{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Haenel, Rafael","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2019-08-26T18:24:14Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2019","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Periodic networks composed of capacitors and inductors have been demonstrated to possess topological properties with respect to incident electromagnetic waves. In this thesis, we develop an analogy between the mathematical description of waves propagating in such networks and models of Majorana fermions hopping on a lattice. Using this analogy we propose simple electrical network architectures that realize Chern insulating phases for electromagnetic waves. Such Chern insulating networks have a bulk gap for a range of signal frequencies that is easily tunable and exhibit topologically protected chiral edge modes that traverse the gap and are robust to perturbations. The requisite time reversal symmetry breaking is achieved by including a class of weakly dissipative Hall resistor elements whose physical implementation we describe in detail.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/71465?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"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\u00a9 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, \u03b3 =RH\u221aC\/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 \u03b3 = RH\u221aC\/L = 0 while the solid line corre-sponds to \u03b3 = 0.25 which gives a gap \u2206 = \u03c90, where \u03c90 =1\/\u221aLC. (b) Spectrum of a strip of width W = 10 with openboundary conditions along y for \u03b3 = 0.25. Boundary condi-tions are chosen as indicated in Fig. 3.2(b). The color scaleindicates the average distance \u3008y\u3009 measured from the centerof the strip of the eigenstate belonging to the eigenvalue \u03c9k.The states inside the bulk gap \u2206 are localized near the oppositeedges of the system. . . . . . . . . . . . . . . . . . . . . . . 20ixFigure 3.4 Spectral function (3.9) for a circuit with parameters \u03b3 = 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 \u03b7 = 0.03\u03c90. . . . . . . . . . . . . . . . . . . . . 23Figure 3.5 Voltage response V respr (\u03c9) induced by a current with in-gapfrequency \u03c9 = \u2206\/2 injected at a node marked by green crossof the 10\u00d710 network with \u03b3 = 0.25 for various values of thedissipative resistance R characterized by parameter \u03b5 = R\/RH . 25Figure 3.6 (left) Spectrum of circuit for \u03b3 = 0.25 and \u03b5 = 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 (\u2206\u03c9)\/\u03c90 = 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\/\u221a2, with (solidlines) and without (dashed lines) the Hall element. The gap pa-rameter is \u03b3 =\u221aC\/LRH = 5\u221a2 and we have defined an over-all frequency scale \u03c90 =\u221a2\/LC. Strip-diagonalization of thenetwork with g= 1\/\u221a2 and \u03b3 = 5\u221a2. Colorscale indicates theaverage distance \u3008y\u3009 measured from the center of the strip ofthe eigenstate belonging to the eigenvalue \u03c9k. . . . . . . . . 29xFigure 3.11 Voltage response V respr (\u03c9) of a 15\u00d715 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 \u03b5 = 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 \u03b5 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 y\u02c6-direction for \u03b3 = RH\u221aC\/L = 10\u221a2,where \u03c90 =\u221a1\/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\u22a5 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 \u2016 E and there is novoltage drop VH between the top and the bottom terminal. Forweak fields (middle) |E| |j|> j \u00b7E > 0 and a small Hall voltageVH j \u00b7E > 0 and a small Hall voltageVH < V is observed. At high fields (right) j\u22a5E and VH ' V . In theinfinite B\u22a5-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\u2207 \u00b7\u03c3\u2207V = 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\u22a5 = 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 \u03b8H and one measures a finite Hall voltage VH . For con-stant current flow, VH increases linearly with B\u22a5 as shown in Fig. 4.2(a). Naively,one could expect that VH \u001d V for large enough field B\u22a5. 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 \u001feld in T00.10.20.30.40.50.60.70.80.9Hall-voltage in V0 0.5 1 1.5 2B \u001feld 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 \u001d V . In fact, it hasbeen shown that, on general grounds, VH \u2264 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\u22a5E . Afterall, the dissipated power is P =\u222bE \u00b7 d j and should vanish when j\u22a5E. 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\u22a5E has the formR\u02c6(\u03c9) = RH(\u2212icot(12\u03c9CLRH) 1\u22121 \u2212icot(12\u03c9CLRH)). (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 \u201cgyration\u201d frequencies\u03c9n =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 \u03c9 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\u2019sfunction 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 \u0393 MXM \u0393 MXFigure 4.4: Plot of log |detY (\u03c9,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\/\u221a2, CL = 1\/5,RH = 5\/4 in units where L=C = 1.responding admittance tensor is given byY\u02c6 =\uf8eb\uf8ec\uf8ed\u2212i\u03c9Cg2 \u2212 4i\u03c9L 1+e\u2212ikxi\u03c9L1+e\u2212ikyi\u03c9L1+eikxi\u03c9L \u2212i\u03c9C\u2212 2i\u03c9L + i tan(\u03c9CLRH)2RH \u03beysec(\u03c9CLRH)\u221212RH\u03b71+eikyi\u03c9L \u2212 sec(\u03c9CLRH)\u221212RH \u03b7\u2217 \u2212i\u03c9C\u2212 2i\u03c9L + itan(\u03c9CLRH)2RH\u03bex\uf8f6\uf8f7\uf8f8(4.4)38where \u03bei = 2(1\u2212 coski) and \u03b7 = e\u2212ikx + eiky \u2212 e\u2212ikx+iky \u2212 1. In the bottom leftpanel of Fig. 4.4, we show a plot of log\u2223\u2223detY\u02c6 (\u03c9,k)\u2223\u2223. Values of \u03c9,k, for which theeigenmode equation detY\u02c6 = 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 \u2019gyrators\u2019 and various other implemen-tations have been conceived [42\u201346]. 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 \u201dnegative impedance converter\u201d, 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 =\u2212Iout as a consequence. Onthe other hand Iout = Vin\u2212VoutRH , so that the full description is(IinIout)=1RH(\u22121 11 \u22121)(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\u2019s 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. In addition, the edge modes are robust against a moderate amount ofbulk disorder, as realized, e.g., by a random spread in the parameters characteriz-ing the individual network elements.Chern insulating EM networks provide a highly tunable experimental environ-ment. Scale invariance of Maxwell\u2019s equations allows for engineering of bandgaps and edge modes in a wide frequency range. Moreover, the flexible graph na-ture of such networks removes any restriction on dimensionality or locality. Con-sequently, exotic synthetic materials of arbitrary dimension and connectivity maybe designed. In addition to possible engineering applications, Chern insulating EMnetworks may be established as a teaching resource in university laboratory coursesand demonstrations.43Bibliography[1] C. K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, \u201cClassification oftopological quantum matter with symmetries,\u201d Reviews of Modern Physics,vol. 88, no. 3, pp. 1\u201363, 2016. \u2192 pages viii, 4, 9, 12, 13[2] T. Hofmann, T. Helbig, C. H. Lee, M. Greiter, and R. Thomale, \u201cChiralVoltage Propagation and Calibration in a Topolectrical Chern Circuit,\u201dPhysical Review Letters, vol. 122, p. 247702, jun 2019. \u2192 pages xii, 39, 40[3] M. Z. Hasan and C. L. Kane, \u201cColloquium: Topological insulators,\u201d Reviewsof Modern Physics, vol. 82, no. 4, pp. 3045\u20133067, 2010. \u2192 pages 1, 3, 7[4] X.-L. Qi and S.-C. Zhang, \u201cTopological insulators and superconductors,\u201dReviews of Modern Physics, vol. 83, no. 4, pp. 1057\u20131110, 2011.[5] M. Franz and L. Molenkamp, Topological Insulators. Burlington: ElsevierScience, 2013. \u2192 pages 1, 3, 7[6] F. D. M. Haldane and S. Raghu, \u201cPossible realization of directional opticalwaveguides in photonic crystals with broken time-reversal symmetry,\u201d Phys.Rev. Lett., vol. 100, p. 013904, Jan 2008. \u2192 page 1[7] L. Lu, J. D. Joannopoulos, and M. Soljac\u02c7ic\u00b4, \u201cTopological photonics,\u201d NaturePhotonics, vol. 8, no. 11, pp. 821\u2013829, 2014. \u2192 page 1[8] E. Prodan and C. Prodan, \u201cTopological Phonon Modes and Their Role inDynamic Instability of Microtubules,\u201d Physical Review Letters, vol. 103,p. 248101, dec 2009. \u2192 page 1[9] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang,\u201cTopological acoustics,\u201d Physical review letters, vol. 114, no. 11, p. 114301,2015. \u2192 page 1[10] C. L. Kane and T. C. Lubensky, \u201cTopological boundary modes in isostaticlattices,\u201d Nature Physics, vol. 10, pp. 39\u201345, jan 2014. \u2192 page 144[11] S. D. Huber, \u201cTopological mechanics,\u201d Nature Physics, vol. 12, p. 621,2016.[12] Y. Barlas and E. Prodan, \u201cTopological classification table implemented withclassical passive metamaterials,\u201d Physical Review B, vol. 98, p. 094310, sep2018. \u2192 page 1[13] T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael, \u201cTopologicalPolaritons,\u201d Physical Review X, vol. 5, p. 031001, jul 2015. \u2192 page 1[14] J. Ningyuan, C. Owens, A. Sommer, D. Schuster, and J. Simon, \u201cTime- andsite-resolved dynamics in a topological circuit,\u201d Physical Review X, vol. 5,no. 2, pp. 1\u20137, 2015. \u2192 pages 1, 2[15] C. H. Lee, S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp,T. Kiessling, and R. Thomale, \u201cTopolectrical Circuits,\u201d CommunicationsPhysics, vol. 1, p. 39, dec 2018. \u2192 pages 1, 2[16] E. Zhao, \u201cTopological circuits of inductors and capacitors,\u201d Annals ofPhysics, vol. 399, pp. 289 \u2013 313, 2018.[17] K. Luo, R. Yu, and H. Weng, \u201cTopological Nodal States in Circuit Lattice,\u201dResearch, vol. 2018, pp. 1\u201310, sep 2018. \u2192 page 1[18] W. Zhu, S. Hou, Y. Long, H. Chen, and J. Ren, \u201cSimulating quantum spinhall effect in the topological lieb lattice of a linear circuit network,\u201d Phys.Rev. B, vol. 97, p. 075310, Feb 2018. \u2192 page 1[19] Y. Lu, N. Jia, L. Su, C. Owens, G. Juzeliu\u00afnas, D. I. Schuster, and J. Simon,\u201cProbing the Berry curvature and Fermi arcs of a Weyl circuit,\u201d PhysicalReview B, vol. 99, p. 020302, jan 2019. \u2192 page 1[20] M. Ezawa, \u201cNon-Hermitian higher-order topological states in nonreciprocaland reciprocal systems with their electric-circuit realization,\u201d PhysicalReview B, vol. 99, p. 201411, may 2019. \u2192 page 1[21] M. Ezawa, \u201cHigher-order topological electric circuits and topological cornerresonance on the breathing kagome and pyrochlore lattices,\u201d PhysicalReview B, vol. 98, p. 201402, nov 2018.[22] S. Imhof, C. Berger, F. Bayer, J. Brehm, L. W. Molenkamp, T. Kiessling,F. Schindler, C. H. Lee, M. Greiter, T. Neupert, and R. Thomale,\u201cTopolectrical-circuit realization of topological corner modes,\u201d NaturePhysics, vol. 14, pp. 925\u2013929, sep 2018. \u2192 page 145[23] Haldane, \u201cModel for a quantum hall effect without landau levels:Condensed-matter realization of the \u201dparity anomaly\u201d,\u201d Physical reviewletters, vol. 61, no. 18, pp. 2015\u20132018, 1988. \u2192 pages 2, 31, 32[24] H. Shen, B. Zhen, and L. Fu, \u201cTopological band theory for non-hermitianhamiltonians,\u201d Physical review letters, vol. 120, no. 14, p. 146402, 2018. \u2192page 2[25] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Higashikawa, andM. Ueda, \u201cTopological phases of non-hermitian systems,\u201d Phys. Rev. X,vol. 8, p. 031079, Sep 2018.[26] K. Luo, J. Feng, Y. X. Zhao, and R. Yu, \u201cNodal Manifolds Bounded byExceptional Points on Non-Hermitian Honeycomb Lattices andElectrical-Circuit Realizations,\u201d arXiv:1810.09231, 2018.[27] N. Moiseyev, Non-Hermitian quantum mechanics. Cambridge UniversityPress, 2011. \u2192 page 2[28] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, \u201cClassificationof topological insulators and superconductors in three spatial dimensions,\u201dPhysical Review B, vol. 78, p. 195125, nov 2008. \u2192 page 4[29] A. Kitaev, \u201cPeriodic table for topological insulators and superconductors,\u201d inAIP Conference Proceedings, vol. 1134, pp. 22\u201330, AIP, may 2009. \u2192 page4[30] K. Wilson and J. B. Kogut, \u201cThe renormalization group and the \u03b5expansion,\u201d Physics Reports, vol. 12, pp. 75\u2013199, aug 1974. \u2192 page 5[31] J. von Delft and H. Schoeller, \u201cBosonization for beginners - refermionizationfor experts,\u201d Annalen der Physik, vol. 7, pp. 225\u2013305, nov 1998. \u2192 page 5[32] U. Schollwo\u00a8ck, \u201cThe density-matrix renormalization group in the age ofmatrix product states,\u201d Annals of Physics, vol. 326, pp. 96\u2013192, jan 2011. \u2192page 5[33] A. Altland and M. R. Zirnbauer, \u201cNonstandard symmetry classes inmesoscopic normal-superconducting hybrid structures,\u201d Physical Review B,vol. 55, no. 2, pp. 1142\u20131161, 1997. \u2192 page 7[34] C.-K. Chiu, \u201cThe classification of topological insulators andsuperconductors for non-spatial and spatial symmetries,\u201d Doctoraldissertation, 2013. \u2192 page 946[35] C. K. Chiu, H. Yao, and S. Ryu, \u201cClassification of topological insulators andsuperconductors in the presence of reflection symmetry,\u201d Physical Review B- Condensed Matter and Materials Physics, vol. 88, no. 7, pp. 1\u201324, 2013.\u2192 page 11[36] C.-K. Chiu and A. P. Schnyder, \u201cClassification ofreflection-symmetry-protected topological semimetals and nodalsuperconductors,\u201d Physical Review B, vol. 90, p. 205136, nov 2014. \u2192 page11[37] C. Weeks and M. Franz, \u201cTopological insulators on the Lieb and perovskitelattices,\u201d Physical Review B, vol. 82, p. 085310, aug 2010. \u2192 page 27[38] B. D. Tellegen, \u201cThe gyrator, a new electric network element,\u201d Philips Res.Rep, vol. 3, no. 2, pp. 81\u2013101, 1948. \u2192 page 28[39] N. W. Ashcroft and N. D. Mermin, Solid state physics. Holt, Rinehart andWinston, 1976. \u2192 page 34[40] R. F. Wick, \u201cSolution of the field problem of the germanium gyrator,\u201dJournal of Applied Physics, vol. 25, no. 6, pp. 741\u2013756, 1954. \u2192 pages35, 36[41] G. Viola and D. P. DiVincenzo, \u201cHall effect gyrators and circulators,\u201dPhysical Review X, vol. 4, no. 2, p. 81, 2014. \u2192 pages 36, 39[42] C. L. Hogan, \u201cThe Ferromagnetic Faraday Effect at Microwave Frequenciesand its Applications,\u201d Bell System Technical Journal, vol. 31, pp. 1\u201331, jan1952. \u2192 page 39[43] B. Shenoi, \u201cPractical Realization of a Gyrator Circuit and RC-GyratorFilters,\u201d IEEE Transactions on Circuit Theory, vol. 12, no. 3, pp. 374\u2013380,1965.[44] A. Fabre, \u201cGyrator implementation from commercially availabletransimpedance operational amplifiers,\u201d Electronics Letters, vol. 28, no. 3,p. 263, 1992.[45] M. Castellanos-Beltran, K. Irwin, L. Vale, G. Hilton, and K. Lehnert,\u201cBandwidth and Dynamic Range of a Widely Tunable Josephson ParametricAmplifier,\u201d IEEE Transactions on Applied Superconductivity, vol. 19,pp. 944\u2013947, jun 2009.47[46] J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin,\u201cTime-reversal-symmetry breaking in circuit-QED-based photon lattices,\u201dPhysical Review A, vol. 82, p. 043811, oct 2010. \u2192 page 39[47] L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and nonlinear circuits.McGraw-Hill, 1987. \u2192 page 3948","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. There is no restriction on the nature of this information, e.g., it could be plain text, hypertext, or an image; it could be a definition, information about the scope of a concept, editorial information, or any other type of information."}],"Genre":[{"label":"Genre","value":"Thesis\/Dissertation","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","classmap":"dpla:SourceResource","property":"edm:hasType"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","explain":"A Europeana Data Model Property; This property relates a resource with the concepts it belongs to in a suitable type system such as MIME or any thesaurus that captures categories of objects in a given field. It does NOT capture aboutness"}],"GraduationDate":[{"label":"GraduationDate","value":"2019-09","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","classmap":"vivo:DateTimeValue","property":"vivo:dateIssued"},"iri":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","explain":"VIVO-ISF Ontology V1.6 Property; Date Optional Time Value, DateTime+Timezone Preferred "}],"IsShownAt":[{"label":"IsShownAt","value":"10.14288\/1.0380599","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","classmap":"edm:WebResource","property":"edm:isShownAt"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","explain":"A Europeana Data Model Property; An unambiguous URL reference to the digital object on the provider\u2019s website in its full information context."}],"Language":[{"label":"Language","value":"eng","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/language","classmap":"dpla:SourceResource","property":"dcterms:language"},"iri":"http:\/\/purl.org\/dc\/terms\/language","explain":"A Dublin Core Terms Property; A language of the resource.; Recommended best practice is to use a controlled vocabulary such as RFC 4646 [RFC4646]."}],"Program":[{"label":"Program","value":"Physics","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","classmap":"oc:ThesisDescription","property":"oc:degreeDiscipline"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the program for which the degree was granted."}],"Provider":[{"label":"Provider","value":"Vancouver : University of British Columbia Library","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","classmap":"ore:Aggregation","property":"edm:provider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","explain":"A Europeana Data Model Property; The name or identifier of the organization who delivers data directly to an aggregation service (e.g. Europeana)"}],"Publisher":[{"label":"Publisher","value":"University of British Columbia","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/publisher","classmap":"dpla:SourceResource","property":"dcterms:publisher"},"iri":"http:\/\/purl.org\/dc\/terms\/publisher","explain":"A Dublin Core Terms Property; An entity responsible for making the resource available.; Examples of a Publisher include a person, an organization, or a service."}],"Rights":[{"label":"Rights","value":"Attribution-NonCommercial-NoDerivatives 4.0 International","attrs":{"lang":"*","ns":"http:\/\/purl.org\/dc\/terms\/rights","classmap":"edm:WebResource","property":"dcterms:rights"},"iri":"http:\/\/purl.org\/dc\/terms\/rights","explain":"A Dublin Core Terms Property; Information about rights held in and over the resource.; Typically, rights information includes a statement about various property rights associated with the resource, including intellectual property rights."}],"RightsURI":[{"label":"RightsURI","value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","attrs":{"lang":"*","ns":"https:\/\/open.library.ubc.ca\/terms#rightsURI","classmap":"oc:PublicationDescription","property":"oc:rightsURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#rightsURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the Creative Commons license url."}],"ScholarlyLevel":[{"label":"ScholarlyLevel","value":"Graduate","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","classmap":"oc:PublicationDescription","property":"oc:scholarLevel"},"iri":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the scholarly level of the author(s)\/creator(s)."}],"Title":[{"label":"Title","value":"Chern insulators for electromagnetic waves in electrical circuit networks","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/title","classmap":"dpla:SourceResource","property":"dcterms:title"},"iri":"http:\/\/purl.org\/dc\/terms\/title","explain":"A Dublin Core Terms Property; The name given to the resource."}],"Type":[{"label":"Type","value":"Text","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/type","classmap":"dpla:SourceResource","property":"dcterms:type"},"iri":"http:\/\/purl.org\/dc\/terms\/type","explain":"A Dublin Core Terms Property; The nature or genre of the resource.; Recommended best practice is to use a controlled vocabulary such as the DCMI Type Vocabulary [DCMITYPE]. To describe the file format, physical medium, or dimensions of the resource, use the Format element."}],"URI":[{"label":"URI","value":"http:\/\/hdl.handle.net\/2429\/71465","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#identifierURI","classmap":"oc:PublicationDescription","property":"oc:identifierURI"},"iri":"https:\/\/open.library.ubc.ca\/terms#identifierURI","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the handle for item record."}],"SortDate":[{"label":"Sort Date","value":"2019-12-31 AD","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/date","classmap":"oc:InternalResource","property":"dcterms:date"},"iri":"http:\/\/purl.org\/dc\/terms\/date","explain":"A Dublin Core Elements Property; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF].; A point or period of time associated with an event in the lifecycle of the resource.; Date may be used to express temporal information at any level of granularity. Recommended best practice is to use an encoding scheme, such as the W3CDTF profile of ISO 8601 [W3CDTF]."}]}