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Chern insulators for electromagnetic waves in electrical circuit networks Haenel, Rafael 2019

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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 <V is observed. At high fields (right) j⊥E and VH 'V . Inthe infinite B⊥-field limit the electric field diverges at the twocontact points marked by red arrows. . . . . . . . . . . . . . 35Figure 4.2 (left) Linear dependence of Hall-voltage as a function of mag-netic field. (right) The longitudinal voltage shows a depen-dence on the magnetic field that is linear in the strong-field limit. 36xiFigure 4.3 Illustration of capacitively contacted Hall elements in (a) four-terminal (two-port) and (b) three-terminal configuration. Thecapacitance of each contact is CL, directionality of currents isindicated by black arrows. . . . . . . . . . . . . . . . . . . . 37Figure 4.4 Plot of log |detY (ω,k)| for ideal Hall-resistor (top row) andcapacitively coupled Hall-resistor (bottom row) for infinite ge-ometry (left column) and strip-geometry (right column), re-spectively. Dark colors show divergences of the log where theeigenmode equation detY = 0 is satisfied. For the top row thesame parameters as in Fig. 3.10 are assumed, for the bottomrow calculation we have g = 1/√2, CL = 1/5, RH = 5/4 inunits where L=C = 1. . . . . . . . . . . . . . . . . . . . . 38Figure 4.5 (a) Circuit element called ”negative impedance converter”, com-posed of three resistors and one operational amplifier, intro-duced in Ref. [2]. It can be used to construct an ideal Hallelement in two-port configuration (b) as implemented in thesquare lattice Chern insulating network discussed Sec. 3.2, orin a three-terminal configuration (c) required in the honeycombnetwork of Sec.3.2. . . . . . . . . . . . . . . . . . . . . . . . 40xiiGlossaryCAZ Cartan-Altland-ZirnbauerEM electromagneticRLC Resistor-Inductor-CapacitorSPT Symmetry Protected TopologicalxiiiAcknowledgmentsI would like to thank Prof. Marcel Franz for his fabulous guidance, advice, andsteady support during the two years of my degree.Further I would like to thank the members of my research group E´tienne Lantagne-Hurtubise, Chengshu Li, Og˘huzan Can, Tarun Tumurru, and Stephan Plugge formany inspiring discussions and brilliant help.Research described in this thesis was supported by NSERC and by CIFAR.Software that facilitated this research was provided by CMC Microsystems.xivChapter 1IntroductionTopological states of matter in electronic systems exhibit topologically non-trivialbulk band structures accompanied by protected edge or surface modes [3–5]. Moregenerally, the insights gained from the study of electrons in crystalline solids withnon-trivial topology can be applied to any physical system whose degrees of free-dom are governed by a wave equation. If bulk solutions of the wave equation donot exist in some range of frequencies, the system may be viewed as insulatingfor these frequencies and may in addition possess topologically protected propa-gating modes at its boundary. This realization has led to a theoretical study andphysical implementation of a wide variety of periodic systems in which topologi-cal properties analogous to electronic topological insulators, superconductors, andsemimetals are manifest. Most prominent examples of these efforts include pho-tonic [6, 7], acoustic [8, 9], mechanical [10–12], polaritonic [13], and electricalsystems [14–17].In this thesis, we focus on the latter class of topological systems, more specif-ically, periodic networks comprised of inductors, capacitors, and resistors. Thesestructures, also referred to as topoelectrical circuits [15], have been demonstratedto possess topological properties with respect to the incident electromagnetic (EM)wave signals. In close analogy to electronic tight-binding models, various circuitmodels realizing classical analogs of quantum spin Hall states [14, 18], Dirac andWeyl semimetals [15, 19], and higher order topological insulators [20–22] havebeen proposed and some of them have been experimentally characterized.1Conspicuously absent from this list has been the Chern insulator – the ana-log of the most basic electronic topological phase, the quantum Hall insulator intwo dimensions [23]. The reason is simple: networks composed of capacitors andinductors are governed by Maxwell equations which are fundamentally invariantunder the time reversal operationT . A Chern insulator, on the other hand, requiresbroken T symmetry.We note that ordinary resistors in a Resistor-Inductor-Capacitor (RLC) networkcause dissipation and therefore break T . This, however, hinders the comparisonto isolated quantum systems where dynamics are unitary. On the other hand, dis-sipative networks can provide useful examples of systems studied in the rapidlyadvancing field of non-Hermitian quantum mechanics [24–27].In the present work, we circumvent this problem by employing a class ofweakly dissipative Hall resistors. These are linear circuit elements whose voltageresponse to a longitudinal current is predominantly transverse. An ideal Hall re-sistor introduces strong T breaking into the circuit without significant dissipationand thus enables construction of the Chern insulator.The class of EM Chern insulators we introduce here has a bulk gap for EMwaves in a range of frequencies but exhibits chiral propagating edge modes thattraverse the gap. The edge modes are topologically protected by a non-zero Chernnumber defined by the bulk band structure of the network and are robust againstany imperfections in the network that do not close the gap.This thesis is organized as follows: In Ch. 2 we give a brief overview ofthe classification of Symmetry Protected Topological (SPT) matter and discuss theapplicability of these concepts outside of the field of condensed matter physics.In Ch. 3 we introduce three specific circuit network architectures that real-ize a Chern insulating phase. We do this by taking advantage of a novel mappingthat connects the dynamics of a certain class of periodic RLC networks to HermitianBloch Hamiltonians describing Majorana fermions in a crystal lattice. Such Hamil-tonians are well known to possess Chern-insulating phases. While the possibilityof non-trivial topological structure of Kirchhoff’s equations has been previouslyrecognized, it is usually discussed in terms of admittance bands or mapped ontonon-Hermitian eigenvalue problems [14, 15]. The description developed in thiswork offers a more direct analogy to crystalline solids and thus a more transparent2physical interpretation in terms of well understood topological band theory [3–5].The key physical element required in the realization of our EM Chern insulatorarchitecture is the Hall resistor. We discuss its physical implementation in Ch. 4.3Chapter 2Symmetry protected topologicalmatterIn this chapter, we review the concept of classification of SPT phases based onhomotopy classification of Dirac mass gaps in the presence of global discrete sym-metries. I have tried to convey the idea behind this topic in the simplest manner.None of the ideas in this chapter are new and excellent reviews can be found in[1, 28, 29].2.1 Tight-binding modelsA general condensed matter system is modelled by the second-quantized Hamilto-nianH =∑i jψ†i hi jψ j+∑i jklVi jklψ†i ψ†jψkψl (2.1)and the canonical commutation relations of fermionic creation and annihilationoperators {ψi,ψ†j}= δi j ,{ψi,ψ j}= 0 . (2.2)4The dynamics of the system is given by the Schro¨dinger equationi∂t |ψ〉= H|ψ〉 (2.3)where states |ψ〉 are built by application of creation operators ψ†i on the ground-state |0〉 and can be in any superposition due to linearity of (2.3).The quartic term proportional to Vi jkl in Eq. (2.1) describes two-fermion in-teractions. The solution of interacting many-body systems defines the forefrontof present-day condensed matter research. For weak Vi jkl , solutions are usuallyapproximated analytically using mean-field techniques, (diagrammatic) perturba-tion theory, or field theoretical methods such as the Renormalization Group [30].Strong interactions may be solved by Bosonization [31] in one dimension or in ar-bitrary dimensions using numerical methods, such as Exact Diagonalization, Den-sity Matrix Renormalization Group [32], and Quantum Monte Carlo computations.However, a plethora of interacting phenomena remain elusive to this date.For many laboratory or real-world materials it is an excellent approximation toneglect the Vi jkl-terms altogether. We then arrive at the tight-binding modelH =∑i jψ†i hi jψ j . (2.4)Given a basis of single-particle eigenstate ψ†k |0〉 corresponding to eigenenergies{εk}, any N-particle eigenstate can be constructed asψ†k1ψ†k2 · · ·ψ†kN |0〉 , (2.5)where the corresponding eigenenergy is simply given by the sum of single-particleenergies Ek1...kN = ∑Ni=1 εki .Applying an arbitrary single-particle state ∑n φknψ†n |0〉 on Eq. (2.3), we geti∂t∑nφnψ†n |0〉=∑i jnhi jψ†i ψ jφnψ†n |0〉=∑i jhi jφ jψ†i |0〉 (2.6)5from which we extract the conditioni∂tφi =∑jhi jφ j . (2.7)This is the Schro¨dinger equation for a single particle in its first-quantized form.The full solution of the energy spectrum therefore reduces to computation of theeigenvalues εk of the single-particle Hamiltonian hi j, which can then be added inall possible combinations to yield the many-body spectrum.2.1.1 Majorana tight-binding modelsFor future reference to our circuit models, we will introduce a special type of tight-binding model that describes Majorana Fermions hopping on a lattice. Majoranafermions are particles that are their own anti-particles. In second quantized notationthis is expressed by the self-adjointness of the corresponding operatorsγ†j = γ j . (2.8)The γi satisfy the commutation relations{γi,γ j}= 2δi j . (2.9)A general model for non-interacting Majorana fermions on a lattice is defined sim-ilarly to Eq. (2.4) by a Hamiltonian of the formH =∑i jhi jγiγ j. (2.10)Hermiticity of H together with relations (2.8) and (2.9) implies that hi j is purelyimaginary and antisymmetric. We henceforth write it as hi j = iti j, where ti j is areal antisymmetric matrix.In complete analogy to Eq. (2.7), time evolution of an arbitrary state ∑i φiγi|0〉is governed by the corresponding Schro¨dinger equationi∂tφi(t) =∑i jhi jφ j(t) . (2.11)6Using the property hi j = iti j we see that Eq. (2.11) becomes a purely real-valuedwave equation which, therefore, admits purely real solutions φ(t). Real-valued,dispersing solutions of the Schro¨dinger equation are referred to as Majorana fermions.They should, however, not be confused with Majorana zero modes, which are local-ized real-valued solutions, whose corresponding quasi-particles obey non-Abelianexchange statistics.2.2 Classification of symmetry protected topologicalstates of matterGapped symmetry protected topological (SPT) states of matter are gapped bulkstates with linearly dispersing gapless edge states that are immune to symmetrypreserving perturbations as long as they do not close the bulk gap [3, 5]. Remark-ably, these states of matter are characterized by a topological invariant that is abulk property only - a fact that is commonly referred to as the ’bulk-boundary cor-respondence’.In this section, we will outline the fundamental concept behind SPT phases andsketch the classification of these phases in the so called 10-fold way [33]. We dothis by restricting ourselves from the set of arbitrary Hermitian Hamiltonians tothe set of Dirac Hamiltonians, to be defined below. Auspiciously, this restrictiondoes not affect the classification outcomes as gapped SPT phases can always beapproximated by a Dirac model at low energies.2.2.1 Dirac model methodA general Dirac Hamiltonian in d dimensions that describes an insulator can beexpressed asH = mγ0+d∑i=1kiγi , (2.12)where the ki are momentum variables. The γi, i = 1, ..., d˜ ≥ d, are elements of theClifford algebra Cl0,d˜ that satisfy the anticommutation relation{γi,γ j}= 2δi j1.The spectrum of (2.12) is given by the gapped bands E =±√∑i k2i +m2. Each of7these bands is N/2-fold degenerate depending on the dimension N = 2(d˜−1)/2 ofthe matrix representation of the γi. We see that the point m= 0 marks a topologicalphase transition where the gap closes.We now want to examine the boundary-physics of this minimal model. Withoutloss of generality we introduce a topological phase boundary perpendicular to thex j-direction by requiring m= m(x j) to be a monotonic function withm(xk) =−m0 for x j→−∞0 for x j = 0m0 for x j→ ∞ .(2.13)Since this choice of mass-term m(x j) breaks translational invariance along x j, wereplace k→−i∂ j and arrive at the field theoryH = γ0 (m(x j)− i∂ jγ0γ j)+∑i 6= jkiγi . (2.14)For ki = 0 we find a zero-mode|ψ0〉= e−∫ x j0 m(x′j)dx′j |−〉 (2.15)that is localized at the phase-boundary. Here, iγ0γ j|−〉=−|−〉. Furthermore iγ0γ jcommutes with Hk =∑i6= j kiγi. This means we can choose |−〉 to be an eigenvectorof Hk so thatH|ψ0〉= [γ0 (m(x j)− i∂ jγ0γ j)+Hk] |ψ0〉= Hk|ψ0〉=±√∑i6= jk2i |ψ0〉 . (2.16)We have found an edge-state that linearly traverses the bulk gap. Note that theexistence of this gapless state relies on the absence of a second mass term m′γ ′0with {γ ′0,H}= 0. In the presence of such a term, the edge spectrum is gapped out:E =±√∑i6= jk2i +m′2 . (2.17)In the minimal Dirac-model approach the concept of topological protection there-8fore boils down to the question if such a second mass-term can be found [1, 34].In practice, it is sensible to first restrict oneself to a certain symmetry class, i.e. byrequiring that all terms in the Hamiltonian satisfy a specified set of symmetries.The topological classification will naturally depend on the symmetry class.In the following, we will introduce and motivate the set of symmetries under-lying the classification in the so-called tenfold way and discuss the question ofsecond mass terms in the mathematical framework of homotopy classes.2.2.2 SymmetriesWe consider tight-binding models introduced above,H =∑i jψ†i hi jψ j , (2.18)where the creation- and annihilations operators satisfy the anti-commutation re-lation{ψi,ψ†j}= δi j. The symmetries underlying the Cartan-Altland-Zirnbauer(CAZ) classification scheme are called time-reversal symmetry T , particle-holesymmetry C , and chiral symmetry S = T C which is the product of the formertwo. Chiral symmetry S is sometimes also referred to as sublattice symmetryin the literature. We will define the corresponding symmetry transformations asa similarity transformation of the second quantized operators ψi. The symmetryoperations then read as follows:T : ψi→T ψiT −1 = (UT ) ji ψ j (2.19)C : ψi→ CψiC−1 = (UC ) ji ψ†j (2.20)S =T C : ψi→SψiS −1 = (UCUT ) ji ψ†j . (2.21)Additionally, we impose T to be antiunitary, T iT =−i. Particle-hole symmetryC is a unitary symmetry, hence the productS =T C is antiunitary.A system H respects the symmetry P if [H,P] = 0. It is often useful torewrite this as a condition on the first-quantized part hi j of H only. We will do thisexplicitly for the case of time-reversal symmetry T . From9T HT −1 = ∑i jT ψ†i hi jψ jT−1 =∑i jT ψ†i T−1 (h∗)i jT ψ jT −1 (2.22)= ∑i jnmψ†i(U†T)in(h∗)nm (UT ) jmψ j!= H =∑i jψ†i hi jψ j (2.23)we find the conditionU†T h∗UT = h . (2.24)Similarly, relations for C andS can be derived. In summary, we haveT : U†T h∗UT = h (2.25)C : U†C h∗UC =−h (2.26)S : U†S hUS =−h . (2.27)If the Hamiltonian h can be expressed in the momentum basis h(k), Eqs. (2.25-2.27) must be modified as inT : U†T h∗(−k)UT = h(k) (2.28)C : U†C h∗(−k)UC =−h(k) (2.29)S : U†S h(k)US =−h(k) , (2.30)where the sign in −k is a consequence of the anti-unitarity of T and C in thefirst-quantized representation.The CAZ classification scheme now distinguishes between the symmetry classess listed in Tab. 2.1. ’0’ indicates the absence of the corresponding symmetry,whereas ’1’ indicates its presence. For T and C one additionally distinguishesbetween the cases UT /CU∗T /C =±1. These are labeled as ’±’ in Tab. 2.1.We see that there are 10 symmetry classes, therefore this classification schemeis sometimes referred to as the ’10-fold’ way. T and C alone give rise to 9 classes,since either one can be 0,±1. The tenth class results from the ambiguity when bothT and C are absent. Here, the product S = T C is no longer well-defined and10S can be either absent or present.The CAZ symmetry classification is only well-defined in the absence of anyconserved quantities. Consider, for example, the case where a system has twodistinct time-reversal symmetries T and T ′. The combined symmetry T T ′ actsasT T ′ : (U∗TUT ′)† h(U∗TUT ′) = h (2.31)and we notice that the matrix U∗TUT ′ defines a conserved quantity. This means hcan be block-diagonalized, with blocks corresponding to the eigenvalues ofU∗TUT ′ .The CAZ scheme should then be applied to each block separately.For the CAZ classification, T , C , andS have been selected as the underlyingset of symmetries. In principal, an arbitrary set of symmetries can be chosen. Infact, Hamiltonians have been classified with respect to various spatial symmorphicand non-symmorphic symmetries [35, 36]. Looking back at Eqs. (2.25-2.27) wecan, however, appreciate the motivation behind the particular choice of T , C ,and S . They are the three global symmetries that give rise to (anti-)commutationrelations{US ,h}= 0 (2.32)[UT K,h] = 0 (2.33){UCK,h}= 0 , (2.34)where K denotes the complex conjugation operation. Eqs. (2.32-2.34) are the sim-plest (anti-)commutation relations that one could write down apart from the usualcommutation relation [U,h] = 0 which trivially affects the classification scheme asdiscussed above.112.2.3 Classification of Dirac mass gapsThe presence of T , C , andS impose conditions on the γi of the Hamiltonian 2.12as can been seen from Eqs. (2.28-2.30). For the mass term they areT : UT γ∗0U†T = γ0 (2.35)C : UC γ∗0U†C =−γ0 (2.36)S : US γ0U†S =−γ0 . (2.37)For the γ-matrices of the kinetic terms kiγi, i≥ 1, the signs must be flipped for Tand C since these symmetries are anti-unitary in the first-quantized representation.This yieldsT : UT γ∗i U†T =−γ0 (2.38)C : UC γ∗i U†C = γ0 (2.39)S : US γiU†S =−γ0 , i≥ 1 . (2.40)Equations (2.38-2.40) and the requirement {γ0,γi} = 0 for i = 1, . . . ,d restrict themass-term mγ0 to some parameter space Rs,d . Consider two real-space regionsA,B separated by a domain wall. We pick two points rA ∈ A,rB ∈ B and examinethe mass terms mγ0(rA),mγ0(rB) ∈Rs,d at these points. The two domains A,B aretopologically equivalent if we can smoothly deform the two masses into each other,i.e. if there exists a continuous pathmγ(t) ∈Rs,d ∀t ∈ [0,1] (2.41)with mγ(0) =mγ0(rA) and mγ(1) =mγ0(rB). If the masses are not path connected,the two domains are topologically distinct. The number of distinct topologicalphases is therefore equivalent to the number of path connected regions in Rs,d .Mathematically, this is expressed by the 0th homotopy of the parameter spacepi0 (Rs,d) [1].The results of the computation of homotopy classes for all ten symmetry classesand for dimensions d = 0 to 3 are listed in Tab. 2.1. Here, 0 indicates that theparameter space Rs,d is path connected, i.e. only the topologically trivial phase12Class s T P S d = 0 1 2 3A 0 0 0 Z 0 Z 0AIII 0 0 1 0 Z 0 ZAI + 0 0 Z 0 0 0BDI + + 1 Z2 Z 0 0D 0 + 0 Z2 Z2 Z 0DIII - + 1 0 Z2 Z2 ZAII - 0 0 2Z 0 Z2 Z2CII - - 1 0 2Z 0 Z2C 0 - 0 0 0 2Z 0CI + - 1 0 0 0 2ZTable 2.1: CAZ classification of symmetry protected topological matter [1].exists. Z2-classification implies the existence of a single topological phase next tothe trivial one, i.e. Rs,d has two separate path connected regions. Phase boundariesnecessarily exhibit gapless modes as discussed in Sec. 2.2.1. Z2 phases allow fora single pair of such protected edge modes. Z and 2Z have an infinite number oftopologically distinct phases and integer or even-integer number of edge modesmay be present, respectively.2.2.4 Mass classification for trivial insulators and Chern insulatorsThe homotopy classification of Dirac mass gaps leading to Tab. 2.1 is a ratherabstract concept. To make the topological classification more explicit, we willcome back to the end of Sec. 2.2.1 where we concluded that the edge state spectrumof a gapped Dirac Hamiltonian is stable if no second mass terms is allowed bysymmetry. This agrees with the homotopy classification in the following way. Letus set the mass at point rA in domain A to mγ0(rA) = mγ0 and at rB in domain B tomγ0(rB) = −mγ0. If no second mass term exist, the only way to deform mγ0 into−mγ0 is by slowly changing m to −m. At some point, however, m must vanish.But since 0 /∈ Rs,d the condition (2.41) is violated at that point and the two gapscan not be path-connected. If, on the other hand, a second mass term γ ′0 exists such13that γ ′0 ∈Rs,d , the pathmγ0→(mγ0+ γ ′0)→ γ ′0→ (−mγ0+ γ ′0)→−mγ0 (2.42)becomes possible. In this case, domains A,B are topologically equivalent.Let us now explicitly verify some results of the classification table 2.1 that wewill refer back to when we discuss circuit models in Ch. 3. We start with classBDI in d = 2. Without loss of generality we choose the time-reversal operator tobe represented byUT = σx and particle-hole symmetry byUC = 1. Note that bothsymmetries square to +1. Chiral symmetry is necessarily present with US = σx.The simplest 2-dimensional Dirac Hamiltonian requires a 4×4 representationof the γ-matrices. We choose them as {σx,σy,σzτx,σzτy,σzτz}. The Dirac modelsatisfying T ,C , andS can hence only beh= k1σzτx+ k2σzτz . (2.43)Of the remaining three γ-matrices σy and σzτy satisfy the mass term conditionsEqs. (2.38-2.40). Therefore, two distinct mass terms exist and we conclude thatclass BDI is topologically trivial in d = 2.We now relax the symmetry constraints by breaking time-reversal symmetryT as well as chiral symmetry S . This brings us into class D. A minimal DiracHamiltonian satisfying C with UC = 1 ish= k1σx+ k2σz . (2.44)The only mass-term available is mσy. Therefore, class D possesses a topologicalphase in two dimensions. To differentiate between Z and Z2 classification, we haveto examine if more than one pair of stable edge modes can exist. We do this byincluding multiple copies of the Hamiltonian,h′ = (k1σx+ k2σz)⊗1n , (2.45)and then checking if additional, higher-dimensional mass terms can be found nextto mσy⊗1n. It is clear that Eq. (2.45) does not allow for any second symmetry-14preserving mass term, independent of the dimension of the identity 1n. Conse-quently, any number of edge modes is topologically protected and we assign theclassification Z to class D. Note that above arguments also hold in the absence ofthe symmetry C , so that class A is expected to have the same topological classifi-cation. Comparing with Tab. 2.1, we confirm that our arguments indeed lead to thecorrect result.Topologically non-trivial Hamiltonians in class A or D are referred to as Cherninsulators. Their main characteristic is the unidirectional propagation of their edgemodes. The topological phase of a bulk model can be identified by computation ofthe Chern number, a topological invariant, which is given by the integralQ(n) =− 12pi∫BZdk(∂A(n)y∂kx− ∂A(n)x∂ky)of the Berry connectionA(n)j (k) = i〈Ψn(k)|∂k j |Ψn(k)〉, for j = x,yover the Brillouin zone.2.3 Do we need quantum mechanics?We have started this chapter by introducing a fermionic tight-binding model (2.4)using second-quantized fermionic operators. These operators ψi,ψ†i obey anti-commutation relations, Eq. (2.2), that make the problem inherently quantum-mechanical.However, for the homotopy classification we simply made use of the first quan-tized single particle Hamiltonian hi j. The only two ingredients were an extensiveset of conserved quantities k originating from translational invariance of the under-lying system and a spectral gap in the eigenvalues εk of the matrix hi j. In fact, thefirst condition can even be relaxed as topological protection also holds in the caseof weakly translational-invariance-breaking perturbations.We can therefore conclude that quantum mechanics, i.e. the non-commutativebehavior of fermionic operators, is not a necessary ingredient for the scheme of15topological classification. For that reason SPT classification extends to a wide va-riety of problems that possess a dispersion relation and can be formulated as aneigenvalue problem.16Chapter 3Chern insulators from RLCnetworksWe have motivated in Ch. 2 that SPT phases occur outside the realm of condensed-matter physics. We now enter the main part of this thesis and study three modelsof periodic circuit networks that realize Chern insulating phases.3.1 General setup and a toy modela bFigure 3.1: (a) Square RLC lattice toy model realizing the Chern insulator forEM waves. A unit cell is marked by by gray background. (b) The Hallresistor element with four side terminals and one central terminal.The simplest RLC network capable of exhibiting non-trivial topology is de-picted in Fig. 3.1(a). It consists of an array of five-terminal Hall elements, denoted17by gray diamonds, arranged in a square lattice. The central terminal of each Hallelement is connected to ground via a capacitor C while the side terminals connectto neighboring Hall resistors through inductors L.The five-terminal Hall element is characterized by its resistance tensor Rˆ, de-fined by the relation V1V2V3V4=R1 R4 R3 R2R2 R1 R4 R3R3 R2 R1 R4R4 R3 R2 R1I1I2I3I4 . (3.1)Here, the voltages Vi are measured with respect to the central terminal and thedirectionality of currents Ii is indicated in Fig. 3.1(b). We note that Eq. (3.1) is themost general parametrization of Rˆ under fourfold rotational symmetry.The description of the EM signal propagating through the circuit requires thedefinition of three dynamical variables: voltage Vr(t) across each capacitor, andtwo currents Ixr (t) and Iyr (t) flowing through the inductors in each unit cell labeledby vector r. They are denoted by red and green labels in Fig. 3.1(a), respectively.Then, Kirchhoff’s laws yield the following coupled system of linear differentialequations:C∂Vr∂ t= Ixr−xˆ+ Iyr−yˆ− Ixr − Iyr ,Vr −Vr+xˆ = L∂ Ixr∂ t+[Rˆ · I r ]3− [Rˆ · I r+xˆ]1,Vr −Vr+yˆ = L∂ Iyr∂ t+[Rˆ · I r ]4− [Rˆ · I r+yˆ]2.(3.2)The first of these equations expresses current conservation for each Hall elementand the remaining two relate the voltage differences between neighboring unit cellsto the corresponding currents, through the usual constitutive relations for inductorsand resistors. I r = (Ixr−xˆ, Iyr−yˆ,−Ixr ,−Iyr )T is a vector of currents flowing into theHall resistor at position r.We begin by considering the case of a non-resistive network, i.e. Rˆ = 0. ThenEqs. (3.2) exhibit invariance underT which sends t→−t and reverses all currents,18a b...... ... ... ......... ... ... ...............i iiiiɀiɀacbFigure 3.2: (a) Effective Majorana tight-binding model corresponding to theRLC network toy model with ideal Hall elements. Tunneling matrixelements between sublattices a,b,c are labeled by straight lines, arrowsindicate directionality. Here, γ = RH√C/L. (b) Sketch of boundaryconditions used for calculations in the strip geometry.(Ixr , Iyr )→ (−Ixr ,−Iyr ). In addition, because voltages and currents are by definitionreal-valued, Eqs. (3.2) are trivially invariant under complex conjugation.Equations (3.2) can be recast in the form of a Schro¨dinger equation i∂tφi =∑ j hi jφ j with the wavefunction φi containing voltages and currents and hi j the Her-mitian Hamiltonian matrix. We can further exploit translational invariance of thenetwork by expanding currents and voltages in terms of plane wavesVr(t) =∑kei(ωt−k·r)Vk/√C,Iαr (t) =∑kei(ωt−k·r)I αk /√L ,(3.3)where α = x,y and the rescaling is made for convenience. Equations (3.2) reduceto a 3×3 Hermitian eigenvalue problem ∑ j (hk)i j φk j = ωkφki, whereφk =VkI xkI yk , hk = 1√LC 0 Γx ΓyΓ∗x 0 0Γ∗y 0 0 , (3.4)and Γα = i(1− eikα ).hk is formally identical to a tight-binding model of Majorana fermions (cf. Sec.2.1.1) on the Lieb lattice. The underlying Bravais lattice of the Lieb lattice is thesquare lattice with lattice constant a. It has three sublattices. One sublattice is19a bM Γ X M- 3- 2- 10123Γ XMFigure 3.3: (a) Bulk band structure of the circuit network. The dashed linecorresponds to γ = RH√C/L = 0 while the solid line corresponds toγ = 0.25 which gives a gap ∆= ω0, where ω0 = 1/√LC. (b) Spectrumof a strip of width W = 10 with open boundary conditions along y forγ = 0.25. Boundary conditions are chosen as indicated in Fig. 3.2(b).The color scale indicates the average distance 〈y〉 measured from thecenter of the strip of the eigenstate belonging to the eigenvalue ωk. Thestates inside the bulk gap ∆ are localized near the opposite edges of thesystem.placed on the Bravais lattice points, the remaining two sublattices are located onlinks of neighboring Bravais lattice sites, i.e. they are shifted by (0,a/2), (a/2,0),respectively. The effective electronic unit cell with imaginary hopping parametersis sketched in Fig. 3.2(a). The correspondence with Majorana as opposed to com-plex fermions follows from the fact that the original wave equation (3.2) is purelyreal-valued as is the time-domain Schro¨dinger equation for Majorana fermions ofEq. (2.11). We would like to emphasize that non-trivial Majorana physics in thecondensed matter context relies on the existence of an exponentially large many-body Hilbert space. Here, braiding of Majoranas is represented by non-Abelianunitary operations acting on states in that Hilbert space. However, in the case of aclassical circuit, no such many-body Hilbert space exists.The spectrum of hk consists of one zero mode ωk,0 = 0, and two non-zero20eigenvalues of the formωk,± =± 1√LC√|Γx|2+ |Γy|2=± 2√LC√sin2 (kx/2)+ sin2 (ky/2).(3.5)It can be checked that the states belonging to the ωk,0 = 0 eigenvalue correspond tostatic patterns of currents in the network consistent with current conservation andzero voltages. These will be damped in the presence of arbitrary resistance and areof no interest to us. The two branches in Eq. (3.5) define the propagating modesof the system. They are gapless and linearly dispersing near k = 0, as illustratedin Fig. 3.3(a). Only the positive-frequency branch is physical; the negative branchappears because the ansatz in Eq. (3.3) permits complex-valued solutions whilevoltages and currents are strictly real.In the Bloch Hamiltonian formulation time reversal symmetry T and chargeconjugation symmetry C may be expressed asT : UT h∗−kU†T = hk ,C : h∗−k =−hk .(3.6)with UT = diag(1,−1,−1). Both T and C square to +1 and thus define the BDIclass in the CAZ classification. In two spatial dimensions class BDI supports onlytopologically trivial gapped phases, as listed in Tab. 2.1. Therefore, we must breaktime reversal symmetry to enable a topological phase in this system. (The C sym-metry derives from real-valuedness of Eq. (3.2) and therefore, like the analogoussymmetry present in a generic superconductor, cannot be broken by a physical per-turbation.) When T is broken, the system belongs to class D which has an integertopological classification in d = 2. The corresponding topological invariant is theChern number c and its non-zero values label distinct Chern insulating phases.To proceed, we now include a non-zero resistance tensor defined by Eq. (3.1).21The Bloch Hamiltonian describing the network becomeshk =1√LC 0 Γx ΓyΓ∗x Lxk Mk+NkΓ∗y M∗k −N∗k Lyk , (3.7)withLαk = 2i√CL(R3 coskα −R1),Mk =− i2√CL(R4−R2)(1+ eiky)(1+ e−ikx),Nk =− i2√CL(R4+R2)(1− eiky)(1− e−ikx).(3.8)Time-reversal is explicitly broken whenever Rˆ is non-zero. We observe that theHamiltonian (3.7) remains Hermitian only when Lαk and Nk both vanish for all k.This requires R1 = R3 = 0 and R4 = −R2. Under these conditions the resistancetensor (3.1) becomes purely off-diagonal and antisymmetric. This form signifies apurely transverse, non-dissipative response – an “ideal Hall resistor”. It is impor-tant to note that the resistance tensor, Eq. (3.1), is not invertible in this limit. As aconsequence, the current response to applied voltages is ill-defined. However, wecan still achieve sensible results by keeping a small non-zero dissipative compo-nent R1 = R3 = R. This causes the network Hamiltonian to become non-Hermitianand results in weak damping of the ac signal. Topological properties of the sys-tem should not be affected as we explicitly illustrate below. Large non-Hermitiancomponents could lead to new interesting topological phases.We now focus on the approximately Hermitian limit and define the Hall param-eter RH = R4 =−R2. The bulk spectrum corresponding to the Hamiltonian (3.7) isillustrated by blue lines in Fig. 3.3(a). It develops a gap ∆= 4RH√C/Lω0 at k = 0,where ω0 = 1/√LC. Since the term Mk, responsible for the gap formation, is oddunder time reversal, we expect the gapped phase to be topologically non-trivial.An explicit calculation indeed indicates a non-zero Chern number c = sgnRH forthe negative frequency band. Numerical calculation of the spectrum in a strip ge-ometry confirms the existence of a single chiral edge mode traversing the gap, as22Figure 3.4: Spectral function (3.9) for a circuit with parameters γ = 0.25 inthe absence (left) and presence (right) of 30 percent box disorder forR,L,C parameters. The lifetime broadening parameter is chosen as η =0.03ω0.shown in Fig. 3.3(b).To examine the finite-size behaviour of the system and to test the stability ofthe edge modes against disorder, we numerically compute the spectral functionA(k,ω) =−2Im∑jGRj (k,ω) , j = a,b,c (3.9)for a finite circuit of 10× 10 unit cells and the same parameters as in the finiteHall-resistor computation of Fig. 3.3(a). Here, GRc is the retarded Green’s functionGRj (k,ω) =−i∫ ∞0dteiωtφk j(t)φ †k j(0) (3.10)and j indexes current- and voltage degrees of freedoms of the wavefunction. Theresult, plotted in the left panel of Fig. 3.4, reveals a bulk-bandstructure in agree-ment with the diagonalization of the translationally invariant model. Additionally,we notice the quantization of energy levels due to the finite size of the circuit, and23observe linearly dispersing edge modes close to the Γ-point. The spectral functionfor the same parameters with an additional assumption of 30 percent randomnessin R,L,C parameters is plotted in the right panel of Fig. 3.4. Evidently, the dis-order washes out the bulk bands but has little effect on the edge states. This is aconsequence of topological protection.Experimental characterization of a finite size network can be given throughtwo-point impedance measurements which are conveniently described by the cir-cuit Green’s function formalism. To this end one writes the frequency-domainKirchhoff law for current conservation in the matrix form0 =∑r ′Yrr ′(ω)Vr ′ , (3.11)which defines the admittance tensor Yrr ′(ω) and the voltage distribution Vr corre-sponding to an eigenmode. Solutions to this equation exist only for detY (ω) = 0,which yields the resulting eigenspectrum ωk equivalent to Eq. (3.5). The circuitGreen’s function Gr,r ′(ω) =[Y (ω)−1]rr ′ describes the voltage response of the net-work at point r to a driving current profile Idriver ′ (ω) at frequency ω according toV respr (ω) =∑r′Gr,r ′(ω)Idriver ′ (ω) . (3.12)In analogy to condensed matter systems, where the complete characterization ofa non-interacting system is contained in the time-ordered two-point correlationfunction 〈T ψ(r, t)ψ†(r ′, t ′)〉, full experimental knowledge of Gr,r ′(ω) providesa complete characterization of the electrical circuit. We can therefore expect topo-logically non-trivial behavior to be evident in a circuit’s two-point impedance.In Fig. 3.5 we demonstrate this explicitly by plotting the voltage profile V resprinduced by a current with frequency ω injected at the boundary of a 10× 10 net-work. As an example of possible dissipative dynamics we include a non-zero Rcomponent of the resistance tensor Rˆ and quantify the strength of dissipation bya dimensionless parameter ε = R/RH . For the frequency inside the bulk bandgapthe signal is seen to propagate along the boundary of the system and in one direc-tion only, consistent with the chiral nature of the gapless edge mode. Parameter εclearly controls the lengthscale over which the signal is damped.24Figure 3.5: Voltage response V respr (ω) induced by a current with in-gap fre-quency ω = ∆/2 injected at a node marked by green cross of the 10×10network with γ = 0.25 for various values of the dissipative resistance Rcharacterized by parameter ε = R/RH .3 2 1 0 1 2 3kx0. 0 2Vresp (arb.)BulkEdgeFigure 3.6: (left) Spectrum of circuit for γ = 0.25 and ε = 0.1. (right) Voltageas measured in the bulk (black) or at the edge (blue) after current hasbeen injected at an edge-site.A more quantitative analysis of this behavior is shown in Fig. 3.6. Here, weplot the voltage measured at a bulk site (black) and an edge site (blue) after acurrent of frequency ω has been injected at an edge side. As expected, the edge-to-bulk signal is close to zero for frequencies inside the bulk gap, as such signals canonly propagate around the edge and not enter the bulk. The edge-to-edge signalis prevalent for the whole frequency range since propagating modes are availablethroughout the whole plotted frequency range. Dips and spikes of the two-pointsignal are the result of finite size energy level quantization.Finally, we investigate the propagation of such signals in the time domain. To25Figure 3.7: Time evolution of a localized Gaussian wave packet of frequencywidth (∆ω)/ω0 = 0.35 excited at the boundary. The simulation mod-els disorder by assuming a capacitor and inductor device tolerance of30%. Colorscale corresponds to the weight of the wavefunction on thecircuit node. The signal travels along the boundary and circumvents theboundary defect indicated in white.Circuit boundaryFigure 3.8: Plot of voltage profile along boundary sites as function of timethat shows the constant group velocity of the wave packet.this end we excite a Gaussian wave packet with the frequency width (∆ω)/ω0 =0.35, spatially localized around an edge site, and unitarily evolve it in time withthe propagatorU = exp(−iht). The corresponding simulation for a non-dissipativenetwork with γ = 1 and assuming ±30% randomness in L and C values is shownin Fig. 3.7. The edge signal propagates unidirectionally along the circuit bound-ary, even in the presence of boundary defects. A plot of voltage profile along thenetwork boundary as a function of time in Fig. 3.8 reveals approximately constantgroup velocity of the wave packet.26The circuit described above illustrates the mathematical correspondence be-tween periodic RLC networks and tight-binding Hamiltonians with non-trivial topol-ogy. Our approach allows for the mapping of the differential equations governingthe RLC network onto a simple Bloch equation with similar models analyzed in thecondensed-matter literature [37]. The non-trivial ingredient required to break timereversal symmetry is the five-terminal Hall element described by the resistance ten-sor, Eq. (3.1). However, as we will discuss in Ch. 4, its experimental realizationis not straightforward. For this reason, we may regard the above network as aninstructive but unphysical toy model. Next, we will describe two different networkarchitectures which have well-defined experimental implementations and are onlyslightly more complex.3.2 Chern insulator on the square latticeFigure 3.9: Square RLC lattice network with four-terminal Hall elements.Voltage nodes (red) and currents (green) are labeled for the unit cell(gray background) at position r.Consider the network depicted in Fig. 3.9(a). It has a square lattice symmetryand contains four inductors, three capacitors, and one Hall resistor per unit cell.The Hall resistor is now in a four-terminal configuration. We characterize it by the27Hall admittance tensor Yˆ that relates input currents to terminal voltages via I = YˆV .In its idealized version it isI1I2I3I4= 1RH0 −1 0 11 0 −1 00 1 0 −1−1 0 1 0V1V2V3V4 . (3.13)Currents and voltages are labeled as shown previously in Fig. 3.1(b) with the dif-ference that no central terminal exists. We note that Yˆ has rank 2 and is thereforenot invertible. We can reduce (3.13) to a set of two linearly independent equationsby realizing that it conserves current for pairs of opposing terminals, that is, for anyvoltage input the currents satisfy I1 =−I3 and I2 =−I4. In electrical circuit theorythis is known as the port condition. Two opposing terminals define a port. A fulldescription of the Hall element is then achieved in terms of two currents throughthe ports, I1 and I2, and two voltages across the ports, V1−V3 and V2−V4. Thecorresponding resistance tensor isRˆ= Yˆ−1 =(0 RH−RH 0). (3.14)We note that the circuit element corresponding to the above resistance matrix isin fact well known in electrical engineering literature as the gyrator [38]. Thisdevice, together with the resistor and the capacitor, defines a basis of linear circuitelements. All other network elements can be composed from the aforementionedthree.The degrees of freedom describing the network in Fig. 3.9(a) can be chosen asthree voltages on the capacitors and four currents flowing through the inductors,forming a seven-component vector Ψr = (VAr ,VBr ,VCr , I1r , I2r , I3r , I4r )T . To preservethe fourfold rotational symmetry of the network, we take capacitances on B and Csublattices to be equal, CB =CC =C, and further set CA =C/g2 with g a dimen-sionless parameter. All inductors have inductance L.The corresponding Bloch Hamiltonian follows from current conservation forall nodes and Kirchhoff’s second law for the potential difference between two28nodes connected through an inductor. It can be represented as a 7× 7 matrix ofthe formhk =1√LC(Mk PkP†k 0ˆ), (3.15)where 0ˆ is a 4×4 matrix with all elements zero and Pk denotes the 4×3 matrixPk = i−g ge−ikx −g ge−iky1 −1 0 00 0 1 −1 . (3.16)The 3× 3 matrix Mk contains time reversal breaking terms due to the presence ofthe Hall element,Mk =0 0 00 0 mk0 m∗k 0 , (3.17)with mk = iRH√LC (1− eikx)(1− e−iky).M Γ X M0. bFigure 3.10: Eigenmode spectrum of the network for g = 1/√2, with (solidlines) and without (dashed lines) the Hall element. The gap parameteris γ =√C/LRH = 5√2 and we have defined an overall frequency scaleω0 =√2/LC. Strip-diagonalization of the network with g= 1/√2 andγ = 5√2. Colorscale indicates the average distance 〈y〉measured fromthe center of the strip of the eigenstate belonging to the eigenvalue ωk.The mode spectrum of the circuit consists of seven bands. Charge-conjugation29a dcbFigure 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. Frequencies ofthe injected currents are denoted in plot titles. For all plots we assumea small resistance of the inductors ε = 0.005 responsible for dampingof the signal. Bottom panels include a defect where white sites havebeen removed. For the bottom right panel we additionally model 17%randomness in L, C, RH , and ε values.symmetry C constraints the bands to come in pairs of opposite frequency and theunpaired band to be confined to ωk = 0. In the absence of the Hall resistor, timereversal symmetry enforces degeneracies at k = (0,0) and (pi,pi) as followsω(0,0) = ω0(0,±0,±1,±√1+2g2),ω(pi,pi) = ω0(0,±1,±1,±√2g).(3.18)Here, we have defined ω0 =√2/LC. The Hall resistor breaks T and splits thedegeneracy at (pi,pi). The quadratic band crossing thus acquires a gap and thetwo bands become topologically non-trivial with the Chern number c=±sgn(RH).Since M(0,0) = 0 the degeneracy at the Γ point remains intact.For an arbitrary g and RH one thus expects the network to realize a Chern insu-lator. A situation of special interest occurs for g= 1/√2. In the absence of the Hallresistor, three bands then touch at (pi,pi) and the middle band is completely flat; seeFig. 3.10(a). The Hall resistor separates the three bands and makes the top and bot-tom bands topological with Chern number c = ±sgn(RH). The flat band remainstrivial with c= 0. This is confirmed by numerical diagonalization of HamiltonianEq. (3.15) on a strip geometry with translational invariance along xˆ, shown in Fig.3.10(b). We clearly observe chiral edge modes. We further analyze the admittance30properties of the network by calculating the circuit Green’s function in a finite sys-tem and plotting the voltage response to a current injected at a single node. Forthese calculations, we assume that the inductors are weakly resistive and charac-terize their resistance RL by a parameter ε = RL/RH . The resulting Hamiltonianbecomes weakly non-Hermitian and the propagating waves are damped.Fig. 3.11(a) shows the voltage response to a current of in-gap-frequency ωthat is injected at a bulk site. The voltage profile is localized around the node ofinjection. If we tune the frequency out of the bulk gap, the voltage signal propa-gates through the whole circuit, independent of the point of injection, as shown inpanel (b). To demonstrate the topological nature of the edge transport, we includea defect on the circuit’s left boundary and excite the edge mode of the circuit atan in-gap frequency, cf. panel (c). As expected the signal propagates around thedefect by following the distorted edge. This does not change qualitatively when weintroduce bulk disorder (panel d), which we model by including a 17% randomnessin L, C, ε , and RH values, larger than typical tolerances of commercially availableelectronic components.3.3 Chern insulator on the honeycomb latticeThe graph structure of RLC networks in principle allows for engineering of arbi-trary lattice models. Here, we briefly discuss a circuit whose tight-binding analogis similar to the Haldane model on the honeycomb lattice [23], which was histor-ically the first model realizing the Chern insulator in electronic systems. A unitcell is schematically shown in Fig. 3.12(a). Each of the two sublattices of the hon-eycomb lattice contains a node that is connected to ground through a capacitor C.Nearest-neighbor nodes are connected by inductors L. Second neighbors within ahexagonal plaquette each connect to a three-terminal Hall resistor.The three-terminal Hall resistor is described by a three-fold rotationally sym-metric resistance tensor whose idealized, non-dissipative form is defined by therelation (V1V2)=(0 RH−RH 0)(I1I2). (3.19)31a b..................Figure 3.12: (a) Unit cell of a network realizing the honeycomb lattice model.Three next-nearest neighbors within each plaquette connect to a three-terminal Hall element. (b) Band structure of a strip with zig-zag termi-nation in yˆ-direction for γ = RH√C/L= 10√2, where ω0 =√1/LC.Colorscale shows mean value of the distance of the correspondingeigenfunction from the center of the strip.While the above resistance tensor is formally equivalent to the matrix in Eq. (3.14),the port condition does not apply for a triangular Hall element. Instead Rˆ relatesinput currents at two of the three terminals to the corresponding terminal voltages.The potential at the third terminal is gauged to zero and the corresponding currentis determined by current conservation.The network in Fig. 3.12(a) has a tight-binding representation and topologicalstructure closely related to Haldane’s celebrated lattice model of the Chern insula-tor [23]. The Bloch Hamiltonian is a 5× 5 matrix which takes the same form asEq. (3.15), where nowMk =(mk 00 −mk), Pk = i(−1 −1 −11 eik·a1 e−ik·a2). (3.20)Here, mk = 2R−1H√LC ∑3α=1 sin(k · aα) and ai are Bravais lattice vectors as shownin the inset of Fig. 3.9(b). Similar to graphene the band structure has a pair ofDirac points located at the corners of the hexagonal Brillouin zone but they nowoccur at non-zero frequency. The band crossings are protected by a combinationof T and the lattice inversion symmetry. Inclusion of the Hall resistors breaks T32and creates a Chern insulator. Numerical diagonalization of the Hamiltonian in thestrip geometry confirms the existence of the chiral edge modes traversing the gap;see Fig. 3.12(b).33Chapter 4Hall resistor implementationNon-trivial physics in the models studied in Ch. 3 above relies on Hall resistorscharacterized by a transverse voltage response to a longitudinal current that is oddunder time reversal. We now discuss concrete physical implementations of these el-ements using the classical Hall effect in metals and simulated Hall effects achievedthrough active circuit elements.4.1 Hall-resistor implementation by classical Hall effect4.1.1 Hall effect, galvanic couplingWe consider a metal or semiconductor film in a perpendicular magnetic field B⊥. Insuch a setting the microscopic current response is accurately described by Ohm’slaw, j = σE, where the material’s conductivity takes the form [39]σ =σ01+σ20R2HB2(1 −σ0RHB⊥σ0RHB⊥ 1). (4.1)Here, σ0 is the zero-field conductivity andRH is the Hall coefficient. For σ0RHB1, the microscopic current response to a potential gradient is predominantly trans-verse. One might be tempted to assume that a device depicted in Fig. 4.1 couldtherefore serve as a near-ideal Hall resistor. As our simulations below illustrate,this unfortunately is not the case because of the phenomenon of geometric magne-34B=0 B=0.5T B=10TVFigure 4.1: Finite element simulation of the current and voltage distributionsin a two-dimensional Hall plate in a perpendicular magnetic field B⊥with current density j driven by potential difference V = 1V from leftto right terminal. White lines follow the electric field E, black arrowsdenote the direction of the current flow j. At zero field (left panel) j ‖ Eand there is no voltage dropVH between the top and the bottom terminal.For weak fields (middle) |E| |j| > 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. in V0 0.5 1 1.5 2B eld in T0. 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. 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. -2 -1 0 1 2 -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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