Demonstration of a quantized microwave quadrupole insulator with topologically protected corner states
Christopher W. Peterson, Wladimir A. Benalcazar, Taylor L. Hughes, Gaurav Bahl
DDemonstration of a quantized microwavequadrupole insulator with topologicallyprotected corner states
Christopher W. Peterson , Wladimir A. Benalcazar ,Taylor L. Hughes , and Gaurav Bahl ∗ University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering Department of Physics Department of Mechanical Science and Engineering ∗ To whom correspondence should be addressed; [email protected]
October 11, 2017
The modern theory of electric polarization in crystals associates thedipole moment of an insulator with a Berry phase of its electronic groundstate [1, 2]. This concept constituted a breakthrough that not only solvedthe long-standing puzzle of how to calculate dipole moments in crystals,but also lies at the core of the theory of topological band structuresin insulators and superconductors, including the quantum anomalousHall insulator [3, 4] and the quantum spin Hall insulator [5–7], as wellas quantized adiabatic pumping processes [8–10]. A recent theoreticalproposal extended the Berry phase framework to account for higher elec-tric multipole moments [11], revealing the existence of topological phasesthat have not previously been observed. Here we demonstrate the firstmember of this predicted class – a quantized quadrupole topological in-sulator – experimentally produced using a GHz-frequency reconfigurablemicrowave circuit. We confirm the non-trivial topological phase throughboth spectroscopic measurements, as well as with the identification ofcorner states that are manifested as a result of the bulk topology. Weadditionally test a critical prediction that these corner states are pro-tected by the topology of the bulk, and not due to surface artifacts, bydeforming the edge between the topological and trivial regimes. Our a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t esults provide conclusive evidence of a unique form of robustness whichhas never previously been observed. The simplest model of a system with a quantized dipole moment is atwo-band insulator in 1D [12] which, due to the presence of chiral or in-version symmetries [13, 14], exhibits quantized fractional edge charges of ± e/ q xy = e/ edge-localized dipolemoments tangent to the edge, and corner-localized charges, both of magni-tude e/ λ ) a
2D bulkquadrupoleq xy edge dipole p x edge d i po l e p y edge d i po l e p y edge dipole p x c D en s i t y o f s t a t e s d Topological edge λ e > γ e Intermediate phase λ e = γ e Trivial edge λ e < γ e b λ � � � � �� � � � �� � � � �� � � � �� � � � � γ Protectedmid-gapcorner modes Q uad r upo l e t opo l og i c a l pha s e Q uad r upo l e t opo l og i c a l pha s e Figure 1: Quadrupole topological insulator – (a)
A 2D bulk quadrupole topological insulator (blue square)manifests edge-localized topological dipoles (orange lines) and corner-localized charges of ± e/ (b) The tight-binding representation of the model that realizes a quadrupole topological insulatorhaving four sites per unit cell. λ are couplings between unit cells and γ are couplings within unit cells. Dashedlines indicate a − π per plaquette. The model is in the quadrupole topological phase for λ > γ and in the trivial phase for λ < γ . (c) Theoretically calculated density of states for the quantized quadrupole insulator (5 × (d) Theoretically calculated probabilitydensity functions of the four in gap modes as the unit cells on the lowest edge are reconfigured from λ e /γ e = 4 . λ e /γ e = 1 (center), and to λ e /γ e = 1 / . γ e is changed, λ e = λ = 1and γ = 1 / . γ and λ describe couplingbetween resonators within the same unit cell and between adjacent unitcells, respectively. Each plaquette, a square of any four adjacent resonators,contains a single negative coupling term (dashed lines in Fig. 1b) whichamounts to the generation of a synthetic magnetic flux of π puncturing theplaquette. The existence of this non-zero flux opens both the bulk and edgeenergy gaps, which are necessary to protect the corner-localized mid gapmodes.Each resonator in our experimental array is implemented using an H -shaped microstrip transmission line that has a fundamental resonance at f = 2 .
08 GHz, having typical linewidth ∼
20 MHz, with a spatial voltagedistribution as illustrated in Fig. 2a (bottom). At the center of the crosspiece lies a voltage node while each end-point of the H -shape is a quarter-wavelength from the center and is therefore an anti-node. Adjacent tips ofthe H are separated by a half-wavelength and thus differ in phase by π rad,and the pinched-in ends are designed to bring the anti-nodal points hav-ing opposing phase physically close together. The unique resonator geom-etry facilitates the coupling of adjacent resonators either in-phase (positivecoupling) or out-of-phase (negative coupling). To produce the quadrupoletopology, in each plaquette we arrange three couplings as positive and onecoupling as negative as shown in Fig. 2a.We first experimentally confirm that a π flux threads each plaquette byexamining the limiting cases of λ → γ →
0. This experimental verifi-cation of π flux is necessary to ensure that the spectral features we measureare due to the bulk quadrupole topology, as corner modes themselves, eventopologically protected ones, are not unique to the quadrupole topologicalinsulator [21, 26, 27]. In the λ → γ between all resonators, and a π flux threading the plaquette,the eigenfrequencies are ±√ γ , and each of these are two-fold degenerate(see Supplement § S1). Since the non-trivial topology of the full array is fullymanifest in either the upper or lower band, we choose to characterize onlythe lower band at −√ γ . 4 -1+1R1 R2R3R4 R R R R ( + ) bc Driving R3Driving R4 d T heo r y M ea s u r ed R1 R3R2R4 T heo r y M ea s u r ed T heo r y T heo r y M ea s u r ed Driving R3 Driving R4R1R3 R4R2
R1R2R3 R4 M ea s u r ed R4 (-)f = 2.08 GHz Positive coupling(+ γ )Negative coupling(- γ )1 cm1 cm Positive coupling (+ λ )Negativecoupling (- λ ) π R3R4 R3R41800 1900 2000 2100 2200Frequency (MHz)0.20.40.6 A b s o r p t an c e ( A ) A b s o r p t an c e ( A ) Figure 2: Verification of microwave quadrupole lattice bulk topology – (a)
A unit cell of the quadrupoletopological insulator (photograph on top-left) is composed of four capacitively coupled H-shaped microstrip res-onators, each having a fundamental mode at 2.08 GHz as illustrated (colors represent voltage amplitude). Thecoupling between R4 and R1 is set as negative, as shown in the detailed schematic, in order to produce π fluxthrough the unit cell plaquette. γ ≈
35 MHz. (b)
Eigenmode verification for the unit cell plaquette – The resonatorfrequencies are shifted to ∼ (c) A 2x2 test array of unit cellswith γ →
0. Negative coupling is set between resonators R2 and R3 as illustrated in the schematic. λ ≈
150 MHz. (d)
Eigenmodes for the plaquette formed by the 4 central resonators of a 2x2 array are similar as to those of theunit cell due to the π flux. When driving R3, resonator R2 is seen to be in-phase, confirming the negative couplingbetween these resonators. Here, the resonator frequencies are shifted to ∼ . √ γ ≈
100 MHz. The spa-tial distribution of the lower pair of modes is measured through the voltageamplitude and phase response at each resonator within the plaquette wheneither resonator R3 or R4 is stimulated (see Methods). We find good agree-ment between the magnitudes and phases of the theoretical and measuredmodes (Fig. 2b). Characteristic mode shapes appear due to destructive in-terference, caused by the π flux, between counter-circulating paths aroundthe plaquette. Specifically, when R4 is excited the mode vanishes on thediagonal resonator R3 (and vice versa). In Supplement § S1, we discuss theclear contrast of this observation against the anticipated modes of plaquetteshaving zero flux, although the cases can exhibit spectral similarities.In the γ → λ ) is larger. We experimen-tally verify that the eigenmodes of this inter-unit cell plaquette also havethe features expected for π flux by performing similar measurements to thesingle unit cell case (Fig. 2d). For this measurement, the capacitors thatoriginally coupled the resonators within the unit cells are removed to ensure γ = 0. We also find good agreement between the theoretical and measuredmode shapes, although the lower pair of modes are not perfectly degeneratedue to asymmetric capacitive loading (see Supplement § S2). The measuredfrequency separation (2 √ λ ≈
430 MHz) between the two pairs of modes isapproximately 4 . λ/γ ≈ . × λ/γ ≈ . bc (1) Lower band (2) In-gap (3) Upper band f Positive inter-cell coupling (+ λ )Positive intra-cell coupling (+ γ )4 cm Negative inter-cell coupling (- λ )Negative intra-cell coupling (- γ ) Corner R4 Corner R2Corner R3Corner R1 Corner R4Corner R2 Corner R1Corner R3 d e (1) Lower band (2) In-gap (3) Upper band N o r m a li z ed a v e r age ab s o r p t an c e ( A a v g ) A b s o r p t an c e ( A ) Figure 3: Demonstration of microwave quadrupole topological insulator – (a)
Photograph of the experi-mental array of coupled resonators that form the quadrupole lattice. The array has 5 × λ/γ ≈ .
3. The schematic shows the connectivity of a bulk unit cell. (b)
Normalized averageabsorptance spectrum (ratio of absorbed microwave power to incident power) of all the resonators in the array (seeMethods for details). We observe two large bands (blue) separated by a band gap containing in-gap modes (green). (c)
Spatial distribution of absorptance summed over the lower frequency band indicated in (b). Within this band,the response is dominated by bulk and edge resonators. Circle areas are correspond to local absorptance. (d)
Spatialdistribution of absorptance summed over the in-gap band indicated in (b). The in-gap modes are localized only onthe corner resonators, which are not excited in the lower or upper band. (e)
Spatial distribution of absorptancesummed over the upper band indicated in (b), which again shows excitation of the bulk and edge resonators. (d)
Individual absorptance spectra of the corner resonators reveal that each corner resonator only supports a singlemode. § S2.2). Themain spectroscopic effect of the systematic disorder is a splitting of the lowerband, which manifests in isolated plaquettes as a lifting of the degeneracyof the lower pair of modes (Fig. 2). Despite such disorder and asymmetries,we find that the robust spectral features of the quadrupole topological in-sulator remain, e.g. the spectral bands are gapped, with only 4 resonancesat positions close to mid-gap. Furthermore, we have verified that these fourmid-gap modes are tightly confined to the corners (Fig. 3e).To demonstrate that the corner-localized modes are not the result oflocal effects particular to the physical edges of the array, we tune the unitcells on the lowest row from the topological regime ( γ < λ ) to the trivialregime ( γ > λ ). This experiment begins with the entire array in the originaltopological phase ( λ/γ ≈ .
3) as shown previously in Fig. 3. For this config-uration, we plot the average absorptance spectra of the bottom two rows ofunit cells separately, revealing that both rows are gapped but that the bot-tom row supports the mid-gap modes (Fig. 4a). As previously shown, thesemid-gap modes are localized on the corners of the array (Fig. 4b). Next,we adjust the coupling rates on the bottom row of unit cells to be equal,i.e. λ e /γ e = 1. This is achieved simply by replacing the coupling capacitorswithin the network. This modification narrows the band gap of the bottomtwo rows (Fig. 4c), and the two lower corner modes delocalize from the origi-nal corners into the surrounding unit cells (Fig. 4d). Due to the finite size ofthe experimental array, the corner modes couple to each other at this pointin the process and their degeneracy is lifted. Finally, we make the bottomedge unit cells trivial by setting λ e /γ e ≈ / .
3, broadening the band gap toits original width (Fig. 4e). Though the physical bottom edge of the array isnow in the trivial regime, the corner modes are not destroyed but simply re-cede to the new topological phase boundary. This experimental observation8 ca bdf T opo l og i c a l pha s e λ / γ = . T opo l og i c a l pha s e λ e / γ e = . T opo l ogog i c a l pha s e λ / γ = . I n t e r m ed i a t e pha s e λ e / γ e = T opo l og i c a l pha s e λ / γ = . T r i v i a l pha s e λ e / γ e = / . Corner modesCorner modesCorner modes A a v g A a v g A a v g A a v g A a v g A a v g Corner modes
Figure 4: Experimental test of topologically protected corner states during edge deformation – (a)
The entire array is initially set in the topological phase with λ/γ = 4 .
3. The bottom two rows of unit cells display abandgap, with mid-gap modes – the topological corner modes – appearing only on the bottom row. (b)
Measuredspatial distribution of modes within the bandgap, summed over the shaded band in (a). (c)
The unit cells on thebottom edge are now set at a transition point between the topological and trivial regimes, with λ e /γ e = 1 (blue lines).The bandgap along the bottom edge narrows but remains open. Due to the finite size of the array, the corner modescouple to each other and their degeneracy is lifted. (d) Measured spatial distribution of modes within the bandgap,summed over the shaded band in (c). The in-gap modes are delocalized between the unit cells in the bottom tworows. (e)
The unit cells in the bottom row are finally brought into the trivial regime with λ e /γ e = 1 / .
3, whilethe rest of the array remains topological. The mid-gap modes are shifted one row up towards the new quadrupoletopological phase boundary. (f )
Measured spatial distribution of modes corresponding to (e). The mid-gap modeslocalize on the new corners of the quadrupole topological phase.
Note added:
During preparation of this manuscript we learned of twoparallel efforts to realize quadrupole topological insulators. Imhof et al (Ref.[30]) demonstrate an electronic system at MHz frequencies. Serra-Garcia etal (Ref. [31]) demonstrate a phononic system at kHz frequencies.
Methods
Design of the quadrupole topological insulator lattice –
Each unit cell is fab-ricated individually on Rogers RT/duroid 5880 substrate, with 35 µ m thickcopper on each side. Within each unit cell, we select the coupling param-eter γ by connecting the resonators through a 0 . λ by connecting the resonatorsthrough a 1 pF capacitor. The coupling rate between resonators is a sub-linear function of capacitance such that λ/γ ≈ . γ and λ respectively.The coupling capacitors also capacitively load the resonators, increasing10heir effective length and therefore reducing the resonance frequency. Forbulk resonators the capacitive loading is similar and does not affect thebulk spectral characteristics. However, the reduced capacitive loading foredge and corner resonators is compensated (to match the bulk loading) byadding capacitance to ground of 0 . . § S2.
Spectrum and eigenmode measurements –
We measure the power absorp-tance spectrum at each resonator within the tested networks by means of1-port reflection ( S ) measurements using a microwave network analyzer(Keysight E5063A). The reflection probe is composed of a 50 Ω coaxial ca-ble terminated in a 0 . A = 1 − (cid:12)(cid:12) S (cid:12)(cid:12) . We also define the aver-age absorptance for an array of N resonators as A avg = N (cid:80) Nn A n , where A n is the absorptance of the n th resonator. In this calculation, we apply aminimum threshold to remove probe induced background absorption.The eigenmodes of the unit cell and 2 × S ). The measurement is performed using a pair ofprobes as specified above, with one probe used for stimulus and the othermeasuring response. The S transfer function at the resonant frequencythus produces a direct measurement for the amplitude and phase responsefor the corresponding eigenmode. Acknowledgements
The authors would like to thank Prof. Jennifer T. Bernhard for access tothe resources at the UIUC Electromagnetics Laboratory. This project wassupported by the US National Science Foundation (NSF) Emerging Fron-tiers in Research and Innovation (EFRI) grant EFMA-1627184. C.W.P.additionally acknowledges support from the NSF Graduate Research Fel-lowship. G.B. additionally acknowledges support from the US Office ofNaval Research (ONR) Director for Research Early Career Grant. W.A.B.and T.L.H. additionally thank the U.S. National Science Foundation undergrant DMR-1351895. 11 uthor contributions
C.W.P. designed the microwave quadrupole topological insulator, performedthe microwave simulations and experimental measurements, and producedthe experimental figures. W.A.B. guided the topological insulator designand performed the theoretical calculations. T.L.H. and G.B. supervised allaspects of the project. All authors jointly wrote the paper.
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Observation of a phonon quadrupole topological insu-lator. arXiv:1708.05015 (2017). upplementary Information:Demonstration of a quantized microwavequadrupole insulator with topologically protectedcorner states Christopher W. Peterson , Wladimir A. Benalcazar ,Taylor L. Hughes , Gaurav Bahl , ∗ University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering Department of Physics Department of Mechanical Science and Engineering ∗ To whom correspondence should be addressed; [email protected]
S1 Comparison of unit cell threaded with π fluxand 0 flux A unit cell of our quadrupole topological insulator is a square of four res-onators threaded with π flux, as illustrated in the main manuscript Fig. 2a.In Fig. 2b, we show the measured eigenmodes of a single unit cell, whichmatch well with the theoretically predicted modes. In this section we dis-cuss all four eigenmodes of this system, and also establish a contrast againstunit cells threaded with 0 flux. We find that, without flux, the modes differsignificantly in their spatial distribution but that their energy spectra canbe similar if C symmetry is broken.The calculated energy spectrum and eigenmodes of a unit cell with π fluxand equal coupling magnitudes γ are shown in Fig. S1a. As described inthe main manuscript, there are two pairs of degenerate eigenmodes. Thesemodes can be described by the orthonormal basis vectors u = (cid:2) / − / √ / (cid:3) ,u = (cid:2) − / − / √ / (cid:3) ,u = (cid:2) − / / √ / (cid:3) ,u = (cid:2) / / √ / (cid:3) , (S1)where each vector corresponds to the complex amplitudes of the resonators (cid:2) R1 R2 R3 R4 (cid:3) . u and u are the degenerate pair of lower energy15 Mode index E n e r g y ( un i t s o f γ ) u u u u b E n e r g y ( un i t s o f γ ) Mode index1 2 3 4 v v v v Unit cell containing π flux Unit cell containing flux R1 R3R2R4 R1 R3R2R4 √2−√2 γ γ02−2 c E n e r g y ( un i t s o f γ x ) Mode index1 2 3 4 w w w w Unit cell containing flux γ x > γ y R1 R3R2R4 γ x y Figure S1: Comparison of unit cells threaded with π and 0 flux – (a) Energy spectrum and eigenmodesof a unit cell with π flux. (b) Energy spectrum and eigenmodes of a unit cell with 0 flux, with γ x = γ y . (c) Energyspectrum and eigenmodes of a unit cell with 0 flux, but having unequal coupling rates γ x > γ y . Energy separationbetween the lower pair of modes (and upper pair of modes) is proportional to γ y . modes, and u and u are the degenerate pair of higher energy modes. Inthe main manuscript Fig. 2b we specifically measure the modes u , u . Dueto destructive interference arising from the π flux within the plaquette, whenone resonator is excited (here R3 or R4) the resonator on the opposite corneris not excited. This property leads to the uniquely identifiable modes of thisunit cell. We also find that the location of the negative coupling does affectthe relative phase between the resonators, leading to the opposite relativephase between resonators with and without negative coupling.We can now contrast the above case against the calculated energy spec-trum and eigenmodes of an identical unit cell having 0 flux, as shown in16ig. S1b. These modes can be described by the orthonormal basis vectors v = (cid:2) / / − / − / (cid:3) ,v = (cid:2) −√ / √ / (cid:3) ,v = (cid:2) √ / −√ / (cid:3) ,v = (cid:2) / / / / / (cid:3) . (S2)In this unit cell, only the modes v and v are degenerate, while v has lowerenergy and v has higher energy. Since there is 0 flux threading the unitcell, when one resonator is excited, the resonator on the opposite corner isalways excited as well.While a unit cell with 0 flux and identical horizontal and horizontalcoupling rates ( γ x = γ y ) is not gapped, a bandgap can be opened by setting γ x > γ y (i.e. breaking C symmetry). The calculated energy spectrum andeigenmodes for this case are shown in Fig. S1c. The modes can be describedby the orthonormal basis vectors w = (cid:2) / / − / − / (cid:3) ,w = (cid:2) / − / / − / (cid:3) ,w = (cid:2) / − / − / / (cid:3) ,w = (cid:2) / / / / (cid:3) . (S3)None of these modes are degenerate, but the lower pair (and upper pair)can be brought arbitrarily close for a large ratio γ x /γ y . However, the spatialdistribution of these eigenmodes clearly differs from a unit cell with π flux,since all four resonators are equally excited in each mode. S2 Microwave circuit implementation
Here we discuss specifics of the circuit implementation for our quadrupoletopological insulator design. A transmission line model is provided andthe connections between resonators are detailed. We also discuss two casesof capacitive loading that are representative of situations encountered byresonators in our quadrupole topological insulator array.
S2.1 Resonator design and coupling
A transmission line representation of our resonator is shown in Fig. S2a.The resonator is H -shaped, with the individual sections approximately the17 R R R R ( + ) R4 (-) π -1+10 ℓℓ ℓℓ b ℓ π R1 R2R3R4 N e g a ti v e c o up li n g Z = 110 Ω Z = Ω Z = Ω Z = Ω Z = Ω c R1 R3R2R4 R1 R2R3R4 R1R4R2R4 R4 π Voltage distribution
R1R3 R4R2
Figure S2: Transmission line model – (a)
Transmission line diagram of an individual microstrip resonator.Each section is approximately the same length (cid:96) ≈ . Z = 110 Ω,leading to a fundamental resonance frequency of 2 . (b) The coupling that links resonators within the unitcell is implemented as two 0 . π flux threading the plaquette. (c) The coupling that links resonators betweenunit cells is implemented as two 2 pF capacitors in series. The capacitors between R2 and R3 are connected to theout-of-phase anti-node of R3 to produce the required π flux. same length (cid:96) ≈ w = 0 . Z ≈
110 Ω. This resonator design leads to an unloaded resonance frequencyof approximately 2 . .
08 GHz).To create a unit cell, four microstrip resonators are capacitively coupledas shown in Fig. S2b. Each capacitive coupling is implemented as two 0 . . γ and λ areextracted from the measured data in the limiting cases λ → γ → .
3, which implies the couplingrate ratio λ/γ ≈ . S2.2 Systematic and random disorder in the coupling rates
Small differences in the capacitive loading of resonators inside our quadrupoletopological insulator array implies disorder in both the resonance frequen-cies and coupling rates. The impact of this disorder is seen in the measuredeigenmodes in the limits γ → λ → f = 1.4 GHz a f = 1.3 GHz b Figure S3:
Comparison of resonators loaded with identical total capacitance. Resonance frequencies are calculatedthrough simulation using Keysight ADS. (a)
A resonator with 2 pF loading on a single arm: the resonance frequencyis shifted from 2 . . (b) A resonator with 2 pF loading distributed to two opposite polarity arms: the resonance frequency is shifted from2 . .
19n Fig. S3a, we examine the case where both capacitors are on the samearm of the resonator. This is the case for the intra-unit cell coupling ofresonators R1, R2, R3 (Fig. S2b) and for the inter-unit cell coupling ofresonators R1, R2, R4 (Fig. S2c). The addition of 2 pF capacitance toground on one arm of the resonator causes a frequency shift to 1 . . . . ± .
05 pF and the 2 pF capacitors have atolerance of ± ..