 114 115116 117 118 119 120 121 122 123124 125 126 127 128 129 130 131 C A C B C A C B (a) (b) L LC A C B L LC B C A C A = FIG. 1: Illustration of two artificial Kekul´e LC circuits with (a) molecule-zigzag and (b) partially-bearded edge terminations. Each node isgrounded by inductors and capacitors with the configuration shown in the inset. Dashed red hexagon and rhombus represent the approximateunit cells for the two di ff erent edge shapes. deed observe a strong voltage response along both directionsof the device edge. It is noted that the voltage signal decaysvery fast away from the voltage source, because of the lowquality factor ( Q = −
50) of the inductors. In Figs. 3(b)and 3(e), we plot the theoretical steady-state voltage distribu-tions with higher Q -factor inductors (we set Q = J I ( ω ) and J II ( ω ), we obtain the admit-tance spectrum j n and the corresponding wave functions ψ n , m ,shown in Fig. S1 in Supplemental Material [30]. For circuitsof molecule-zigzag edge, with C A / C B = .
1, isolated statesemerge in the gap of the bulk admittance spectrum, whichcorrespond to the edge states. When C A / C B =
10, only arebulk states identified. For circuits of partially-bearded edge,on the contrary, we find that the edge states emerge in the op-posite capacitance ratio, i.e., C A / C B =
10. For C A / C B = . / edge state. However, this surface / edgestate is trivial because it is sensitive to impurities, defects, anddisorder, which is not compatible with our experimental find-ings. There thus must be a topological reason for the emergingbidirectional edge states we observed. To justify this point ofview, we employ the mirror winding number ( n + , n − ) definedin the unit cell with n ± = − π (cid:73) ddk ⊥ arg(det Q k ±⊥ ) dk ⊥ (3)in the presence of chiral symmetry. The analytical expressionof matrices Q k ±⊥ can be found in Sec. II of Supplemental Ma-terial [30]. The choice of the unit cell depends on the shape ofsample edge. As shown in Figs. 1(a) and 1(b), the dashed redhexagon and rhombus represent the approximate unit cells forthe two di ff erent edge geometries, respectively. For the circuitwith molecule-zigzag edge, we obtain ( n + , n − ) = (1 , −
1) when C A / C B < ,
0) when C A / C B >
1. Therefore, we canobserve the topological edge states when C A / C B <
1. For thecircuit with partially-bearded edge, the case is adverse to theformer: ( n + , n − ) = (0 ,
0) when C A / C B < , −
1) when C A / C B >
1, indicating that the topological edge states arise inthe region of C A / C B > C A / C B <
1, wefind three isolated modes in the band gap [see Fig. 4(b)].The red and blue spectrums represent the helical edge statesbecause of the opposite group velocity. Interestingly, wecan define the circulating bond currents inside the unit cell: i m → n = Im[ ψ ∗ m ψ n ] [35–37] with their flowing direction plot-ted in the right side of Figs. 4(a) and 4(d). We find that the (f) Z/Z max (c) (d)(a) (b)(e) C A / C B =0.1C B C A C A / C B =10 C A / C B =0.1 C A / C B =10 L Ω Z 1425 Ω Z 04 k Ω Z45 Ω Z FIG. 2: Printed circuit boards with (a) molecule-zigzag and (b) partially-bearded edges. Yellow stars indicate the position of signal sourcesin the voltage measurements. (c)-(f) Experimental measurements of the spatial distribution of impedance between each node and the ground.Insets: numerical results. C A / C B = . C A / C B = v/v max (b)(a) (e)(d) Theory (Q=1000) (c)(f)
S SS S SS Simulation (Q=1000)Experiment (Q=25~50)
FIG. 3: Experimental measurements of the steady-state voltage distribution in the devices with (a) molecule-zigzag ( C A / C B = .
1) and (d)partially-bearded ( C A / C B =
10) edges. (b)(e) Theoretical calculation with a higher Q -factor ( Q = ff erent times, with the blue star indicating the position of the signal source, and the red and blue arrows representing thepropagation direction of the voltage signal with pseudospin up and down, respectively. chirality of the circulating current in the unit cell are oppositefor the in-gap red and blue bands, which mimics the electronspin-up and spin-down states, respectively. This observationis reminiscent of the spin-momentum locking in the QSH ef-fect. Brown line denotes the flat band localized in the bot- tom zigzag edge of the ribbon [38]. For C A / C B >
1, there isno in-gap energy spectrum expect for the flat band, see Fig.4(c). For the ribbon with partially-bearded edge, the edgemodes with flipped helicity however only appear in the regionof C A / C B > C A / C B <1 (a) (b) (c)(d) (e) (f) pseudospin uppseudospin down pseudospin up pseudospin downflat band flat bandflat band flat bandxy pseudospin downpseudospin uppseudospin downpseudospin up C A / C B >1 FIG. 4: (a) Schematic plot of a ribbon with molecule-zigzag edge (top) and graphene-zigzag edge (bottom). The ribbon is periodic along ˆ x direction and contains 40 unit cells along ˆ y direction. Insets: the pesudospin is denoted by the chirality of the circulating current in the unitcell. The band structure of the ribbon with two di ff erent capacitor ratios: (b) C A : C B = . C A : C B = .
9. Red and blue linesrepresent the dispersive edge states with pesudospin up and down counterpropagating along the top edge. Brown line denotes the localizededge mode in the bottom boundary. (d) Illustration of a ribbon with partially-bearded edge (top) and graphene-zigzag edge (bottom). The bandstructure of the ribbon with two di ff erent capacitor ratios: (e) C A : C B = . C A : C B = . explain the numerical calculations and experimental measure-ments.To understand the helicity flipping, we map the six-band circuit model to the four-band Bernevig-Hughes-Zhang(BHZ) model originally proposed for HgTe quantum wells[5, 6]. To this end, we express J ab ( ω ) = i H ab ( ω ), in which H ( ω ) can be viewed as a hermitian tight-binding Hamilto-nian. Taking the molecule-zigzag unit cell as an example, onecan write the Hamiltonian of an infinite Kekul´e circuit at res-onant frequency as below: H = − ω C A (cid:88) (cid:104) i , j (cid:105) c † i c j − ω C B (cid:88) (cid:104) i (cid:48) , j (cid:48) (cid:105) c † i (cid:48) c j (cid:48) , (4)where c i is the annihilation operator at site i , and (cid:104) i , j (cid:105) and (cid:104) i (cid:48) , j (cid:48) (cid:105) run over nearest-neighboring sites inside and betweenhexagonal unit cells, respectively. Diagonalizing Hamiltonian(4), we obtain six bands, two of which are high-energy bandswith the phase transition point C A / C B = Γ point, as shown in Fig. S3 in Supplemental Material [30].We further note that the high-energy parts are irrelevant to thetopological phase transition. By performing a unitary trans-formation H (cid:48) = U † H U on H around the Γ point [30], weseparate the two high-energy orbits and obtain the low-energye ff ective BHZ-type Hamiltonian as: H e ff ( k ) = − ω (cid:32) H ( k ) 00 H ∗ ( − k ) (cid:33) , (5)with H ( k ) = (cid:32) M − Bk Ak − A ∗ k + − M + Bk (cid:33) , where M = C B − C A , A = − iC B , B = C B , k = k x + k y , and k ± = k x ± ik y .For the circuit with partially-bearded unit cell, we get thesimilar low-energy e ff ective Hamiltonian, but with a di ff er-ent M = C A − C B . The sign of parameter M is opposite forthe two edge geometries, leading to the helicity flipping ofthe edge states in the opposite parameter regions based on theband inversion mechanism. We thus conclude that, althoughKirchho ff ’s law is rather di ff erent from the Sch¨ordinger equa-tion, the underlying physics between our circuit model and thequantum well model is actually quite similar. The parameter M can be viewed as an e ff ective spin-orbit coupling (SOC)associated with the pseudo spin, which is di ff erent from theintrinsic one originating from the relativistic e ff ect. Whereas,the SOC in circuit is more controllable and can be very large,enabling the observation of the quantum pseduo-spin Hallstates at room temperature.In summary, we reported an edge-dependent quantum pseu-dospin Hall e ff ect in topolectric circuits. We showed that thepesudospin is represented by the chirality of the circulatingcurrent in the unit cell. Through the impedance measurementand spatiotemporal voltage signal detection assisted by cir-cuit simulations, we directly identified the helical nature ofthe edge states. The emerging topological phases were char-acterized by mirror winding numbers, which depend on theshape of device edge. Our work uncovers the importance ofthe edge geometry on the QSH e ff ect, and opens a new path-way of using circuits to simulate the spin-dependent topolog-ical physics, that may inspire research in other solid-state sys-tems in the future.This work was supported by the National Natural ScienceFoundation of China (Grants No. 12074057, No. 11604041,and No. 11704060). X. R. Wang acknowledges the finan-cial support of Hong Kong RGC (Grants No. 16300117,16301518, and 16301619). ∗ Corresponding author: [email protected] † Corresponding author: [email protected][1] M. Z. Hasan and C. L. Kane, Colloquium: Topological insula-tors, Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Topological insulators and supercon-ductors, Rev. Mod. Phys. , 1057 (2011).[3] C. L. Kane and E. J. Mele, Quantum Spin Hall E ff ect inGraphene, Phys. Rev. Lett. , 226801 (2005).[4] C. L. Kane and E. J. Mele, Z Topological Order and the Quan-tum Spin Hall E ff ect, Phys. Rev. Lett. , 146802 (2005).[5] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum SpinHall E ff ect and Topological Phase Transition in HgTe QuantumWells, Science , 1757 (2006).[6] M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buhmann,L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Quantum SpinHall Insulator State in HgTe Quantum Wells, Science , 766(2007).[7] M. K¨onig, M. Baenninger, A. G. F. Garcia, N. Harjee, B.L. Pruitt, C. Ames, P. Leubner, C. Br¨une, H. Buhmann, L.W. Molenkamp, and D. Goldhaber-Gordon, Spatially ResolvedStudy of Backscattering in the Quantum Spin Hall State, Phys.Rev. X , 021003 (2013).[8] P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian,A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Topo-logical surface states protected from backscattering by chiralspin texture, Nature (London) , 1106 (2009).[9] H. Chen, H. Nassar, A. N. Norris, G. K. Hu, and G. L. Huang,Elastic quantum spin Hall e ff ect in kagome lattices, Phys. Rev.B , 094302 (2018).[10] A. Roth, C. Br¨une, H. Buhmann, L. W. Molenkamp, J. Ma-ciejko, X.-L. Qi, and S.-C. Zhang, Nonlocal Transport in theQuantum Spin Hall State, Science , 294 (2009).[11] C. Br¨une, A. Roth, H. Buhmann, E. M. Hankiewicz, L. W.Molenkamp, J. Maciejko, X.-L. Qi, and S.-C. Zhang, Spin po-larization of the quantum spin Hall edge states, Nat. Phys. ,485 (2012).[12] S. Hart, H. Ren, T. Wagner, P. Leubner, M. M¨uhlbauer, C.Br¨une, H. Buhmann, L. W. Molenkamp, and A. Yacoby, In-duced superconductivity in the quantum spin Hall edge, Nat.Phys. , 638 (2014).[13] S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi,R. J. Cava, and P. Jarillo-Herrero, Observation of the quantumspin Hall e ff ect up to 100 kelvin in a monolayer crystal, Science , 76 (2018).[14] S. E. Freeney, J. J. van den Broeke, A. J. J. Harsveld van derVeen, I. Swart, and C. M. Smith, Edge-Dependent Topology inKekul´e Lattices, Phys. Rev. Lett. , 236404 (2020).[15] L. Fu, Topological Crystalline Insulators, Phys. Rev. Lett. ,106802 (2011).[16] R.-J. Slager, A. Mesaros, V. Juriˇci´c, and J. Zaanen, The spacegroup classification of topological band-insulators, Nat. Phys. , 98 (2013).[17] T. Kariyado and X. Hu, Topological States Characterized by Mirror Winding Numbers in Graphene with Bond Modulation,Sci. Rep. , 16515 (2017).[18] T. Cao, F. Zhao, and S. G. Louie, Topological Phases inGraphene Nanoribbons: Junction States, Spin Centers, andQuantum Spin Chains, Phys. Rev. Lett. , 076401 (2017).[19] Y.-L. Lee, F. Zhao, T. Cao, J. Ihm, and S. G. Louie, Topologi-cal Phases in Cove-Edged and Chevron Graphene Nanoribbons:Geometric Structures, Z Invariants, and Junction States, NanoLett. , 7247 (2018).[20] C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W.Molenkamp, T. Kiessling, and R. Thomale, Topolectrical Cir-cuits, 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,and R. Thomale, Topolectrical-circuit realization of topologicalcorner modes, Nat. Phys. , 925 (2018).[22] T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, and R. Thomale,Chiral Voltage Propagation and Calibration in a TopolectricalChern Circuit, Phys. Rev. Lett. , 247702 (2019).[23] W. Zhu, Y. Long, H. Chen, and J. Ren, Quantum valley Halle ff ects and spin-valley locking in topological Kane-Mele circuitnetworks, Phys. Rev. B , 115410 (2019).[24] Y. Lu, N. Jia, L. Su, C. Owens, G. Juzeli¯unas, D. I. Schuster,and J. Simon, Probing the Berry curvature and Fermi arcs of aWeyl circuit, Phys. Rev. B , 020302(R) (2019).[25] Y. Yang, D. Zhu, Z. H. Hang, Y. D. Chong, Observation ofantichiral edge states in a circuit lattice, arXiv:2008.10161.[26] H. Yang, Z.-X. Li, Y. Liu, Y. Cao, and P. Yan, Observation ofsymmetry-protected zero modes in topolectrical circuits, Phys.Rev. Research , 022028(R) (2020).[27] L. Song, H. Yang, Y. Cao, and P. Yan, Realization of the square-root higher-order topological insulator in electric circuits, NanoLett. , 7566 (2020).[28] M. Ezawa, Braiding of Majorana-like corner states in electriccircuits and its non-Hermitian generalization, Phys. Rev. B ,045407 (2020).[29] M. Ezawa, Electric circuits for non-Hermitian Chern insulators,Phys. Rev. B , 081401(R) (2019).[30] See Supplemental Material at http: // link.aps.org / supplemental / for the form of the circuit Laplacian (Sec. I), the derivation ofthe mirror winding number (Sec. II), and the mapping to theBHZ model (Sec. III), which includes Refs. [5, 17, 31].[31] Y. Yang, Z. Jia, Y. Wu, Z.-H. Hang, H. Jiang, and X. C. Xie,Gapped topological kink states and topological corner states ingraphene, Sci. Bull. / LTspice.[33] I. Tamm, ¨Uber eine m¨ogliche Art der Elektronenbindung anKristalloberfl¨achen, Phys. Z. Sowjetunion , 849 (1932).[34] W. Shockley, On the surface states associated with a periodicpotential, Phys. Rev. , 317 (1939).[35] Y. Zhang, J.-P. Hu, B. A. Bernevig, X. R. Wang, X. C. Xie, andW. M. Liu, Quantum blockade and loop currents in graphenewith topological defects, Phys. Rev. B , 155413 (2008).[36] Y. Zhang, J.-P. Hu, B. A. Bernevig, X. R. Wang, X. C. Xie, andW. M. Liu, Localization and the Kosterlitz-Thouless Transitionin Disordered Graphene, Phys. Rev. Lett. , 106401 (2009).[37] L.-H. Wu and X. Hu, Topological Properties of Electrons inHoneycomb Lattice with Detuned Hopping Energy, Sci. Rep. , 24347 (2016).[38] M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, Pe-culiar localized state at zigzag graphite edge, J. Phys. Soc. Jpn. , 1920 (1996). Supplemental Material
Experimental observation of edge-dependent quantum pseudospin Hall e ff ect Huanhuan Yang , Lingling Song , Yunshan Cao , X. R. Wang , ∗ and Peng Yan † School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University ofElectronic Science and Technology of China, Chengdu 610054, China and Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
I. I. CIRCUIT LAPLACIAN
In this section, we show the circuit Laplacian of the two circuits in the main text. For the circuit with molecule-zigzag edgegeometry: J I ( ω ) = ω J − J A − J A . . . J − J A . . . − J A J . . . − J A J . . . J . . . − J A J . . .... ... ... ... ... ... . . . × , (6)with J = C A + C B − / ( ω L ), J A = C A , and J B = C B . For the circuit with partially-bearded edge geometry: J II ( ω ) = ω J − J A . . . J . . . J . . . J . . . J . . . − J A J . . .... ... ... ... ... ... . . . × . (7)Diagonalizing J I ( ω ) and J II ( ω ), we obtain the admittance spectrum j n and the corresponding wave functions ψ n , m . To directlycompare with the experimental results, we adopt C A = L = µ H, and C B =
10 nF or 0 . C A / C B = .
1, we find a series of isolated states in the gap of the admittance spectrum (blue dots), whichcorrespond to the helical edge states, shown in Fig. 5(a). We confirm that all blue dots are edge states (not shown). In the regimeof C A / C B =
10, only the bulk states exist, see Fig. 5(b). However, for the circuits with partially-bearded edges, we find that theedge states emerge in the opposite region, i.e., C A / C B =
10, as shown in Fig. 5(d). In the case of C A / C B = .
1, we can only seethe bulk states, see Fig. 5(c).
II. II. MIRROR WINDING NUMBER
In this section, we calculate the topological invariant mirror winding number to characterize the helical edge states. If weexpress J ab ( ω ) = i H ab ( ω ), H ( ω ) can be viewed as a tight-binding Hamiltonian. With the appropriate unit cells in Fig. 6 (unitcell I for the circuit with molecule-zigzag edge, and unit cell II for the circuit with partially-bearded edge), one can write theHamiltonian of an infinite Kekul´e circuit as: H = ω h h − Q k h h − Q † k h
00 0 h , (8) (a) (b) (c) (d) C A / C B =0.1 C A / C B =10 C A / C B =0.1 C A / C B =10 FIG. 5: Admittance spectrum at di ff erent edges and parameters. The blue and black dots denote the edge states and bulk states, respectively.Insets: spatial distribution of wave functions with the number of state indicated by the arrows. (a)(b) molecule-zigzag edge with C A / C B = . C A / C B =
10. (c)(d) partially-bearded edge with C A / C B = . C A / C B = with the matrix elements h = C A + C B − / ( ω L ), Q I k = C B XY C A C A C A C B XY C A C A C A C B Y (9)for molecule-zigzag edge, where X = e i k · a , Y = e i k · a with a = √ x and a = √ ˆ x + ˆ y being the two basic vectors, and Q II k = C B C A C A C A Y C B C A XYC A XY C A Y C B (10)for partially-bearded edge. C A C B a a II
26 1 354 I (a) (b) II IIII
FIG. 6: (a) Appropriate unit cells for molecule-zigzag and partially-bearded edges. The orange arrows indicate the two basic vectors. (b) Themirror winding numbers ( n + , n − ) as a function of the capacitance ratio C A / C B . At resonant frequency ω = / √ (2 C A + C B ) L , the diagonal element h vanishes, and the Hamiltonian can be simplified as: H = − ω (cid:32) Q k Q † k (cid:33) , (11)where Q k is Q I k (Eq. 9) for molecule-zigzag edge, and Q II k (Eq. 10) for partially-bearded edge.Regarding the momentum k parallel to the unit vector a defined as a free parameter, the system can be viewed as an e ff ective1D model, to which one can assign the winding number as: n ( k (cid:107) ) = − π (cid:73) ddk ⊥ arg(det Q k (cid:107) , k ⊥ ) dk ⊥ (12)For k (cid:107) =
0, the mirror symmetry with the mirror plane perpendicular to a enables us to decompose the Hamiltonian (11)into even and odd sectors H k ±⊥ , where k is replaced by k ⊥ . Concretely, Q k can be decomposed into even and odd sectors Q k ±⊥ .Then, we can assign winding numbers for the even and odd sectors separately by substituting Q k + ⊥ and Q k −⊥ into Eq. 12, whichconstitutes the mirror winding number ( n + , n − ) [17].At k (cid:107) = Q I k is decomposed into Q I k + ⊥ = C B Y √ C A √ C A C A + C B Y , Q I k −⊥ = C B Y − C A , (13)and Q II k is decomposed into Q II k + ⊥ = (cid:32) C B √ C A √ C A Y C B + C A Y (cid:33) , Q II k −⊥ = C B − C A Y . (14)Using Eq. 12, we can compute the mirror winding number ( n + , n − ) immediately, with the results plotted in Fig. 6(b). Forthe circuit with molecule-zigzag and partially-bearded edge, the topological edge states appear in the region of C A / C B < C A / C B > III. III. ANALOGY TO THE QUANTUM SPIN HALL EFFECT
In this section, we map our six-band circuit model to the four-band Bernevig-Hughes-Zhang (BHZ) model forCdTe / HgTe / CdTe quantum wells. (a) (b) (c)
FIG. 7: Admittance spectrum for di ff erent capacitance ratio. (a) C A / C B = .
9, (b) C A / C B =
1, and (c) C A / C B = . We calculate the energy spectrum of Eq. 8 for three capacitance ratios, plotted in Fig. 7. The spectrum are gapless when C A / C B =
1, and the phase transition point is at the Γ point. Two bands of the spectrum are high-energy parts, which areirrevelant to the topological phase transition. Therefore, the six-band Hamiltonian (8) can be downfolded into the four-band oneby omitting the two high-energy bands [26].Taking the circuit with molecule-zigzag edge geometry as an example, we impose a unitary transformation H (cid:48) = U † H U onHamiltonian H (8) to separate the high-energy parts of the Hamiltonian with the matrix: U = √ e i π e i π e i π e i π e i π e i π e i π e i π − e i π e i π e i π e i π e i π e i π e i π e i π − e i π e i π e i π e i π e i π e i π e i π e i π − . (15)Then, imposing Taylor expansion on each matrix element of H (cid:48) around the Γ point to 2nd-order terms, we obtain: H Γ = − ω δ C − C B k − iC B k − h − iC B k + h iC B k + − δ C + C B k h h − C B k − h ∗ δ C − C B k − iC B k + − iC B k − h h ∗
24 32 iC B k − δ C + C B k h
45 32 C B k + iC B k − h ∗
25 32 iC B k + h ∗ − C A − C B + C B k h ∗ − C B k + h ∗