Chern Insulators for Electromagnetic Waves in ElectricalCircuit NetworksbyRafael HaenelB.Sc., Technical University of Berlin, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2019c© Rafael Haenel, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Chern Insulators for Electromagnetic Waves in Electrical Circuit Net-workssubmitted by Rafael Haenel in partial fulfillment of the requirements for the de-gree of Master of Science in Physics.Examining Committee:Marcel Franz, PhysicsSupervisorDouglas Bonn, PhysicsSupervisory Committee MemberiiAbstractPeriodic networks composed of capacitors and inductors have been demonstratedto possess topological properties with respect to incident electromagnetic waves. Inthis thesis, we develop an analogy between the mathematical description of wavespropagating in such networks and models of Majorana fermions hopping on a lat-tice. Using this analogy we propose simple electrical network architectures thatrealize Chern insulating phases for electromagnetic waves. Such Chern insulatingnetworks have a bulk gap for a range of signal frequencies that is easily tunableand exhibit topologically protected chiral edge modes that traverse the gap and arerobust to perturbations. The requisite time reversal symmetry breaking is achievedby including a class of weakly dissipative Hall resistor elements whose physicalimplementation we describe in detail.iiiLay SummaryAn electrical circuit comprised of an inductor and capacitor exhibits a single reso-nance frequency. When driven at that frequency, the circuit can generate highervoltages than fed into it. We propose circuits of a periodically repeated two-dimensional pattern. Instead of a single resonance, these circuits possess a numberof resonance frequencies that is proportional to the area of the two-dimensionalnetwork. We uncover certain topological properties of these resonance spectra.For a range of frequencies, the existence of resonances relies on the presence ofedges in the circuit network. Here, the voltage response to a resonant source islarge only at the boundaries. Pulses at these frequencies travel in one directionalong the boundary only. Distortions of the boundary and component tolerancesdo not yield qualitatively different results. These findings are in analogy to thephysics of so called Chern Insulators in the context of condensed matter physics.ivPrefaceThis thesis is based on the publicationChern insulators for electromagnetic waves in electrical circuit networks,Rafael Haenel, Timothy Branch, and Marcel Franz, Physical Review B 99 235110(2019).The publication resulted from a combined effort of the author and Prof. MarcelFranz. Experimental efforts, that are not part of this thesis, were undertaken byTimothy Branch.Chapters 1, 3, 4, 5 are taken from above publication and were amended withfour additional figures (Figs. 3.4, 3.6, 4.2, 4.4) and slightly more detailed formu-lations by the author. Chapter 2 has been drafted by the author to supplement theintroduction with a broader context.All figures were prepared by the author and all underlying data was computedby the author.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Symmetry protected topological matter . . . . . . . . . . . . . . . . 42.1 Tight-binding models . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Majorana tight-binding models . . . . . . . . . . . . . . . 62.2 Classification of symmetry protected topological states of matter . 72.2.1 Dirac model method . . . . . . . . . . . . . . . . . . . . 72.2.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Classification of Dirac mass gaps . . . . . . . . . . . . . 122.2.4 Mass classification for trivial insulators and Chern insulators 13vi2.3 Do we need quantum mechanics? . . . . . . . . . . . . . . . . . . 153 Chern insulators from Resistor-Inductor-Capacitor (RLC) networks 173.1 General setup and a toy model . . . . . . . . . . . . . . . . . . . 173.2 Chern insulator on the square lattice . . . . . . . . . . . . . . . . 273.3 Chern insulator on the honeycomb lattice . . . . . . . . . . . . . 314 Hall resistor implementation . . . . . . . . . . . . . . . . . . . . . . 344.1 Hall-resistor implementation by classical Hall effect . . . . . . . . 344.1.1 Hall effect, galvanic coupling . . . . . . . . . . . . . . . 344.1.2 Hall effect, capacitive coupling . . . . . . . . . . . . . . . 364.2 Ideal Hall-resistor from operational amplifiers . . . . . . . . . . . 395 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . 42Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44viiList of TablesTable 2.1 Cartan-Altland-Zirnbauer (CAZ) classification of symmetry pro-tected topological matter [1]. . . . . . . . . . . . . . . . . . . 13viiiList of FiguresFigure 3.1 (a) Square RLC lattice toy model realizing the Chern insulatorfor electromagnetic (EM) waves. A unit cell is marked by bygray background. (b) The Hall resistor element with four sideterminals and one central terminal. . . . . . . . . . . . . . . 17Figure 3.2 (a) Effective Majorana tight-binding model corresponding tothe RLC network toy model with ideal Hall elements. Tun-neling matrix elements between sublattices a,b,c are labeledby straight lines, arrows indicate directionality. Here, γ =RH√C/L. (b) Sketch of boundary conditions used for calcu-lations in the strip geometry. . . . . . . . . . . . . . . . . . . 19Figure 3.3 (a) Bulk band structure of the circuit network. The dashed linecorresponds to γ = RH√C/L = 0 while the solid line corre-sponds to γ = 0.25 which gives a gap ∆ = ω0, where ω0 =1/√LC. (b) Spectrum of a strip of width W = 10 with openboundary conditions along y for γ = 0.25. Boundary condi-tions are chosen as indicated in Fig. 3.2(b). The color scaleindicates the average distance 〈y〉 measured from the centerof the strip of the eigenstate belonging to the eigenvalue ωk.The states inside the bulk gap ∆ are localized near the oppositeedges of the system. . . . . . . . . . . . . . . . . . . . . . . 20ixFigure 3.4 Spectral function (3.9) for a circuit with parameters γ = 0.25 inthe absence (left) and presence (right) of 30 percent box disor-der for R,L,C parameters. The lifetime broadening parameteris chosen as η = 0.03ω0. . . . . . . . . . . . . . . . . . . . . 23Figure 3.5 Voltage response V respr (ω) induced by a current with in-gapfrequency ω = ∆/2 injected at a node marked by green crossof the 10×10 network with γ = 0.25 for various values of thedissipative resistance R characterized by parameter ε = R/RH . 25Figure 3.6 (left) Spectrum of circuit for γ = 0.25 and ε = 0.1. (right)Voltage as measured in the bulk (black) or at the edge (blue)after current has been injected at an edge-site. . . . . . . . . . 25Figure 3.7 Time evolution of a localized Gaussian wave packet of fre-quency width (∆ω)/ω0 = 0.35 excited at the boundary. Thesimulation models disorder by assuming a capacitor and in-ductor device tolerance of 30%. Colorscale corresponds to theweight of the wavefunction on the circuit node. The signaltravels along the boundary and circumvents the boundary de-fect indicated in white. . . . . . . . . . . . . . . . . . . . . . 26Figure 3.8 Plot of voltage profile along boundary sites as function of timethat shows the constant group velocity of the wave packet. . . 26Figure 3.9 Square RLC lattice network with four-terminal Hall elements.Voltage nodes (red) and currents (green) are labeled for the unitcell (gray background) at position r. . . . . . . . . . . . . . 27Figure 3.10 Eigenmode spectrum of the network for g= 1/√2, with (solidlines) and without (dashed lines) the Hall element. The gap pa-rameter is γ =√C/LRH = 5√2 and we have defined an over-all frequency scale ω0 =√2/LC. Strip-diagonalization of thenetwork with g= 1/√2 and γ = 5√2. Colorscale indicates theaverage distance 〈y〉 measured from the center of the strip ofthe eigenstate belonging to the eigenvalue ωk. . . . . . . . . 29xFigure 3.11 Voltage response V respr (ω) of a 15×15 circuit with bandstruc-ture as in (c) to current injected at green marked sites. Frequen-cies of the injected currents are denoted in plot titles. For allplots we assume a small resistance of the inductors ε = 0.005responsible for damping of the signal. Bottom panels includea defect where white sites have been removed. For the bottomright panel we additionally model 17% randomness in L, C,RH , and ε values. . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 3.12 (a) Unit cell of a network realizing the honeycomb lattice model.Three next-nearest neighbors within each plaquette connect toa three-terminal Hall element. (b) Band structure of a strip withzig-zag termination in yˆ-direction for γ = RH√C/L = 10√2,where ω0 =√1/LC. Colorscale shows mean value of the dis-tance of the corresponding eigenfunction from the center of thestrip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.1 Finite element simulation of the current and voltage distribu-tions in a two-dimensional Hall plate in a perpendicular mag-netic field B⊥ with current density j driven by potential differ-ence V = 1V from left to right terminal. White lines followthe electric field E, black arrows denote the direction of thecurrent flow j. At zero field (left panel) j ‖ E and there is novoltage drop VH between the top and the bottom terminal. Forweak fields (middle) |E| |j|> j ·E > 0 and a small Hall voltageVH j ·E > 0 and a small Hall voltageVH < V is observed. At high fields (right) j⊥E and VH ' V . In theinfinite B⊥-field limit the electric field diverges at the two contact pointsmarked by red arrows.toresistance [40].We use Comsol Multiphysics finite element software to numerically solve thecurrent conservation equation∇ ·σ∇V = 0 for the four-terminal geometry depictedin Fig. 4.1. We choose insulating boundary conditions for the edges as well as topand bottom terminals and drive a longitudinal current I by a voltage difference Vfrom the left to the right terminal. The resulting potential distribution is plotted asa color scale, electric field lines are white, and the current flow is denoted by blackarrows. For B⊥ = 0, the current flow is parallel to E and the potential differencebetween top and bottom terminals is VH = 0. As one increases the magnetic field,j and E span the Hall angle θH and one measures a finite Hall voltage VH . For con-stant current flow, VH increases linearly with B⊥ as shown in Fig. 4.2(a). Naively,one could expect that VH V for large enough field B⊥. This is the necessarycondition for the realization of an ideal Hall element. However, as one can see inthe right panel of Fig. 4.1, the Hall voltage saturates at VH =V .This effect is commonly known as two-terminal resistance and may be inter-preted as a geometrical magnetoresistance. For the diamond geometry it estab-350 0.5 1 1.5 2B eld in T00.10.20.30.40.50.60.70.80.9Hall-voltage in V0 0.5 1 1.5 2B eld in T0.20.40.60.811.21.4Longitudinal voltage in Va bFigure 4.2: (left) Linear dependence of Hall-voltage as a function of mag-netic field. (right) The longitudinal voltage shows a dependence on themagnetic field that is linear in the strong-field limit.lishes magnetic field dependence of the longitudinal resistance that is linear forlarge fields, see Fig. 4.2(b). For constant current, VH and V then show the samelinear behavior at high fields, precluding the desired limit VH V . In fact, it hasbeen shown that, on general grounds, VH ≤ V for arbitrarily shaped three-, four-,and six-terminal geometry and arbitrary magnetic field [40].It may seem puzzling that a Hall element is dissipative in the limit j⊥E . Afterall, the dissipated power is P =∫E · d j and should vanish when j⊥E. But Pcan still be non-zero if the electric field strength diverges at some point in thesample. In fact, it is known that the two-terminal resistance arises at two points nearthe terminals where the boundary conditions change from galvanic to electricallyinsulating. At these points, the electric field diverges. In our setup the points withdivergent field strength are marked by red arrows in the rightmost panel of Fig. 4.1.4.1.2 Hall effect, capacitive couplingViola and DiVincenco proposed an elegant way [41] to circumvent the problemof diverging electric fields outlined above. They showed that a near ideal Hallresistor can be achieved by replacing galvanic contacts by capacitive coupling tothe terminals. The resulting setup, illustrated in Fig. 4.3, yields solutions of the EM36a bFigure 4.3: Illustration of capacitively contacted Hall elements in (a) four-terminal (two-port) and (b) three-terminal configuration. The capaci-tance of each contact is CL, directionality of currents is indicated byblack arrows.field equations that are well behaved on the whole resistor geometry. Explicitly,their resulting impedance tensor for a two-port geometry Fig. 4.3(a) in the limitj⊥E has the formRˆ(ω) = RH(−icot(12ωCLRH) 1−1 −icot(12ωCLRH)). (4.2)Here, CL is the capacitance of a single contact. All contacts are assumed to havethe same capacity for simplicity. The anti-symmetric structure of the tensor impliesthat no energy is dissipated. For a discrete set of perfect “gyration” frequenciesωn =piCLRH(2n+1), n= 0,1, . . . , (4.3)diagonal elements vanish and the above tensor describes an ideal Hall resistor.The impedance tensor of the capacitively contacted Hall element is intrinsi-cally dependent on the drive frequency ω and this dependence is fundamentallynon-linear. This prevents a description in terms of the Bloch equation with a sim-ple frequency-independent Hamiltonian but one can still use the circuit Green’sfunction method to describe a periodic LC network with these elements. The cor-370.00.51.01.50.00.51.01.5-3 -2 -1 0 1 2 30.00.51.01.50.00.51.01.5-3 -2 -1 0 1 2 3M Γ MXM Γ MXFigure 4.4: Plot of log |detY (ω,k)| for ideal Hall-resistor (top row) and ca-pacitively coupled Hall-resistor (bottom row) for infinite geometry (leftcolumn) and strip-geometry (right column), respectively. Dark colorsshow divergences of the log where the eigenmode equation detY = 0is satisfied. For the top row the same parameters as in Fig. 3.10 areassumed, for the bottom row calculation we have g= 1/√2, CL = 1/5,RH = 5/4 in units where L=C = 1.responding admittance tensor is given byYˆ =−iωCg2 − 4iωL 1+e−ikxiωL1+e−ikyiωL1+eikxiωL −iωC− 2iωL + i tan(ωCLRH)2RH ξysec(ωCLRH)−12RHη1+eikyiωL − sec(ωCLRH)−12RH η∗ −iωC− 2iωL + itan(ωCLRH)2RHξx(4.4)38where ξi = 2(1− coski) and η = e−ikx + eiky − e−ikx+iky − 1. In the bottom leftpanel of Fig. 4.4, we show a plot of log∣∣detYˆ (ω,k)∣∣. Values of ω,k, for which theeigenmode equation detYˆ = 0 is satisfied, are plotted by dark lines. We recognizethree bulk bands that are distorted from the ideal Hall resistor calculation, shownfor comparison in the top left panel. In the right column, we repeat the calculationfor an infinite strip that is 10 unit cells wide. In both, the case of ideal Hall resistorand capacitively coupled Hall element, we observe edge modes between the secondand third band. This suggests the realization of Chern phase.A simple physical description of the Viola-DiVincenco setup that gives Eq.(4.2) relies on the dynamics of the magnetoplasmon edge mode in the Hall effectdevice [41]. It will work equally well in the four- and three-terminal configuration,but the five-terminal configuration that is required for our toy model analyzed inSec. 3.1 cannot be realized in this manner.4.2 Ideal Hall-resistor from operational amplifiersFrom the discussion above we infer that the realization of Hall resistors by theclassical (or quantum) Hall effect can be achieved by capacitively coupling theHall resistors in a strong magnetic field to the circuit. Nevertheless, this realizationis not entirely practical if one wishes to operate the network at room temperatureand use moderate magnetic fields.On the other hand, four-terminal Hall resistors that satisfy the port condition arewell known among electrical engineers as ’gyrators’ and various other implemen-tations have been conceived [42–46]. A notable example is the realization usingoperational amplifiers. Such gyrating circuits are discussed in standard textbooks[47].Here we describe a specific realization of the simulated ideal Hall resistor in-spired by the recent work of Hofmann et al. [2]. It can be used in either four- orthree-terminal configuration required for the Chern insulating networks discussedin Secs.3.2 and 3.3, but not in the five-terminal configuration. Construction of theHall element is based on the building block depicted in Fig. 4.5(a). It consists ofan operational amplifier and three resistors.Assuming the operational amplifier behaves as an ideal op-amp, i.e. it has infi-39+cbaFigure 4.5: (a) Circuit element called ”negative impedance converter”, com-posed of three resistors and one operational amplifier, introduced in Ref.[2]. It can be used to construct an ideal Hall element in two-port con-figuration (b) as implemented in the square lattice Chern insulating net-work discussed Sec. 3.2, or in a three-terminal configuration (c) re-quired in the honeycomb network of Sec.3.2.nite input impedance and infinite open-loop gain, we can easily derive the behaviorof the circuit element in Fig. 4.5(a). Since the op-amp is operated in feedback con-figuration, it will hold the potentials at its inputs equal, Vin =V1. Moreover, due toits infinite input impedance, it will only inject a current at its output and no currentwill flow into its inputs. We immediately see that Iin =−Iout as a consequence. Onthe other hand Iout = Vin−VoutRH , so that the full description is(IinIout)=1RH(−1 11 −1)(VinVout).Remarkably, arranged in a two-port configuration as depicted in Fig. 4.5(b) orthree-terminal configuration in Fig. 4.5(c), these elements precisely realize the re-spective ideal Hall resistors required for our proposed Chern insulator networks.Operational amplifiers are commercially available at low cost and can operatein a wide range of frequencies, voltages and power settings. Experimental realiza-40tion of the Chern-insulating networks using the simulated Hall elements depictedin Fig. 4.5 should therefore be easily achievable.41Chapter 5Discussion and summaryIn this thesis, we proposed periodic RLC networks that function as Chern insulatorsfor electromagnetic signals in a broad range of frequencies tunable by adjustingthe values of inductance L, capacitance C, and Hall resistance RH of the circuitelements. The design is guided by exploiting an analogy between equations gov-erning the EM fields in periodic RLC networks and tight-binding models for Ma-jorana fermions which are known to possess topologically non-trivial phases. Ourapproach maps Kirchhoff’s laws describing the network onto a Hermitian eigen-value problem in crystal momentum space where the eigenvalues correspond tofrequency modes of the network. Topological properties of the network are theninferred transparently in direct analogy to condensed matter Hamiltonians.Explicitly, we have proposed three different network architectures realizingChern insulating phases for EM signals. The required time reversal symmetrybreaking is achieved by including Hall resistors which are non-reciprocal circuitelements also known in engineering literature as gyrators. These may be imple-mented as capacitively contacted metallic or semiconductor films in an externalmagnetic field or as simple circuits with resistors and off-the-shelf operational am-plifiers. In the latter implementation, the time reversal symmetry is broken by theexternal source of power required to operate the amplifiers. Nevertheless, due tothe feedback structure, the operational amplifiers are operated in the linear responseregime and the simulated Hall devices can be regarded as linear circuit elements.Topological properties of the networks proposed in this work are manifest in42the chiral edge modes traversing the gap in the bulk spectrum. These edge modesgive rise to unidirectionally propagating voltage and current signals along the net-work boundary. They are topologically protected by the bulk topological invari-ant (the integer Chern number) and cannot be removed by any deformation of theboundary. 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