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Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits

Abstract

The study of the laws of nature has traditionally been pursued in the limit of isolated systems, where energy is conserved. This is not always a valid approximation, however, as the inclusion of features such as gain and loss, or periodic driving, qualitatively amends these laws. A contemporary frontier of metamaterial research is the challenge open systems pose to the characterization of topological matter1,2. Here, one of the most relied upon principles is the bulk–boundary correspondence (BBC), which intimately relates the surface states to the topological classification of the bulk3,4. The presence of gain and loss, in combination with the violation of reciprocity, has been predicted to affect this principle dramatically5,6. Here, we report the experimental observation of BBC violation in a non-reciprocal topolectric circuit7, which is also referred to as the non-Hermitian skin effect. The circuit admittance spectrum exhibits an unprecedented sensitivity to the presence of a boundary, displaying an extensive admittance mode localization despite a translationally invariant bulk. Intriguingly, we measure a non-local voltage response due to broken BBC. Depending on the a.c. current feed frequency, the voltage signal accumulates at the left or right boundary, and increases as a function of nodal distance to the current feed.

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Fig. 1: Reciprocity breaking in a non-Hermitian linear circuit.
Fig. 2: BBC breakdown in the admittance spectrum.
Fig. 3: Non-local voltage response.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

Circuit simulations were performed using LTspice, available at https://www.analog.com/en/design-center/design-tools-and-calculators/ltspice-simulator.html#.

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Acknowledgements

The work in Würzburg is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 258499086–SFB 1170 and through the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter–ct.qmat Project-ID 39085490–EXC 2147. This research was supported in part by the National Science Foundation under grant no. NSF PHY-1748958. A.S. thanks the Krupp von-Bohlen-and-Halbach foundation for financial support.

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Contributions

T. Helbig, T. Hofmann, C.H.L., M.G. and R.T. conceptualized the project, all authors designed the experimental set-up and theoretical modelling. S.I., M.A. and T.K. carried out the circuit experiments and data analysis, T. Helbig and T. Hofmann performed the LTspice simulations. R.T. supervised the investigation and wrote the paper with key contributions from T. Helbig, T. Hofmann, C.H.L., A.S. and M.G. The manuscript reflects the contributions of all authors.

Corresponding author

Correspondence to R. Thomale.

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The authors declare no competing interests.

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Extended data

Extended Data Fig. 1 Parameter fit.

Frequency-dependent fit of inductances and their corresponding serial resistances \({L}_{1},{L}_{0},{R}_{{L}_{1}},{R}_{{L}_{0}}\) at each experimentally studied frequencies.

Extended Data Fig. 2 Impedance converter.

Setup of the negative impedance converter with current inversion used in the intracell AB connection of the circuit based on an operational amplifier to generate the non-Hermitian, non-reciprocal coupling γ.

Extended Data Fig. 3 Admittance spectral flow.

Spectral flow of the admittance in the complex plane at the frequency f = 80 kHz for the effective model (left) and the aberration-corrected fit model (right). The flow is quantified by the parameter η, which describes the attenuation factor of the 1-20 bond.

Extended Data Fig. 4 Imaginary momentum fit.

Theoretically calculated and experimentally fitted magnitude of the imaginary momentum κ for the experimentally studied frequencies. Experimental values are obtained as a mean value of the exponential decay profile of all eigenmodes, that have been numerically computed from the measurement data. Their error is given by the standard deviation of the mean value while considering all eigenmodes of the system and therefore quantifies the agreement of the localization of all bulk states.

Extended Data Fig. 5 Intracell coupling.

Top: Variation of the intracell coupling v(ω) as a function of frequency. The two OBC exceptional points for the ideal model at v = ± γ and the transition point at v = 0 with delocalized OBC modes are marked by black dashed lines. Bottom: Localization length ξ of the bulk OBC eigenmodes for the ideal effective model as a dashed red curve and for the full-scale fit model as a blue joint line. The circuit components are taken from Fig. 1 of the main text for both curves, while the parasitic resistance is chosen to RL1 = 0 mΩ for the red curve and to RL1 = 320 mΩ for the blue curve.

Extended Data Fig. 6 Non-local voltage response.

Logarithmic plot of the absolute voltage response in a measurement involving an external current feed with a.c. driving frequency f = 98.5 kHz at the left edge of the sample at node 1 in red and at node 7 in blue. The voltage increases exponentially to the right side of the circuit due to a unit-wise amplification. The expected voltage profile normalized to the voltage at the feed node is shown as a straight line, where the slope has been numerically calculated as the average decay profile of the OBC bulk eigenmodes.

Supplementary information

Supplementary Information

Captions of all Extended Data figures, Supplementary Figs. 1–4 and discussion.

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Helbig, T., Hofmann, T., Imhof, S. et al. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys. 16, 747–750 (2020). https://doi.org/10.1038/s41567-020-0922-9

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