Reciprocal skin effect and its realization in a topolectrical circuit

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Reciprocal skin effect and its realization in a topolectrical circuit

Tobias Hofmann, Tobias Helbig, Frank Schindler, Nora Salgo, Marta Brzezińska, Martin Greiter, Tobias Kiessling, David Wolf, Achim Vollhardt, Anton Kabaši, Ching Hua Lee, Ante Bilušić, Ronny Thomale, and Titus Neupert

Phys. Rev. Research 2, 023265 – Published 2 June 2020

Abstract

A system is non-Hermitian when it exchanges energy with its environment and nonreciprocal when it behaves differently upon the interchange of input and response. Within the field of metamaterial research on synthetic topological matter, the skin effect describes the conspiracy of non-Hermiticity and nonreciprocity to yield extensive anomalous localization of all eigenmodes in a (quasi) one-dimensional geometry. Here, we introduce the reciprocal skin effect, which occurs in non-Hermitian but reciprocal systems in two or more dimensions: Eigenmodes with opposite longitudinal momentum exhibit opposite transverse anomalous localization. We experimentally demonstrate the reciprocal skin effect in a passive RLC circuit, suggesting convenient alternative implementations in optical, acoustic, mechanical, and related platforms. Skin mode localization brings forth potential applications in directional and polarization detectors for electromagnetic waves.

Received 20 September 2019
Accepted 11 May 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023265

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Classical transport Edge states Electromagnetism Electronic structure Impedance Metrology Photonics Skin effect Topological materials

Tight-binding model

Condensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsGeneral Physics

Authors & Affiliations

Tobias Hofmann¹, Tobias Helbig¹, Frank Schindler^2,3, Nora Salgo², Marta Brzezińska^4,2, Martin Greiter¹, Tobias Kiessling⁵, David Wolf², Achim Vollhardt², Anton Kabaši ⁶, Ching Hua Lee^7,8, Ante Bilušić ^6,9, Ronny Thomale ¹, and Titus Neupert²

¹Institute for Theoretical Physics and Astrophysics, University of Würzburg, Am Hubland, D-97074 Würzburg, Germany
²Department of Physics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
³Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
⁴Department of Theoretical Physics, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
⁵Physikalisches Institut and Röntgen Research Center for Complex Material Systems, Universität Würzburg, D-97074 Würzburg, Germany
⁶Centre of Excellence STIM, University of Split, Poljička cesta 35, HR-21000 Split, Croatia
⁷Department of Physics, National University of Singapore, Singapore 117542
⁸Institute of High Performance Computing, A*STAR, Singapore 138632
⁹University of Split, Faculty of Science, Ruđera Boškovića 33, HR-21000 Split, Croatia

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Vol. 2, Iss. 2 — June - August 2020

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Figure 1
Theory of the reciprocal skin effect. (a) Left: unit cell and spectrum of the $π$ -flux model defined in Eq. (1) with periodic boundary conditions. Solid and dashed lines correspond to hopping amplitudes $t$ and $- t$ , respectively. The spectrum shows two Dirac cones. Right: $π$ -flux model with a non-Hermitian diagonal hopping (pink) included, as defined in Eq. (2). Each Dirac point splits into a pair of exceptional points (only the real part of the band eigenvalues is plotted). (b) Real and imaginary part of eigenvalues in the vicinity of an exceptional point, at which two complex eigenvalues are degenerate. (c) Schematic of the reciprocal skin effect for OBC in $x$ and PBC in $y$ direction: Near two opposite momenta, $k_{y} = π / 2$ and $k_{y} = 3 π / 2$ , all eigenstates are exponentially localized with localization length $ξ$ at the left and right boundary, respectively. At $k_{y} = 0, π$ the modes are completely delocalized. Right: The reciprocal skin effect could serve as a direction detector for incident electromagnetic waves: Dependent on the propagation direction and polarization, a voltage will build up on the left or right edge of the system, as schematically shown in the figure.
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Figure 2
Experimental topolectrical circuit realization of the reciprocal skin effect and exceptional points. (a) Bulk unit cell and boundary terminations of the circuit that realizes the model defined in Eq. (2) and photograph of six unit cells of the assembled circuit board. Appendix pp3 details the connection of this circuit cell to the model in Eq. (2) using the Laplacian formalism. (b) Measured spectra with PBC of the circuit Laplacian and schematic unit cell of the circuit that corresponds to the $π$ -flux model (left) and the non-Hermitian model (right) from Eq. (2) [cf. Fig. 1]. Only the imaginary part of the eigenvalues is plotted. The Dirac points are deformed into branch cuts, which connect the exceptional points. (c) Left: Imaginary part and phase of the band with smaller imaginary part. The phase of the eigenvalue along the path indicated is plotted in the right panel. It shows a clear jump by $π$ , indicative of an enclosed exceptional point. (Stars denote an LTspice simulation of the circuit, while dots correspond to the measured spectrum.) (d) Measured spectra of the circuit Laplacian as a function of $k_{y}$ for PBC and OBC along the $x$ direction. Localization properties of each eigenstate are indicated by color and determined from the inverse participation ratio (IPR). The bottom panel shows the localization properties of representative individual eigenstates at $k_{y} = π / 2$ and $k_{y} = 3 π / 2$ , both for PBC and OBC. The fact that all OBC and PBC eigenstates differ nonperturbatively constitutes the reciprocal skin effect, with those around $k_{y} \sim π / 2$ right-localized and those around $k_{y} \sim 3 π / 2$ left-localized.
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Figure 3
Numerical calculation of the complex eigenspectrum of the non-Hermitian, nonreciprocal Hamiltonian in Eq. (C7) for $r = 1$ . (a) Real part of the PBC spectrum, (b) real part of the OBC spectrum, (c) imaginary part of the PBC spectrum, (d) imaginary part of the OBC spectrum. There is a nontrivial spectral flow from PBC to OBC, which manifests the breakdown of bulk-boundary correspondence associated with the extensive localization of eigenmodes.
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Figure 4
Representation of the circuit Laplacian spectra for periodic boundary conditions (PBC) in the complex plane as a function of $k_{y}$ . (a) Theoretically computed spectrum showing a genus three surface with four exceptional points. (b) Slice plots obtained from (a) for three different $k_{y}$ . (c) Measured spectra, where each of the four transitions from a single circle in the complex plane to two circles marks an exceptional point (EP1–EP4) as a function of $k_{y}$ .
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Figure 5
Reciprocal skin effect voltage response due to a localized bulk driving current with phase shift. The sites where the current is applied are framed in black; the zoom-ins show how the relative phase is implemented. Importantly, for a phase shift of $+ π / 2$ ( $- π / 2$ ) we find a nonlocal voltage response at the far right (left) side of the circuit with OBC in x direction, which is absent for phase shifts 0 and $π$ . The arrows show the direction of voltage accumulation.
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