A quantized microwave quadrupole insulator with topologically protected corner states

Abstract

The theory of electric polarization in crystals defines the dipole moment of an insulator in terms of a Berry phase (geometric phase) associated with its electronic ground state1,2. This concept not only solves the long-standing puzzle of how to calculate dipole moments in crystals, but also explains topological band structures in insulators and superconductors, including the quantum anomalous Hall insulator3,4 and the quantum spin Hall insulator5,6,7, as well as quantized adiabatic pumping processes8,9,10. A recent theoretical study has extended the Berry phase framework to also account for higher electric multipole moments11, revealing the existence of higher-order topological phases that have not previously been observed. Here we demonstrate experimentally a member of this predicted class of materials—a quantized quadrupole topological insulator—produced using a gigahertz-frequency reconfigurable microwave circuit. We confirm the non-trivial topological phase using spectroscopic measurements and by identifying corner states that result from the bulk topology. In addition, we test the critical prediction that these corner states are protected by the topology of the bulk, and are not due to surface artefacts, by deforming the edges of the crystal lattice from the topological to the trivial regime. Our results provide conclusive evidence of a unique form of robustness against disorder and deformation, which is characteristic of higher-order topological insulators.

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Figure 1: Quadrupole topological insulator.
Figure 2: Verification of microwave quadrupole lattice bulk topology.
Figure 3: Demonstration of microwave quadrupole topological insulator.
Figure 4: Experimental test results for topologically protected corner states during edge deformation.

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Acknowledgements

We would like to thank J. T. Bernhard for access to the resources at the UIUC Electromagnetics Laboratory. This project was supported by the US National Science Foundation (NSF) through the Emerging Frontiers in Research and Innovation (EFRI) grant EFMA-1627184. C.W.P. acknowledges support from an NSF Graduate Research Fellowship. G.B. acknowledges support from the US Office of Naval Research (ONR) through a Director for Research Early Career Grant. W.A.B. and T.L.H. thank the US NSF for grant DMR-1351895.

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Contributions

C.W.P. designed the microwave quadrupole topological insulator, performed the microwave simulations and experimental measurements and produced the experimental figures. W.A.B. guided the topological insulator design and performed the theoretical calculations. T.L.H. and G.B. supervised all aspects of the project. All authors jointly wrote the paper.

Corresponding author

Correspondence to Gaurav Bahl.

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The authors declare no competing financial interests.

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Reviewer Information Nature thanks Y. Kivshar and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Transmission line model.

a, Transmission line model of an individual microstrip resonator. Each section has approximately the same length and the same characteristic impedance Z0 = 110 Ω, which give a fundamental resonance frequency of 2.1 GHz. b, Resonators coupled within the unit cell by two 0.2 pF capacitors in series. The capacitors linking R1 and R4 are connected to the out-of-phase anti-node of R4, creating π-flux threading in the plaquette. c, Coupling of resonators in different unit cells by two 2-pF capacitors in series. The capacitors between R2 and R3 are connected to the out-of-phase anti-node of R3 to produce the required π flux.

Extended Data Figure 2 Comparison of unit cells threaded with π and 0 flux.

a, Energy spectrum and eigenmodes of a unit cell with π flux. b, Energy spectrum and eigenmodes of a unit cell with 0 flux and γx = γy. c, Energy spectrum and eigenmodes of a unit cell with 0 flux and unequal coupling rates γx > γy. The energy separation between the lower two (and upper two) modes is proportional to γy.

Extended Data Figure 3 Comparison of resonators loaded with equal total capacitance.

The resonance frequencies are calculated from simulations using Keysight ADS. a, A resonator with 2-pF loading on a single arm. The resonance frequency is shifted from 2.1 GHz to 1.4 GHz because of the loading. This situation corresponds to that of the intra-unit-cell coupling of resonators R1, R2 and R3 (Extended Data Fig. 1b) and the inter-unit-cell coupling of resonators R1, R2 and R4 (Extended Data Fig. 1c). b, A resonator with 2-pF loading distributed to two opposite-polarity arms. The resonance frequency is shifted from 2.1 GHz to 1.3 GHz owing to the loading. This situation is representative of the intra-unit-cell coupling of resonator R4 and the inter-unit-cell coupling of resonator R3.

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Peterson, C., Benalcazar, W., Hughes, T. et al. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018). https://doi.org/10.1038/nature25777

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